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Essays in Mechanism Design

A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY

Field of Economics

By Toomas Hinnosaar

EVANSTON, ILLINOIS August 2012

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c Copyright by Toomas Hinnosaar 2012

All Rights Reserved

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ABSTRACT

Essays in Mechanism Design

Toomas Hinnosaar

This thesis consists of three essays in mechanism design and its applications. The main goal of mechanism design is to characterize optimal mechanisms, given preferences and information of the agents. However, the fully optimal mechanisms may not be useful, if they are too compex, too costly to implement, or require too strong commitment power from the seller. This is the main theme of the thesis. For specific environments, we would like to know what is the best the seller could do under no restrictions and how the situation changes if he has only limited set of possible mechanisms to use. The first two chapters analyze optimal and restricted mechanisms in dynamic environments. The third chapter studies the equilibrium properties of a particular unusual type of mechanism, called penny auction. Chapter 1 studies a repeated mechanism design problem where a revenue-maximizing monopolist sells a fixed number of service slots to randomly arriving buyers with private values and increasing exit rates. In addition to characterizing the fully optimal mechanism, it describes the optimal mechanisms in two restricted classes. First, the pure calendar

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mechanism, where the seller allocates future service dates instead of general promises. The unique optimal pure calendar mechanism is characterized in terms of the opportunity costs of allocating additional service slots. Second, it analyzes the waiting list mechanism, where promises of delayed service can depend on future arrivals, but the seller cannot discriminate among buyers who are offered the same position in the waiting list. Both the waiting list and the fully optimal mechanism are implemented by non-standard auctions with a scoring rule where the distance between buyers’ bids affects the allocation. A novel property of these auctions is that for buyers it is better to win by a close margin and it is worse to lose by a close margin. Finally, it models partial commitment power as a penalty that the seller has to pay when forfeiting a promise. All the results are given for general partial commitment and therefore include full commitment and no commitment as special cases. Chapter 2 considers optimal pricing policies for airlines when passengers are uncertain at the time of ticketing of their eventual realized willingness to pay for air travel. Auctions at the time of departure efficiently allocate space and a profit maximizing airline can capitalize on these gains by overbooking flights and repurchasing excess tickets from those passengers whose realized value is low. Nevertheless profit maximization entails distortions away from the efficient allocation. In order to encourage early booking, passengers who purchase late are disadvantaged. In order to capture the information rents of passengers with high expected values, ticket repurchases at the time of departure are at a subsidized price, sometimes leading to unused capacity. Chapter 3 analyzes a particular unusual type of mechanism called penny auctions. In these auctions every bid increases the price by a small amount, but it is costly to place

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a bid. The auction ends if more than some predetermined amount of time has passed since the last bid. There are many websites that implement this auction format and the outcomes are often surprising. Even selling cash can give the seller an order of magnitude higher or lower revenue than the nominal value. Sometimes the winner of the auction pays very little compared to many of the losers at the same auction. The unexpected outcomes have led to the accusations that the penny auction sites are either scams or gambling or both. Chapter 3 shows that the high variance of outcomes is a property of the mechanism. Even absent of any randomization, the equilibria in penny auctions are close to lotteries from the buyers’ perspective.

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Acknowledgements First of all, I would like to like to express my deep gratitude to Jeffrey Ely for his continued support and guidance. Regular meetings with him kept me on the right track and taught a lot, both how to think about economic theory and how to present it. I am deeply indebted to my dissertation committee members, Asher Wolinsky and Alessandro Pavan, whose insights have helped me to write better papers. I have also greatly benefited from discussions with Eddie Dekel, Rakesh Vohra, Dan Garrett, Todd Sarver, Marciano Siniscalchi, Wojciech Olszewski, and many other professors and students at Northwestern University. I am also grateful for comments and discussions from seminar participants at Aalto University, Bocconi University, Collegio Carlo Alberto, University of Bonn, University of Exeter, HEC Paris, McGill University, Northwestern University, University College London, and Vanderbilt University. For financial support I am grateful to Northwestern University, the Center for Economic Theory, and Archimedes Foundation in Estonia. There are many other people who have had major influence to my academic career. I am especially thankful to Helje Kaldaru, Raul Eamets, Otto Karma, Karsten Staehr, Ott-Siim Toomet, Andres Võrk, and the Department of Economics and EuroCollege in University of Tartu for pointing me to the right direction and supporting me in the early stages of my career.

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Finally, I am most grateful to my biggest supporter, most meticulous discussant, co-author, inspiration, and wife, Marit Hinnosaar. I would not have been able to do it without her.

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Table of Contents ABSTRACT

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Acknowledgements

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Chapter 1. Calendar mechanisms

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1.1. Introduction

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1.2. Illustrative example

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1.3. General analysis

53

1.4. Discussion

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Chapter 2. Overbooking (joint with Jeffrey Ely and Daniel Garrett)

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2.1. Introduction

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2.2. Model

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2.3. Pricing Mechanisms

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2.4. General Analysis

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2.5. Unrestricted (multiple fare class) mechanism

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2.6. Conclusions

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Chapter 3. Penny auctions are unpredictable

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3.1. Introduction

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3.2. Stylized facts

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3.3. Model

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3.4. Auction with zero price increment

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3.5. Auction with positive price increment

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3.6. Discussion

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References

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Appendix A. Proofs of Results in Chapter 1

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A.1. Calendar mechanism

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A.2. Waiting list

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A.3. Fully optimal mechanism

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Appendix B. Proofs of Results in Chapter 2

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Appendix C. Proofs of Results in Chapter 3

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C.1. Symmetric Stationary Subgame Perfect Nash Equilibrium

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C.2. Properties of ΨN (q)

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C.3. A penny auction with multiple equilibria

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CHAPTER 1

Calendar mechanisms 1.1. Introduction I study a mechanism design problem where a monopolist sells services to randomly arriving privately informed forward-looking customers. Typical applications include Internet ad-auctions, shuttle or concert ticket sales, restaurant reservations or the assignment of service slots by service providers ranging from hair salons to hospitals. The common features in all these examples are that 1) goods are perishable—once the corresponding date is passed, the good disappears; 2) the situation is repeated; and 3) demand is random, with customers willing to wait for some time but not for very long. In addition, in real life buyers generally interact with the seller only once: for example buy a ticket for the closest available date. In most cases continuous involvement would be too costly. Furthermore, if promises are for the too distant future, buyers may find alternative ways to solve their problems. Since the seller makes promises about future service, the promises are credible only if the seller has sufficient commitment power. In the model there is a monopolistic seller who has a fixed number of identical service slots every period. Buyers arrive randomly in time and have independent private values. Buyers discount the value of service, but not necessarily at the same rate as the seller does. The seller’s goal is to maximize the expected discounted revenue1 by choosing the 1As

it is often the case in mechanism design literature, maximizing efficiency is not fundamentally different. In particular, all the results presented in this chapter can be interpreted as efficient mechanisms when virtual values in the expressions are replaced by true values.

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optimal mechanism from a potentially restricted class of mechanisms. In addition to the fully optimal mechanism, I study mechanisms where the seller has only a limited set of allocation rules available. First, the pure calendar mechanism where the seller can give only promises of future service that are fulfilled with certainty. In other words, the seller has a calendar according to which the service is supplied. In each period, the seller assigns available dates from the calendar to new buyers. Selling tickets and making appointments are examples of this kind of mechanism. Second, with the waiting list mechanism the seller can give promises that are conditional on future arrivals but cannot personalize the promises to buyers’ types. This restriction operates as if the seller keeps a list of buyers who were promised delayed service and serves them according to some rule. Customers in different positions in the list may be served differently, but only the position in the list—and not the buyers type—will determine the allocation she gets. In some sense this is a natural relaxation of the pure calendar mechanism assumption—the seller may want to schedule appointments but with an understanding that when a very valuable buyer arrives in the future, some of the appointments will be canceled. Finally, in the fully optimal mechanism, the seller is free to choose any kind of promises about the future service. In contrast to the waiting list the promises will be personalized to each buyer according to their valuations. To study commitment power we assume that the seller is able to commit only to promises of delayed services that are not too tempting to break in the future. This is modeled using a partial commitment constant that can be interpreted as a cost or fine—if the seller does not serve a customer who was promised service, she must pay a penalty. All

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the results presented in this chapter are for general partial commitment constant; therefore full commitment and no commitment come as special cases. This chapter takes a different approach from the related literature by assuming that each buyer interacts with the seller only once. This assumption is motivated by applications, where buyers do not interact with the seller for a very long time. This is presumably because it would be too costly for them to participate in the mechanism more than necessary or that they prefer to have the information about the future service as soon as possible (for example to purchase complementary services, such as to reserve hotel rooms at the same time as buying plane tickets). Note that if the class of mechanisms that the seller can use is unrestricted, the one-time-interaction assumption is without loss. For any multiple-interactions mechanisms, there exist a corresponding one-time-interaction mechanism, where the seller gives each buyer a complete history-dependent allocation rule as a promise. The restrictions to the mechanism designer, including pure calendar mechanism and partial commitment, are defined relative to the one-time-interaction assumption. There are two special cases of this model, where the fully optimal mechanism is relatively simple to characterize. First, if the number of objects is large enough to serve all potential customers at their arrival, the optimal mechanism would be a constant posted-price mechanism. This is a version of Stokey’s (1979) no pure price-discrimination result. Here is the reason. From the buyers’ perspective, being offered delayed service is equivalent to getting fractional allocation. Without the need for rationing and without taking the continuation values into account, the optimal solution would use only corner

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solutions2 (Myerson, 1981). Offering delayed service can only decrease the continuation values, so the dynamic feasibility considerations would not change the conclusion. This result does not hold when at least in some periods there are fewer objects than new buyers, so that rationing is needed. Second, when the buyers discount the future service at a constant rate and the seller has the option to give promises of arbitrary length, it is optimal to give the current service to the buyers who have the highest values3 among all buyers who have arrived but not yet received service. This result follows from Said (2008). The reason is that two buyers who have arrived in different periods look at the future the same way, which makes it easy to compare buyers—only their valuations matter. The result fails, however, if buyers do not discount the future at a constant rate or, equivalently, if they exit from the model at an increasing rate. In these cases the seller needs to treat buyers from different periods differently. The buyers’ types would be multi-dimensional, characterized by both the valuation and the exit process. This chapter concentrates on the remaining case, where rationing is needed and buyers’ continuation values are dependent on their arrival time. There are many applications that fall in this category. First, there could be practical or legal reasons why buyers interact with the seller once and the seller cannot give promises about service in the too distant future. Many types of selling tickets and making appointments fall into this situation. Second, it may be because of buyers’ preferences—it is possible that buyers accept service if it is offered in the near future, but they cannot wait for a long time. Getting a haircut, appointments for emergency services, and transportation seem to satisfy this description. 2In

particular give a buyer the instant service if her virtual surplus is positive and no service otherwise. positive virtual surplus if the goal is to maximize revenue.

3Highest

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Finally, it could be because of competition that we do not model explicitly. It is possible that if buyers expect to get the service soon they would wait, but if the delay is long they would look for alternatives. We will focus attention on the extreme case where the characterizations given above fail. In particular, we assume that buyers interact with the seller only at their arrival, and the seller cannot give promises for more than one period ahead. In this case two buyers who have arrived in two consecutive periods are different, since the buyer who has already waited one period must be served now or will not be served, whereas a new buyer can be offered either current or delayed service. The heterogeneity of customers is now two-dimensional—not only their values but also their exit rates differ. As noted, there are at least three mathematically equivalent interpretations of this assumption. First, the seller simply cannot contract service more than one period ahead. Second, buyers have time preferences such that they get a positive value from current and next-period consumption, but they do not get any value from the service after that. Third, buyers exit the model two periods after their arrival. As we would expect, the optimal mechanism in this framework offers delayed service with strictly positive probability and treats buyers from different periods differently. We show that the fully optimal mechanism is implemented by a non-standard auction with scoring rule, which favors buyers who are waiting and in which the allocation that new buyers receive depends not only on the order of their values, but also on the differences in their values. In particular we show some types of buyers strictly prefer to win in the auction by a small margin, because this is the only way they receive instant service.

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Similarly, some types of buyers prefer to lose by a large margin rather than a small margin, since in this case they would be refused service rather than be offered delayed service. Characterizing the fully optimal mechanism is one contribution of the chapter. In addition to that I derive the optimal in the two classes of restricted mechanisms described above. The optimal pure calendar mechanism is characterized by a unique vector of opportunity costs of the corresponding service slots. It can be implemented by a simple two-stage mechanism. The qualitative properties of the optimal waiting list mechanism are similar to the fully optimal mechanism, but the delayed service contracts cannot be personalized to the buyers receiving the delayed service. The optimal contracts are therefore designed for the “average” buyer who will be offered delayed service. The optimal waiting list will have non-trivial dynamics even if the arrival process is constant. Since the value of an average buyer to whom the seller expects to assign each contract depends on previous promises, it is possible that the optimal contracts change every period. Finally, I characterize all three classes of mechanisms under the general partial commitment assumption, which shows how the optimal mechanism changes with commitment power. In particular there is a sufficient and, in most cases, necessary level of commitment that allows the seller to use the optimal mechanism in all three classes. If the commitment power is relatively low, the optimal calendar mechanism is static, whereas the revenue from the optimal waiting list mechanism and the unrestricted mechanism are strictly increasing in commitment power. As with most of the mechanism design literature, this modeling approach is similar to the seminal papers by Myerson (1981) and Riley and Samuelson (1981). In particular, Myerson’s optimal mechanism is the optimal static mechanism in the model analyzed in

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this chapter. Static mechanism belongs to all three classes of mechanisms that we consider, and it is therefore a benchmark against which to compare all the dynamic mechanisms. We show, however, that it will not generally be optimal. This chapter belongs to a growing body of literature on dynamic mechanism design. A related branch of literature studies the sale of durable goods, where the seller has a fixed number of durable goods and a deadline by which the goods must be sold. Gershkov and Moldovanu (2009) characterize the optimal online mechanism where buyers must be assigned an object at their arrival, and Board and Skrzypacz (2010) characterize the optimal mechanism when the buyers are patient. Pai and Vohra (2008) consider a model where buyers’ unobservable types include their arrival and departure times and give partial characterization of the optimal mechanism. In these papers the most important trade-offs are between extraction of rents and the option value—giving away an object before the deadline means that the object cannot be assigned to potentially higher-valuing buyers who arrive later. The option value decreases in time, however, since it becomes less likely that such buyers will arrive. In contrast to these papers, our model is designed to analyze the situation where the seller has a new set of goods every period, and one set of goods perishes every period. This situation creates a different set of trade-offs. Bergemann and Välimäki (2010) and Pavan, Segal, and Toikka (2011) give results for general classes of dynamic mechanisms, where both adverse selection and moral hazard issues could be studied. Pavan, Segal, and Toikka give the envelope formula for a large class of dynamic mechanism design questions and show how to compute transfers. One application of their general result is the sale of a durable good to buyers with changing types. Similarly to our results they show that the optimal mechanism is implementable

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with a non-standard auction with scoring rule, where the differences between bids affect the allocation. However, in their model, winning by a large margin is good and losing by a large margin bad, which is the opposite result from the implications of our model. Bergemann and Välimäki’s main result is the characterization of transfers that implement any efficient mechanism. All the papers noted above study the fully optimal mechanism. The pure calendar mechanism and waiting list mechanism are new to this thesis. The particular way we model partial commitment is also new, but analogous to renegotiation-proofness models—for example Hart and Tirole (1988). In this literature the mechanism-designer cannot offer a contract if it is known that it will be mutually beneficial to renegotiate this contract in the future. In our model buyers interact with the seller only once, which means that renegotiation is impossible, but the approach we take is the same: the seller with partial commitment power can give only those promises that she will not find it optimal to break in the future. For simplicity, we assume that the cost of commitment is exogenous4. Section 1.2 shows that most of the trade-offs of the problem can be studied in a simple illustrative example, with only one object and two new buyers every period. In particular, Section 1.2.1 characterizes the optimal pure calendar mechanism, Section 1.2.2 the optimal waiting list mechanism, and Section 1.2.3 the fully optimal mechanism. Section 1.2.4 introduces the partial commitment assumption, and shows how all the previous results generalize for partial commitment. Section 1.3 generalizes the results for an arbitrary number of service slots and random arrivals. Section 1.4 concludes and discusses the limitations of the model. Proofs are in Appendix A. 4There

buyers.

would be several interesting ways to endogenize it, including reputation or direct compensation to

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1.2. Illustrative example In this section we discuss an example of the model with just one object and two new buyers each period. The results in this section are special cases of the general results in Section 1.3, and therefore formal statements and full proofs are omitted. A monopolistic seller has one service slot each period t ∈ {0, 1, 2, . . . }. It could be a time slot for providing service, a seat, or a perishable good produced on the same day. The seller maximizes expected discounted revenue, with discount factor δ ∈ (0, 1). Each period two new buyers arrive. Each buyer i has unit demand for the service, with independent private value vi ∈ [0, 1]. Value vi is a random variable with cumulative density function F and probability density function f (vi ) > 0 for all vi ∈ [0, 1]. Function (vi ) denotes the standard static virtual surplus function, which is assumed w(vi ) = vi − 1−F f (vi )

to be strictly increasing5. To simplify the notation, we rename the buyers so that buyer 2 is the one with higher value and buyer 1 with lower value. Denote the CDF6 of buyer i by Fi and corresponding PDF7 by fi . Note that we are not following this convention in the figures, where we allow v1 and v2 to be arbitrary. Buyer i with type vi who is served si periods after her arrival and pays pi gets payoff β si vi − pi . In particular, if she is served at the same period she arrives, she gets payoff vi − pi . Constant β ∈ (0, 1) is the same for all buyers and is referred to as buyers’ discount factor. We allow buyers’ discount factor β to be different from the seller’s discount factor δ.

5Monotone

hazard rate condition is a sufficient condition. (v ) = F (vi )2 , F1 (v1 ) = F (v1 )[2 − F (v1 )]. 2 2 7f (v ) = 2F (v )f (v ), f (v ) = 2[1 − F (v )]f (v ). 2 2 2 2 1 1 1 1 6F

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We concentrate on mechanisms where a buyer who is not served one period after arrival is never served. As argued in the introduction, this may be the case for several reasons— restriction on the mechanism, preferences, or exit process. It captures the situation where in the short run the dates are imperfect substitutes for the buyers, but in the long run the value decreases faster than a constant rate. Each buyer interacts with the seller only at arrival. This means that each period the two new buyers announce simultaneously their types, and the seller responds by giving both of them either instant service or promises about future service and requests transfers. A promise of future service could be a refusal of service or delayed service conditional on next period arrivals. A promise of delayed service is a complete description of the allocation that the buyer gets conditional on the future arrivals. In the illustrative example it is characterized ˆ ⊂ [0, 1]2 , such that if buyer i receives promise D ˆ she will be served in the by a set D next period if and only if the new buyers who arrive in the next period announce values ˆ This means that the expected payoff from promise D ˆ to buyer i with v = (v1 , v2 ) ∈ D. ˆ ˆ ˆ ˆc ˆ ˆ value vi is P r(D)βv i + P r(D )0 − pi = β dvi − pi , where d = P r(D). The set D can be interpreted as a contract or a ticket that the seller can give to a buyer one period before the service in exchange for money. A state captures all relevant details from the history up to a particular period. In particular, it characterizes the promises of delayed service given to buyers in the previous period. In the illustrative example a state is characterized by set D ⊂ [0, 1]2 , which includes all the combinations of types v for which the instant service is unavailable. It is clear that a state is equivalent to a promise given to a buyer in the previous period—the

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current object is unavailable because it was promised to a buyer in the previous period.8 ˆ Unless explicitly stated otherwise, we denote the state by D and a contract by D. The goal of the mechanism design problem is to find mechanisms that are Bayesian incentive-compatible and ex-ante individually rational. As it turns out, all mechanisms that we will discuss below are implementable in dominant-strategy incentive-compatible and ex-post individually rational transfers. As an illustration of possible dynamics, consider the following example. In period 0 the seller has not made any promises, so the initial state D0 = ∅. Two new buyers arrive and announce their types. The seller serves one of them instantly (at period 0) and promises the other that she will be served in the next period if and only if all new buyers announce ˆ 0 = [0, v]2 , which is the state in period values below v. This is described by a contract D ˆ 0 = [0, v]2 . Now, in period 1 two new buyers arrive and announce their values 1, D1 = D v = (v1 , v2 ). If v ∈ D1 , the seller has to serve the buyer from the previous period and therefore cannot serve any of the new buyers instantly, whereas if v ∈ / D1 , the instant service is available. When the seller does not offer delayed service to either of the new buyers, we return to the initial state. One possible mechanism in this framework is the repetition of the static mechanism, where the good is always sold to one of the buyers arriving in the particular period. In other words, the seller never gives promises of delayed service, so the state is D = ∅ in all periods. By Myerson (1981) the optimal static mechanism assigns instant service to the

8There

is a minor abuse of notation here, since it is possible that two of the buyers from the previous period received promises of delayed service, say in sets D1 and D2 respectively. At the optimum this happens with zero probability, but is feasible when D1 ∩ D2 = ∅ and D1 ∪ D2 ⊂ [0, 1]. In this case we can simply redefine the state as D = D1 ∪ D2 .

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buyer with the higher value if his virtual valuation is positive, and refuses service to both buyers otherwise. This allocation rule is illustrated by Figure 1.1.

v2 1

(R, I)

p∗

(R, R)

0 0

(I, R)

p∗

1

v1

Figure 1.1. Optimal static allocation

The optimal static mechanism can be implemented by the second price auction with ∗

(p ) reserve price p∗ such that w(p∗ ) = p∗ − 1−F = 0. This ensures expected revenue f (p∗ ) R1 π s = p∗ w(v2 )dF2 (v2 ). This mechanism will typically not be optimal in a dynamic setting.

1.2.1. Pure calendar mechanism A pure calendar mechanism is a mechanism where the promise of delayed service is unconditional on next period arrivals. It is like marking dates on a calendar and then providing services according to this calendar, or selling tickets that guarantee the service. This means that a promise of delayed service needs to be fulfilled with certainty, and the

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seller cannot serve anyone instantly when a buyer from the previous period was promised delayed service. In the notation introduced above, the promise of delayed service can be only D = ∅ or D = [0, 1]2 , which correspond to probabilities 0 and 1 respectively. There are only two possible states: the instant service is either available, or it is already promised to someone and is therefore unavailable. To shorten the notation, we denote the first state by 0 and the second by 1, both of which correspond to the promises given to buyers in the previous periods. The pure calendar mechanism assumption also means that the seller cannot personalize the promise of delayed service depending on the type of buyer to whom it is promised. In the next subsections we relax both assumptions. First, the waiting list mechanism relaxes the assumption that the promises have to be unconditional on the next period arrivals; later, the fully optimal mechanism also allows personalized promises. We will now characterize the optimal allocation rule. There are two cases to consider. First is the interior case, where the seller sometimes promises delayed service. Second is the corner solution, where delay is never promised. The latter case is therefore a static mechanism design problem that we already know how to solve. We start with the interior solution and will later discuss the corner solution. Suppose that the parameters are such that the seller promises delayed service for some types of buyers. There are two states—the instant service is either available, or it was promised to a buyer who arrived in the previous period and is therefore unavailable. Denote the revenue from the state when the instant service is available by π(0) and when

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it is unavailable by π(1). Clearly, the expected revenue from the former is higher than from the latter. Denote this difference by ∆ = π(0) − π(1) > 0. The difference in revenues in the two states is an opportunity cost of a promise one period ahead. In particular, if the seller promises delayed service to a current buyer, the discounted loss is

δ∆ . 1−δ

The denominator 1 − δ is needed to compare the difference in

continuation values, ∆, to the difference in flow values, which is the revenue extracted from current buyers. Consider the state where instant service is unavailable. Then the only object that can be allocated is a promise of next-period service. Buyers’ values of the object are βv1 and βv2 , and allocating this object costs

δ∆ 1−δ

to the seller. The optimal mechanism therefore

assigns the object to the buyer with the higher value, if his discounting-adjusted virtual surplus βw(v2 ) is greater than the opportunity cost

δ∆ . 1−δ

It is illustrated by Figure 1.2(a).

This allocation rule can be implemented by a standard second price auction for the delayed service with reserve price v, such that βw(v) =

δ∆ . 1−δ

Consider now the state where instant service is available, so that the seller sells two goods to two buyers. The instant service is more valuable to both buyers and costless to allocate, whereas allocating delayed service changes the state exactly as in the case studied above and has the same opportunity cost

δ∆ 1−δ

as above. This means that the

instant service is allocated often, and the delayed service is promised only if both buyers have sufficiently high values. This is a demand-smoothing argument. When in some period both buyers have high values, the demand is high. Since it is unlikely that in the next period the values would

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v2

v2

1

1

(D, I)

(R, D) v

v

(I, D)

(R, I)

p∗ (R, R)

(D, R) (R, R)

0 0

v

(a) Instant service unavailable

1

v1

0

(I, R)

p∗

0

v

1

v1

(b) Instant service available

Figure 1.2. Optimal pure calendar mechanism

be as high, it is optimal to extract more revenue from the current high-valuing buyers and lose some revenue from tomorrow’s buyers (whose values are uncertain). In particular, the optimal mechanism allocates the instant service to the buyer with higher value when his virtual surplus is positive. It promises delayed service to the buyer with lower type, when his discounting-adjusted virtual surplus exceeds the opportunity cost. This allocation rule is illustrated by Figure 1.2(b). This allocation rule is implemented by a simple two-stage auction mechanism. At the first stage, the instant service is allocated at a standard second price auction with reserve price p∗ . In the second stage, the loser is offered the next period service at a fixed price βv. When the loser accepts the offer, the winner of the first-round auction gets a discount β(vl − v) > 0, where vl is the loser’s bid in the auction.

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Offering the delayed service to the loser of the second price auction creates surplus for the loser. This means that the winner must be compensated, since losing is not as bad as it would be without the second stage. If types of buyers are v2 ≥ w1 ≥ v, the high type 2 would prefer winning to losing in the first round if compensation p is high enough to satisfy v2 − v1 + p ≥ β(v2 − v). In particular, in the critical case when v1 = v2 , we get the compensation given above, whereas if v2 > v1 , this compensation is sufficient and even gives some extra surplus to the buyer. Note that the threshold v that we used to determine whether or not to allocate the delayed service is the same in both states. The opportunity cost of a promise comes from being in a less restrictive state in the next period. This cost is independent of the type who receives the delayed service. The discussion above took the opportunity cost

δ∆ 1−δ

as given, but the difference ∆

between the two states is endogenous and yet to be determined. The difference in revenue in the two states can be decomposed to two parts. First, if the instant service is available, it is assigned to the higher-valuing buyer whenever her virtual valuation is positive. This is exactly the same as static optimum and thus gives the static revenue π s . The second difference is dynamic and comes from the promises of delayed service. When the instant service is unavailable, the delayed service is promised to the higher buyer whenever his discount-adjusted virtual surplus βw(v2 ) is higher than the opportunity cost, which is equal to βw(v). In this case the seller gets the difference. This gives R1 expected dynamic flow revenue π2 (v) = v [w(v2 ) − w(v)]dF2 (v2 ). In the state where the instant service is available, the delayed service is offered under the same condition, but

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only to the lower-valued buyer. This ensures expected dynamic flow revenue π1 (v) = R1 [w(v1 ) − w(v)]dF1 (v1 ), which is smaller than π2 (v). v Using these observations, we can express the opportunity cost

δ∆ 1−δ

in two ways as a

function of v to get equation (1.1) that uniquely determines v and therefore ∆, assuming that the interior solution exists.

(1.1)

βw(v) =

δ∆ = δ (π s + β[π1 (v) − π2 (v)]) . 1−δ

We can now address the corner solution. To see why equation (1.1) may not have a solution in v ∈ [0, 1], consider the case when the buyers’ discount factor β is very low compared to the seller’s discount factor δ. With sufficiently small β, the value of delayed service is negligible to the buyers and therefore it is never optimal to offer delayed service. In this case the optimal calendar mechanism is static. We already know that the optimal static mechanism is the second price auction with reserve price p∗ , which guarantees revenue π s . Suppose now that β is slightly higher, so that the seller is exactly indifferent between offering delayed service to a very small set of potential buyers and using the optimal static mechanism. The incentive to offer delayed service is clearly highest for highest types. In particular, the gain from offering delayed service to a buyer with type vi = 1 would be β, but the cost would be lost revenue π s next period, with discounted value δπ s . This gives β ≥ δπ s as a necessary and sufficient condition for optimal delay. If β/δ > π s , it is optimal to offer delayed service with positive probability, whereas when β/δ < π s the optimal pure calendar mechanism is always static.

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1.2.2. Waiting list

Another commonly used method to allocate future service is a waiting list. In this chapter we model the waiting list as a restriction to promises, where the seller can choose to delay service conditionally to some buyers but cannot discriminate between two types of buyers who are offered delayed service. In particular, the waiting list assumption means that at any state D, possible promises are such that each buyer gets either instant service, no service, or a promise of delayed service in the next period, when the new arrivals are in ˆ ⊂ [0, 1]2 . The waiting list assumption is that this set is the same for all some fixed set D ˆ be different for all states D buyers who are offered delayed service. Note that we allow D or even for all periods. There are potentially many ways for how a waiting list could be organized. The interpretation of our assumption is that people who wait are put on a list, and once the person is on this list in a certain position, the type is forgotten and the process becomes the same for all types buyers who wait in the same position. In the case when the number of objects is one, the optimal waiting list will include at most one individual, which is a very simple list. In the general model we will analyze in Section 1.3 the optimal waiting list includes up to m individuals. We have already studied two cases of waiting lists. At one extreme, when the promise for delayed service is always the empty set, we get the static mechanism. At the other extreme, where the promise of delayed service is unconditional on arrivals, we get the pure calendar mechanism. As we will show below, the pure calendar mechanism can be the optimal waiting list mechanism with certain parameters of the model, whereas static

28

ˆ mechanisms will never be optimal. In particular, we show that the optimal contract D always includes the set9 [0, p∗ ]2 as a strict subset. We will show that the optimal waiting list mechanism has some interesting properties. Under some combinations of parameters10, the optimal waiting list mechanism is implemented by non-standard auctions with scoring rule where the services that buyers are offered will depend on how similar their bids are to their opponent’s bids. In particular, it is sometimes better to win by a little than by a lot, and it is sometimes better to lose by a lot than by a little. After we have characterized the optimal mechanism we will make these statements precise, but the general intuition is that distortions from serving a buyer from a past period depends on the values of both new buyers. Therefore whether or not the seller uses instant service to serve new buyers or a buyer from the previous period will depend on the values of both new buyers. Another interesting observation will be the dynamics of waiting list, which is non-trivial. It is possible that the optimal contracts are different each period. The distribution of buyers who receive the promise of delayed service will depend on the contract given to a buyer in the previous period. It is possible that in each period this contract is different from the previous contract and therefore the new optimal contract is also going to be different. As long as the new arrivals are such that one of them receives delayed service, the mechanism could therefore be in a new state every period. The characterization of the optimal waiting list proceeds in five steps. Initially we assume that the expected discounted revenue at each state D, denoted by π(D), is welldefined. We start by characterizing the optimal allocation rule for each state D when the ∗

(p ) that p∗ is defined so that the virtual surplus w(p∗ ) = p∗ − 1−F f (p∗ ) = 0. 10For example when buyers’ discount factor β equals the seller’s discount factor δ.

9Remember

29

ˆ is fixed. This gives us the allocation rule for contract offered at this state, denoted by D, ˆ The second step is to characterize the each v = (v1 , v2 ) at each state D for a given D. ˆ when the probability of service, dˆ = P r(D), ˆ is known. The third step optimal contract D ˆ For this it is convenient to optimize with respect to is to find the optimal probability d. another variable b, which we define later and that describes the maximum distortion from giving a promise. The fourth step is to use the first order condition with respect to b and the characterization of optimal allocation to fully characterize the optimal promise and allocation rule or to observe that we get the corner solution. The fifth and final step is to show that the functions π(D) exists, which will be verified using the contraction mapping theorem. Once we have characterized the optimal waiting list mechanism, we discuss its implementation and the properties mentioned above. Before continuing with the characterization we can make a simple observation. Note that the seller never wants to offer service to buyers with negative surplus. This means ˆ such that P r(D) ˆ < F2 (p∗ ) can never be optimal. Instead of this contract, that a contract D ˆ 0 = [0, p∗ ]2 , which ensures strictly higher revenue from all the seller could offer contract D current buyers who are promised this contract and is not restrictive in the next period. ˆ for optimal Therefore in the following analysis we assume that dˆ ≥ F2 (p∗ ) and [0, p∗ ]2 ⊂ D ˆ contract D. Step 1: allocation rule. Suppose that the expected revenue function π(D) is a known function and denote π(d) = maxD:P r(D)≥d π(D). Let the state be D and suppose that ˆ with corresponding probability the optimal contract of delayed service in this state is D, ˆ Suppose that the two buyers who arrived announced values v = (v1 , v2 ), dˆ = P r(D). where we reorder the values so that v1 ≤ v2 .

30

When buyers announced types v ∈ D, the seller has to serve the waiting buyer. This means that the only remaining question is when to allocate the delayed service with ˆ and when not to do so. The trade-off is analogous to the state where the contract D instant service was unavailable in the pure calendar mechanism. The seller has one object, ˆ 1 and β dv ˆ 2. ˆ to allocate. The value of the contract D ˆ to buyers is β dv namely contract D, The cost of allocating the contract is the opportunity cost that comes from being in a more restrictive state in the next period. In particular, the discounted opportunity cost is

δ [π(∅) 1−δ

ˆ The optimal allocation rule gives the contract D ˆ to the buyer with − π(D)].

higher valuation if and only if his discount-adjusted virtual valuation is higher than the opportunity cost. Equivalently, there is a threshold v so that the higher type receives promise of delayed service if and only if his type is higher than the threshold, v2 > v. The threshold v must be such that it equalizes the discount-adjusted virtual surplus to the opportunity cost, which gives equation (1.2). ˆ = β dw(v)

(1.2)

δ ˆ [π(∅) − π(D)]. 1−δ

When the new buyers have types v ∈ / D, the seller has both instant service and delayed service to allocate to them. The trade-offs are analogous to the state when the instant service is available in the pure calendar mechanism case. Again, instant service is more ˆ has valuable for both buyers and costless to produce, whereas promising the contract D an opportunity cost

δ [π(∅) 1−δ

ˆ Therefore the buyer with higher value receives the − π(D)].

instant service if and only if her virtual surplus is positive, and the buyer with lower value gets the delayed service if and only if her value is above v, which satisfies (1.2).

31

The new question is compared to the pure calendar mechanism case is the choice of ˆ Since buyers’ values depend only on probability of getting the service, optimal contract D. ˆ This is as if the seller describes we can first compute the optimal probability dˆ = P r(D). ˆ and there is agreement that only the probability of service dˆ instead of the full contract D the promise is going to be fulfilled in the next period. Then in the next period the seller ˆ according to which serve the waiting has dˆ as the constraint and chooses optimal set D customer. ˆ was given optimally Step 2: fulfilling promises optimally. Suppose that contract D to previous a period customer. Remember that this implies the probability of service dˆ ≥ F2 (p∗ ). From the allocation rule above we can calculate the expected revenue for each v in both of the two cases discussed. Denote the expected revenue from v such that the instant service is unavailable by π(v) and the expected revenue when the instant service is available by π(v). The difference between the two, π(v) − π(v) can be interpreted as the ˆ The expected revenue from probability dˆ opportunity cost of adding v to the contract D. is therefore ˆ = π(d)

max

ˆ r(D)≥ ˆ dˆ D:P

n h i h io ˆ v π(v)|v ∈ D ˆ v π(v)|v ∈ D ˆ + (1 − d)E ˆc dE

= Ev [π(v)] − dˆ

min

ˆ r(D)≥ ˆ dˆ D:P

h i ˆ . Ev π(v) − π(v)|v ∈ D

ˆ must include points where the distortion from including It is clear that in optimum D ˆ computed as π(v) − π(v), is the smallest. It is easy to see this profile of promises to D, that when v2 ≥ p∗ , distortion π(v) − π(v) is strictly increasing in v2 , and therefore the constraint must be binding.

32

ˆ is characterized by the upper bound for the distortion that we Then the contract D denote by b. That is, the optimal way to fulfill promised probability dˆ is to serve the buyer from the previous period if and only if the distortion from doing so is less than b. ˆ ˆ = {v : π(v) − π(v) ≤ b} for b such that P r(D) ˆ = d. Therefore we can write11 D Step 3: optimal probabilities of delayed service. The third step is to find the optimal probability dˆ that will be promised to the current buyers. We have already discussed ˆ = {v : π(v) − π(v) ≤ b} for b such that how the promise will be fulfilled—in the set D ˆ ˆ = d. P r(D) ˆ we have to find how it Before optimizing with respect to the probability of delay d, affects the expected revenue. An increase in probability dˆ has two opposite effects on revenue. First, the seller extracts a higher proportion of discounting-adjusted virtual surplus βw(vi ) from each buyer who receives the promise. Second, the contract will be more restrictive and therefore it decreases the continuation revenue whenever delayed service is promised. To study both effects we have to know the probability of assigning the contract to someone and the distribution of buyers’ types who receive the promise. We already showed that inside set D the delayed service is promised to the high type and outside the set D it is promised to the low type. In both cases the condition to assign the delayed service is that the corresponding type must be higher than threshold v. Let us

ˆ as fixed. In optimum it depends on the state that this analysis took the contract offered in state D ˆ ˆ with D, but as a necessary condition for optimality, the analysis is still valid. Suppose that at the state D ˆ ˆ does not have the form described above. Then we could increase the ˆ the set D the optimal contract D revenue slightly while keeping the probability of service fixed. Because of the envelope theorem the effect ˆˆ through the change in optimal contract D is negligible.

11Note

33

define probability density function12 gv (·|D) as Z gv (vi |D) =

vi

Z

1

1[(vi , v2 ) ∈ / D]f1 (vi )f2 (v2 )dv2 .

1[(v1 , vi ) ∈ D]f1 (v1 )f2 (vi )dv1 + vi

0

Let the corresponding cumulative density function be Gv (vi |D) =

R vi 0

gv (vi0 |D)dvi0 .

The distribution Gv that we defined here plays a crucial role in the analysis. Remember that the seller cannot discriminate between buyers who receive the delayed service contract. The distribution Gv characterizes the distribution of the type of buyer who receives the delayed service contract. It is a mixture of f1 and f2 , the distributions of high and low type, where the precise mixture depends on the current state D. In particular, if the seller did not give any promises in the previous period, D = ∅, then it is the same as the distribution of low type, whereas if the seller promised delayed service with certainty (as in pure calendar mechanism), it is the distribution of high type. From the allocation rule above we know that the seller assigns the delayed service contract to a corresponding buyer if and only if her type vi > v. This happens with probability 1 − Gv (v|D). The expected revenue extracted from the buyer who receives the R1 ˆ i )dGv (vi |D). promise of delayed service is now v β dw(v ˆ and in We can now characterize expected revenue function for a given contract D ˆ It includes two parts. First, the static component of the revenue particular, probability d. ˆ is simply that is independent of D (1 − δ)(1 − d)Ev [max{0, w(v2 )}|v ∈ / D] + δπ(∅).

12The

function gv is a PDF, since by construction gv (vi |D) ≥ 0 for each vi and P r(D) + P r(Dc ) = 1.

R1 0

gv (vi |D)dvi =

34

ˆ i ) from each buyer who Second, the dynamic part, where the seller extracts β dw(v ˆ in continuation revenue. This gives receives delayed service, but loses δ[π(∅) − π(D)] (1 − δ)β dˆ

Z

1

ˆ w(vi )dGv (vi |D) − δ[1 − Gv (v|D)][π(∅) − π(D)].

v

The last step before optimization is a change of variables. Instead of working with ˆ it is more convenient to work with maximum distortion b that characterizes probability d, ˆ We will first argue why we can do it and then ˆ and therefore probability d. the contract D ˆ more conveniently using maximum distortion b. show how to compute dˆ and [π(∅) − π(D)] ˆ = {v : π(v) − π(v) ≤ b} for b such Remember that for each dˆ ≥ F2 (p∗ ), we have D ˆ The distortion function π(v) − π(v) is strictly monotone in v2 and the ˆ = d. that P r(D) ˆ This means that there distribution is continuous, so b must be strictly increasing with d. is one-to-one relationship between dˆ ∈ [F2 (p∗ ), 1] and b which will be13 in [0, 1 − δ]. The value of b that corresponds to dˆ can be interpreted as the maximum distortion that the ˆ Instead of choosing optimal d, ˆ seller must incur in the next period to fulfill the promise d. it is convenient to optimize with respect to b. In particular, let Gb (b) = P r(π(v) − π(v) ≤ b). Then dˆ = Gb (b). Moreover, we also ˆ Compared to non-constrained optimization problem know how to compute π(∅) − π(D). ˆ forces the seller to serve the previous buyer at set D ˆ and therefore at state ∅, state D lose π(v) − π(v) at all these realizations. A convenient way to compute this value is to R ˆ = b bdGb (b). integrate over Gb . We get that π(∅) − π(D) 0

13Easy

to verify: distortion is minimal if both virtual surpluses are close to 0, which gives lower bound 0; and distortion is maximal if both virtual surpluses are 1, which gives 1 − δ.

35

We can therefore rewrite the contract design problem in terms of b to obtain the R1 dGv (vi |D) following maximization problem, where EGv [w(vi )|vi ≥ v] = v w(vi ) 1−G is the v (v|D) expected surplus of a buyer who receives the delayed service contract. ( max b∈[0,1−δ]

Z

(1 − δ)βGb (b)EGv [w(vi )|vi ≥ v] − δ

)

b

bdGb (b) , 0

which gives a first order condition14

(1.3)

(1 − δ)βEGv [w(vi )|vi ≥ v] − δb = 0

Equation (1.3) balances the trade-off from extracting slightly more revenue from the ˆ and the loss from added distortion in the next average type who receives the contract D period. If the first term is large enough to be higher than b even at its upper bound 1 − δ, we get the corner solution where b = 1 − δ or equivalently dˆ = 1. Since the first term is non-negative and the lower bound of b is 0, we can never have the lower corner solution. Step 4: finding v and b. The fourth step of the analysis is to combine the results above to characterize the interior and corner solutions of the optimal mechanism design problem. We have now characterized the optimal mechanism at state D with two variables, the ˆ and threshold v that characterizes the opportunity maximum distortion b from contract D ˆ The two variables are related by equations (1.2) and cost of promising the contract D. (1.3), which can be rewritten as (1.2’) and (1.3’) respectively.

(1.2’)

14Note

Rb bdGb (b) δb δ δb δb 0 w(v) = = EGb ≤ , 1 − δ βGb (b) β(1 − δ) β(1 − δ) β(1 − δ)

that we can ignore the effect though v by using the envelope theorem.

36

δb = EGv [w(vi )|w(vi ) ≥ w(v)] . β(1 − δ)

(1.3’)

The first condition (1.2’) characterizes the trade-off between promising a delayed service ˆ to marginal type v and some revenue next period. In particular, the opportunity cost of D is the average distortion it creates next period. The distortion has conditional distribution function

gb (b) , βGb (b)

which gives us the condition. The second condition (1.3’) characterizes

ˆ slightly and therefore extracting slightly more the trade-off between increasing the set D ˆ and the additional distortion this revenue from average type who receives the promise D increase creates. Both equations (1.2’) and (1.3’) describe strictly increasing continuous relationships between

δb β(1−δ)

and w(v), so the remaining question is when these two equations can

and cannot be simultaneously satisfied. Note that by assumptions w(v) ∈ [0, 1] and b ∈ [0, 1 − δ]. If the solution to these equations is such that b > 1 − δ, then it means that the revenue is strictly increasing for each b, so that we get the corner solution where b = 1 − δ or equivalently dˆ = 1. Depending on the ordering of β and δ we get one of two cases illustrated by Figure 1.3. Let’s first consider the case when the seller’s discount factor δ is not smaller than the buyers’ discount factor β. Intuitively, when β is close to 0, we should find that it is optimal ˆ This case is illustrated by Figure 1.3(a). In this case there is always to choose very low d. an interior solution (b, v). Indeed, when β is very low, the solution has to be such that b is ˆ close to [0, p∗ ]2 . very small, which corresponds to dˆ close to F2 (p∗ ) and therefore D ˆ includes strictly more Note that whenever β > 0 the optimal b > 0, and therefore D values than just [0, p∗ ]2 . To see why this is the case suppose that the proposed contract is

37

δb β(1−δ) δ β

δb β(1−δ)

δEGb b β(1−δ)

1

δEGb b β(1−δ)

δ β

1 EGv w(v) EGv w(v)

1

w(v)

(a) Case δ ≥ β: only interior solutions.

1 w(v)

(b) Case δ < β: corner solution possibility.

Figure 1.3. Plots illustrating the characterization of the optimal waiting list mechanism

[0, p∗ ]. By increasing the set marginally and therefore increasing the probability marginally, the seller extracts additional revenue from all types of buyers who receive the promise of delayed service, which is strictly positive gain. The loss from this change is that the seller loses an option to offer instant service next period to the types of buyers who were added to the set. But these buyers have virtual valuation close to 0, which means that the lost revenue is close to 0. Consider now the case when δ < β, which is illustrated by Figure 1.3(b). Now there are two possibilities. Either there is an interior solution (b, v) satisfying both equations, or we get the corner solution where the contract ensures service in the next period with certainty. If this is the case both at states D = ∅ and D = [0, 1]2 we get the calendar mechanism studied in Section 1.2.1.

38

The condition for getting the corner solution is relatively simple. Let v cm be the threshold used to offer delayed service in the calendar mechanism case. The average discount-adjusted virtual surplus of a buyer who receives a promise of delayed service with threshold v cm is βEGv [w(vi )|vi ≥ v cm ]. If this value is higher than δ, then even at the highest possible distortion b, the revenue would be increasing in b, so we would be in the corner solution. Note that by definition, βw(v cm ) =

δ [π(∅) 1−δ

− π([0, 1]2 )] =

δ E [b], 1−δ Gb

which gives the formal condition for the corner solution in (1.4). (1.4)

δ ≤ βEGv

δEGb [b] w(v) w(v) ≥ = βEGv [w(v) |v ≥ v cm ] β(1 − δ)

Step 5: the existence of π(D). We show that the function π(d) exists. Since in the characterization above, only the continuation values in terms of probabilities were needed, this is sufficient to characterize π(D) for arbitrary state. For a given function π(d) the characterization above defines expected revenues for each possible state and therefore for each probabilistic promise d0 . Let T π(d0 ) denote the profit for any given d0 . We want to argue that there is a unique fixed point π = T π. We use Blackwell’s sufficient conditions to argue that T is a contraction with rate δ and then apply the Contraction mapping theorem to get the existence result. First, monotonicity. If π ˆ (d) ≥ π(d) for all d, then any allocation rule with continuation value function π ˆ ensures higher revenue than with π and therefore the optimum cannot give lower expected revenue. Second, to verify discounting, suppose π ˆ (d) = π(d) + a. Now, since the dynamic part of the revenue depends only on π(0) − π(d), the optimal allocation is unchanged. The only remaining dependence is through static continuation value δˆ π (1) and therefore Tπ ˆ (d) = T π(d) + δa for all d.

39

Implementation. We have now showed how to choose optimal probability dˆ of delayed ˆ ˆ that fulfills probabilistic promise d, service at each state D, how to design a contract D ˆ Combining the and how to allocate the promises at each state D with a given contract D. results we can describe the shape of optimal contracts and conclude how to implement the optimal mechanism. There are two possible types of allocation rules in terms of qualitative results. One ˆ is characterized by a simple threshold rule, D ˆ = [0, v]2 possibility is that the contract D as described by Figure 1.4(a). This threshold is defined such that (1 − δ)w(v) = b. It is optimal to use this type of contract only if optimal b and therefore probability of service is small enough. The second possibility is slightly more complex contract described in Figure 1.4(b). Next we will discuss each case separately.

v2

v2

1

1

(D, I) (R, I)

(R, I)

(I, D)

(D, I) (I, D)

(R, D)

v

v (D, R) (R, R)

(R, R) (I, R)

0

v

0

(a) Low β/δ

(I, R)

1

v1

0

v

0

(b) High β/δ

Figure 1.4. Optimal waiting list mechanism

1

v1

40

Before studying the two cases, note that from step 4 of the analysis and in particular from Figure 1.3 we see that

δb β(1−δ)

≥ w(v). Using the definition of v we get βw(v) ≤ δw(v).

Therefore v can be higher than v only if buyers’ discount factor β is sufficiently larger than seller’s discount factor δ. A sufficient condition for being in the “high β/δ” region is when β ≤ δ, whereas when β is close to 0 we are clearly in the low “β/δ” region. In the first case, when β/δ is relatively low (Figure 1.4(a)), the optimal mechanism promises delayed service with relatively low probability and it is possible to fulfill this ˆ = [0, v]2 . In this case promise in a region characterized by a simple threshold v, so that D the implementation of the mechanism is analogous to the pure calendar mechanism, but with a different reserve price. The mechanism is the following: In the first stage the instant service is sold at a second price auction with reserve price v. If neither of the buyers bid above this value, the waiting buyer is served and new buyers are refused service. However, if both buyers bid above the reserve price, the loser is offered delayed service with price ˆ If he accepts the offer, the winner will be given discount β d[v ˆ l − v], where vl is the β dv. loser’s bid in the second price auction. The reason for this discount is the same as in pure calendar mechanism case—the second stage creates surplus for the loser of the auction, therefore extra surplus must be given to the winner to avoid bid shading. In the second case, when β/δ is relatively high (Figure 1.4(b)), the optimal contract is more complex than a single threshold. The exact transfers can be computed from ex-post envelope condition. The optimal mechanism in this case will be implemented by a non-standard auction with scoring rule, where whether or not buyers receive instant or delayed services depends not only on their own types and order of types, but also on the similarity of their types.

41

In particular, there are some types of buyers who receive delayed service unless the other buyer has only slightly smaller value (in this case they get instant service). There are also types of buyers who receive delayed service unless the other buyer has only slightly higher value. Figure 1.5 focuses on these types of buyers. We will now discuss the reasons for these effects. v2 1

D

R

0 0

D

v 0 v 00

I

1

v1

Figure 1.5. Allocation given to buyer 1 under optimal waiting list mechanism when β/δ is not small

Some types of buyers get instant service only if they win by a close margin, receiving delayed service otherwise. An example of such type is v 00 in Figure 1.5. This can be interpreted via opportunity costs. Suppose one buyer has value v 00 and the opponent has negative virtual surplus. In this case it it is optimal to serve the waiting buyer and offer the high type delayed service. This is a distortion to current optimum—instead of serving a buyer with positive virtual surplus instantly the seller is offering him delayed service. But ex-ante, in the previous period, this distortion was low enough to include it in the

42

optimal contract that was given to the waiting buyer. Now, when the value of the lower type increases, then at some point there will be extra distortion from serving the waiting customer—in addition to delaying service to buyer with v 00 , the seller is losing the value potentially extracted from low type. In other words, in this case delaying the service to high type comes with increased opportunity cost of not being able to serve the low type. Therefore when the opponent’s value gets close enough to v 00 , the waiting buyer is not served and instead both new buyers are served, and the one with high value gets instant service. Of course, when the opponent’s value rises even above v 00 , the opponent will be the one receiving instant service and therefore the buyer with v 00 again gets delayed service. There are also some types of buyers who are refused service only if they lose by a close margin, but who receive delayed service otherwise. An example of such type is v 0 in Figure 1.5. The reason is similar. If a buyer has type v 0 and the opponent has even lower value, the distortion is low enough so that the seller serves the waiting buyer and offers the higher of the types delayed service. That is, when the opponent’s value rises above v 0 , the buyer with v 0 will be refused service, since delayed service goes to the opponent and there are no more goods available. Now, when the opponent’s value raises even higher, the total distortion increases and at some point becomes high enough so that the seller will serve not the waiting buyer but both new buyers instead. Therefore the buyer with v 0 will get delayed service even when being the low type. Finally note that this mechanism has potentially non-trivial dynamics in the following sense. Suppose the initial state is such that no promises have been given, D0 = ∅. Then the seller designs some optimal contract D1 that is offered to some types of buyers, which will always be the lower of the two. Now, at the next period the state is D1 , which is

43

different from D0 . Therefore the types of buyers who now receive a promise of delayed service come from a different distribution, which is a mixture of high and low types of buyers, characterized by a function Gv (vi |D1 ). The optimal contract is now different, say D2 . If this contract is given to a new buyer, in period 2 the state is D2 , which is again different from D0 and D1 . Continuing the same way, we get a sequence of promises D1 , D2 , . . . , which are all different and optimal, given that there has been a sequence of arrivals such that one buyer always got a promise of delayed service.

1.2.3. Fully optimal mechanism In this section we analyze the fully optimal mechanism, where the seller can choose any allocation rule she might like. Compared to the waiting list this means removing the last remaining restriction—at each state with any vector of types the seller is now allowed to give the optimal promise of delayed service to each buyer. The relaxation of this assumption affects the mechanism design problem. The fully optimal mechanism is similar to the optimal waiting list with the exception that the promise of delayed service will depend on the buyer’s type who receives the promise. In particular, the probability of receiving the service in the next period is an increasing function of the type. This simplifies the analysis, since instead of designing a contract that works for the average buyer who gets the contract, the seller can now design a separate contract for each type. There are some straightforward observations that make the analysis simpler. First, the seller will promise delayed service to only one buyer15. Therefore a state is still simply 15Instead

of promising D1 and D2 two two current buyers, promising D1 ∪ D2 to the higher of them would increase flow revenue while keeping .

44

D, the promise given to one of the previous buyers. Second, we can again separate the ˆ and contract D ˆ which optimal assignment decision to probability of delayed service d, will be optimal given this probability. Note that initial state may be sub-optimal, but continuation states are always optimal. Even if the initial state is such that there are two buyers waiting, their transfers are already sunk, so we can treat of them as a single buyer. We proceed by characterizing the fully optimal mechanism in four steps. First we take the expected revenue functions π(d) as given and characterize the optimal allocation rule. ˆ which at The optimal allocation rule will now include the optimal promise function d(·), this point is characterized as a function of continuation values. Second, we discuss how the probabilities of delay are fulfilled optimally. Third, we characterize the optimal promise ˆ i ) explicitly and discuss its properties. Finally, we use contraction mapping function d(v theorem to show that functions π(d) exist. Since the analysis is similar to the waiting lists, we are omitting some of the details and emphasizing the differences. Step 1: allocation rule. Again, suppose π(D) and π(d) functions exist and fix a state D. As before, we denote the optimal contract of delayed service offered to a buyer in the ˆ (which will then be the state in the next period). Since the buyers current period by D are interested in probability of service, we are again characterizing the optimal promises ˆ and later will describe how the seller optimally fulfills in terms of probability dˆ = P r(D) these promises. Suppose the new buyers announce types v ∈ D. Then the seller has to serve the previous buyer and therefore can offer delayed service to only one of the buyers. This is ˆ The optimal choice of dˆ offered to the highest of the two buyers with some probability d.

45

solves the following maximization problem n o ˆ 2 ) − δ[π(0) − π(d)] ˆ , π(v) = max δπ(0) + (1 − δ)β dw(v dˆ

where the first component is the static continuation revenue that the seller gains by not giving any promises; the second term is the revenue extracted from the high type by giving a promise of service with probability dˆ in the next period; and the third term is the loss in continuation value from giving this promise. The revenue extracted from the high type is now his standard static virtual surplus w(v2 ) adjusted by discount factor β and probability ˆ The first order condition from the maximization problem is d. ˆ ≥ 0. (1 − δ)βw(v2 ) + δπ 0 (d)

(1.5)

We will come back to the function dˆ once we know how to compute π(d) and differentiate ˆ 2 ) is only a function of v2 and it balances the trade-off from it. From (1.5) we see that d(v extracting revenue from the buyer and having higher opportunity cost because of being in ˆ = 0 whenever it is promised a more restrictive state in the next period. In particular, d(v) to a buyer with negative virtual surplus and dˆ ≥ F2 (p∗ ) for all v2 > p∗ . When buyers announced v ∈ / D, the instant service is available, so that the higher of the new buyers is served instantly if and only if he has strictly positive virtual surplus, ˆ that solves the whereas the lower new buyer is offered delayed service with probability d, following optimization problem n o ˆ ˆ π(v) = max (1 − δ) max{0, w(v2 )} + δπ(0) + (1 − δ)β dw(v1 ) − δ[π(0) − π(d)] . dˆ

46

There are two differences with the previous maximization problem. First, the static revenue is increased, since some revenue is extracted from serving the high type instantly. Second, the delayed revenue comes now from the low type rather than the high type, which means lower dynamic revenue in expectation. The first order condition is the same as previously given with (1.5), but for v1 instead of v2 . Therefore the optimal promise is ˆ i ) in both cases; the difference is that it is promised to a different the same function d(v buyer in each of the two cases. Graphical illustration of the combined allocation rule is by Figure 1.6.

v2 1

(R, I)

(D, I) (I, D)

(R, D) p∗

(R, R)

0 0

(D, R)

p∗

(I, R)

1

v1

Figure 1.6. Optimal allocation in fully optimal mechanism

Step 2: fulfilling promises optimally. As with the waiting list, the difference of revenues in the two cases, π(v) − π(v), can be interpreted as the distortion from serving a buyer from the previous period rather than serving new buyers optimally. This difference is ˆ so that it strictly increasing in v2 for v2 ≥ p∗ . It is again optimal to choose contract D

47

minimizes π(v) − π(v) for a given probability of delayed service dˆ ≥ F2 (p∗ ), so we get ˆ = {v ∈ [0, 1]2 : π(v) − π(v) ≤ b} such that P r(D) = b. D Step 3: optimal probabilities of delayed service. As with waiting lists, it is more convenient to optimize with respect to maximum distortion b than the probability of ˆ Let b(vi ) denote the maximum distortion corresponding to a promised probability service d. ˆ i ). d(v In this notation, a slight increase in promised probability dˆ means that a new set of ˆ where the instant service will be unavailable next values must be added to the contract D period. If the change is marginal, all those added values have distortion that is close to ˆ i )) = −b(vi ). the maximum distortion b. That is, π 0 (d(v We can now analyze equation (1.5), the first order condition that characterizes optimal ˆ i ), which takes the form promise d(v (1 − δ)βw(vi ) = δb(vi ). Here we see that whenever w(vi ) > 0, the maximum distortion b(vi ) > 0 and so the ˆ i ) > F2 (p∗ ). Moreover, maximum distortion b(vi ) and therefore optimal probability d(v ˆ i ) are strictly increasing functions of vi for all interior solutions. Finally, we probability d(v ˆ i ) = 1, or equivalently maximum get the upper corner solution where the probability d(v distortion b(vi ) = 1 − δ whenever w(vi ) ≥ βδ . If δ = β, each type of buyer is promised different probability, with type vi = 1 being the only type who receives delayed service with certainty. If δ > β, then the buyers value the future service relatively lower than the seller and therefore the probabilities are relatively low. In particular, all promises are strictly lower probability than 1, and in the limit when

48

ˆ i ) very close to F2 (p∗ ). On the other β is very small, all promises are with probability d(v hand, if δ < β, the buyers value the delayed service highly and in this case even some buyers with value strictly below 1 will receive delayed service with certainty. Step 4: existence of π(d). This step is exactly analogous to step 5 in the waiting list mechanism—we can again verify that Blackwell’s sufficient conditions are satisfied and therefore the mapping we characterized above is a contraction (with rate δ). By the contraction mapping theorem there exists a fixed point, so that π(d) is indeed well defined. Implementation: The allocation rule described above is again implemented by a nonstandard auction with scoring rule. The new aspect compared to the waiting list is that a promise of delayed service is personalized. It means that each type of buyer who gets delayed service pays a transfer that is a strictly increasing function of her own type16. Similarly to the waiting list case the scoring rule favors close winners and punishes close losers. The reason for both effects is the same as with waiting list mechanisms—when deciding whether or not to serve a buyer from the previous period, the seller needs to compute the total distortion from doing so. This distortion is increasing in both new buyers’ values, and therefore the allocation that a buyer gets also depends on the opponent’s bid.

1.2.4. Partial commitment Since this chapter focuses on promises of delayed service, commitment power plays a crucial role. Whenever a promise is restrictive, the seller would have incentive to break it in the future. In this section and in the general model in the next section, we are extending the results from the three classes of mechanisms studied thus far to a general partial 16In

particular, the transfer with value vi pays when receiving delayed service with probability R a buyer ˆ i ) is pi (v) = β d(v ˆ i )vi − v∗i d(v ˆ 0 )dv 0 by the ex-post envelope condition d(v i i p

49

commitment assumption. The full commitment analysis we did above and no commitment are special cases of the general assumption. There are many reasons why the promises are kept: legal rules and fines, reputation of the seller, or perhaps intrinsic motivation. In this chapter we are taking a reduced-form approach and define partial commitment as an exogenous cost of breaking a promise to one buyer. This could be interpreted as a fine that has to be paid to a government agency. Of course, the model could be extended to make the commitment constant endogenous, for example by modeling reputation explicitly or allowing the seller to pay the compensation directly to the consumer, but this is not the focus of the current chapter. As always in mechanism design literature, we assume that the sets of states and promises are such that the seller is able to fully commit to any promise she has made. To study the lack of commitment, we are restricting the set of possible promises. We define partial commitment with constant c as a restriction to possible promises and therefore states. A seller cannot give any promise with which at any future realization the benefit from breaking a promise to one buyer would increase revenue by more than c. Then c can be interpreted as a commitment cost or fine for breaking the promise. If c = 0 we say that the seller has no commitment power and if c → ∞ she has full commitment power. Benchmark: no commitment. As a benchmark, let’s consider the case when the seller does not have any commitment power. This will be the limiting solution for both the waiting list and the unrestricted mechanism. Then the commitment constant is c = 0, which means that breaking any promise is costless. This is as if the next period seller is a separate player and the current seller does not have any power to enforce promises. In

50

this situation, promising delayed service is limited. Since the transfers from the previous period are already sunk, the seller always gives the priority to new arrivals. Whenever a new buyer has positive virtual surplus, the seller would like to serve her as soon as possible. Feasibility implies that higher type is offered instant service and lower type is offered delayed service with highest feasible probability, whenever their types are above p∗ . This puts an upper bound to the promises that can be given about the future service. ˆ = [0, p∗ ]2 , where p∗ is defined so that virtual The seller can promise service only in set D surplus w(p∗ ) = 0. Therefore if the buyer with lower type has positive virtual surplus, she ˆ = [0, p∗ ]2 . is offered to be served with probability F2 (p∗ ) and will be served when v ∈ D This allocation rule is illustrated by Figure 1.7.

v2 1

(D, I) (R, I) (I, D)

(R, R)

0 0

(I, R)

p∗

1

v1

Figure 1.7. Optimal no-commitment allocation

The optimal no-commitment mechanism can be implemented by the following indirect mechanism. First, both types are asked to pay F2 (p∗ )βp∗ for a chance to be served

51

tomorrow, if the seller has free time. After this, an upgrade to instant service is sold at a second price auction with reserve price [1 − βF2 (p∗ )]p∗ . Pure calendar mechanism with partial commitment. As long as commitment constant c is higher than 1 − δ, commitment level is sufficient to sustain the optimal pure calendar mechanism described in Section 1.2.1. However, when c < 1 − δ, the only possible calendar mechanisms are static. The discreteness of the result comes from the pure calendar mechanism assumption, where the seller either does or does not use unconditional promises of delayed service; there are no intermediate cases. Promising delayed service with certainty decreases the revenue in the next period, which means that the pure calendar mechanism requires some commitment power. The question is how much commitment is sufficient. The incentive to break a promise to a previous buyer would be the highest when both new arrivals have very high type, so the limiting case is when v = (1, 1). In this case when keeping the promise, the seller would get βw(1) = β from promising delayed service to one of the buyers and would have to refuse service to the other. When breaking the promise, he could instead get 1 + β, with the same continuation value. This means that the maximal gain from breaking the promise in terms of normalized revenue is (1 − δ)w(1) = 1 − δ. This is the necessary commitment level c∗ that ensures that the seller is able to keep any promises. Waiting list mechanism with partial commitment. Above we computed the optimal no-commitment mechanism, which is a waiting list mechanism, since the delayed service is ˆ = [0, p∗ ]2 and therefore does not discriminate between buyers always with a fixed set D who are offered delayed service. As the opposite benchmark we have also characterized the optimal waiting list with full commitment in Section 1.2.2.

52

The optimal waiting list with full commitment at state D gave an optimal promise in ˆ = {v : π(v) − π(v) ≤ b}, where b ∈ [0, 1 − δ] is the maximum distortion that the form D we know how to compute, and π(v) − π(v) is the distortion from serving a buyer from the previous period to a current revenue when arriving buyers announce v (the difference between revenues when the instant service is and is not available). Therefore with arrivals v, if the seller breaks the promise to a previous buyer, the additional revenue is exactly ˆ is the maximum π(v) − π(v). This means that the maximal incentive to break a promise D distortion b. If the maximum distortion b is lower than commitment constant c, the seller ˆ whereas when b > c, it is known that the promise will be broken can give promise D, under some realizations and therefore this promise is not possible. From this argument we see that finding the optimal waiting list mechanism for general partial commitment c is not very different from what we did with full commitment. The only difference is that maximum distortion b now has an additional upper bound c. It is illustrative to consider the two extremes. In the limit when commitment constant c converges to 0 we obviously get the no-commitment solution described above. However, the no-commitment solution is optimal only in the limit. Whenever c > 0, the seller would ˆ at least slightly larger than [0, p∗ ]2 . Increasing the probability of make the contract D service is valuable for all buyers who receive delayed service and therefore strictly increases revenue. Near the no-commitment solution it is almost costless, since the first types added ˆ have virtual surplus close to zero and therefore there is almost no distortion from to D not serving them. The upper bound of b in the full commitment analysis was 1 − δ. Therefore, if c ≥ 1 − δ, the commitment constraint is never binding. This is the same sufficient commitment level

53

that was needed to commit to the optimal pure calendar mechanism. When c < 1 − δ, we will sometimes get a corner solution where maximum distortion b = c and therefore the probability of service dˆ is always strictly below 1. In this sense a waiting list can be seen as a relaxation of a pure calendar mechanism. Unrestricted mechanism with partial commitment. The effect of adding partial commitment to the fully optimal mechanism (discussed in Section 1.2.3) is similar as to the waiting list. Again, it means that any promise of delayed service must be such that the distortion created by this promise to the next period’s revenue is not larger than c. The only difference is that in this case we allow the promises to depend on the type of the buyer who receives the promise, so that the upper bound must hold for each type. Formally b(vi ) ∈ [0, min{1 − δ, c}]. Again, the limit when c → 0 is the no-commitment mechanism (but it is optimal only if c = 0) and when c ≥ 1 − δ, the commitment level is sufficient to sustain any mechanism.

1.3. General analysis In this subsection we give formal results for arbitrary fixed number of service slots and random number of buyers. The main trade-offs are the same as with the illustrative example, but obviously the notation and characterizations of optimal mechanisms are somewhat more complex. In contrast to the previous section, we are prove the results with the partial commitment assumption, so that full commitment is a special case.

54

1.3.1. Model Seller has fixed number m ∈ N service slots each period t ∈ {0, 1, . . . } and maximizes expected discounted revenue with discount factor δ ∈ (0, 1). Every period random number n new buyers arrive such that the maximum number of buyers arriving with positive probability, N > m. Each buyer has unit demand and independent private value vi which is i.i.d. draw from CDF F and PDF f (v) > 0. Let Fk:n and fk:n denote the CDF17 and PDF18 of k’th smallest value out of n (k’th order statistic). Then for example, Fn:n is the CDF of the maximum of the values of all arrived customers. We maintain two important assumptions. First, each buyer interacts with the seller only at arrival. Buyer i who is promised service si periods after her arrival and pays pi at arrival gets payoff of β si vi − pi . Second, we concentrate on mechanisms where the seller never makes promises of delayed service more than one period ahead. The promise of delayed service to buyer i is a set Di ⊂ {(n, v) : n ∈ {0, . . . , N }, v ∈ [0, 1]n } and a state is a vector D = (D1 , . . . , DN ), a collection of promises given to buyers from the previous period. If buyer i receives a promise of delayed service with a contract Di , she will be served in the next period if and only if the number of buyers n and their announcements v are such that (n, v) ∈ Di . Since buyers’ utility functions are linear in value and quasilinear in transfers, the only variables affecting buyer i’s payoff are her transfer pi (v) and discounted expected quantity, denoted by φi (v), which is 1 if i receives instant service, 0 if refused service, and βP r(Di ) if is promised delayed service in set Di . These variables are functions of the allocation the Pn = j=k nj F (vi )j [1 − F (vi )]n−j . n! 18 fk:n (vi ) = (k−1)!(n−k)! F (vi )k−1 [1 − F (vi )]n−k f (vi ).

17F

k:n (vi )

55

seller chooses as a response to vector of types v announced by the buyers in this period. Let p˜i (ˆ vi ) and φ˜i (ˆ vi ) denote the expected values of the respective variables over the other buyers’ types assuming they report their types truthfully. We are looking for (Bayesian) incentive-compatible and individually rational mechanisms. In this notation, the individual rationality constraint (IR) for buyer i with type vi is φ˜i (vi )vi − p˜i (vi ) ≥ 0. The incentive-compatibility constraint (IC) is φ˜i (vi )vi − p˜i (vi ) ≥ φ˜i (ˆ vi )vi − p˜i (ˆ vi ) for all vˆi ∈ [0, 1]. Using the standard methods developed by Myerson (1981) it is straightforward to verify that a mechanism satisfies IC and IR constraints if and only if (1) the discounted expected quantity φ˜i is weakly increasing in buyer’s own type, (2) buyer with type vi = 0 gets payoff at least 0 (an in optimum exactly 0), (3) the transfers are computed from ex-ante envelope condition

(1.6)

p˜i (vi ) = vi φ˜i (vi ) −

Z

vi

φ˜i (ˆ vi )dˆ vi + p˜i (0).

0

Although we are looking for optimal mechanisms that induce truthful behavior when buyers do not know the other buyers’ types, it turns out that all the optimal mechanisms are implementable even when they know the other types. Let’s define ex-post individual rationality constraint (EPIR) for buyer i when the type vector is v = (v1 , . . . , vn ) by φi (v)vi − pi (v) ≥ 0. The dominant-strategy incentive-compatibility constraint (DSIC) is defined as φi (v)vi − pi (v) ≥ φi (ˆ vi , v−i )vi − pi (ˆ vi , v−i ) for all i’s announcements vˆi ∈ [0, 1] and vectors of opponents’ types v−i ∈ [0, 1]n−1 . Again, it is straightforward to verify using standard methods that a mechanism satisfies DSIC and EPIR constraints if and only if (1) the ex-post discounted expected quantity

56

φi is weakly increasing in buyer’s own type vi , (2) buyer with type vi = 0 always gets a payoff at least 0 (and in optimum exactly 0), and (3) the transfers are computed from ex-post envelope condition vi

Z (1.7)

pi (vi ) = vi φi (vi ) −

φi (ˆ vi )dˆ vi + pi (0). 0

We are using these results for two purposes. First, the ex-ante envelope condition (1.6) and the fact that pi (0) must be 0 are used to express the seller’s expected revenue at state z as " π(z) = max En,v (1 − δ)

n X

# φi (v)w(vi ) + δπ(z 0 ) ,

i=1

where the seller maximizes over feasible promises, w(vi ) = vi 1−Fvi(vi ) is the standard static virtual surplus, and z 0 is the continuation state. This expression is separable in (n, v). That is, the seller can choose optimal promises for each possible combination (n, v) separately and the value of being in a particular state can then be computed as expectation over all realizations. We will find the optimal allocation rule for different restrictive classes of mechanisms that restrict feasibility. In all the cases, the optimal allocation rule turns out to be monotone with respect to buyers’ own types. Therefore, as the second use of the results above, we can apply ex-post envelope condition (1.7) to compute the transfers that implement this mechanism under even stronger conditions (EPIR, DSIC). Before studying the dynamic mechanisms, there is one special mechanism in this model that serves as a useful benchmark and for which the allocation rule and implementation are well known. We are call a mechanism static, if the seller never gives promises for

57

delayed service. That is, at each period, each buyer either gets instant service or is refused service. This mechanism does use delayed service, so the restrictions for possible promises do not apply and it is therefore a feasible mechanism in all the cases studied below. Analogously to Myerson (1981), the static optimum in this framework is the allocation rule that assigns objects to m buyers with highest types, given that their virtual valuations are positive. This is implemented under DSIC and EPIR constraints by a standard uniform price auction with reserve price p∗ such that w(p∗ ) = 0.

1.3.2. Pure calendar mechanism The pure calendar mechanism is a mechanism where delayed service is unconditional on new arrivals. Each buyer is either assigned instant service, refused service, or promised service in the next period with certainty. The general set of states D = (D1 , . . . , DN ) can be simplified to a number z ∈ {0, . . . , m}. This is true since instead of a promise set we only have to keep track of which buyers received the delayed service and which did not. Moreover, from the seller’s point of view, who did and who did not receive the delayed service is not payoff-relevant—the only relevant variable is how many people were promised delayed service. Therefore a state z ∈ {0, . . . , m} denotes how many service slots are unavailable, and thus m − z slots are still available. The main result in this section is Theorem 1.3.1, which characterizes the optimal pure calendar mechanism (with partial commitment). The result has the same general trade-offs as the illustrative example in Section 1.2.1. First, delay is optimal only if buyer’s discount factor β is high enough compared to δ. In particular,

β δ

must be at least

En,v [max{w(vn:n ), 0}], which is the average of expected revenue that can be extracted from

58

the highest type buyer in a static mechanism. If β is strictly less than δ times this value, then it would not be optimal to give delayed service even to buyers with highest possible values of 1. The static mechanism would be strictly better. Note that the requirement is satisfied, for example, when β ≥ δ. Second, the seller’s commitment power must be high enough. The sufficient commitment power is the same as in the illustrative example, c ≥ 1 − δ for the same reason. If this condition is satisfied, even if all new arrivals have types very close to 1, the seller would prefer not to break previous promises. But in general it is necessary only if N ≥ 2m. When N < 2m, there can be only N < 2m buyers with high positive surplus next period. This limits the cost of current promises, since the seller is never forced to refuse service to very high customers (he can extract part of revenues by promises of future service). The main trade-off is again between extracting revenue from current relatively high buyers and the opportunity cost of having one less service slot available next period. Giving away each additional unit of next period service will be associated with opportunity due to being in a more restrictive state next period. The optimal pure calendar mechanism balances extraction from current buyers by promising some of them delayed service and the opportunity costs of each additional slot that is promised away. In particular, the optimal pure calendar mechanism is characterized by m thresholds, denoted by w = (w1 , . . . , wm ), where wk can be interpreted as the opportunity cost of the k’th object. The actual choice of promises will be non-trivial. We will describe below how to make this choice and Corollary 1.3.3 describes a special case where the allocation rule is much simpler.

59

Theorem 1.3.1. Suppose m ∈ N and n ∼ Γ. Under sufficient conditions β ≥ δEn,v [max{w(vn:n ), 0}] and c ≥ 1 − δ, the optimal pure calendar mechanism mechanism with partial commitment uses the following allocation rule at each state z = {0, . . . , m} (i) m − z new buyers with highest strictly positive virtual surplus will get instant service. (ii) If there are more than m − z buyers with positive virtual surplus, the next k highest of them will be promised service in the next period, where k is defined as

k = arg max

m X

[w(vn+1−(m−z)−i:n ) − wi ]

k∈{0,...,m} i=1

where constants w1 , . . . , wm are uniquely determined by wj

" δ = En,v max{0, w(vn−(m−j):n )} β + β

k k X X max w(vn+1−(m−j)−i:n ) − wi − max w(vn−(m−j)−i:n ) − wi k

i=1

k

!# .

i=1

The result covers the situations where it is optimal to offer delayed service and commitment level is sufficient to do so. The following Corollary 1.3.2 shows that these sufficient conditions are also almost necessary. In particular, the condition relating β and δ is necessary, whereas the condition on commitment level is necessary when the probability that at least 2m new buyers arrive is positive.

Corollary 1.3.2. If one of or neither of the sufficient conditions do not hold: (i) When β < δEn,v [max{w(vn:n ), 0}], the optimal mechanism is static. (ii) When N ≥ 2m and c < 1 − δ, the optimal mechanism is static.

60

(iii) When m < N < 2m and c < 1 − δ, depending on details the optimal mechanism could be static or with some delay. The proofs for both are in Appendix A.1. Here is the overview of the proofs. We start by assuming that β and c are high enough, so that there is some optimal delay. We can then make some instant observations: (1) buyers with negative virtual surplus are never served, (2) if there are enough buyers with positive virtual surplus, all current objects will be allocated, and (3) promises are monotone in types (higher types will be served earlier). This gives an almost full description of optimal allocation rule. The remaining question is for situations where there are more buyers with strictly positive virtual surplus than current service slots. The question is, to how many of them should the seller promise delayed service. To do this, we define constants w1 , . . . , wm where wk can be interpreted as the opportunity cost of giving away kth unit of tomorrow’s service. Now, vector w is defined in terms of continuation values and is therefore endogenous. To show that w exists and is uniquely defined we write the equation system for w in the form of w = Φ(w) and show that there exists a fixed point. The solution—if it exists—will be unique because Φ will be strictly monotone. The final part of the proof is about commitment power. We compute the cost of keeping a promise in all possible situations for each additional buyer. Value 1 − δ will be an upper bound of these costs, therefore if c ≥ 1 − δ, commitment power is sufficient. When N ≥ 2m, this upper bound is the lowest upper bound, since it is possible that 2m buyers arrive and all have values close to 1, which gives the bound. When N < 2m, this situation is not possible and therefore it is possible to have optimal delay even when c is slightly below 1 − δ.

61

Theorem 1.3.1 implies that the profit from calendar mechanism at state z can be written as " π(z) = En,v

(1 − δ)

m−z X

max{0, w(vn+1−i:n )} + δπ(m)

i=1

+

(1 − δ)β max k

k X

# w(vn+1−(m−z)−i:n ) − wi

.

i=1

The first two terms add up to the maximum revenue that can be extracted by static promises—the revenue from highest m − z buyers plus continuation value of having all m object available next period. The last term is positive and captures the extra revenue from delayed service to buyers with lower valuations. In particular, it is the sum of virtual surpluses extracted from the k buyers who are promised delayed service minus the opportunity costs wi that are lost due to the fact that fewer objects are available in the next period. Suppose (n, v) is such that there are enough buyers with positive surplus to allocate all current and next period objects, that is, vn−2m:n > p∗ . In this case, the number of buyers who are served instantly z, which is clearly increasing in z. In the appendix we prove Lemma A.1.1 which shows that the optimal number of buyers that are offered delayed service is weakly decreasing in m − z. This is true since with higher m − z more of the high-value buyers will receive instant service and therefore the remaining buyers have lower values. In general the optimal mechanism is non-monotone in terms of assigning delayed service. It is possible, for example, that at some state with some vector of values two buyers are offered delayed service, but after weakly increasing all the values only one buyer is offered

62

delayed service. There is one case where the mechanism is relatively easier to interpret and implement—if the thresholds characterized by Theorem 1.3.1 happen to be ordered. In this case the assignment is monotone and therefore each of the following buyers is served if and only if her value is higher than the corresponding threshold. Corollary 1.3.3 gives the formal result. Corollary 1.3.3. If the assumptions in Theorem 1.3.1 hold and the implied vector w is such that w1 ≤ · · · ≤ wm , then the optimal mechanism uses the following allocation rule at each state z (i) m − z buyers with highest strictly positive virtual surplus will get instant service. (ii) If there are more than z buyers with positive virtual surplus, k highest of them will be promised service in the next period, where k is such that w(vn+1−(m−z)−i:n ) > wi if and only if i ≤ k. The expected revenue at state is " π(z) = En (1 − δ)

m−z X

πn+1−i (0) + δπ(0) + (1 − δ)β

i=1

where πk (w) ˆ =

R1

w−1 (w) ˆ

m X

# πn+1−(m−z)−i (wi ) .

i=1

[w(vk:n ) − w]dF ˆ k:n (vk:n ).

Proof is in Appendix A.1.

1.3.3. No commitment As a benchmark, we are considering the limiting case where the seller does not have any commitment power. That is, commitment constant c = 0. In this case the seller always gives priority to new arrivals, since the payments from previous buyers is sunk. That is,

63

whenever a new buyer with strictly positive virtual surplus arrives, she will be served before all waiting buyers. This implies that promises can be made only for the cases when there are sufficiently few new buyers with positive virtual surpluses. The following Proposition 1.3.4 formalizes this. Proposition 1.3.4. Suppose m ∈ N and n ∼ Γ. The optimal no-commitment mechanism is characterized as follows. Let k be the number of new buyers with strictly positive virtual surpluses. (i) If k ≥ m, then m new buyers with highest strictly positive virtual surpluses are served instantly and no waiting buyers are served. If k < m then all k new buyers with strictly positive virtual surplus as well as m − k waiting buyers with highest positive surpluses are served. (ii) Buyers with strictly positive surpluses who were not served instantly are promised that they will be served if sufficiently few buyers with strictly positive surpluses arrive next period. In particular, n + 1 − m − ith highest buyer will be served next period if and only if his virtual surplus is strictly positive and next period k 0 ≤ m − i new buyers with strictly positive virtual surplus arrive.

Proof follows from the discussion above.

1.3.4. Waiting list The waiting list mechanism is a mechanism where the promise depends only on the position and not the type of buyer who is in the position. Formally, at each state D = (D1 , . . . , DN ), ˆ = (D ˆ 1, . . . , D ˆ N ) that are fixed the waiting list is characterized by a vector of contracts D

64

ˆ i is a set D ˆ i ⊂ {(n, v) : n ∈ {0, . . . , N }, v ∈ before the arrival of buyers. Each contract D [0, 1]n } that describes under what conditions the receiver of the contract will be served. ˆ being fixed before the number of buyers and their values In addition to contracts D being observed, the waiting list assumption says that when k new buyers are promised ˆ 1, . . . , D ˆ k . This is the “list” part of the assumption. delayed service, they receive contracts D ˆ k , it is as if he is added to some list in the For example if buyer i was promised set D position k, after k − 1 other buyers. He will be served if and only if n new buyers who ˆ k. arrive next period announce values v such that (n, v) ∈ D Notice that we do not explicitly assume that the waiting list is a priority ranking in the sense that being higher in the list ensures more likely service than being in a lower position. This is a property of an optimal waiting list. In particular, by Theorem 1.3.5 ˆ1 ⊃ D ˆ2 ⊃ · · · ⊃ D ˆ m and D ˆ k = ∅ for all k > m. the optimal waiting list is such that D Another property of optimal waiting list will be that each contract is determined separately from all other contracts. It will balance the same trade-offs that the illustrative example. On one hand, the assignment of jth contract balances the marginal surplus extracted from a buyer who gets this contract (analogously to the pure calendar mechanism) and on the other hand the precise composition of the contract balances the trade-off between average surplus extracted from a buyer who gets the contact and increased distortion from this contract (which is analogous to the fully optimal case). Theorem 1.3.5. The optimal waiting list mechanism with partial commitment is such that at each state D = (D1 , . . . , DN ), when n buyers arrive and announce values v, then (i) z = #{i : (n, v) ∈ Di } waiting buyers and m − z new buyers with highest positive virtual surpluses are served instantly.

65

(ii) The next m highest buyers with positive virtual surpluses are offered delayed services ˆ= D ˆ 1, D ˆ 2, . . . , D ˆm . with sets D ˆ are characterized as follows. where contracts D ˆ j = {(n, v) : π j−1 (n, w) − π j (n, w) ≤ bj }, D

m−j j

π (n, w) = (1 − δ)

X

max{0, wn+1−i } + δπ(0, . . . , 0)

i=1

+ (1 − δ)β

k X

max k∈{0,...,m}

dˆi [wn+1−(m−j)−i − wi ],

i=1

and wj ∈ [0, 1], bj ∈ [0, min{1 − δ, c}] are characterized by δ wj = (1 − δ)β

Z 0

bj

dGbj (bj )

δbj and ≥ bj (1 − δ)β Gbj (bj )

Z

1

wˆj wj

dGwj (wˆj ) , 1 − Gwj (wj )

where Gbj (bj ) = P r({(n, v) : π j−1 (n, w) − π j (n, w) ≤ bj }) and Gwj (wˆj ) =

R wˆj 0

gwj (wˆj0 )dwˆj0 ,

st " gwj (wˆj ) = En

1[D1c ]fn+1−(m−j) (wˆj )

+

m−1 X

1[Di \ Di+1 ]fn+1−(m−j)+i (wˆj )

i=1

# +

1[Dm ]fn+1−(m−j)+m (wˆj ) .

Proof is in Appendix A.2 and follows the same steps as the illustrative example. Proof structure: Step 0 We assume that functions π(d) are well-defined and make some immediate observations regarding the contracts and the allocation rule.

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ˆ we characterize the optimal allocation rule. It Step 1 For a fixed vector of contracts D will be characterized by a vector of constants w = (w1 , . . . , wm ). Constant wj can be interpreted as the opportunity cost of allocating the jth contract. Step 2 For a fixed vector of probabilities of delayed service dˆ = (dˆ1 , . . . , dˆm ) we show how to fulfill these promises with these probabilities optimally. In particular, each ˆ j minimizes the distortion from not having one extra object j optimal contract D available. Step 3 We derive the condition for optimal probability of delayed service for jth contact.To do this, we change the decision variables from probabilities of delay dˆ to maximum distortions b and find the first order condition with respect to this. The optimum balances the revenue extracted from the average person who gets the contract and the marginal increase in the distortion from giving this contract. Step 4 We argue then that we have characterized the optimal contracts. At this point we have derived two equations and of two variables that each contract j has to satisfy. The two variables are opportunity cost of allocating jth contract, wj , and the maximum distortion characterizing the contract j, bj . The equations are the corresponding first order conditions. We argue that each contract is well defined, either it is in the interior or in the upper corner. The upper bound for bj is either 1 − δ or commitment constant c. Step 5 Finally, the existence of π(d) functions can be verified using Blackwell’s sufficient conditions and the Contraction mapping theorem.

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1.3.5. Fully optimal mechanism In this section again, a state is a vector D = (D1 , . . . , DN ), where Dk denotes the realizations in which the seller has to serve a particular buyer corresponding to k from the ˆ = (D ˆ 1, . . . , D ˆ N ), but previous period. Promises of delayed service are again denoted by D now this vector can depend on the announcements of current buyers. The optimal mechanism has to fulfill all previous promises and assigns the remaining instant service slots to buyers with highest positive virtual values. If there still remains one or more buyers with positive surplus, the seller must decide what to promise. The result states that the promises they get are ordered in terms of both probabilities and sets. ˆ 1 ) that depends The highest of the remaining buyers is promised a probability dˆ1 = P r(D only on her valuation and is increasing in her type. Now, the seller would want to assign all the possible delayed promises to the same buyer, since she is the remaining buyer with the highest value, but she cannot use more than one service slot next period. Therefore the seller reserves the first slot for her and uses this first slot to serve only her or new buyers. The next slot is matched with the next highest remaining buyer with the positive valuation—he will receive a promise that is increasing in his type, but since his value is lower than the first buyer’s value, this probability is lower. The assignment continues until all m future slots are matched with buyers (some of whom may receive probability zero as a promise). The optimal way to fulfill the promise is such that the difference between maximized expected revenue when this slot is available relative to when it is unavailable is minimized.

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Theorem 1.3.6. The optimal unrestricted mechanism with partial commitment is such that at each state D = (D1 , . . . , DN ), when n buyers arrive and announce values v, then (i) z = #{i : (n, v) ∈ Di } waiting buyers and m − z new buyers with highest positive virtual surpluses are served instantly. (ii) The next m highest buyers with positive virtual surpluses are offered delayed services ˆ ˆ 1 (vn+1−(m−j)−1:n ), D ˆ 2 (vn+1−(m−j)−2:n ), . . . , D ˆ m (vn+1−(m−j)−m:n ) , with sets D(v) = D ˆ where contracts D(v) are characterized as follows. ˆ j (vi ) = {(n, v) : π j−1 (n, v) − π j (n, v) ≤ b(wi )} D

j

π (n, v) = (1 − δ)

j X

max{0, w(vn+1−i:n )}

i=1

( + max (1 − δ)β dˆ

π(d) = En,v [π 0 ] −

m X

m X

) ˆ , dˆi w(vn+1−(m−j)−i:n ) + δπ(d)

i=1

En,v 1[π j−1 (n, w) − π j (n, w) ≤ bj ][π j−1 (n, w) − π j (n, w)] ,

j=1

b(wi ) =

0

∀wi ≤ 0,

(1−δ)β δ wi ∀0 < wi < (1−δ)β min{1 − δ, c}, δ min{1 − δ, c} ∀wi ≥ δ min{1 − δ, c}. (1−δ)β

Proof is in Appendix A.3. The structure is the same as in the illustrative example and parts of it are analogous to the waiting list. Proof structure: Step 0 We assume π(d) is well-defined function and make the same immediate observations regarding the allocation rule and the optimal contracts as in the waiting list case.

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Step 1 We derive the optimal allocation rule. In this case optimal delayed services are personalized, so we get first-order conditions relating probabilities of service dˆk and buyer’s virtual valuation who gets the promise. Step 2 Next we describe how to fulfill promises optimally. The solution minimizes distortions as in the waiting list case. Step 3 We can now analyze the first-order conditions we got in Step 1. We show that the optimal probability of delayed service depends only of buyer’s own virtual surplus and is strictly increasing in the interior. Step 4 Finally to prove the existence of π(d) we use the Contraction mapping theorem.

1.4. Discussion I studied the problem of selling service slots. From previous literature we already knew the solution to the problem in two special cases. First, when the capacity is high, the optimal mechanism is a constant posted price mechanism. Second, when the buyers’ discount rate and exit rate are constant, the fully optimal allocation rule always assigns the objects to the highest-valuing19 buyers among all remaining buyers. This chapter shows how to solve the optimal mechanism design problem in the remaining case, where rationing is necessary and buyers are heterogeneous not only in their valuations for service, but also in their exit rates, so that the buyers from different periods should be treated differently. This introduces a new set of trade-offs which are balanced optimally by a new type of auction mechanism. 19If

the goal is to maximize revenue, then the mechanism assigns service only to buyers with positive virtual surplus.

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In addition to the fully optimal mechanism, we analyzed two classes of restricted mechanisms where the possible contracts of delayed service are limited: pure calendar mechanisms and waiting list mechanisms. The optimal pure calendar mechanism was a simple mechanism, characterized by a vector of opportunity costs. The qualitative properties of waiting lists were similar to the fully optimal mechanisms. Finally, we showed how the optimal mechanisms change in all three classes when the seller has less than perfect commitment power. We made several simplifying assumptions that could be relaxed without changing the conclusions. We assumed that the arrivals of buyers are observable both for the other buyers and for the seller. This assumption is without loss in the sense that if buyers had a chance to choose whether to enter the mechanism at their arrival or later—without knowing anything about the other buyers—they would always enter immediately. It would also be relatively straightforward to introduce more general arrival processes. We assumed that the number of objects m is constant and the number of buyers n is an independently and identically distributed random variable. We could assume instead that m and n are independent draws from some joint distribution, where the distribution can be different each period. This is a generalization that includes, for example, seasonal fluctuations in demand and supply. In real life applications the most common mechanisms to allocate service are posted price mechanisms. As we argued, whenever there is a need for rationing, the optimal mechanism would involve an auction. The assumption that the seller has to use a posted price mechanism would be another restriction to the allocation rule. Some related papers20 20See

Gershkov and Moldovanu (2009) and Board and Skrzypacz (2010) for example.

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have shown that if the arrival process is continuous, then the optimal mechanism would be implementable with posted prices, since two buyers never arrive at the same time. The same could be done with the model here, by assuming arrival in continuous time and leaving the production unchanged. Alternatively, we could assume that buyers in one period arrive sequentially and the seller has to assign the allocation to each buyer at the moment of their arrival, before observing the other new buyers’ arrivals and their types. Perhaps the most restrictive assumption was concentrating on contracts only one period ahead. A generalization of current models would involve a common discounting process β, where payoff from being served eventually at date si is β(si )vi − pi , such that β(si ) is decreasing in si and lims β(si )/δ si → 0. Special cases of this model would be β(si ) = β si which follows from Said (2008) and β(0) = 1, β(1) = β, β(si ) = 0, which was studied in this chapter.

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CHAPTER 2

Overbooking (joint with Jeffrey Ely and Daniel Garrett) 2.1. Introduction Overselling limited seating space is standard practice among airlines. According to US Department of Transportation reports, between the months of April and June of 2011, 1 out of every 100 passengers ticketed on a US domestic flight was denied boarding on that flight. One commonly cited rationale for the widespread use of overbooking is as a form of ex-post demand management. Because a certain number of passengers can be expected not to show up (on time) for a flight, overbooking capitalizes on slack capacity. In this chapter, we explore a different rationale for overbooking, one based on price discrimination among passengers who face uncertainty about their value for flying on the date of travel. When a flight is overbooked, airlines typically use some form of auction mechanism to repurchase excess tickets. A common practice is for an airline agent to ask for volunteers, raising the level of compensation until enough volunteers have been found willing to delay their travel. As a result of these kinds of approaches, only about 1% of the passengers who were denied boarding to oversold flights are bumped involuntarily.1 Auction-type reallocation mechanisms seem to offer airlines the ability to improve ex-post efficiency at the same time as improving the value of a ticket. Such mechanisms

1Auction-like

schemes are becoming more sophisticated. For instance, Delta Airlines has recently implemented a system in which passengers nominate their value in terms of travel vouchers that they would be willing to receive to give up their seat and take a later flight.

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therefore must be designed jointly with the ticket pricing and any ticket rationing decisions. This chapter considers how an airline optimally uses these instruments to maximize profits. We view tickets as contracts that arise when airlines have a limited opportunity to interact with passengers before the date of travel, where these passengers have only imperfect information about their future value for being seated.

Passengers can be

offered a single kind of contract which they may purchase or pass up.

In this sense,

communication at the ticketing stage is limited. The restriction to a single kind of ticket is a simplifying assumption which allows us to focus on key trade-offs, but it also favors a simpler implementation which seems both closer to current practice and more likely to be useful for guiding We show (Theorem 2.4.1) that airlines optimally use this restricted opportunity to sell some passengers contracts which (almost surely) guarantee their right to travel, but which also permit them to participate in an auction-type mechanism where they may be compensated for giving up their seat. The opportunity to contract with (i.e., sell tickets to) passengers before the date of travel is exploited by the airline because passengers have only noisy information about their eventual value for being seated on the flight. As such, the value passengers place on rights to be seated are less dispersed before the date of travel arrives; timely contracting therefore allows the airline to extract more surplus than if contracting is delayed (this observation is by now well understood in the dynamic mechanism design literature, see e.g. Courty and Li (2000) and Eso and Szentes (2007). On the other hand, in our setting, because passengers can only communicate acceptance of a ticket, tickets cannot be tailored finely to passengers’ (heterogeneous) priors about their future values. Optimal screening of these priors requires determining which kinds of passengers should receive tickets.

74

We illustrate how the airline may find it optimal not to ration tickets, but rather to sell tickets to all passengers with sufficiently favorable information about their final value for flying. In this case, the number of tickets may exceed the number of available seats (the airline’s fixed capacity level) with positive probability, and we interpret this as overbooking. Ticketed passengers who are not seated must be compensated, which is of course costly for the airline; however the anticipated value of such compensation can be recovered through the price of the ticket. We also study how airlines can most profitably use the available instruments – tickets and subsequent auctions. In particular, we seek to understand a novel interplay between screening ex ante via prices and screening ex post by auctions.

As noted above, we

show (Theorem 2.4.1) that involuntary bumping of passengers (akin to selecting ticketed passengers not to fly at random) is never optimal. Efficiency can always be enhanced by allowing those passengers with the highest values for flying to be seated. In particular, ticketed passengers should be given the option to be seated. In addition, we characterize the optimal treatment of ticketed and unticketed passengers (see Theorem 2.4.3). Passengers may be unticketed either because they turn down the opportunity to purchase a ticket or because they arrive late to the market. We argue that the terms on which unticketed passengers can receive seats will tend to be biased against them. For instance, we consider a “handicapped” double auction (see Theorem 2.4.5), where ticketed passengers may sell their seats and unticketed passengers may purchase them. We argue that the airline is likely to gain by introducing a wedge between the prices at which these transactions occur. The airline gains from unfavorable treatment of unticketed passengers because such treatment reduces the surplus available from not

75

purchasing a ticket. Ticket prices can then be raised without sacrficing the demand for tickets. Of course, providing unfavorable treatment to non-ticket holders adversely affects ex-post efficiency, and we argue that such treatment becomes more pronounced the smaller the probability that (potential) passengers are unticketed. We also note that profit-maximizing airlines may find it optimal to repurchase seating rights also if there is excess capacity. We thus extend familiar ideas in optimal monopoly screening to the dynamic contracting environment described above. The rest of the chapter unfolds as follows. After discussing related literature in the rest of this section, we introduce a model in Section 2. Section 3 introduces the mechanisms where a single kind of ticket is sold before the date of travel as described above.

To

illustrate how this mechanism operates, we provide a stylized example. Section 4 provides an analysis of pricing in a more general framework. Finally, Section 5 derives the optimal mechanism when there is no restriction on the kinds of contracts the airline offers.

2.1.1. Related literature Auction mechanisms as a response to overbooking in the airline industry were first suggested by Julian Simon in the mid-1960s (see Simon (1994)). These suggestions were implemented by the industry in the US from 1978, at least in the sense that airlines commenced informal reverse auctions at this time, with volunteers being selected on the basis of willingness to give up their seat, rather than arbitrarily.2 Vickery (1972) proposed the use of an efficient (Vickrey) auction to resolve the allocation of seats in the event of overbooking. In the same paper, Vickrey proposed extending this mechanism to include dynamic flexible 2Simon

(1994) notes that, before 1978, United Airlines, for instance, followed a practice of bumping “old people and armed services personnel, on the assumption that they would be least likely to complain.”

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pricing schemes which would allow prices to respond dynamically to the level of realized demand and provide ticketed passengers with an option for compensation in case choosing not to take a flight. In this chapter we go one step further and suggest a model with which dynamic pricing policies can be studied, a model in which overbooking arises as part of an optimal pricing policy. In our model, the airline optimally allocates seats via auction in the event of overbooking. However, given the airline’s market power, the profit-maximizing auction will typically not be efficient. We find support for Vickrey’s conjecture that an airline could profit from a “rising price expectation ... maintained to encourage early reservations”: indeed, in our model, higher fares for latecomers make early ticket purchasing more attractive and this allows the airline to profit more from these early purchases. We also provide a framework for analyzing Vickrey’s question about whether a “monopolist’s reaction would be such as to impose an additional burden on consumers greater or less than the gain to the monopolist”. We suggest that a dynamic pricing policy involving auctions would often contribute to efficiency even if selected by a monopolist. This chapter also contributes to the more recent literature on optimal selling policies in dynamic environments where consumers learn about their valuations over time. An important question on which the chapter sheds light is: When should the firm allocate available capacity, before or after the buyer has learned his value? Dana (1998), Möller and Watanabe (2010) and Nocke, Peitz, and Rosar (2011) provide models where sellers choose to discriminate between consumers based on their prior information, for instance by offering “advanced purchase discounts”. DeGraba (1995) and Courty (2003) study related models where there is no role for intertemporal discrimination due to buyers lacking

77

private information at the initial stage. The key distinction relative to our work is that capacity constraints in these papers are either respected (e.g., Möller and Watanabe) — tickets sold do not exceed the available capacity — or completely absent (e.g., Courty and Nocke, Peitz, and Rosar). We instead demonstrate how a seller (airline) may find it optimal to over-allocate capacity and then use an appropriate mechanism for ensuring capacity constraints are respected on the date of consumption. In this chapter we consider a restricted class of mechanisms where the airline gives up some of its ability to screen the initial information of passengers in favor of contractual simplicity. By allowing passengers only to accept or reject contracts offered before the date of travel (see the definition of “tickets” above), we focus on the extent to which the airline induces contracting at passengers’ first opportunity as opposed to at the date of travel. The airline’s decision must balance trade-offs which do not arise in the literature on “fully-optimal” mechanisms (e.g., Courty and Li (2000), Battaglini (2005), Eso and Szentes (2007) and Pavan, Segal, and Toikka (2011)) where the seller optimally contracts with all agents at the time of commitment to the mechanism.3 Indeed, in our setting, by assumption, selling tickets is the only way to directly distinguish those passengers who believe their values for flying will be high from those who do not. We do, however, investigate the fully-optimal or “unrestricted” mechanism in Section 5. Selling tickets in our model also provides a way for the airline to distinguish timely arrivers from those who arrive to the market late. This chapter is therefore related to the literature on “revenue management” which studies optimal pricing to buyers who arrive 3Typically,

inducing immediate contracting is strictly optimal. However, Nocke, Peitz, and Rosar (2011) provide a restricted environment where the optimal mechanism can be implemented also with delayed contracting by some buyers.

78

over time but who face no uncertainty about their future valuations (see, Bergemann and Said (2010) for an overview as well as McAfee and Velde (2006) for a review focused on applications to the airline industry).

In contrast to that literature, we study an

environment with both dynamic arrivals and valuation uncertainty. We thus build on recent work by Garrett (2011, ????) who studies the unrestricted optimal mechanism for a durable-goods monopolist facing buyers who arrive over time and whose values evolve stochastically. 2.2. Model The monopolist airline is selling m seats on a flight that departs at date 1. There are n potential passengers who are ex ante anonymous and symmetric. A passenger may enter the market at date 0 or only at date 1. A passenger i arriving at date 0 receives partial information about his value for flying at date 1, vi , which is captured by a signal θi . Vectors of values are denoted in bold font, i.e. v = (vi )ni=1 , with v−i0 = (vi )i6=i0 for any individual passenger i0 . Both a passenger’s time of arrival and information about his value for flying are determined independently of the other passengers’ realizations and are his private information. The signals θ˜i are drawn from a distribution with CDF F whose support is Θ = [0, 1] ∪ {∅} (for any S ⊂ Θ, F (S) = Pr θ˜i ∈ S ).4 The signal θi = ∅ indicates that the passenger is out of the market, and unavailable for ticket purchases at date 0. For notational convenience we will adopt the convention that ∅ < 0.

A signal θi ∈ [0, 1]

indicates that the passenger has entered the market and we assume that F admits a density f over that range and include the possibility of an atom at ∅. 4Throughout,

random variables are denoted using tildes.

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Conditional on his signal θi , a passenger i’s (non-negative) eventual willingness to pay v˜i is distributed according to the CDF G(vi |θi ) with density g(vi |θi ). The support of the marginal distribution of v˜i is [v, v¯] (the distribution of v˜i conditional on a realization of the signal θi may be a strict subset). Abusing notation, we also let G(v|S) and G(v−i0 |S−i0 ) denote the joint distributions of passenger values conditional on the events (θi )ni=1 ∈ S ⊂ Θn and (θi )i6=i0 ∈ S−i0 ⊂ Θn−1 . The stochastic process described by F and G necessarily admits an “independent-shock” representation as follows (see Eso and Szentes (2007) and Pavan, Segal, and Toikka (2011)). For each passenger i, ε˜i is a random variable distributed according to H and independently of θ˜i with the property that that, for all θi , v˜i = z (θi , ε˜i ) is distributed according to G(·|θi ). We assume in addition that this is true for a continuous distribution H with interval support and admitting a density h, and with z (·, ·) a differentiable and increasing function. Thus we consider at first instance a large class of processes exhibiting first-order stochastic dominance in the sense that θi0 > θi implies G(vi |θi0 ) ≤ G(vi |θi ) for all vi . There is no discounting. At date 0 the airline will sell tickets at some announced price. Passengers realize utility vi if they are allocated a seat and zero otherwise, net of any payments made to the airline across the two periods. So a passenger whose (possibly negative) total payment to the airline is ρ earns utility vi − ρ if he is seated and −ρ if he is not. Given a fixed number of available seats and a zero cost of seating a passenger in an available seat, the airline maximizes the expected total payment by passengers.

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2.3. Pricing Mechanisms We consider pricing mechanisms ΩT of the following form. At date 0, the airline sets a price p for tickets, and also commits to a family of re-allocation mechanisms to be used at date 1 depending on the number of tickets sold. In particular, the mechanism will be used to determine which passengers will sell back their tickets in the event of overbooking, to broker the possible transfer of tickets from the early purchasers to those unticketed passengers who are willing to pay to fly, and possibly to sell additional seats to passengers who wish to purchase tickets at date 1. Intuitively, selling a ticket is a way of dividing customers among those who are present and date zero and have certain beliefs over their willingness to pay, and those who do not. When the set of possible customer date-zero beliefs is sufficiently rich, a finer partition will often be optimal. Thus, for instance, the airline may wish to sell different fare classes (e.g., high and low priority tickets) to distinguish between these beliefs. We delay our discussion of the mechanism with an unrestricted number of fare classes until Section 5. This restriction ruling out finely screening customers with different signals is, however, the only restriction which reduces the airline’s profit.

In particular, selling tickets is

optimal in any environment where the communication of passengers with the airline at date zero is restricted to a single message from a singleton message space. The mechanism will be required to satisfy standard constraints. First, at most m passengers can be seated on the plane. Second, the mechanism must be implementable in perfect Bayesian equilibrium: whether or not a passenger purchases a ticket, his continuation play in period 1 must be optimal given updated beliefs. In general, passengers act in date 1 without knowing how many other tickets have been sold and the types of the

81

other passengers. Nevertheless, we will show below that any Bayesian incentive-compatible mechanism can be implemented using a mechanism that informs all passengers of the total number of tickets sold and that gives all passengers dominant strategies in date 1. At date 0, a passenger’s decision whether or not to purchase a ticket must be optimal given the anticipated continuation play. Our treatment of participation constraints reflects an assumption about the nature of ticketing. We are assuming that a ticket is a contract between the airline and the passenger. This contract specifies the re-allocation mechanisms that will be used at date 1 and the purchase of a ticket entails the passenger’s agreement to participate. On the other hand we are assuming that the ticket is the only contract formed at date zero and therefore passengers who have not purchased a ticket are not compelled to participate. Note, however, that the assumption that a ticket is a contractual commitment on the part of the passenger is never crucial, because, as we discuss below, the airline has a degree of freedom to shift payments across time. By increasing the price of the ticket and reducing any period-1 payment (equivalently, increasing any period-1 subsidy) to ticket holders, the airline can ensure that ticket holders find period 1 participation optimal even if they are not contractually bound. We adopt the convention that the airline does not limit ticket sales but instead sells to all passengers willing to purchase, which will be those with signals in some set ΘT ⊂ [0, 1]. This is without loss of generality: Any outcome achievable by limiting the number of tickets sold can be achieved by committing to dishonour some proportion of the tickets sold.

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The airline designs the date 1 mechanisms taking into account not just the revenues they generate, if any, but also how they affect the passengers’ willingness to pay for a ticket in date 0. To formalize the tradeoffs we will need some notation. Let π s denote the airline’s date 1 expected revenue from the mechanism used when s ∈ {0, . . . , n} is the number of passengers holding tickets purchased at date 0. We will use subscripts j to denote passengers that hold tickets at date 1 and subscripts k to denote those who do not.

Thus, we can order passengers so that passengers j = 1, . . . , s hold tickets and the

remaining passengers k = s + 1, . . . , n do not. We let let S s denote this event, i.e. S s = {θj ∈ ΘT , j = 1, . . . , s; θk ∈ / ΘT , k = s + 1, . . . n} s and we denote by S−i the corresponding marginal event for the passengers other than i s (S−i = projΘ−i S s ).

For each possible number of purchased tickets s ≤ n, and each vector of possible valuations v, let Ui (v;s) denote the payoff to passenger i from flying at date one, net of any transfers made at date 1 (i.e., payments made by passenger i excluding the price of the ticket if any). We denote the expected utility of a passenger participating in the s

s-mechanism as a function of his date 0 signal by V (·) and when he is not holding a ticket by V s (·). Thus, for the signals of ticketed passengers θj and of non-ticketed passengers

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θ k ,5 h i s V (θj ) = E Uj (˜ v;s)|θ˜j = θj h i v;s)|θ˜k = θk V s (θk ) = E Uk (˜ The expected payoff to a buyer i with signal θi who purchases a ticket at price p is Er˜V

r˜+1

(θ) − p, where the number of ticket-holders r˜ is determined according to a binomial

distribution with parameters (n − 1, F (ΘT )). The expected payoff to this buyer if not purchasing a ticket is Er˜V r˜(θi ). Given that signals are continuously distributed on [0, 1], we must then have

(2.1)

Er˜V

r˜+1

(θ∗ ) − p = Er˜V r˜(θ∗ )

whenever θ∗ is a boundary point of ΘT .

This is a necessary condition for incentive

compatibility of the ticket purchasing strategy defined by ΘT whenever boundary points exist (i.e. ΘT is not equal either to ∅ or [0, 1]). However, optimality of ticket pricing dictates that it must also hold when ΘT = [0, 1], letting θ∗ = 0 (otherwise, the ticket price could be raised, still inducing all passengers arriving at date zero to purchase). Equation 2.1 states that any “threshold” or “marginal” ticketholder must be indifferent between purchasing a ticket and not purchasing (again, this threshold is defined to be zero in case all period-0 arrivers purchase). Incentive compatibility then requires additionally

5These

expectations are taken with respect to the conditional distribution G(·|θi ) over v˜i and with respect to the unconditional distribution over v ˜−i and include any expected payments made at date 1, excluding any payments already made at date 0.

84

that passenger i purchases a ticket (θi ∈ ΘT ) if and only if Er˜V

r˜+1

(θi ) − p ≥ Er˜V r˜(θi ) for

all θi ∈ ΘT . The threshold condition allows us to express the ticket price in terms of the marginal ticket-buying signal. Abusing notation, let F (ΘT ) be the probability that a passenger purchases a ticket. Then, for any threshold signal θ∗

(2.2)

p=

n−1 X n−1 r

r=0

n−1−r r+1 ∗ F (ΘT )r 1 − F (ΘT ) [V (θ ) − V r (θ∗ )].

Using the assumption that the mechanism treats each passenger identically from an s

ex-ante perspective, let Π and Πs denote the expected revenue earned from a ticket holder and from a non-ticket holder, respectively, in the s-mechanism. The airline’s expected profit is s˜

π = nF (ΘT )p + Es˜[˜ sΠ + (n − s˜)Πs˜], where s˜ has binomial distribution with parameters (n, F (ΘT )). After substituting Equation 2.2 and rearranging, expected profit can be expressed in terms of the date 1 mechanisms and a threshold ticket-buying signal θ∗ ,

(2.3)

!# T F Θ s s φ(s) s Π + V (θ∗ ) + (n − s) Πs − V s (θ∗ ) , T) 1 − F (Θ s=0

n X

"

where6 n−s s n φ(s) = 1 − F ΘT F ΘT . s The date 1 mechanism. At date 1 each passenger observes his realized willingness to pay as well as whether or not he is holding a ticket. A direct revelation mechanism is described 6See

Appendix B for derivation.

85

by a collection of functions (q, t), such that, for each s ∈ {0, . . . , n}, q(v; s) = (qi (v; s))ni=1 gives the probability that each passenger i is seated while t(v; s) = (ti (v; s))ni=1 gives the payment that each passenger i makes to the airline at date 1. All ticket-holders are treated symmetrically, as are all non-ticket holders, and passengers are ordered as described above such that the first s purchase tickets while the remaining n − s do not. Without loss of optimality, each ti (v; s) is deterministic. While each qi (v; s) may specify a probability of being seated, the realized allocations q ∈ {0, 1}n must satisfy the feasibility constraint P i qi ≤ m. We do not rule out that Supp [G (·|θi )] = 6 [v, v¯] for some θi , in which case the message space associated with the period-1 direct mechanism may differ for ticket-holders and non-ticket holders. In such cases, equilibrium reports of ticket-holders must lie in the support of G (·|ΘT ) while non-ticket holder equilibrium reports lie in the support of G (·|Θ\ΘT ). We find it convenient nonetheless to consider “extended” direct mechanisms where allocations and transfers are defined for vectors of reports v such that vi0 ∈ [v, v¯] for some i0 while vj ∈ Supp G ·|θj ∈ ΘT , vk ∈ Supp [G (·|θk ∈ Θ)] , for all j, k 6= i0 . This comes with no loss of optimality relative to the restricted message space, since the range of outcomes that different messages can induce need not be expanded. The advantage of considering extended mechanisms is that it allows a passenger who has not followed equilibrium ticket-purchasing strategies (e.g., failed to purchase a ticket while θi ∈ ΘT ) to nonetheless report his value truthfully at date 1.

86

The date 1 mechanism (q, t) is Bayesian incentive compatible if, for each ticketed passenger j, all vj and all reports vˆj , E [qj (vj , v ˜−j ; r˜ + 1)vj − tj (vj , v ˜−j ; r˜ + 1)] ≥ E [qj (ˆ vj , v ˜−j ; r˜ + 1)vj − tj (ˆ vj , v ˜−j ; r˜ + 1)] and, for each unticketed passenger k, all values vk and all reports vˆk , E [qk (vk , v ˜−j ; r˜)vk − tk (vk , v ˜−k ; r˜)] ≥ E [qk (ˆ vk , v ˜−k ; r˜)vk − tk (ˆ vk , v ˜−k ; r˜)] where, as above, r˜ follows a binomial distribution with parameters (n − 1, F (ΘT )). The mechanism is ex-post incentive compatible if qi (vi , v−i ; s)vi − ti (vi , v−i ; s) ≥ qi (ˆ vi , v−i ; s)vi − ti (ˆ vi , v−i ; s) for all s, vi , vˆi and v−i . Thus in an ex-post incentive compatible mechanism the airline can publicly announce the number of ticket-holders and truth-telling would remain ex-post optimal for all passengers.7 The following lemma shows that any Bayesian incentive-compatible mechanism can be replaced by an ex-post incentive compatible mechanism that generates the same expected s˜

revenue and interim expected utilities Es˜V (θ), Es˜V s˜(θ). The proof follows from essentially the same arguments made in Goeree and Kushnir (2011).

7By

ex-post incentive compatibility, we mean incentive compatibility given other passengers play a strategy which can arise in equilibrium. This restriction makes sense in light of the extended message space. However, as discussed above, the extension of the message space is merely for convenience and we could just as well consider the restricted message space and dominant strategy implementation.

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Lemma 2.3.1. There is no loss of optimality in restricting attention to ex-post incentive-compatible mechanisms, i.e. mechanisms in which the airline announces the number of tickets sold and truth-telling is ex-post optimal for all passengers. Below we demonstrate how the optimal allocation of seats can be derived, assuming that it is optimal for ticket sales to follow a simple threshold policy where passengers purchase tickets if and only if their signals θi exceed some threshold θ∗ . We first ascertain the optimal allocation conditional on a given θ∗ and then illustrate how it may also be possible to find θ∗ and verify the optimality of the simple threshold policy for ticket sales.

2.3.1. Illustrative Example In this section we will illustrate the main trade-offs faced by the airline in deciding the price for the ticket, how seats will be allocated and whether to overbook. We do so by means of a simple discrete example with two passengers and one seat to be allocated. We will assume that the buyer either arrives in the first period with a signal in θ, θ , θ < θ, or does not arrive until the second period, i.e θ = ∅. In the second period each passenger i realizes willingness to pay vi in {v, v} where 0 < v < v. For this example we will suppose that θ represents an inflexible passenger: with probability 1 his willingness to pay will be v. On the other hand, θ represents a passenger with uncertainty about his eventual willingness to pay, g(v|θ) = g(v|θ) = 1/2 The passenger is unavailable for early purchases with probability f (∅). We will focus our analysis on the optimal date 1 mechanisms when tickets are priced so that early arriving passengers with either signal θ or θ purchase tickets. As we discuss below, this turns out to be optimal; indeed the mechanism we derive here turns out to

88

achieve the highest profit for the airline among all possible mechanisms (this is a result of the simplifying assumption that θ indicates a value v¯ with certainty). We discuss these “unrestricted” optimal mechanisms more generally in Section ?? It turns out that we can also restrict attention to mechanisms whose final allocation has “no distortion at the top”. That is, if there is at least one passenger with the high value v, then regardless of how many tickets were purchased in the first period the seat will not be left empty and at least one of the high value passengers will be seated. For example if both high-value passengers are ticket-holders (so that the flight was overbooked), one of them will be induced to relinquish his ticket while the other will be seated. Likewise if neither is holding a ticket, then the seat will be sold to one of them. In both of these cases the tie-breaking can be arbitrary. When only one of the two high-value passengers has a ticket there is no loss in assuming that it is the ticketholder who keeps his seat. To see why it is never optimal to leave the seat empty when there are high value passengers note that excluding a high value buyer is dominated by seating him (when possible) and extracting all of his rents by charging him (an additional) v for the seat in the second period. It clearly increases profits for the airline and since it leaves all passengers’ first-period expected payoffs unchanged it remains incentive-compatible. Thus, in this example, the problem reduces to deciding how to treat passengers when both have low realized values (v) depending on which of the two (if any) purchased tickets in the first period. Let’s begin by considering the case in which both passengers have low values and are holding tickets so that the flight is overbooked. The repurchase mechanism. With an overbooked flight, at least one of the tickets must be repurchased by the airline. If the mechanism repurchases exactly one ticket, i.e. an

89

efficient mechanism, then since both passengers are holding tickets and both have low realized values, the airline has no reason to treat them asymmetrically. One of them will be chosen at random to sell back their ticket. What’s interesting is that it will often be optimal for the airline to instead repurchase both tickets and leave the seat empty. To see why, note that from Equation 2.3, the airline’s repurchase mechanism is chosen to maximize 2

2

Π + V (θ∗ ).

(2.4)

Thus, the airline is maximizing a hybrid welfare function which is the sum of producer 2

surplus Π , conditioned on the average ticket holder θ ∈ {θ, θ} plus consumer surplus 2

V (θ∗ ) conditioned on the marginal ticket holder θ∗ , in this case θ. Intuitively, repurchasing tickets on an overbooked flight is a transfer to ticketholders and potentially raises the value of holding a ticket which in turn increases the price the airline can charge for tickets in the first period. On the other hand such a transfer directly reduces the airline’s profits. However, the fact that these two terms are conditioned on different events creates a wedge which implies that these costs and benefits are not offset one-for-one. 2

To illustrate, consider first the effect on V (θ) of a one dollar cash transfer in the event that both passengers have low values. Conditional on having signal θ, a passenger expects to receive this transfer with probability α given by α = g(v|θ)f (θ|θ ≥ θ)g(v|θ).

90

2

2

The transfer thus increases V (θ) by α. On the other hand, it reduces Π by β = [g(v|θ)f (θ|θ ≥ θ)]2

because β is the probability, conditional on two tickets purchased, that both ticketholders have low values and therefore that the airline will have to make the transfer. Note that β = αf (θ|θ ≥ θ) so that the net effect on the airline’s objective is positive. A one dollar 2

2

transfer increases Π + V (θ∗ ) by α (1 − f (θ|θ ≥ θ)). It follows that the airline has an incentive to increase the size of the subsidy as much as possible. The size of the subsidy is constrained by incentive compatibility. If type θ is receiving a large transfer then a ticketed passenger with type v will be induced to claim to be v. How seats are allocated affects the incentive constraint. For example, consider an efficient mechanism where only one ticket is repurchased and the remaining passenger is seated. In that case, the transfer τ cannot be larger than v/2. To see why, note that if a passenger with value v reports value v, he would fly with probability 1/2 (due to a tie in values), receive the subsidy τ and obtain payoff τ + v/2. Incentive compatibility requires that this payoff be no larger than the payoff from reporting v truthfully, flying with probability 1 and receiving no subsidy, τ + v/2 ≤ v.

As an alternative the airline could use an inefficient mechanism which seats no passenger when both ticketholders report low values. This enables the subsidy to be raised to v

91

without violating the incentive compatibility of a high-value passenger. The high value passenger is indifferent between a seat which is worth v and subsidy equal to v. On the other hand, the cost of increasing the subsidy is that type v will not be seated, and this 2

reduces the expected welfare of the marginal ticket holder, V (θ∗ ). We can calculate the net effect on the airline’s objective as follows. Increasing the 2

2

subsidy from v/2 to v decreases Π by βv/2 but it increases V (θ∗ ) by α [v/2 − v/2]

because α is the conditional probability in the eyes of the marginal ticketholder that the subsidy will be triggered, and in this event his subsidy increases by v/2 but his utility from flying decreases by v/2 (in the efficient mechanism he would have had an equal chance of retaining his seat). Thus, the distorted, high subsidy mechanism is preferred by the airline whenever (α − β) v/2 − αv/2 > 0 which, since β = αf (θ|θ ≥ θ) is equivalent to (1 − f (θ|θ ≥ θ)) v > v.

We see that when v < v/2, it is profitable to leave the seat empty and increase the transfer to the low value passenger.

92

The Spot Mechanism. Now consider the case in which no tickets were sold in the first period. In this case the airline may sell the seat in a spot auction to passengers arriving in the second period. According to Equation 2.3 it will use a mechanism which maximizes8

(2.5)

f (∅)Π0 − [1 − f (∅)] V 0 (θ).

Notice that the airline is trading off increased profits from the spot mechanism against reduced welfare of a passenger with the marginal signal. The key observation is that, in equilibrium, a passenger with the marginal signal θ will purchase a ticket and therefore will not be present in the spot market. Thus the welfare V 0 (θ) represents the hypothetical value that she would realize if she were to deviate and wait to purchase a ticket in the spot market. The airline has the incentive to distort the terms of trade in the spot market in order to make it unattractive to the marginal ticket holder. To build intuition for the effects of this new incentive, it helps to first recall the tradeoffs present in a conventional spot auction not preceded by an initial round of ticket sales. The seller must decide whether to sell the seat in the event that the two potential buyers in the spot market both have willingness to pay v. If the airline sells the seat to, say, buyer 1 this affects profits Π0 in two ways. First, the direct increase in profits is the additional revenue v from the sale. The countervailing effect is the loss in revenue from the infra-marginal scenario when buyer 1 has the high value v and buyer 2 has value v. The price in that scenario would have to be reduced from v to v in order to maintain incentive compatibility for type v. Thus, conditional on the event that buyer 2 has value v, the effect on profits 8To

see that Equation 2.3 applies to the case with discrete signals (and values) it is enough to note that Equation 2.1 applies with θ∗ = θ. This is a necessary condition for optimal optimal ticket pricing.

93

results from the usual tradeoff between these two effects: g(v)v − g(v)(v − v).

where g(v) and g(v) represent the unconditional distribution of valuations since in the present scenario all buyers participate in the spot auction. Thus, an airline that does not use a sequential mechanism would seat the low value passenger if and only if

(2.6)

v g(v) − ≥0 v − v g(v)

With advance ticket sales the criterion changes in two ways. First, the distribution of passenger values, conditional on arriving to date 1 with zero tickets sold is given by g(·|∅), since in our mechanism types θ and θ purchase tickets. Second, the airline now internalizes the effect on V 0 (θ). By selling to type v and thereby conceding surplus v − v to type v, the airline makes the option of waiting for the spot market more attractive to early-arriving passengers. In particular, the marginal signal θ attaches probability g(v|θ) to the event that waiting would earn him these rents. Thus, sale to v would increase V 0 (θ) by g(v|θ)(v − v) and thus the overall effect on the airline’s objective in the spot mechanism is f (∅) [g(v|∅)v − g(v|∅)(v − v)] − [1 − f (∅)] g(v|θ)(v − v).

94

After re-arranging we see that the airline seats the low value passenger in the spot mechanism if and only if v g(v|∅) 1 − f (∅) g(v|θ) − − ≥0 v − v g(v|∅) f (∅) g(v|∅) Thus, by comparison to a standard spot auction there are two effects that operate in potentially opposing directions. First, if the relative probability of high values among unticketed passengers is smaller than among ticketed passengers, then the airline has a reduced incentive to distort. This is captured by a comparison of the second terms above and in Equation 2.6. On the other hand, the incentive to distort is magnified in order to increase the incentive to book early as captured by the third term above. Note that this last effect is decreasing in the probability of late-arriving passengers: when the probability of late arrival is high, information rents expected by the buyer with marginal signal (θ) are less important relative to the efficiency losses experienced by the low-value unticketed passenger. The Re-allocation Mechanism. Finally, consider the case in which a single ticket was purchased in the first period. Then, in the second period, the airline may wish to repurchase the ticket and possibly re-sell it to an unticketed passenger. According to Equation 2.3, this re-allocation mechanism should maximize 1 − F (θ∗ ) 1 ∗ Π + Π + V (θ ) − V (θ ) F (θ∗ ) 1

1

1

∗

Consider the event that the two passengers have low values and suppose that in this event the airline re-allocates the seat from the original ticket holder to the new arrival. First, 1

1

calculate the effect on the revenue from and surplus of the ticket holder, i.e. Π + V (θ∗ ).

95

For the same reasons as in the re-purchase mechanism, the airline will want to buy back the ticket at the highest possible price consistent with incentive compatibility, i.e. at a price of v. Thus, g(v|∅) [(−v)g(v|θ ≥ θ) + (v − v)g(v|θ)] 1

1

measures the effect of the repurchase on Π +V (θ∗ ) because in the event that the newcomer has value v, the airline will pay v to a low-valued ticketholder, and will thereby increase the rents of the low-valued ticketholder by (v − v). We can subtract and add g(v|θ ≥ θ)v and re-arrange to obtain g(v|∅) {(−v)g(v|θ ≥ θ) + (v − v) [g(v|θ) − g(v|θ ≥ θ)]} , or, by substituting g(v|·) = 1 − g(v|·),

(2.7)

g(v|∅) {(−v)g(v|θ ≥ θ) + (v − v) [g(v|θ ≥ θ) − g(v|θ)]} ,

which is an expression that can be interpreted as the virtual utility of the low-value ticket holder. Note that the coefficient on the rent term (v − v) accounts for the usual infra-marginal loss g(v|θ ≥ θ), but in addition adjusts for the benefit associated with increased rents to a buyer with signal θ. Next the effect on the remaining two terms, the revenue and surplus associated with the non-ticket holder is measured by

1 − f (∅) g(v|θ)(v − v) , g(v|θ ≥ θ) v g(v|∅) − (v − v)g(v|∅) − f (∅)

96

since in the event that the ticketholder has value v the airline has additional revenue v from a low-valued new arrival, loses revenue (v − v) from the infra-marginal sale, and raises the rent expected by θ from foregoing the early purchase of a ticket. Combine terms to yield

(2.8)

g(v|θ)(v − v) g(v|θ ≥ θ) v g(v|∅) − (v − v) [g(v|∅) − g(v|θ)] − f (∅)

.

The value to the airline of transferring the ticket from the current ticketholder to the newcomer is thus measured by the sum of the cost, as measured in Equation 2.7 and the benefit as measured in Equation 2.8. We can see from comparing these expressions that as long as g(v|∅) ≥ g(v|θ ≥ θ), i.e. late arriving passengers have equal or higher values than the average ticketholder, then the overall effect on the arline’s revenue is negative and the ticket would not be transferred. (In order for such a transfer to be optimal the comparison must be sufficiently strong in the opposite direction to offset the negative term in Equation 2.8.) This does not however imply that the ticketed passenger will always be seated in this event. A third option is to repurchase her ticket and leave the seat empty. If the virtual utility in Equation 2.7 is negative then the airline prefers to repurchase the ticket and leave the seat empty. Indeed, the motivation for this is identical to the overbooked case (s = 2), and the virtual utility essentially replicates the tradeoffs calculated explicitly there. First Period Ticket Sales. We have analyzed the optimal allocation conditional on the assumption that the airline sells tickets to any buyer who arrives in the first period. In particular the price of a ticket is given by Equation 2.1 where θ∗ = θ. As mentioned above, in the particular example studied here, the “unrestricted” optimal mechanism coincides

97

with the mechanism derived here. The reason is twofold. First, in any mechanism which induces different treatment of passengers with different period-0 signals, it must be that the passenger with signal θ¯ earns rent at least the rent obtained by taking the date-0 contract designed for the passenger with signal θ. Under mechanism we derived here, these two rents coincide. Second, it is optimal for the mechanism to treat any passenger with value v¯ identically (and to seat that passenger whenever possible). In particular, this is true regardless of whether his date-0 signal is θ¯ or θ, and so the optimum does not require a distinction in the allocations for signals θ and θ¯ in case value v¯ is realized. For a similar reason, the airline cannot improve profits (indeed, often achieves strictly ¯ In that case, the lower profit) if it induces ticket purchase only by passengers with signal θ. passenger with signal θ¯ must be left at least as much surplus as he can earn by imitating the passenger with signal θ, i.e. by not purchasing a ticket. This kind of ticket mechanism can hence do no better than by contracting with all passengers only at date 1. ¯ it is of interest to Having observed the optimality of selling tickets to both θ and θ, understand when this mechanism strictly improves on what is achievable by contracting only in the spot market at date 1. Rather than determining the precise parameter values, we instead illustrate some general advantages of early ticket sales and show some simple sufficient conditions that they imply. Consider first the case in which the optimal spot auction is efficient, i.e. always sells the seat to the passenger with the highest value (breaking ties at random). In such a mechanism, an early-arriving passenger with signal θ will earn rents in expectation. This is because there is a positive probability that his realized value will be v and the other

98

passenger will have value v. In that case the auction price will be v and he will earn rents equal to v − v. Consider now a two-stage mechanism in which (1) Tickets are sold at a price equal to θ’s expected rents from the spot auction. (2) Non-ticket holders are excluded. (3) Otherwise the allocation and transfer rules are the same as the spot auction. (4) In particular the value of a non-ticket holder is still used to determine the allocation and price paid by a ticket holder. In this mechanism, buying a ticket is equivalent to buying admission to the spot auction. The price extracts the expected rents of θ, and reduces the rents of θ by an equal amount but leaves the welfare from trade with ticketholders unchanged. These effects raise profits relative to the spot auction. Any losses come from exclusion of non-ticketholders. Thus, if the probability of ∅ is sufficiently low, then this two-stage mechanism improves upon the spot auction. Exclusion of non-ticket holders ensures that an early arriver would obtain zero utility if he were to refuse to buy a ticket. This increases the marginal passenger’s surplus from holding a ticket and this surplus will be extracted through the price. Total exclusion of non-ticket holders is not necessary to achieve this. It is enough that the airline refuses to seat a low-value passenger unless he is holding a ticket. This guarantees that high-value non-ticket holders earn no information rents and again the expected utility of a non-ticket holder is zero. Next, suppose that the optimal spot auction excludes low-value buyers. Then a mechanism involving ticket sales to both types of early-arrivers can potentially improve upon a spot auction by improving the efficiency of the allocation. There are two instruments

99

available for this. First, consider the two-stage mechanism described above. By comparison to the inefficient spot auction, it increases the efficiency of the allocation while still giving zero rents to a passenger with signal θ. On the other hand, it gives information rents to a passenger with signal θ. Thus, a sufficient condition for the two-stage mechanism to give greater revenue is that θ has sufficiently low probability. A further improvement can be obtained via the second instrument, subsidizing lowvalue ticket holders in the event they are not allocated the seat (as shown in Subsection 2.3.1). Increasing the price at which tickets are repurchased from low-value ticket holders increases the value of holding a ticket to the marginal ticket purchaser, θ and the increase can be directly re-captured by increasing the ticket price. Because the increased ticket price captures some of the information rents of the infra-marginal ticket purchaser θ, the overall effect is an increase in the profits from the mechanism. Here the effect on profits is larger when the probability of θ is large.

2.4. General Analysis General properties of the optimal mechanism. Before describing how we approach the problem of characterizing the optimal allocation rule, we propose a way to achieve any allocation which can be implemented by selling tickets. Let ΩT be any (dynamic) mechanism which allocates seats via the sale of a single fare class of tickets (i.e., with restricted communication). One can determine, for any s, corresponding allocation and payment rules (qj (v; s) , qk (v; s)) and (tj (v; s) , tk (v; s)) for all j and k defined on reports v in the extended message space described above.

These coincide with equilibrium

100

outcomes under ΩT , as well as the outcomes in case a single buyer acts inconsistently with the equilibrium ticket-purchase strategy but acts optimally at period 1. By the envelope theorem, the direct mechanism defined above must satisfy

qi (v; s)vi − ti (v; s) = Ui (v;s) Z

v

= Ui (v, v−i ; s) −

qi (˜ vi , v−i ; s)d˜ vi vi Z vi

(2.9)

= Ui (v, v−i ; s) −

qi (˜ vi , v−i ; s)d˜ vi . v

Note that the share of the surplus a passenger i expects to obtain at date one is determined for each s and v−i , up to a constant, by his value and the allocation rule up. For ticket holders, adjusting this constant does not affect expected payoffs at date zero provided the ticket price is adjusted correspondingly according to Equation 2.1. On the other hand, adjusting period-1 payoffs for non-ticket holders Uk (v;s) by a constant does affect ex-ante payoffs. In particular, if the payoff earned by the passenger with the minimum value Uk (v, v−k ; s) is greater than zero the airline can profit by increasing transfers by non-ticket holders by an appropriate constant. This not only reduces the rent left to non-ticket holders but also allows the airline to increase ticket prices according to Equation 2.1. For any fixed (implementable) allocation rule, a mechanism that maximizes profit subject to that allocation rule must therefore satisfy Uk (v, v−k ; s) = 0. Without reducing profit, the airline can therefore replace any mechanism ΩT with the corresponding extended direct mechanism, with transfers and ticket prices adjusted so that Uj (¯ v , v−k ; s) = v¯, Uk (v, v−k ; s) = 0 and Equation 2.1 holds. The corresponding transfers

101

are given by Z

v¯

tj (vj , v−j ; s) = vj qj (vj , v−j ; s) − v¯ +

(2.10)

qj (˜ vj , v−j ; s) d˜ vj , vj

Z tk (vk , v−k ; s) = vk qk (vk , v−k ; s) −

vk

qk (˜ vk , v−k ; s) d˜ vk . v

Given Uk (v, v−k ; s) = 0, for all s, all k, and all v−k , the airline’s profit depends only on the allocation rule. Note also from (2.9) that adjusting the allocation to ensure passengers with values v¯ fly wherever possible (i.e., setting qi (¯ v , v−i ; s) = 1 wherever possible and adjusting transfers accordingly) does not affect date-1 payoffs. Moreover, since a buyer i’s allocation remains monotonic in his valuation (this monotonicity is necessary for the ex-post implementability of the allocations) the allocation remains implementable. In other words, it is possible and costless to provide the passenger with the option to fly by reporting v¯ whenever the number of passengers reporting v¯ is less than the available seats m. This observation, together with the optimality of the transfer rule given in (2.10) implies the following result. Proposition 2.4.1. It is optimal to structure the mechanism so that (1) A ticket is an option to fly and involuntary bumping is never used. (2) Ticketholders who are seated make (and receive) no payments. (3) Any ticketholder who does not fly receives a payment at least his value in compensation. Part 1 of the proposition states that, at least in the above framework, airlines cannot profit from using involuntary bumping. In particular, there is no loss to airlines in allowing those with the greatest value of flying to do so. Indeed, the arrangement can be optimally

102

structured so that ticketed passengers do not pay anything to keep their seat (Part 2 of the proposition) but are compensated for giving up a seat. Under this arrangement, enticing passengers with high values to give up their seats (as required in case if the flight is overbooked and the passengers with tickets have high values) while still seating those with the very highest valuations requires a high compensation to the unseated ticket holders. While the compensation provided by the airline in such cases may be large, the surplus provided to ticket-holding passengers can be recouped through the date-zero ticket price. Note that the suboptimality of “involuntary bumping” follows from the use of sufficiently sophisticated auction mechanisms for resolving which passengers will fly. For instance, if the flight is overbooked by passengers with high values for being seated, the level of refund offered responds to their reported values and is correspondingly high. A possible disadvantage of such mechanisms in practice is that the available refund as well as the price of a seat to unticketed passengers is completely determined only at date 1 after passengers report their values. Airlines may favor simpler mechanisms where prices are specified before passengers have learned their final values for flying. If these mechanisms cannot respond to excess demand (e.g. when more ticket-holders than there are seats realize high values for flying), then involuntary bumping would be necessary for reconciling demand with the number of seats. While a more detailed analysis is left for future work, we believe the analysis below sheds some light on optimal pricing under less flexible or responsive mechanisms. Optimal allocations. We now derive the optimal allocation for the airline supposing that ticket-purchasing follows a simple threshold rule, namely that ΘT = [θ∗ , 1] for some threshold (“marginal”) signal θ∗ ∈ [0, 1].

From Equation 2.3, the airline’s profits are

103

proportional to 1 − F (θ∗ ) s ∗ s s Π + V (θ ) + (n − s) Π − V (θ ) . F (θ∗ ) s

(2.11)

s

∗

From (2.9), the revenues earned from a ticketed passenger j and non-ticketed passenger k through the s-mechanism are

s

Z

Π =

Z

s

tj (v)dG(v|S ) = v

vj qj (v; s) − Uj (v; s)dG(v|S s )

v

and s

Z

vk qk (v; s) − Uk (v; s)dG(v|S s )

Π = v

and the welfare of the passenger with the marginal signal is s

∗

Z Z

s Uj (v; s)dG(v−j |S−j )dG(vj |θ∗ )

V (θ ) =

(2.12)

vj

v−j

for a ticketed passenger j and s

∗

Z Z

s Uk (v; s)dG(v−k |S−k )dG(vk |θ∗ )

V (θ ) =

(2.13)

vk

v−k

for a passenger k without a ticket. Focusing on the expressions for ticket-holders, we substitute the Mirrlees formula separately into each expression and integrate by parts.

s

Z

Π = v

qj (v; s)vj dG(v|S s ) +

Z v−j

"Z

# s G(vj |θj ≥ θ∗ )qj (v; s)dvj dG(v−j |S−j )

vj

Z − v−j

s Uj (v, v−j ; s)dG(v−j |S−j )

104

and s

V (θ∗ ) =

Z

s Uj (v, v−j ; s)dG(v−j |S−j )−

v−j

"Z

Z v−j

# s G(vj |θ∗ )qj (v; s)dvj dG(v−j |S−j ).

vj

Putting these together and collecting terms we obtain an expression for the virtual surplus of ticket holders.

Ev

( s X

G(vj |θ∗ ) − G(vj |θj ≥ θ∗ ) qj (v; s) vj − g(vj |θj ≥ θ∗ ) j=1

) s S

As anticipated above, all terms involving Uj (v, v−j ; s) are canceled: The constant term in the Mirrlees formula is a free variable in the airline’s maximization implying that individual rationality constraints can be ignored.9 Turning now to the terms in Equation 2.11 involving non-ticket holders, we can carry out the analogous transformations for Πs and V s (θ∗ ) to derive the expected virtual surplus of unticketed passengers. One important difference is that the mechanism must respect an individual rationality constraint because these passengers have no contract with the airline. This constraint will bind so that utility will be zero for the passenger with the lowest value among those not holding a ticket. ( E

n X

k=s+1

" qk (v; s) vk −

1−F (θ∗ ) (1 F (θ∗ ) g(vk |θk < θ∗ )

1 − G(vk |θk < θ∗ ) +

# ) − G(vk |θ∗ )) s S

Summing up, the airline’s optimal date 1 mechanisms are designed based on the types of passengers who purchase tickets and the types who do not and allocates space in order to maximize a virtual surplus which is correspondingly modified to account for the effects 9As

described above, this is intuitive because any constant added to the ticket holder’s ex post utility at date 1 can be recovered via an equal increase in the ticket price at date 0.

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on incentives to purchase tickets at date 1. This virtual surplus is given in the following proposition. Proposition 2.4.2. Suppose that ΩT is a (single fare-class) mechanism implementing, for each s ∈ {0, . . . , n}, the allocation rule qj (v; s) for passengers with signals θ ≥ θ∗ and qk (v; s) for passengers with signals θ < θ∗ . Suppose the non-ticket holder with value v earns zero surplus. Then the expected revenue from this mechanism must be equal to

(2.14)

Ev

( s X

qj (v; s)VS(vj ) +

j=1

n X k=s+1

) qk (v; s)VS(vk ) S s

where VS(vj ) = vj −

G(vj |θ∗ ) − G(vj |θj ≥ θ∗ ) g(vj |θj ≥ θ∗ )

is the virtual surplus of ticket-holder j, and

VS(vk ) = vk −

1−F (θ∗ ) (1 F (θ∗ ) g(vk |θk < θ∗ )

1 − G(vk |θk < θ∗ ) +

− G(vk |θ∗ ))

is the virtual surplus of unticketed passenger k. The transformed expected revenue (2.14) is a familiar expected virtual surplus and as usual the first pass at a solution is to consider the allocation rule which, at every realized valuation profile, allocates seats to the m passengers with the highest (non-negative) virtual surpluses. One can maximize (2.14) pointwise, which simply requires awarding seats to those passengers with the highest virtual surplus, provided this surplus is non-negative. One then verifies that the ticket price defined by Equation 2.1 and transfers defined by Equation 2.10 implement this allocation (that passengers are willing to purchase tickets at date zero if and only if their signals exceed θ∗ , and that it is indeed ex-post incentive

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compatible for them to report their value in the date-1 direct mechanism). The following conditions on the virtual surpluses are sufficient to guarantee this.

Proposition 2.4.3. Fix θ∗ ∈ [0, 1] Suppose in addition that (i) max VS(vj ), 0 and max {VS(vk ), 0} are non-decreasing functions, and (ii) VS(vj ) ≥ VS(vj ) for all vj ∈ Supp [G (·|ΘT )] ∩ Supp [G (·|Θ\ΘT )]. There exists an implementable allocation rule q, defined on the extended message space, such that, in equilibrium, the m seats are awarded to passengers with the highest virtual surpluses, provided these are non-negative.

Conditions (i) and (ii) in the Proposition are strong monotonicity conditions. The first ensures that an extended allocation can be found such that each passenger’s probability of being seated is non-decreasing in his own report (whether or not that report is made in equilibrium). It guarantees the existence of a period-1 mechanism which implements the allocation. The second condition ensures that ticket-holders are favored in terms of the allocation over non-ticket holders. This ensures that the expected payoff from purchasing a ticket is increasing faster in the passenger’s period-0 signal if he purchases a ticket that if he does not. It thus ensures the implementability of the proposed threshold policy for ticket purchasing. An example in which the conditions of Proposition Theorem 2.4.3 are met is given at the end of Section 5, where we also derive the optimal threshold θ∗ . 2.4.0.1. Understanding the trade-offs. We now consider the allocations described in Proposition 2.4.3 and how these reflect the airline’s incentives. The virtual surplus of ticketholders is given by

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VS(vj ) = vj −

G(vj |θ∗ ) − G(vj |θ˜j ≥ θ∗ ) . g(vj |θj ≥ θ∗ )

To interpret this, first consider that the decision not to seat a ticketed passenger j, i.e. by setting qj (v; s) = 0 for some s and realized values v, is an advance commitment to a ticket purchaser j that his ticket will be repurchased in the event that s passengers hold tickets and the profile of all passengers’ realized valuations is v. Such a commitment impacts the airline’s profits via two effects. First, it raises the payment to a ticket holder announcing value vj and thus by incentive compatibility requires the airline to raise by an equal amount the utility of all types lower than vj . This directly reduces the airline’s profits within the date 1 mechanism by G(vj |θ˜j ≥ θ∗ ) and correspondingly lowers the airline’s willingness to repurchase the ticket from passenger j. Note that G(vj |θ˜j ≥ θ∗ ) measures the probability that j’s value will fall below vj conditional on j’s purchase of a ticket at date 0. The second effect operates indirectly via revenues from date 0 ticket sales. Indeed, some of the additional utility provided by the airline through date 1 repurchases can be recouped via increased ticket prices. At the time of ticket purchase, the marginal type θ∗ assesses a probability G(vj |θ∗ ) that his value will fall below vj and he will benefit from the increased utility resulting from the airline’s commitment to repurchase a ticket from type vj . That leads to an increased willingness to pay for a ticket that can be extracted dollar-for-dollar by increasing the price of a ticket. This indirect effect raises the airlines profits by G(vj |θ∗ ) where crucially, this measures the probability that j’s value will fall below vj conditional on i being the marginal ticket purchaser, θ∗ .

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Putting all of this together, the airline is willing to seat a ticketed passenger only if the surplus from doing so, vj , exceeds a measure that accounts for the net effect on the airline’s profits. That measure is proportional to the difference G(vi |θ∗ ) − G(vi |θi ≥ θ∗ ) in the repurchase probability assessed by the marginal and average ticket purchasers (θ∗ and θ ≥ θ∗ respectively.) In light of this wedge between marginal and average ticketholders, overbooking and repurchasing can be seen as an instrument to refine the screening of passengers by willingness to pay at date 0. For illustrative purposes, suppose that the airline practices no overbooking and contemplates switching to a mechanism involving overbooking and repurchases. The ticket price without overbooking will be the marginal ticket purchaser’s willingness to pay for a ticket, and all higher types θ ≥ θ∗ will earn consumer surplus equal to their excess willingness to pay. With ticket sales alone, the only way the airline can capture some of the infra-marginal consumer surplus is to raise ticket prices but this has the cost of losing sales to the marginal types. Overbooking enables the airline to capture consumer surplus without sacrificing ticket sales. The airline raises ticket prices and effectively strikes a deal with the marginal type θ∗ that the price increase will be returned in expectation through repurchases in the event that his valuation turns out to be low. Since this is calculated to be a one-for-one intertemporal transfer for the marginal type θ∗ , it is less favorable for higher types θ > θ∗ because they assess a strictly lower probability of repurchase. Thus, the higher price coupled with repurchases maintains the indifference of the marginal type and strictly lowers the consumer surplus of infra-marginal types.

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Unticketed Passengers. Consider now the virtual surplus of a non-ticket holder.

VS(vk ) = vk −

1−F (θ∗ ) (1 F (θ∗ ) g(vk |θk < θ∗ )

1 − G(vk |θk < θ∗ ) +

− G(vk |θ∗ ))

Just as with ticketed passengers, the incentive to seat an unticketed passenger mixes the direct effect on revenues in the s-mechanism with an indirect effect on revenues from ticket sales. Indeed, the virtual surplus can be seen as the sum of two terms, the first of which, vk −

1 − G(vk |θk < θ∗ ) g(vk |θk < θ∗ )

is the familiar expression for the marginal revenue from selling to buyer k in a standard monopoly problem. It has the one noteworthy difference that the expressions are conditioned on passenger k having a date 0 signal below the threshold θ∗ . The remaining term in VS is a correction which accounts for the effect on first period ticket sales from a date 1 decision to seat an unticketed passenger. A mechanism which allocates space to unticketed passengers reduces the value of holding a ticket and hence revenue from ticket sales. This term conditions on θ∗ because it is the marginal ticketholder’s willingness to pay that determines the ticket price. Accuracy of signals. The usefulness of the ticket sales as an additional screening device depends on the accuracy of the signals buyers receive about their valuations. In particular, in one extreme where the buyers learn their valuations with certainty at date 0, selling tickets does not increase revenue. In the other extreme, when buyers do not have any information about their eventual valuations at date 0, the seller can extract the whole expected surplus from early arriving

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buyers through ticket sales and there is no need to distort the allocation for ticketed passengers. Of course, the seller still has to give the right incentives for the buyers who arrive in date 1. Theorem 2.4.4 shows how the optimal ticket mechanism simplifies in the two extreme cases.

Proposition 2.4.4. (1) When signals are fully informative, i.e. passengers arriving at date zero know their values with certainty at that date, a spot auction at date 1 is the optimal mechanism. The airline does not benefit from selling tickets in advance of the date of travel. (2) When the signals are fully uninformative so that G(·|θi ) = G (·) for all signals θi , the optimal mechanism sells tickets to all buyers in date 0. In date 1, the optimal mechanism allocates the objects to up to m buyers with positive virtual valuations, where virtual valuation of a ticketholder is her true value VS(vi ) = vi and virtual valuation of a non-ticketholder is VS(vi ) = vi −

1−G(vi ) 1 . g(vi ) f (∅)

Implementation via a double auction. In case Proposition 2.4.3 applies, an optimal allocation can be implemented in dominant strategies using a double auction mechanism that handicaps the non-ticket holders. Some new notation will be convenient for formalizing the rules of the auction. For any ticket-holder value vj , define the matching value vk (vj ) for a non-ticket-holder by the following equation:

VS(vj ) = VS(vk (vj )).

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In the regular case, as long as VS(vj ) ≥ 0, the matching value vk (vj ) is uniquely defined. Moreover, vk (vj ) is strictly increasing in vj . Conversely, define vj (vk ) as the matching value of the ticket-holder. The rules of the double-auction are as follows. Each passenger submits a bid and the airline announces the reserve price r defined by VS(r) = 0. Those passengers whose bids exceed the reserve price are ranked in descending order of bids, and the n highest bidding passengers whose bids exceed r are allocated a seat. Payments are determined as follows. Let bn and bn+1 denote the n and n + 1st highest bids. In case the number of bids exceeding r is smaller than either n or n+1 then bn and/or bn+1 are set equal to r. Any ticketed passenger who is not seated receives compensation equal to bn . Any unticketed passenger who wins a seat is charged vk (bn+1 ). The transfers are zero for all ticketed passengers who fly and unticketed passengers who do not.

Proposition 2.4.5. In the regular case, there exists a dominant strategy equilibrium of the double auction which implements the optimal mechanism.

2.5. Unrestricted (multiple fare class) mechanism In this section we derive the unrestricted optimal mechanism and compare it with the mechanism derived in the previous section where date zero communication is limited. With no restriction on date zero communication, the airline finds it optimal to contract with all passengers arriving in the market at date zero. Without loss of optimality, we consider direct mechanisms where each passenger i reports his initial signal θi in case he arrives at date zero and then reports his value vi in the second period. We can think of the report of the signal as the passenger choosing among multiple fare classes available at date zero. For notational simplicity, we denote the null report, where the passenger fails

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to report at date zero, by ∅. Let ΩM denote such a mechanism, comprising allocations q = (qi )ni=1 and transfers t = (ti )ni=1 such that, for all θ = (θi )ni=1 and v, the probability each passenger is seated is given by qi (θ, v) and the total payment by each payment over two periods is given by ti (θ, v) (given the absence of discounting the timing of payments does not affect payoffs). Again, we allow an extended message space to permit truthful reporting of a passenger’s value for flying even after he has misreported his signal. The allocation of seats satisfies the same feasibility constraint introduced above: no more passengers may be seated than availableseats. Denote by W ΩM (θi ) the expected payoff of a passenger with signal θi given the opportunity to participate in the mechanism ΩM at date zero (but with no other information about his value or about the signals or values of other passengers).

By the envelope

theorem (see Pavan, Segal and Toikka (2011)), a necessary condition for the incentive compatibility of truthful reporting of signals is that for all θi ≥ 0, W

ΩM

(θi ) = W

ΩM

θi

Z (0) + 0

∂z (y, ε˜i ) ˜ qi y, θ−i , z (y, ε˜i ) , v ˜−i dy. E ∂θi

We now conjecture that the value of W ΩM (0) is determined by the value a passenger with signal zero can obtain by deviating and reporting only at date 1. A passenger i arriving at date 1 and truthfully reporting his value vi , given that other passengers report θ−i and v−i earn a payoff W

ΩM

(∅, θ−i , vi , v−i ) = W

ΩM

Z (∅, θ−i , v, v−i ) +

vi

qi (∅, θ−i , y, v−i ) dy. v

We may optimally choose W ΩM (∅, θ−i , v, v−i ) = 0 while ensuring individual rationality of a passenger participating for the first time at date 1. A necessary condition for date 0

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participation by a passenger with the zero signal is therefore

W

ΩM

Z (0) ≥ E

v˜i

qi

˜ ˜ ∅, θ−i , y, v ˜−i dy|θi = 0

v

1 − G (˜ vi |0) ˜ qi ∅, θ−i , v˜i , v ˜−i dy|θ˜i = 0 =E g (˜ vi |0) 1 − G (˜ vi |0) ˜ qi ∅, θ−i , v˜i , v ˜−i dy|θ˜i = ∅ . =E g (˜ vi |∅)

We conjecture that this is the only relevant participation constraint at date zero and therefore that the inequality must bind in an optimal mechanism. Taking expectations and integrating by parts, we find that the expected airline profit (expected social surplus less passenger rents) is equal to (2.15) 1−F θ˜ ∂z θ˜ ,˜ε ( ( i i) i) ˜ ˜ ˜ ˜ n ˜−i |θi ≥ 0 (1 − f (∅)) E z θi , ε˜i − f θ˜ qi θ, z θi , ε˜i , v X ∂θi ( i) h i vi |∅) 1−f (∅) 1−G(˜ vi |0) ˜ ˜ i=1 f (∅) E v˜i − 1−G(˜ − q θ, v ˜ , v ˜ | θ = ∅ i i −i i g(˜ vi |∅) f (∅) g(˜ vi |∅)

This is the expected payoff in any optimal mechanism such that the passenger with signal zero is indifferent between participating at date zero and waiting to participate at date 1. We can now conjecture an optimal allocation rule along the same lines as for the single fare class mechanisms. Let (θi ) ∂z(θi ,εi ) z (θi , εi ) − 1−F if θi ≥ 0 f (θi ) ∂θi (2.16) V S (θi , εi ) = . z (θi , εi ) − 1−G(z(θi ,εi )|∅) − 1−f (∅) 1−G(z(θi ,εi )|0) if θi = ∅ g(z(θi ,εi )|∅) f (∅) g(z(θi ,εi )|∅) Let q∗ be the allocation defined by maximizing the virtual surplus V S.

That is, by

letting, for all i, qi∗ (θ, (zi (θi , εi ))ni=1 ) = 1 in case V S (θi , εi ) > V S (θl , εl ) for all l 6= i, and qi∗ (θ, (zi (θi , εi ))ni=1 ) = 0 if V S (θi , εi ) < V S (θl , εl ) for some l 6= i. Ties may be

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broken arbitrarily. The following result gives conditions under which this allocation is implementable by an appropriate system of transfers.

Proposition 2.5.1. Suppose that V S (·, ·) as defined by Equation 2.16 is non-decreasing in both arguments. Then there exists a system of transfers t∗ that implements the allocation q∗ . The mechanism (q∗ , t∗ ) is profit maximizing in the class of all possible mechanisms.

The virtual surplus for the unrestricted mechanism is closely related to those for the single fare-class mechanism studied above. The airline would like to distort downwards the probability a passenger with a low signal is seated so as to reduce the rents that must be left to passengers with higher signals to dissuade them from mimicking the distribution of reports by passengers with the lower signals. This idea is familiar from the other work on dynamic mechanism design. With the single fare-class mechanism studied above, the airline lacks the flexibility to distinguish the allocations for passengers with different signals, but it faces an incentive to achieve a similar end by selling tickets only to passengers with high signals (conferring a higher probability of being seated on the holder). The airline would also like to reduce the probability of flying for passengers who only arrive to the market at date 1 in order to reduce the rents available to them.

The

advantage in doing so is that it permits the airline to reduce the rent that must be left to passengers who arrive at date zero in order to persuade them to participate at that date. This again parallels the airline’s incentive to limit the rents of non-ticket holders in the single fare class mechanism studied above. These ideas are developed in an infinite-horizon setting in Garrett (2011).

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The relationship between the virtual surplus V S (·, ·) identified above and the virtual surpluses of the single fare class mechanism V S (·) and V S (·) for ticketed and unticketed passengers (under a threshold ticket purchasing policy with threshold θ∗ ) turns out to be straightforward. For each vi in the support of G (·|ΘT ), we have h i V S (vi ) = E V S θ˜i , vi |θ˜i ≥ θ∗ while i h ∗ ˜ ˜ V S (vi ) = E V S θi , vi |θi < θ . One can then view the problem of designing the optimal mechanism with a single fare class as equivalent to maximizing Equation 2.15 subject to the constraint that the allocation does not depend on the precise signal θi but whether it lies in the set ΘT = [θ∗ , 1]. Assuming that the allocation is chosen optimally conditional on θ∗ , one can apply the envelope theorem to determine the effect of a marginal increase in θ∗ on expected airline profit. In case m ≥ n so that capacity constraints do not bind, for interior values of θ∗ (i.e., θ∗ ∈ (0, 1)), this is simply

(2.17) h i R ε¯ 1−F (θ∗ ) ∂z(θ∗ ,y) ∗ ∗ n X ε Eθ˜−i ,˜ε−i z (θ , y) − f (θ∗ ) qi θ− , y, θ˜−i , ε˜−i h (y) dy ∂θi R , h i ε¯ 1−F (θ∗ ) ∂z(θ∗ ,y) ∗ ∗ ˜ i=1 z (θ , y) − f (θ∗ ) qi θ+ , y, θ−i , ε˜−i h (y) dy − ε Eθ˜−i ,˜ε−i ∂θi ∗ ∗ where qi are the constrained allocations and θ− represents any signal below θ∗ and θ+

represents any signal above θ∗ . Hence, a necessary condition for θ∗ ∈ (0, 1) to be the optimal threshold for purchasing a ticket in the single fare class mechanism is that (2.17) is

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equal to zero. The virtual surplus V S (θ∗ , ·) of the marginal ticketholder thus determines the optimal choice of the threshold. The following example illustrates the usefulness of this observation. Example Suppose that, for each passenger i, θ˜i is distributed uniformly on [0, 1] while ε˜i is distributed uniformly on [0, a], where a ∈ 0, 41 . Suppose that passengers’ values are determined by z (θi , εi ) = θi + εi . If m ≥ n, the optimal threshold for a single-class mechanism such that ΘT = [θ∗ , 1] is θ∗ =

1 2

− a4 .

While the effect of a condition in case m < n is more complicated than (2.17), one can check that the optimal threshold θ∗ must lie in 21 − 3a , 1 (this is equivalent to saying 4 that the airline profits from selling tickets rather than simply running a spot auction at date 1; i.e. the profit from a mechanism with threshold θ∗ = 1 or θ∗ ≤

1 2

−

3a 4

does no

better than a spot auction). 2.6. Conclusions In this chapter we described an alternative economic reason for overbooking and its novel implications. The conventional wisdom tells that since not all customers will show up at the gate, airlines practice overbooking to ensure that all seats are filled. In our terminology, some customers who had relatively high expected value have low realized value. Seller increases efficiency and/or revenue by selling their seats to someone else. More generally, overbooking and reallocating seats at the date of departure allows the seller to improve efficiency while capturing part of the increased surplus. However, this trade-off between ex-ante screening via ticket prices and ex-post screening via auctions creates new distortions. We showed that it is sometimes optimal to repurchase

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more seats than were overbooked and therefore leave some seats empty rather than seating buyers whose willingness to pay turned out to be low. Moreover, because of the ticket sales motive, the last-minute allocation mechanisms should be distorted to disfavor agents who have not purchased tickets. Finally, we showed that involuntary bumping (which rarely happens in practice) is never optimal.

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CHAPTER 3

Penny auctions are unpredictable 3.1. Introduction A typical penny auction may sell a new brand-name digital camera, at starting price 0 and timer at 1 minute. When the auction starts, the timer starts to tick down and players may submit bids. Each bid costs $1 to the bidder, increases price by $0.01, and resets the timer to 1 minute. Once the timer ticks to 0, the bidder who made the last bid can purchase the object at the current price. Note that the structure of penny auctions is similar to dynamic English auctions, but with one significant difference. In penny auction the bidder has to pay significant price for each bid she makes. Both the name and general idea of the auction is very similar to the dollar auction introduced by Shubik (1971). In this auction cash is sold to the highest bidder, but the two highest bidders will pay their bids. Shubik used it to illustrate potential weaknesses of traditional solution concepts and described this auction as extremely simple, highly amusing, and usually highly profitable for the seller. Dollar auction is a version of all-pay auction, that has used to describe rent-seeking, R&D races, political contests, and job-promotions. Full characterization of equilibria under full information in one-shot (first-price) all-pay auctions is given by Baye, Kovenock, and de Vries (1996). A second-price all-pay auction, also called war of attrition, was introduced by Smith (1974) and has been used to study evolutionary stability of conflicts,

119

price wars, bargaining, and patent competition. Full characterization of equilibria under full information is given by Hendricks, Weiss, and Wilson (1988). Siegel (2009) provides a equilibrium payoff characterization for general class of all-pay contests. Penny auction1 is an all-pay auction, but not a special case of well known auctions mentioned above. None of the auctions mentioned above allow the actual winner to pay less than the losers, but in penny auctions it happens in practice relatively often. There are some special cases, where penny auction is a version of well-known auctions. In two-player case the penny auction is similar to war of attrition, since the game continues only of both players continue being active in bidding and therefore incur costs. When bid cost converges zero, penny auction is converging to a dynamic first price auction. In this chapter we only consider auctions with strictly positive bid costs. We will show that in the limit where bid costs are close to zero, the object is never sold. Penny auctions are not discussed in the economics literature, with the exception of two recent working papers: Platt, Price, and Tappen (2009) and Augenblick (2009). The focus of both papers is on the empirical analysis of penny auctions. Both offer much more detailed description of the penny auctions in practice and are able to bidding behavior relatively well. To be able to use the model on the data both papers make assumptions that are in some sense strategically less flexible than ours. The theoretical model introduced by Platt, Price, and Tappen (2009) assumes that the bidders never make simultaneous decisions, which gives simple unique characterization of 1There

are two kinds of practical auctions where the name penny auctions has been used. First type was observed during the Great Depression, foreclosed farms were sold at the auctions. In these auctions sometimes the farmers colluded to keep the farm in the community at marginal prices. These low sales prices motivated the name penny auctions. Second use of the term comes from the Internet age, where in the auction sites auctions are sometimes started at very low starting prices to generate interest in the auctions. Both uses of the term are unrelated to the auctions analyzed in this chapter.

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equilibria. The theoretical model in Augenblick (2009) is much closer to ours, but with one significant difference in bid costs that will be pointed out when we introduce the model. This gives Augenblick simple equilibrium characterization. The chapter is organized as follows. Section 3.2 describes how penny auctions are used in practice and presents some stylized facts. Section 3.3 introduces theoretical model and discusses its assumptions. The analysis is divided into two parts. Section 3.4 analyzes the case when the price increment, or “the penny” in the auction name is zero, which means that the auction game will be infinite. Section 3.5 discusses the case, where price increment is strictly positive. Section 3.6 gives some concluding remarks and suggests extensions for the future research.

3.2. Stylized facts The data used in this section comes from Swoopo2 , a large penny auctions site. The data about 61,153 auctions were collected directly from their website and includes all auctions that had complete data in the beginning of May 20093 Each auction had information about the auction type, the value of the object (suggested retail price), delivery cost, the winners identity and the number of free and costly bids the winner made (used to calculate “the savings”), and the identities of 10 last bidders with information whether the bid was made using BidButler4 or not (594,956 observations in total). All auctions in Swoopo have the same structure as described in this chapter, but they have several different types of auctions which imply different parameter values. Their 2See

http://www.swoopo.com/what_is.html for details. Auctions that had incomplete data or had not finished were excluded from the dataset. 4 BidButler is an automatic bidding system where user fixes minimum and maximum price and the number of bids between them and the system makes bids for them according to some semi-public algorithm. 3

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main auction types are the the following5. The number of observations and some statistics to compare the orders of magnitude are given in the Table 3.1. • Regular auction6 is a penny auction with price increment of $0.15 and bid cost of $0.757. • Penny auction is an auction where price increment is $0.01 instead of $0.15. • Fixed Price Auction, where at the end of auction the winner pays some preannounced fixed price instead of the ending price of the auction. The Free Auction (or 100% Off Auction) is a special case of Fixed Price Auction where the winner pays only the delivery charges.8Both of these auction have the property that price increment is zero, which means that there is no clear ending point and the auctions could in principle continue infinitely. • NailBiter Auction is an auction where BidButlers are not allowed, so that each bid is made by actual person clicking on the bid button. • Finally there are some variations regarding restrictions about customers who can participate. If not specified otherwise, everyone who has won less than eight auctions per current calendar month can participate. Beginner Auction is restricted to customers who have never won an auction. Open Auction is an

5

Auctions also differ by the length of timer, ie in 20-Second Auction if after the last submitted bid the timer ticks 20 seconds, the auction ends. 6In the calculations below, we call the auction regular if it is not any of the other types of the auctions, but the other types are not mutually exclusive. For example auction can be a nailbiter penny auction with fixed price, so it is included in calculations to all three types. 7In all auction formats, $0.75 is the standard price, which is actually the upper bound of bid cost, since bids can be purchased in packages so that they are cheaper and perhaps also sunk. Also, sometimes bids can be purchased at Swoopo auction at uncertain costs. 8Both Fixed Price Auctions and Free Auctions were discontionued by 2009.

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auction where the eight auction limit does not apply, so the participation is fully unrestricted. Table 3.1. General descriptive statistics about penny auctions Type of auction Regular Penny Fixed price Free Nailbiter Beginner All auctions

Observations 41760 7355 1634 3295 924 6185 61153

Average value 166.9 773.3 967 184.5 211.5 214.5 267.6

Average price 46.7 25.1 64.9 0 8.3 45.8 41.4

Norm. value, v 1044 75919.2 6290.7 1222 1394.1 1358.5 10236.3

Norm. cost, c 5 75 5 5 5 5 13.4

Avg # of bids 242.9 1098.1 2007.2 558.5 580.1 301.6 420.9

Figure 3.1 describes the distribution of end prices in different auction formats. To be able to compare the prices of objects with different values, the plot is normalized by the value of object. For example 100 means that final price equals the retail price. Most auction formats give very similar distributions with relatively high mass at low values and long tails. Penny auctions are much more concentrated on low values, which is to be expected, since to reach any particular price level, in penny auction the bidders have to make 15 times more bids than in other formats. The most intriguing fact in the Figure 3.1 should be the positive mass in relatively high prices, since the cumulative bid costs to reach to these prices can be much higher than the value of the object. This implies that the profit margins to the seller and winner’s payoff are very volatile. Indeed, Figure 3.2(a) describes the distribution of the profit margin910 and there is positive mass in very high profit margins. The figure is somewhat arbitrarily 9Profit

bid costs−Value margin is simply defined as End price+Total · 100. Value 10 To make the plot, we need an approximate for the average bid costs. Official value is $0.75, but it is possible to get some discounts and free bids, so this would be the upper bound. In the dataset we have the number of free and non-free bids that the winners made and it turns out that about 92.88% of the bids are not free, so we used 92.88% of $0.75 which is $0.6966 as the bid cost. The overall average profit

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.

............... ............................................. . (a) Auctions with increasing prices

. . ..... . .. . ...... . .. . ........ . . . ................... . . . . . . . ............. ........ (b) Auctions with fixed prices

Figure 3.1. Distribution of the normalized end prices in different types of auctions truncated at 1000%, there also is positive, but small mass at much higher margins. From the auction formats not presented in this figure, Penny auctions have the highest average profit margin (185.8%) and Nailbiter auctions the lowest (25.2%). Note that the profit margin is calculated relative to suggested retail value, so that zero profit margin should be sufficient profit for a retail company, but mean profit margin is positive for all the auctions.

Similarly, Figure 3.2(b) describes the winner’s savings1112 from different types of auctions. In this plot, 0 would mean no savings compared to retail price, so that on the left of this line even the winner would have gained just by purchasing the object from a retail margin would be 0 at average bid cost $0.345, which is about two times smaller than our approximation of the average bid cost. 11Defined by Swoopo.com as the difference between the value of the object and winner’s total cost divided by the value. Obviously, the losers will not save anything and the winner cannot ensure winning, so the term “savings” can be misleading in ex-ante sense. Note that the reported savings at the website are such that the negative numbers are replaced by 0. 12 Again, the question is what is the right average bid cost to use. For the winners we know the number of free bids, so this is taken into account precisely, but for the costly bids, the we used the official value $0.75. True value may be below it, since there could be some quantity discounts, but it does not take into account any other constraints (like cost of time and effort). However, winner’s average savings are positive for bid costs up to $2.485, which is far above the reasonable upper bounds of the bid cost.

124

. .. ... ...... .. . ........ . ...... . . .......... . . .......... ......... ............... ......... . ............ .. ........................ .......................................................................

(a) Profit margin

.

.. ... . . . ..... . ....... . . . . . . .......... ..............................................................................

(b) Winner’s savings

Figure 3.2. Distribution of the profit margin and winner’s savings in Regular and Free auctions

store. Mostly the winner’s savings are highly positive, which is probably the reason why agents participate in the auctions after all. The density of the winnings is increasing in all auctions with mode near 100%, but the auctions differ. Regular auctions have relatively low mean and flattest distribution, whereas Free auctions (and similarly Penny auctions and Fixed price auctions) have highest mean and more mass concentrated near 100%. This is what we would expect, since in these auctions the cost is relatively more equally distributed between the bidders (if the winner was the one making most of the bids, she would win very early). The final piece of stylized facts we are looking here is the distribution of the number of bids. Figure 3.3 shows the distribution of the total number of bids. The frequencies decrease as the number of bids increases, but except in the very low numbers of bids, this decrease is slow. In Penny auctions there are on average much more bids than in other formats. The same is true for Fixed price auctions (on average 2100.6 bids), which is not

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included in the figure13 The type where auction ends at relatively low number of bids relatively more often is the nailbiter auction (on average 233.4 bids), where the bidders cannot use automated bidding system.

.. . . ..... .. . . .... .. .. .... ...... ..... ....... ......... ............... ...................... .............................. . .... .. . . .. ........ . .. .................................................................................................................................. .. . .. ..

Figure 3.3. Distribution of the number of bids submitted in different types of auctions

3.3. Model In the next three sections is to introduce a simple model that generates the stylized facts from the previous section. We will discuss some extensions in Sections 3.6. The auctioneer sells an object with market price of V dollars. We assume that this is fixed and common value to all the participants. There are N + 1 ≥ 2 players (bidders) participating in the auction, denoted by i ∈ {0, 1, . . . , N }. We assume that all bidders are risk-neutral and at each point of time maximize the expected continuation value of the game (in dollars). 13The

fact that in Free auctions and Fixed Price auctions look different in this figure is somewhat surprising and explaining this would probably require more careful empirical analysis. One possibility is that the objects sold are sufficiently different.

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The auction is dynamic, bids are submitted in discrete time points t ∈ {0, 1, . . . }. Auction starts at initial price P0 . At each period t > 0 exactly one of the players is the current leader and other N players are non-leaders. At time t = 0 all N + 1 bidders are non-leaders. At each period t, the non-leaders simultaneously choose whether to submit a bid or pass. Each submission of a bid costs C dollars and increases price by price increment ε. If K > 0 non-leaders submit a bid, each of them will be the leader in the next period with equal probability,

1 . K

So, if K > 0 bidders submit a bid at t, then Pt+1 = Pt + Kε,

and each of these K players pays C dollars to the seller14. The other non-leaders and the current leader will not pay anything at this round and will be non-leaders with certainty. The current leader cannot anything15. If all non-leaders pass at time t, the auction ends. If the auction ends at t = 0, then the seller keeps the object and if it ends at t > 0, then the object is sold to the current leader at price Pt . Finally, if the game never ends, all bidders get payoffs −∞ and the seller keeps the object. All the parameters of the game are commonly known and the players know the current leader and observe all the previous bids by all the players. We will use the following normalizations. In case ε > 0, we normalize v = and pt =

14This

Pt −P0 . ε

V −P0 , ε

c=

C , ε

In games where ε = 0 we use v = V − P0 , c = C, and pt = Pt − P0 .

is the assumption where our model differs from Augenblick (2009), which assumes that only the submitting bidder who was chosen to be the next leader has to incur the bid cost. This simplifies the game, since whenever there is at least C dollars of surplus available, all non-leaders would want to submit bids. In our model, since even unsuccessful attempts to become the leader are costly, the behavior of opponents is much more relevant. 15 This is a simplifying assumption. However, thinking about the practical auctions, it seems to be a plausible assumption to make. We will assume that the practical design of the auction is constructed so that whenever a current leader submitted a bid, the auctioneer or system assumes that it was just a mistake and ignores the bid.

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Therefore both in all cases p0 = 0. Given the assumption and normalizations, a penny auction is fully characterized by (N, v, c, ε), where ε is only used to distinguish between infinite games that we will discuss in Section 3.4 and finite games in Section 3.5. Assumption 3.3.1. We assume16 v − c > bv − cc and v > c + 1. The first assumption says that v − c is not a natural number. It is just a technical assumption to avoid considering some tie-breaking cases, where the players are indifferent between submitting one last bid and not. The second assumption just ignores irrelevant cases, since c + 1 is the absolute minimum amount of money a player must spend to win the object. So, if the assumption does not hold, the game never starts. To discuss the outcomes of the auction, we will use the following notation. Given a particular equilibrium, the probability that the game ends without any bids (with the seller keeping the object) is denoted by Q0 . Conditional on the object being sold, the probability that there was exactly p bids is denoted by Q(p). The unconditional probability of having p bids is denoted by Q(p), so that Q(0) = Q0 and Q(p) = (1 − Q0 )Q(p) for all p > 0. The normalized revenue to the seller is denoted by R and the expected revenue, conditional on object being sold, by R. As the solution concept we are considering Symmetric Stationary Subgame Perfect Nash Equilibrium (SSSPNE). We will discuss the formal details of this equilibrium in Appendix C.1 and show that in the cases we consider SSSPNE are Subgame Perfect Nash Equilibria that satisfy two requirements. First property is Symmetry, which means that the players’ identity does not play any role (so it could also be called Anonymity). The second property is Stationarity, which means that instead of conditioning their behavior on 16b·c

is the floor function bxc = min{k ∈ Z : k ≤ x}.

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the whole histories of bids and identities of leaders, players only condition their behavior on the current price and number of active bidders. In case ε > 0 this restriction means that we can use the current price p (independent on time or history how we arrived to it) as the current state variable and solve for a symmetric Nash equilibrium in this state, given the continuation values at states that follow each profile of actions. So the equilibrium is fully characterized by a q : {0, 1, . . . } → [0, 1], where q(p) is the probability of submitting a bid that each non-leader independently uses at price p. In case ε = 0 the equilibrium characterization is even simpler, since there are only two states. In the beginning of the game there is N + 1 non-leaders, and in any of the following histories the number of non-leaders is N . So, the equilibrium is characterized by (ˆ q0 , qˆ) where qˆ0 is the the probability that a player submits a bid at round 0 and qˆ is the probability that a non-leader submits a bid at any of the following rounds. The SSSPNE can be found simply by solving for Nash equilibria at both states, taking into account the continuation values. Lemmas C.1.1, C.1.2, and C.1.3 in Appendix C.1 show that any equilibria found in this way are SPNE satisfying Symmetry and Stationarity, and vice versa, any SSSPNE can be found using the described methods. It must be noted that restricting the attention to this particular subset of Subgame Perfect Nash equilibrium, is restrictive and simplifies the analysis. As we will argue later, in general there are many other Subgame Perfect Nash equilibria in these auctions. The restrictions correspond to a situation where the players are only shown the current price. In practice players have more information, but in the case when they for one reason or

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another do not want to put in enough effort to keep track on all the bids (or believe that most of the opponents will not do it), the situation is similar. As an approximation this assumption should be quite plausible. 3.4. Auction with zero price increment We will first look at a case where the price increment ε = 0. This is called “Free auction” (if P0 = 0) or “Fixed-price auction” (if P0 > 0) in Swoopo.com. One could also argue that this could be a reasonable approximation of a penny auction where ε is positive, but very small, so that the bidders perceive it as 0. In this case the auction very close to an infinitely repeated game, since there is nothing that would bound the game at any round17. After each round of bids, bid costs are already sunk and the payoffs for winning are the same. This is a well-defined game and we can look for SSSPNE in this game. As argued above and proved in the Appendix C.1, the SSSPNE is fully characterized by a pair (ˆ q0 , qˆ), where qˆ0 is the probability that a non-leader will submit at round 0 and qˆ the probability that a non-leader submits a bid at any round after 0. Let vˆ∗ , vˆ be the leader’s and non-leaders’ continuation values (after period 0). The following theorem shows that the SSSPNE is unique and gives full characterization for this equilibrium. Theorem 3.4.1. In the case ε = 0, there is a unique SSSPNE (ˆ q0 , qˆ), such that (i) qˆ ∈ (0, 1) is uniquely determined by equality (1 − qˆ)N ΨN (ˆ q ) = vc , (ii) for N + 1 = 2, then qˆ0 = 0; otherwise qˆ0 ∈ (0, 1) is uniquely determined by (1 − qˆ)N ΨN +1 (ˆ q0 ) = vc , 17This

is in contrast to ε > 0 case, where the game always ends in finite time. We will establish this in Lemma 3.5.1 in Section 3.5.

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where18 ΨN (q) =

N −1 X K=0

1 N −1 K . q (1 − q)N −(K+1) K +1 K

Function ΨN (q) is the player i’s probability of becoming the new leader in after submitting a bid when N − 1 other non-leaders submit their bids independently, each with probability q. The

1 K+1

part comes from the fact that if K other non-leaders submit bids,

then each of these players becomes the leader with this probability. Since each make their decision separately, K is Binomially distributed with parameters (q, N − 1), which gives us the expression. Lemma C.2.1 in Appendix C.2 shows that the function ΨN (q) has some nice properties. First, it is strictly decreasing in q — as the opponents bid more actively, it is harder to become the leader. Secondly, it has limits on 1 and

1 . N

This is true since when the

opponents bid with neglicent probabilty, the player who submits a bid will be the next leader with certaintly, whereas when all the opponents submit a bid with certainty, all N non-leaders will be the leaders with equal probability

1 . N

Finally, it is decreasing in N —

for a fixed q, the more opponents there is, the less likely it is to become a leader. These three properties ensure that the equilibrium exists and is unique. Proof First notice that there is no pure strategy equilibria in this game, since if qˆ = 1, then the game never ends and all players get −∞, which cannot be an equilibrium. Also, if qˆ = 0, then vˆ∗ = v and vˆ = 0. This cannot be an equilibrium, since a non-leader would want to deviate and submit a bid to get vˆ∗ − c, which is higher than vˆ, since v > c + 1 > c by assumption. Therefore, in any equilibrium qˆ ∈ (0, 1). 18 N K

is the binomial coefficient,

N K

=

N! K!(N −K)! , ∀0

≤ K ≤ N.

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We will start with the case when N +1 = 2. Since the equilibrium is in mixed strategies, non-leader’s value must be equal when submitting a bid or not. If she submits a bid, she will be the next leader with certainty and the value of not submitting a bid is 0, since the game ends with certainty. Thus vˆ = vˆ∗ − c = 0, and so vˆ∗ = c. Being the leader, there is (1 − qˆ) probability of getting the object and qˆ probability of getting vˆ = 0 in the next round, so vˆ∗ = (1 − qˆ)v = c and therefore qˆ = 1 − vc . At t = 0, if qˆ0 > 0 then expected value from bid is strictly negative19, therefore the only possible equilibrium is such that qˆ0 = 0, ie with no sale. This is indeed an equilibrium, since by submitting a bid alone gives vˆ∗ = c with certainty and costs c with certainty, so it is not profitable to deviate. Note that when N = 1, then Ψ1 (ˆ q ) = 1, so (1 − qˆ)N Ψ1 (ˆ q ) = 1 − qˆ and (1 − qˆ)v = vc v = c = vˆ∗ , so the results are a special case of the claim from the theorem. Suppose now that N + 1 ≥ 3. Look at any round after 0. Again, this is a mixed strategy equilibrium, where qˆ ∈ (0, 1), so non-leader’s value is equal to the expected value from not submitting a bid. The other N − 1 non-leaders submit a bid each with probability qˆ, which means that the game ends with probability (1 − qˆ)N −1 and continues from the same point with probability 1 − (1 − qˆ)N −1 . Therefore vˆ = [1 − (1 − qˆ)N −1 ]ˆ v + (1 − qˆ)N −1 0 ⇐⇒ vˆ = 0,

19The

cost is certainly c, but expected benefit is weighted average c and 0 both with strictly positive probability.

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since 0 < qˆ < 1. This gives the leader (1 − qˆ)N chance to win the object and with the rest of the probability to become a non-leader who gets 0, so vˆ∗ = (1 − qˆ)N v + [1 − (1 − qˆ)N ]ˆ v = (1 − qˆ)N v.

The value of qˆ is pinned down by the mixing condition of a non-leader

vˆ = 0 =

N −1 X K=0

N −1 K 1 K N −1−K ∗ qˆ (1 − qˆ) vˆ + vˆ − c ⇐⇒ K K +1 K +1

N −1 X c N − 1 qˆK (1 − qˆ)2N −(K+1) N = (1 − qˆ) = (1 − qˆ)N ΨN (ˆ q ). v K + 1 K K=0 By Lemma C.2.1, ΨN (ˆ q ) is strictly decreasing continuous function with limits 1 and

1 N

as

qˆ → 0 and qˆ → 1 correspondingly. As qˆ changes in (0, 1), it takes all values in the interval 1 , 1 , each value exactly once. Now, (1− qˆ)N is also strictly decreasing continuous function N with limits 1 and 0, so the function (1 − qˆ)N ΨN (q) is a strictly decreasing continuous function in qˆ and takes all values in the interval (0, 1). Since 0 <

c v

< 1 and there exists

unique qˆ ∈ (0, 1) that solves the equation (1 − qˆ)N ΨN (ˆ q ) = vc . Let us now consider period 0 to find the equilibrium strategy at qˆ0 . Denote the expected value that a player gets from playing the game by vˆ0 . We claim that qˆ0 ∈ (0, 1). To see this, suppose first that qˆ0 = 0, which means that the game ends instantly and all bidders get 0. By submitting a bid, a player could ensure becoming the leader with certainty in the next round and therefore getting value vˆ∗ − c = (1 − qˆ)N v − c. Equilibrium condition says that this must be less than equilibrium payoff 0, but then (1 − qˆ)N v − c ≤ 0 ⇐⇒ (1 − qˆ)N ≤

c = (1 − qˆ)N ΨN (ˆ q ), v

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so ΨN (ˆ q ) ≥ 1. This is contradiction, since ΨN (ˆ q ) < 1 for all qˆ > 0. Suppose now that qˆ0 = 1 is an equilibrium, so that each bidder must weakly prefer bidding to not bidding and getting continuation value of a non-leader, vˆ = 0. This gives equilibrium condition 1 1 (1 − qˆ)N c vˆ∗ − c = (1 − qˆ)N v − c ≥ 0 ⇐⇒ ≥ = (1 − qˆ)N ΨN (ˆ q ), N +1 N +1 N +1 v so ΨN (ˆ q) ≤

1 N +1

<

1 , N

which is a contradiction by Lemma C.2.1.

Thus, in equilibrium 0 < qˆ0 < 1 is defined by 0=

N X N K=0

K

qˆ0K (1 − qˆ0 )N −K

1 vˆ∗ − c K +1

⇐⇒

c (1 − qˆ)N ΨN +1 (ˆ q0 ) = . v

To show that this equation defines qˆ0 uniquely (for a fixed qˆ ∈ (0, 1)), we can rewrite it as follows. (1 − qˆ)N ΨN +1 (ˆ q0 ) =

c = (1 − qˆ)N ΨN (ˆ q) v

⇐⇒

ΨN +1 (ˆ q0 ) = ΨN (ˆ q ). 1 ,1 N

. As argued above (continuous, strictly decreasing) ΨN +1 (ˆ q0 ) takes values in the interval N1+1 , 1 ⊃ N1 , 1 , Now, ΨN (ˆ q ) is a fixed number. By Lemma C.2.1 ΨN (ˆ q) ∈

so the equation must have unique solution qˆ0 .

Corollary 3.4.2. From Theorem 3.4.1 we get the following properties of the auctions with ε = 0: (i) qˆ0 < qˆ. (ii) If N + 1 > 2, then the probability of selling the object is 1 − (1 − qˆ0 )N +1 > 0. If N + 1 = 2, the seller keeps the object.

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(iii) Expected ex-ante value to the players is 0. (iv) Expected revenue to the seller, conditional on sale, is v, Proof We will prove each part and also give some intuition where applicable. (i) By Lemma C.2.1, ΨN +1 (q) ≥ ΨN (q). Since ΨN +1 (ˆ q0 ) = ΨN (ˆ q ) and ΨK (q) is strictly decreasing function of q, we have qˆ0 < qˆ. This is intuitive, since from the perspective of a non-leader, the two situations are identical in terms of continuation values, but at t = 0 there is one more opponent trying to become the leader. (ii) This is just reading from the theorem. By the rules of the game, the seller sells the object whenever there was at least one bid, so the object is not sold only in the case when all bidders choose not to submit a bid at round 0. Therefore, the object is sold with probability P (p > 0) = 1 − (1 − qˆ0 )N +1 . If N + 1 = 2, then qˆ0 = 0, so P (p > 0) = 0. If N + 2 > 2, then qˆ0 > 0, so P (p > 0) > 0. (iii) Let vˆ0 be the expected ex ante value to the players. If N + 1 = 2, then vˆ0 = 0, since players pass with certainty. If N + 1 > 2, then each bidder is at round 0 indifferent between bidding and not bidding, and not bidding gives 0 if none of the other players bid and vˆ = 0 of some bid. Therefore vˆ0 = 0. (iv) There is another way how the ex-ante value to the players, vˆ0 , can be computed. Let the actual number of bids the players submitted in a particular realization of uncertainty be B. Conditioning on sale means that B > 0. Since the value to the winner is v, and collectively all the players paid Bc in bid costs, the aggregate value to the players is v − Bc. By symmetry and risk-neutrality,

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ex-ante this value is divided equally among all players, so

0 = (N + 1)v(0) =

∞ X

[v − Bc]E(B|B > 0) = v − cR.

B=1

Expected revenue to the seller, given that the object is sold, is Bc from all the bids. So R = E(Bc|B > 0) = cE(B|B > 0) = v. The following observations illustrate, that although in expected terms all the payoffs are precisely determined, in actual realizations almost anything can happen with positive probability.

Observation 3.4.3. (i) With probability (N + 1)(1 − qˆ0 )N qˆ0 (1 − qˆ)N > 0 the seller sells the object after just one bid and gets R = c. The winner gets v − c and the losers pay nothing. (ii) When we fix arbitrarily high number M , then there is positive probability that revenue R > M . This is true since there is positive probability of sale and at each round there is positive probability that all non-leaders submit bids. (iii) With positive probability we can even get a case where revenue is bigger than M , but the winner paid just c.

Observation 3.4.4. None of the qualitative results in this case were dependent on the parameter values, so changes in parameters only affect the numerical outcomes.

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(i) In particular, given that Assumption 3.3.1 is satisfied, the expected revenue and the total payoff to the bidders does not depend on the parameter values other than the fact that R = v. q) = (ii) Equilibrium conditions were (1 − qˆ)N ΨN (ˆ

c v

and ΨN +1 (ˆ q0 ) = ΨN (ˆ q ) and functions

(1 − qˆ)N ΨN (q), ΨN (q), and ΨN +1 (q) are strictly decreasing. Therefore, as

c v

increases,

both qˆ and qˆ0 will decrease. This means that for a fixed v, as c decreases, the probability of sale decreases. Note that in the limit as c → 0, we get an auction that can be approximately interpreted as dynamic English auction. The puzzling fact is that in this auction the object is never sold. (iii) As N increases, since ΨN (q) is decreasing in N , both qˆ and qˆ0 decrease.

Remark The discussion above was about SSSPNE. If we do not require stationarity and symmetry, then almost anything is possible in terms of equilibrium strategies, expected revenue to the seller, and the payoffs to the bidders. It is easy to see this from the following argument (i) Fix i ∈ {1, . . . , N + 1}. One possible SPNE is such that player i always bids and all the other players always pass. This is clearly an equilibrium since given i’s strategy, any j 6= i can never get the object and can never get more than 0 utility. Also, given that none of the opponents bid, i wants to bid, since v − c > 0. This equilibrium gives v − c to i and 0 to all the other bidders. (ii) Using this continuation strategy profile as a “punishment” we can construct other equilibria, including one where no-one bids (if i bids at the first round then some

137

j= 6 i will punish him by always bidding in the next rounds that, so that the deviator i pays c and gets nothing, whereas punisher j will get v − c > 0). (iii) Or we can construct an equilibrium where all the players bid bv/cc times and then quit. If the bidding rule is constructed so that all bidders get non-negative expected value and are punished as described above, this is indeed a possible equilibrium. This will be the highest possible revenue from a pure strategy equilibrium with symmetry on the path of play. (iv) With suitable randomizations it is possible to construct equilibria that extract any revenue from c to v. 3.5. Auction with positive price increment As argued above and proved in the Appendix C.1, we can characterize any SSSPNE by a vector q = (q(0), q(1), . . . ), where q(p) is the non-leaders’ probability to bid at price p. We showed that it is both necessary and sufficient to check for stage-game Nash equilibria, given the continuation payoffs induced by the chosen actions. In a given equilibrium, we will denote leader’s continuation value at price by v ∗ (p) and non-leaders’ continuation value by v(p). Define p˜ = bv − cc and γ = (v − c) − bv − cc ∈ [0, 1), so that v = c + p˜ + γ. Note that by Assumption 3.3.1, γ > 0 and p˜ > 0. If price increment is positive and game goes on, the price rises. This means if the game does not end earlier, then sooner or later the price rises to a level where none of the bidders would want to bid. The following Lemma establishes this obvious fact formally and gives upper bound to the prices where bidders are still active.

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Lemma 3.5.1. Fix any equilibrium. None of the players will place bids at prices pt ≥ p˜. That is, q(p) = 0 for all p ≥ p˜. Proof First note that if p > v, then the upper bound of the winner’s payoff in this game is v − p < 0 and therefore any continuation of this game is worse to all the players than end at this price. So, we know that the prices where q(p) > 0 are bounded by v. Let pˆ be the highest price where q(ˆ p) > 0. Suppose by contradiction that pˆ ≥ p˜ = bv − cc. Since q(ˆ p + K) = 0 for all K ∈ N, the game ends instantly if arriving to these prices. Therefore v(ˆ p + K) = 0 < c, and so v ∗ (ˆ p + K) = v − pˆ − K = (c + p˜ + γ) − pˆ − K = p˜ − pˆ + γ − K +c < c, | {z } | {z } ≤0

<0

So, if K − 1 ∈ {0, . . . , N − 1} opponents bid, by submitting a bid the agent gets strictly negative expected value. By not submitting a bid, any non-leader can ensure getting 0. Thus each non-leader has strictly dominating strategy not to bid at pˆ, which is a contradiction. Therefore q(p) = 0 for all p ≥ p˜.

Finally, to get cleaner results the technical Assumption 3.3.1 is not enough in some cases. In these cases we will use the following Assumption 3.5.2, which is slightly stronger.

Assumption 3.5.2. v > c + 2 and v − c < bv − cc + (N − 1)c.

The first assumption says that v − c > 2 which is same as saying p˜ > 1 (instead of p˜ > 0). The second assumption says that γ < (N − 1)c, ie neither c and N are not too small. Both assumptions are mild and easily satisfied in practical applications, where v c > 1, so γ < 1 < (N − 1)c whenever N > 1.

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Corollary 3.5.3. With ε > 0, in any equilibrium: (i) Price level max{˜ p − 1 + N, N + 1} is an upper bound of the support of realized prices. Under Assumption 3.5.2, the upper bound is just p˜ − 1 + N . If p˜ > 1, then the last price where bidders could make bids with positive probability is p˜ − 1 and if all N non-leaders make bids, we will reach the price p˜ + N − 1. If p˜ = 1, then the bidders only make bids at 0 and there are N + 1 non-leaders at this stage, so the upper bound is N + 1. Combination of these two cases gives us the upper bound. Assumption 3.5.2 and specifically the assumption that v − c > 2 ensures that p˜ > 1 and therefore we do not have to us the max operator. (ii) The game is finite and there exists a a point of time τ ≤ p˜ + N , where game has ended with certainty at any equilibrium. This is true since at each period when the game does not end, the price has to increase at least by 1. (iii) All non-leaders have strictly dominating strategy not to bid at prices pt ≥ p˜ and at t + 1 the game has ended with certainty. This means that we can use backwards induction to find any SPNE.

3.5.1. Two-player case The two-player case is very simple, since we have an alternating-move game, where at t > 0, one of the players is always leading and the other (non-leader) can choose whether to bid and become leader or pass and end the game. We can simply solve it by backwards induction. To see the intuition, let us start by solving a couple of backward induction steps before stating the result formally.

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By Lemma 3.5.1, at prices p ≥ p˜, the non-leader would never bid. Therefore, the continuation values values are v ∗ (p) = v − p, v(p) = 0,

∀p ≥ p˜, and in particular

v ∗ (˜ p) = v − p˜ = c + γ. At p = p˜ − 1, non-leader will make a bid since v ∗ (p + 1) − c = v ∗ (˜ p) − c = γ > 0. Therefore v ∗ (˜ p − 1) = v(˜ p) = 0, v(˜ p − 1) = γ. At p = p˜ − 2 > 0, non-leader will not make a bid, since continuation value in the next round is 0 which does not cover the cost of bid. Thus v ∗ (˜ p − 2) = v − (˜ p − 2) = c + γ + 2, v(˜ p − 2) = 0. We can continue this process for all t > 0 and then need to consider the simultaneous decision at stage 0. The following Proposition 3.5.4 characterizes the set of equilibria for two-player case. Proposition 3.5.4. Suppose ε > 0 and N + 1 = 2. There is a unique SSSPNE and the strategies q are such that

q(p) =

0 ∀p ≥ p˜ and ∀p = p˜ − 2i > 0, i ∈ N, 1 ∀p = p˜ − (2i + 1) > 0, i ∈ N,

and q(0) is determined for each (v, c) by one of the following cases. (i) If p˜ is an even integer, then q(0) = 0. (ii) If p˜ is odd integer and v ≥ 3(c + 1), then q(0) = 1. v−(c+1) (iii) If p˜ is odd integer and v < 3(c + 1), then q(0) = 2 v+(c+1) ∈ (0, 1).

Proof As argued above, by Lemma 3.5.1, q(p) = 0 for all p ≥ p˜. For p ∈ {1, . . . , p˜} we are using backwards induction. In particular, we show that q(p) is optimal at p given that

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it is optimal for prices higher than p using mathematical induction. Since q(˜ p) = 0, at p = p˜ − 1 bidding gives v − (p + 1) − c = v − c − bv − cc > 0, so q(p) = 1. This gives us induction basis for i = 0, since then p˜ − 2i = p˜ and p˜ − (2i + 1) = p˜ − 1. Assuming that the claim is true for i, we want to show that it holds for i + 1. Since q(˜ p − 2i) = 0 the game ends and the leader wins instantly, so v ∗ (˜ p − 2i) = v − p˜ + 2i = c + γ + 2i,

v(˜ p − 2i) = 0.

Also, q(˜ p − (2i + 1)) = 1, that is, the price increases by 1 with certainty and the roles are reversed, so v ∗ (˜ p − (2i + 1)) = v(˜ p − 2i) = 0,

v(˜ p − (2i + 1)) = v − p˜ + 2i − c = 2i + γ.

Let p = p˜ − 2(i + 1). Then p + 1 = p˜ − (2i + 1), so submitting a bid would give v ∗ (˜ p − (2i + 1)) − c = −c to the non-leader, which is not profitable. Therefore q(˜ p − 2(i + 1)) = 0 and the leader gets v ∗ (˜ p − 2(i + 1)) = v − p˜ + 2(i + 1) = c + γ + 2(i + 1). Let p = p˜ − (2(i + 1) + 1), so that p + 1 = p˜ − 2(i + 1). Then making a bid would give v ∗ (˜ p − 2(i + 1)) − c = γ + 2(i + 1) > 0 to the non-leader, which means that it is profitable to make a bid. To complete the analysis, we have to consider t = 0, where p = 0 and both players are non-leaders simultaneously choosing to bid or not. In this stage, there three cases to consider.

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First consider the case when p˜ is an even integer, ie p˜ = 2i + 2 for some i ∈ N. Then 2 = p˜ − 2i and 1 = p˜ − (2i + 1), so we get the strategic-form stage game in the Figure 3.4. In this game both players have strictly dominating strategy to pass, ie q(0) = 0. That is,

B P

1 (2i 2

B P + γ − c), 21 (2i + γ − c) −c, 2i + γ 2i + γ, −c 0, 0

Figure 3.4. Period 0 game when n = 2 and p˜ is even

the unique SPNE in the case when p˜ is even, is the one where the seller keeps the object. Suppose now that p˜ is odd number, ie p˜ = 2i + 1, so that 1 = p˜ − 2i and 2 = p˜ − (2i − 1). Then we get strategic form in the Figure 3.5

B P

γ 2

B P γ + i − 1 − c, 2 + i − 1 − c 2i + γ, 0 0, 2i + γ 0, 0

Figure 3.5. Period 0 game when n = 2 and p˜ is odd

Note that 2i + γ = p˜ − 1 + γ = v − c − 1, so 12 (2i + γ − 2) − c = 12 (v − 3(c + 1)). The sign of this expression is not determined by assumptions, so we have to consider two cases. If v ≥ 3(c + 1), then bidding at round 0 is dominating strategy for both players, ie q(0) = 1. Both players will submit a bid at round 0, and the one who will be the non-leader will submit another bid after that. This means that in total players make 3 bids and the price ends up to be 3. This is where the condition v ≥ 3(c + 1) comes from.

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If v < 3(c + 1), then there is a symmetric MSNE20, where both bidders bid with probability q ∈ (0, 1), where q is determined by q

1 (2i + γ − 2) − c + (1 − q)(2i + γ) = 0 ⇐⇒ 2

q(0) =

v − (c + 1) 2(2i + γ) =2 ∈ (0, 1). 2c + 2 + 2i + γ v + (c + 1)

Observation 3.5.5. Some observations regarding the SSSPNE in the two-player case. (i) Equilibrium outcomes are very sensitive to seemingly irrelevant detail — is p˜ even or odd. (ii) For realistic parameter values v 3(c + 1). Then the equilibrium collapses in a sense that R = 3(c + 1) v or the object is not sold. (iii) In a special case when p˜ is an odd integer and v < 3(c + 1), we get the results similar to ε = 0 case: P (p > 0) ∈ (0, 1), E(R|p > 0) = v, v(0) = 0. In this equilibrium the players submit bids with positive probabilities and hope that the other does not submit a bid. But if she does, players actually prefer to be non-leaders, since at price p = 2, non-leader submits one more bid and the game ends at p = 3. Therefore P (0) > 0, P (1) > 0, P (2) = 0, P (3) > 0, P (p) = 0, ∀p ≥ 4.

20There

are also two asymmetric pure-strategy NE in the subgame, (P, B) and (B, P ), where one player makes exactly one bid, so the revenue is c + 1 and the value for this bidder is v − (c + 1).

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3.5.2. More than two players In N + 1-player case (for arbitrary N ≥ 2) the discussion is similar to previous, but at each round we have 2 or more non-leaders choosing to bid or not simultaneously. To see how an equilibrium looks like, consider the Example 3.5.6. Example 3.5.6. Let N + 1 = 3, v = 4.1, c = 2, and ε > 0. The unique SSSPNE for this game is given in the Table 3.2. Since q(0) ∈ (0, 1), the expected utility for all players Table 3.2. Example 3.5.6, solution p q(p) v ∗ (p) v(p) Q(p) Q(p) 0 0.2299 0 0.4567 1 0.0645 2.7129 0 0.358 0.6588 2 0 2.1 0 0.1715 0.3157 3 0 1.1 0 0.0139 0.0255 4 0 0.1 0 0 0 is v(0) = 0 and expected revenue for the seller E(R|p > 0) = v = 4.1. Note that ex-ante expectation of the sales price is going to be non-trivial. In fact, with 2.5% probability we observe price 3, which implies revenue 3(2 + 1) = 9, which is significantly higher than 4.1. From this, c + 3 = 5 > 4.1 = v is paid by the winner and both losers will pay 2. By Lemma 3.5.1 in any game q(p) = 0, ∀p ≥ p˜. When we take p = p˜ + K for K = 0, 1, . . . , then q(˜ p + K) = 0 and v ∗ (˜ p + K) = v − (˜ p + K) = v − c − p˜ + c − K = c + γ − K,

v(˜ p + K) = 0.

So, we can consider the rest of the game to be finite and solve it using backwards induction. Take p ∈ {0, . . . , p˜ − 1}. If p > 0, there are N non-leaders and if p = 0, there are N + 1.

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¯ . Then one of the following three situations Denote the number of non-leaders by N characterizes q(p), v ∗ (p), and v(p). ¯ non-leaders submit bids with certainty. In First, a stage-game equilibrium where all N this case q(p), v ∗ (p), and v(p) are characterized by the three equalities in conditions (C1). This is an equilibrium if none of the non-leaders wants to pass and become non-leader at ¯ − 1 with certainty, which gives us the inequality condition in (C1). price p + N ¯ ), and Conditions 1 (C1). q(p) = 1, v ∗ (p) = v(p + N ¯ 1 ¯ ) + N − 1 v(p + N ) − c ≥ v(p + N ¯ − 1). v(p) = ¯ v ∗ (p + N ¯ N N ¯ non-leaders choose to Secondly, there could be a stage-game equilibrium where all N pass. This is characterized by (C2). Conditions 2 (C2). q(p) = 0, v ∗ (p) = v − p, and v(p) = 0 ≥ v ∗ (p + 1) − c. Finally, there could be a symmetric mixed-strategy stage-game equilibrium, where ¯ non-leaders bid with probability q ∈ (0, 1). This gives us (C3). equilibrium, where all N Conditions 3 (C3). 0 < q(p) < 1,

v(p) =

¯ −1 N X K=0

¯ − 1 N 1 K ¯ −1−K K N ∗ q (1 − q) v (p + K + 1) + v(p + K + 1) − c K K +1 K +1

=

¯ −1 N X

¯ − 1 N ¯ q K (1 − q)N −1−K v(p + K), K

K=1

∗

¯ N

v (p) = (1 − q) (v − p) +

¯ N X ¯ N K=1

K

¯

q K (1 − q)N −K v(p + K).

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Note that every equilibrium each q(p) must satisfy either (C1), (C2), or (C3) and therefore an equilibrium is recursively characterized. However, nothing is saying that the equilibrium is unique. In Appendix C.3 we have example, where at p = 2, each of the three sets of conditions gives different solutions and so there are three different equilibria. ¯ − 1’th order polynomial, so it may Moreover, in (C3) the equation characterizing q is N ¯ − 1 different solutions which could lead to different equilibria. have up to N Theorem 3.5.7. In case ε > 0, there exists a SSSPNE q : N → [0, 1], such that q and the corresponding continuation value functions are recursively characterized (C1), (C2), or (C3) at each p < p˜ and q(p) = 0 for all p ≥ p˜. The equilibrium is not in general unique. Proof N + 1 = 2 is already covered in Proposition 3.5.4 and is a very simple special case of the formulation above. If N + 1 > 2, then the formulation above describes the method to find equilibrium q. The conditions (C1), (C2), and (C3) are written so that there are no profitable one-stage deviations. To prove the existence we only have to prove that there is at least one q that satisfies at least one of three sets of conditions. At each stage, we have a finite symmetric strategic game. Nash (1951) Theorem 2 proves that it has at least one symmetric equilibrium21. Since there conditions are constructed so that any mixed or pure strategy stage-game Nash equilibrium would satisfy them, there exists at least one such q. Finally, Appendix C.3 gives a simple example where the equilibrium is not unique. 21His

concept of symmetry was more general — he showed that there is an equilibrium that is invariant under every automorphism (permutation of its pure strategies). Cheng, Reeves, Vorobeychik, and Wellman (2004) point out that in a finite symmetric game this is equivalent to saying that there is a mixed strategy equilibrium where all players play the same mixed strategy. They also offer a simpler proof for this special case as Theorem 4 in their paper.

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Corollary 3.5.8. With ε > 0, in any SSSPNE, we can say the following about R. (i) R ≤ v, (ii) if q(p) < 1, ∀p, then R = v, (iii) In some games in some equilibria R < v.

Proof (i) Similarly to the proof of Corollary 3.4.2, the aggregate expected value to the players must be equal to v minus the aggregate payments, which is the sum of p and costs pc. The revenue to the seller is exactly the sum of all payments, so (N + 1)v(0) = v − E(p + pc|p > 0)v − R.

Players’ strategy space includes option of always passing, which gives 0 with certainty. Therefore in any SSSPNE, v(0) ≥ 0, so R ≤ v. (ii) If q(p) < 1 for all p, then this mixed strategy puts strictly positive probability on the pure strategy where the player never bids. This pure strategy gives 0 with certainty and so v(0) = 0. (iii) If q(p) = 1 for some p, then the previous argument does not work, since the player does not put positive probability on never-bidding pure strategy. To prove the existence claim, it is sufficient to give an example. We already found in previous subsection that in N + 1 = 2 player case, if p˜ is odd and v > 3(c + 1), then q(0) = 1 and E(R|p > 0) = 3(c + 1) < v. Example in Appendix C.3 gives a more complex equilibrium (details are in the Table C.1) where q(0) ∈ (0, 1), q(1) = 0, but q(2) = 1 and E(R|p > 0) = 8.62 < 9.1 = v.

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The following lemma gives restriction how often the players can pass. It shows that there cannot be two adjacent price levels in {1, . . . , p˜}, where none of the bidders submits a bid. Lemma 3.5.1 showed that p˜ is the upper bound of the prices where bidders may submit bids. Lemma 3.5.9 says that at p˜ − 1 players always bid with positive probability, so that it is the least upper bound. Lemma 3.5.9. With ε > 0, in any SSSPNE, @ˆ p ∈ {2, . . . , p˜} st q(ˆ p − 1) = q(ˆ p) = 0. In particular, q(˜ p − 1) > 0. Proof Suppose ∃ˆ p ∈ {2, . . . , p˜} such that q(ˆ p − 1) = q(ˆ p) = 0. Since q(ˆ p) = 0, the game ends there with certainty and therefore v ∗ (ˆ p) = v − pˆ. q(ˆ p − 1) = 0, so the game ends instantly and all non-leaders get 0. By submitting a bid at pˆ − 1 a non-leader would become leader at price pˆ with certainty. So the equilibrium condition at pˆ − 1 is 0 ≥ v ∗ (ˆ p) − c = v − pˆ − c ⇐⇒ pˆ ≥ v − c = p˜ + γ > p˜. This is a contradiction with assumption that pˆ ≤ p˜. Since q(˜ p) = 0 by Lemma 3.5.1, this also implies that q(˜ p − 1) > 0.

The following proposition says that, conditional on the object being sold, very high prices are reached with positive probability. In fact, with relatively weak additional Assumption 3.5.2, the upper bound of possible prices is reached with positive probability. Proposition 3.5.10. Let ε > 0, fix any SSSPNE where the object is being sold with positive probability, and let p∗ be the highest price reached with strictly probability. Then

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(i) p˜ ≤ p∗ ≤ max{˜ p + N − 1, N + 1}, (ii) Under Assumption 3.5.2, p∗ = p˜ + N − 1. Proof (i) By Corollary 3.5.3, p∗ ≤ max{˜ p + N − 1, N + 1}. Since p∗ is reached with positive probability and the higher prices are never reached, q(p∗ ) = 0. Equilibrium condition for this is v ∗ (p∗ + 1) − c ≤ 0. When arriving to any p > p∗ , the game ends with certainty, so in particular at p∗ + 1 we have v ∗ (p∗ + 1) = v − p∗ − 1. This gives p∗ ≥ v − c − 1 = p˜ − (1 − γ) > p˜ − 1. Since p∗ and p˜ are integers, this implies p∗ ≥ p˜. (ii) With Assumption 3.5.2 Corollary 3.5.3 gives p∗ ≤ p˜+N −1. Suppose by contradiction that p˜ ≤ p∗ < p˜ + N − 1. This can only be true if Q(p∗ − N ) > 0 and q(p∗ − N ) > 0. First, look at case q(p∗ − N ) < 1. This would mean Q(()p) > 0 for all p ∈ p − 1) > 0 and by Lemma 3.5.9 q(˜ p − 1) > 0, so {p∗ − N, . . . , p∗ }. In particular, Q(˜ Q(˜ p −1+N ) > 0, which is contradiction with p∗ < p˜−1+N . Therefore q(p∗ −N ) = 1, so all non-leaders submit bids, knowing that all others do the same and the price rises to p∗ with certainty. This can be an equilibrium action if 1 ∗ ∗ N −1 ∗ v (p ) + v(p ) − c ≥ v(p∗ − 1). N N Since p∗ ≥ p˜, the game ends instantly at this price and therefore v ∗ (p∗ ) = v − p∗ and v(p∗ ) = 0. Finally, v(p∗ − 1) ≥ 0 (since player can always ensure at least 0 payoff by not bidding). This gives the condition v − N c ≥ p∗ ≥ p˜ = v − c − γ

⇐⇒

γ ≥ (N − 1)c.

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This contradicts Assumption 3.5.2. Corollary 3.5.11. When the object is sold and Assumption 3.5.2 is satisfied, (i) R > v with positive probability, (ii) R < v with positive probability

So, we have shown in previous Proposition that sometimes the object is sold at very high prices, and in this Corollary that sometimes the seller earns positive profits and sometimes incurs losses. This means that the auction has the stylized properties described in Section 3.2. Proof (i) By the previous proposition, there is positive probability that the object is sold at price p∗ ≥ p˜ + 1 = v − c + (1 − γ). Therefore, when object is sold at price p∗ , the revenue is R = (c + 1)p∗ ≥ (c + 1)[v − c + (1 − γ)] > v ⇐⇒

c − (1 − γ) v > , c+1 c

which holds as strict inequality, since v > c + 1 and γ < 1. (ii) Since R ≤ v and R > v with strictly positive probability, it must be also R < v with strictly positive probability. With ε > 0 the equilibria are non-trivially related to parameter values. The number of equilibria may increase or decrease as parameter values changes, and the equilibrium

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outcomes may are generally affected non-monotonically. However, we can make some observations regarding the parameter values in the limits. When c is very small, then in the limit we would get a version of Dynamic English auction. Perhaps contrary to the intuition this auction generally ends very soon. The general intuition of this observation is the following. Suppose N + 1 = 3, q(p + 1) < 1, and q(p + 2) < 1; v ∗ (p + 1) > 0, v ∗ (p + 2) > 0, v(p + 1) = v(p + 2) = 0 and c → 0. Then at price p there is certainly a stage-game equilibrium where q = 1 since v ∗ (p + 2) − c > v(p + 1) = 0. There are no equilibria q < 1, since player cannot be indifferent between positive expected value from bid and 0 from no bid. For this reason there will be relatively many prices where q(p) = 1. Now, if q(p) = 1 then being leader at p is in general worse than being non-leader, so at p − 1 the players have lower incentives to bid. In many equilibria this leads to situation where R v. To put it in the other words, when cost of bid is small, then whenever there is positive expected value from bidding, players compete heavily, which drives down the value to the bidders and therefore there are low incentives to bid in earlier rounds. If c is nearly the upper bound v − 1, then the game gives positive utility to the bidders only if there is exactly one bid. q = 0 will not be an equilibrium, since lone bidder would get positive utility. Also, at p > 0 no-one bids. Therefore the unique equilibrium is such that q(0) is a very small number and q(p) = 0 for all p > 0. Then R = v, but probability of sale is very low. As mentioned above, if c ≥ v − 1 or equivalently, 1 ≥ v − c = p˜ + γ, then p˜ = 0 and there can never be any bids. This is obvious, since to get positive payoff one needs to become a leader and minimal possible cost for this is c + 1.

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Increase in v means that the game is getting longer and this means that there are more states with strategic decisions and generally more possible equilibria and non-trivial effect on strategies and revenue. Decrease in v has the opposite effect and as v → c + 1 we get the case described above. If N is very large, then q(p) < 1 for any p just because if q(p) = 1 this would mean that p + N > v and so players cannot get positive value from bidding, whereas they have to incur cost and may ensure 0 by not bidding. Obviously, in q(p) is not always 0, since it would still be good to be a lone bidder. So, in general we would expect to see many p’s with low positive (and sometimes 0) values of q(p). Since q(p) < 1, ∀p we would have R = v.

3.6. Discussion The model introduced in this chapter has some interesting properties of penny auctions. In these auctions, the outcome to the individual bidders and to the seller is very unpredictable and varies in a large interval. We showed that under very mild conditions that are satisfied in all practical auctions, any symmetric and stationary equilibrium must be such that even the highest possible prices are sometimes reached. In particular, we showed that in the fixed price penny auctions there can be unboundedly many bids in equilibrium, therefore the (actual) revenue of the seller is unbounded. In the increasing price auctions, the upper bound of possible prices is p∗ = bv − cc − 1 + N and it is reached with positive probability. This is a very high price where even the winner gets strictly negative payoff22 and to reach this price, players had to make many costly bids. 22The

winner has to pay at least p∗ + c, so her value is at most v − p∗ − c = γ + 1 − N < 0.

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Since under some realizations the number of bids is very high, but the expected revenue is always bounded by v, there is also high probability that the auction ends at low prices. This gives the shapes of the figures we saw in Section 3.2. However, this kind of model is unable to replicate one property that the practical penny auctions seem to have. As shown in Figure 3.2(a), in real auctions the average profit margin seems to be significantly higher than zero. In penny auctions the objects sold have well-defined market value, handing over the object has alternative cost v to the seller. Since in the game above, the expected revenue is less than or equal to v, it means that the seller would always be better off by setting up a supermarket and selling the object at a posted price. This is not a property of the auction, but a general individual rationality argument — since individuals can always ensure at least 0 value by inactivity, it is impossible to extract on average more than the value they expect to get. To achieve an outcome where expected revenue is strictly higher than the value of the object, we would need to add something to the model. A trivial way to overcome (or actually ignore) the problem is to say that the value to the seller is some vs < v = vb . It could be for example that the suggested retail value is by far higher than the cost to the seller, but around the value that the customers expect to get. This would obviously mean that there are expected profits, but it does not explain why the seller would not use alternative selling methods. One explanation from practice seems to be that it is “Entertainment shopping”. This could mean that the bidders get some positive utility from participating, some “Gambling value” vg in addition to v if winning. Then again vb = v + vg > vs . This could be true because winning an auction feels like an accomplishment. In this case this could be an

154

increasing function of N (beating N opponents is great). There are other possible ways to model this entertainment value. (1) For example, modeling it as a lump-sum sum value just from participating or (2) as a positive income that is increasing in the number of bids. (3) Assuming that “Saving” money gives some additional happiness. Then instead of v − p the player would have some increasing function f (v − p). If it is linear, it is a simple transformation of previous. Alternatively, individuals might not consider c to be at the same monetary scale as v and p, since it is partly sunk. In practice people buy “bid packs” with 50 or 100 bids at a time, so the story is complicated, but it is reasonable to think that with some probability an individual has marginal cost of next bid less than c. Suppose the bidders consider the cost of bid cb < c. Then R = (c + 1)E(p|p > 0) > (cb + 1)E(p|p > 0) and 0 ≤ (N + 1)v(0) = v − (cb + 1)E(p|p > 0), so it is possible to earn profit. There could be other ways to affect v, p, c via linear or lump-sum changes to tell other stories. Another approach would be to consider some boundedly rational behavior or uncertainty in the model. A specific property of “penny auctions” seems to be that the price increase is marginal for a bidder. We could consider a case where individuals behave as (at least for a while) that the action is with ε = 0, but with value shrinking as the price increases. Generally this would not be an equilibrium in game-theoretic sense, but it might be realistic in practice and, as shown in this chapter, is computationally easier, since there is always unique and explicitly characterized equilibrium. Another question to consider is the reputation of players. Since in practical auctions the user name of a bidder is public, this could mean reputation effects between the auctions and during one auction. If a player has built a reputation of being “tough” bidder in

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previous auctions, since it is an all-pay auction, it obviously affects the other bidders. Then the first thing to notice is the fact that in this case the equilibrium is in general not symmetric. As we argued in some cases above, there could be (and in some cases are) equilibria, where one bidder always bids and other never bid. This means that there is reputation-type equilibrium even without any costs of reputation building, just some communication between bidders is enough. Of course, in the long run, it may be profitable to invest in building reputation and therefore there could be some types of behaviors to consider outside of our model. Finally, in practice automated bids called Bid butlers are used. The system allows bidders to specify starting and ending prices and the number of bids the system should make on their behalf. Players can always cancel their Bid butler and the opponents see whether bid is made using Bid butler or manually. This may have interesting implications for the game. In most trivial way – just assuming the bidders can start and stop their Bid butlers at any moment of time, it would not affect the game at all, since everyone can replicate any strategy either with Bid butler or without. But when assuming there is some probability that the Bid butler is used while opponent is away from the game for some positive amount of time means that it may have reputation-type effect during the auction. By observing a bid by Bid butler, opponents update their belief about the next move slightly, and this may change their behavior radically. As argued here, this is only the first attempt to characterize this type of auctions in a game-theoretic model. The next steps would involve adding some behavioral aspects that would probably benefit form a careful empirical analysis that would show which

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kind of behaviors or biases are behind of the outcomes that cannot be replicated by a straightforward model.

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APPENDIX A

Proofs of Results in Chapter 1 To shorten the notation, for a given realization (n, v) we denote the kth lowest virtual surplus by wk = w(vk:n ), so that w is always the ordered virtual surplus vector corresponding to (n, v). We denote its distribution function by Fk , so that Fk (w) = P r(wk ≤ w) = P r(vk:n ≤ w−1 (w)) = Fk:n (w−1 (w)).

A.1. Calendar mechanism Proof of Theorem 1.3.1 Remember that state z ∈ {0, . . . , m} denotes how many service slots are already promised to buyers in the previous period. Then at state z the seller can serve up to m − z new buyers instantly. Let π(z) denote maximum revenue from state z. It is clear that π(z) is strictly decreasing in z. We will start by assuming that c is large enough so that the seller can commit to all promises. Fix state z, number of buyers n and the corresponding ordered virtual surplus vector w. Let {I, D, R} be a partition of buyers {1, . . . , n}, such that I is the set of buyers who will be served instantly, D receive delayed service, and R are refused service. Feasibility requires that #I ≤ z, #R ≤ m. We can express the maximum revenue from state z and realization (n, w) by π z (n, w) = (1 − δ)

X i∈I

wi + (1 − δ)

X i∈D

βwi + δπ(#D)

162

We can make some instant observations. First, when wk ≤ 0, buyer k will not get the object, since refusing him service would be feasible and would increase instant revenue without decreasing continuation payoff. Second, if z + #I < m, so that some of the current objects remain unallocated, it must be because there were exactly #I buyers with wi > 0. This is true, since by replacing the policy that assigns wn−(#I+1) current object, the seller would strictly increase flow payoff while still satisfying feasibility and not decreasing continuation payoff. Finally, it must be the case that if i ∈ I, d ∈ D, and r ∈ R, then wi ≥ wd ≥ wr . That is, we can express I, D, and R as I = {n, . . . , n + 1 − #I}, D = {n − #I, . . . , n − #I + 1 − #D}, R = {n − #I − #D, . . . , n − #I − #D + 1 − #R}). If this is not true, the seller could swap the service dates of the buyers for whom the order does not hold, thus preserving feasibility and continuation payoffs, but increasing flow revenue. This means that the optimal policy such that up to m − z buyers with highest positive virtual surpluses will receive the current service. If there are less than m − z buyers with positive surpluses, the rest of the buyers are refused service and the continuation state will be m (and k = 0). Consider the situation where there are more than m − z buyers with strictly positive virtual surpluses, so that wn−(m−z) > 0. Then #I = {n, . . . , n + 1 − (m − z)} and the seller chooses optimal number k of buyers with highest positive virtual surpluses among remaining buyers to whom to offer delayed service, so that D = {n − (m − z), . . . , n + 1 − (m − z) − k}. Denote the revenue difference between having all m service slots available and having one slot unavailable by ∆1 = π(0) − π(1) > 0. Similarly, let ∆k denote the revenue decrease from having kth slot unavailable, so that ∆k = π(k − 1) − π(k). Analogously to the

163

illustrative example, the normalized discounted version of this difference is the opportunity cost, which we denote by wk =

δ ∆k . 1−δ β n X

π z (n, w) = (1 − δ)

Now, k is optimal if it maximizes

wi + (1 − δ)β

k X

wn+1−(m−z)−i + δπ(k).

i=1

i=n+1−(m−z)

Since π(k) = π(0) − π(0) + π(1) − · · · + π(k) = π(0) −

Pk

i=1

∆i , we can rewrite the objective

as

(A.1)

n X

π z (n, w) = (1 − δ)

wi + δπ(0) + (1 − δ)β

i=1

i=n+1−(m−z)

The first part, (1 − δ)

Pn

i=n+1−(m−z)

k X [wn+1−(m−z)−i − wi ].

wi + δπ(0), is the maximum static revenue that the

seller can achieve, whereas the last part is the dynamic revenue, which in optimum is always non-negative. To compute the expected revenue in state z, we have to take expectation of π z (n, w) with respect to the number of buyers n and valuations v. This gives " π(z) = En

(1 − δ)

m−z XZ 1 i=1

wdFn+1−i (w) + δπ(0)

0

# +

(1 − δ)βEv W ((wn−(m−z) , . . . , wn+1−(m−z)−m ), w) ,

where W ((wn−(m−z) , . . . , wn+1−(m−z)−m ), w) = maxk∈{0,...,m}

Pk i=1

wn+1−(m−z)−i − wi . There-

fore for each j ∈ {1, . . . , m},

wj =

δ π(j − 1) − π(j) δ En,w [π j−1 (n, w) − π j (n, w)] = = Ψj (w), 1−δ β 1−δ β

164

where δ Ψj (w) = En β

"Z

1

wdFn−(m−j) (w) 0

# +βEv

W ((wn−(m−j) , . . . , wn+1−(m−j)−m ), w) − W ((wn−(m−j)−1 , . . . , wn−(m−j)−m ), w)

To show that vector w ∈ [0, 1]m exists and is uniquely defined, we show that Ψ is continuous and strictly increasing and Ψ(w) ∈ [0, 1]m for all w ∈ [0, 1]m . Then w exists by Brouwer’s fixed point theorem and is unique because of strict monotonicity. We will need some properties of W functions that are proven separately in Lemmas A.1.1 to A.1.3. Continuity comes directly from the fact that expectation operator is continuous and the function W is continuous (Lemma A.1.1). For boundedness we look at w = 1 = (1, . . . , 1) and w = 0 = (0, . . . , 0). At w = 0, the term maxk∈{0,...,m}

Pk

i=1

wn+1−(m−j)−i includes wn+1−(m−j)−i if and only

if wn+1−(m−j)−i > 0. Therefore1 Ev [W ((wn−(m−j) , . . . , wn+1−(m−j)−m ))] =

m Z X i=1

1

[1 − Fn+1−(m−j)−i (w)]dw.

0

We get an analogous expression for the other W term. Plugging this into Ψi and rearranging gives Z 1 δ Ψj (0) = En 1 − (1 − β)Fn−(m−j) (w) + βFn−(m−j)−m (w) dw ≥ 0. β 0 Next, at w = 1 note that [wk − 1] ≤ 0 for any k > 0, therefore W (·, 1) = 0. Therefore δ Ψj (1) = En β 1Integration

Z

by parts gives

1

0

R1 0

δ [1 − Fn−(m−j) (w)]dw ≤ En β

wdFk (w) =

R1 0

[1 − Fk (w)]dw.

Z

1

[1 − Fn (w)]dw ≤ 1,

0

165

where the first inequality comes from the stochastic dominance of higher order statistics hR i 1 and the second inequality from the assumptions. When βδ En 0 [1 − Fn (w)]dw > 1, not only is the existence not guaranteed, but also optimal calendar mechanism with positive probability of delay cannot exist, since at any (n, v, z), the surplus extracted from delay is less than the expected value from having one more object available next period. Therefore in this case the optimal mechanism would be the optimal static mechanism. Finally, strict monotonicity of Ψj (w) with respect to w follows from Lemma A.1.3. Therefore Ψj (w) ∈ [Ψj (0), Ψj (1)] ⊂ [0, 1] and so we have continuous Ψ : [0, 1]m → [0, 1]m which means that by Brouwer’s fixed point theorem w exists and by strict monotonicity it is unique. To complete the proof we have to verify that when c ≥ 1 − δ, the optimal mechanism described above satisfies commitment conditions. Fix any state z < m (at state m there are obviously no commitment issues), realization (n, w). By fulfilling all promises, the seller gets revenue π z (n, w), whereas leaving a promise to one previous customer unfulfilled will give one additional instant service spot, which therefore gives revenue π j−1 (n, w). Therefore the cost of fulfilling the promise is π j−1 (n, w) − π j (n, w) = (1 − δ)wn−(m−z) # " k k X X [wn−(m−z)−i − wi ] . +(1 − δ)β max [wn+1−(m−z)−i − wi ] − max k

By Lemma A.1.1, maxk

i=1

Pk

i=1 [wn+1−(m−z)−i

k

i=1

− wi ] ≤ maxk

Pk

i=1 [wn−(m−z)−i

− wi ], and

by assumptions wn−(m−z) ≤ w(1) = 1, therefore maximum possible value for π j−1 (n, w) − π j (n, w) is 1 − δ. This means that c ≥ 1 − δ is sufficient for the commitment.

166

It is also necessary if at each z there exists (n, w) such that π j−1 (n, w) − π j (n, w) achieves value 1 − δ. When N ≥ 2m, we have that n ≥ z + m at each z ∈ {0, . . . , m}, and therefore a possible realization is such that N = 2m, wn−(m−z) = · · · = wn−(m−z)−m = 1, which gives the equality. When N ∈ {m+1, . . . , 2m−1} the necessary condition (especially for low states) is even weaker and needs to be computed directly from the expression

above. Pk

− wi ] for all w, w ∈ [0, 1]m such P that w1 ≥ w2 ≥ · · · ≥ wm . Let k ∗ (w, w) = min argmaxk∈{0,...,m} ki=1 [wi − wi ]. Then Lemma A.1.1. Let W (w, w) = maxk∈{0,...,m}

i=1 [wi

(i) W continuous, non-decreasing in w, non-increasing in w, and convex. (ii) k ∗ is non-decreasing in w and non-increasing in w. Remark: k ∗ is defined as the smallest maximizer simply to avoid the complications that arise when there are multiple maximizers and the changes do not affect the maximum. In this case the set of maximizers may change (monotonically), but the maximum does not. Since we are not interested in characterizing all possible tie-breaking cases, we will pick just the smallest one. Proof Since W is the maximum of linear non-decreasing functions of (w, −w), it is continuous, non-decreasing in each argument (thus non-increasing in w), and convex. Suppose k ∗ is not non-decreasing in (w, −w). Then there exist (w, w) and (w, ˆ w) ˆ such that wˆ ≥ w and −wˆ ≥ −w, but k ∗ (w, ˆ w) ˆ < k ∗ (w, w). Optimality of k ∗ (w, w) and k ∗ (w, ˆ w) ˆ requires that k∗ (w, ˆ w) ˆ

k∗ (w,w)

k∗ (w,w)

X

X

X

i=1

[wˆi − wˆ i ], >

i=1

[wˆi − wˆ i ] ⇒ 0 ≥

[wˆi − wˆ i ]

i=k∗ (w, ˆ w) ˆ

167

k∗ (w,w)

k∗ (w, ˆ w) ˆ

X

X

[wi − wi ] >

i=1

k∗ (w,w)

X

[wi − wi ] ⇒

[wi − wi ] > 0.

i=k∗ (w, ˆ w) ˆ

i=1

Since [wˆi − wˆ i ] ≥ [wi − wi ] for all i, this is a contradiction.

Lemma A.1.2. Let W (w, w) = maxk∈{0,...,m}

Pk

i=1 [wi

− wi ] for all w, w ∈ [0, 1]m such

that w1 ≥ w2 ≥ · · · ≥ wm . Fix w0 ≥ w1 ≥ · · · ≥ wm and denote w = (w1 , . . . , wm ) and wˆ = (w0 , w1 , . . . , wm−1 ). Fix w ∈ [0, 1]m , l ∈ {1, . . . , m}, ε > 0 and let wˆ be such that wˆ l = wl + ε and wˆ i = wi for all i 6= l. Then W (w, w) ˆ − W (w, ˆ w) ˆ ≥ W (w, w) − W (w, ˆ w). ˆ w) and k ∗ (w, w) ˆ ≤ k ∗ (w, ˆ w). ˆ Proof First note that by Lemma A.1.2 k ∗ (w, w) ≤ k ∗ (w, Moreover, by the same result k ∗ (w, w) ˆ ≤ k ∗ (w, w) and k ∗ (w, ˆ w) ˆ ≤ k ∗ (w, w). ˆ Suppose l > k ∗ (w, w). Then, since the relevant terms in the summation are not affected by ε, we have W (w, w) ˆ = W (w, w). Formally, k∗ (w,w)

W (w, w) ˆ ≥

X

k∗ (w,w)

[wi − wˆ i ] =

i=1

X

[wi − wi ] = W (w, w) ≥ W (w, w). ˆ

i=1

Since we have W (w, ˆ w) ˆ ≤ W (w, ˆ w), Lemma A.1.1 implies that W (w, w) ˆ − W (w, ˆ w) ˆ ≥ W (w, w) − W (w, ˆ w). Suppose now that l ≤ k ∗ (w, w). Then l ≤ k ∗ (w, ˆ w). Now observe that either k ∗ (w, w) ˆ = k ∗ (w, w) or k ∗ (w, w) ˆ < l, because if this were not the case, it would imply that

W (w, w) ˆ =

k∗ (w,w) ˆ

k∗ (w,w)

k∗ (w,w) ˆ

k∗ (w,w)

X

X

X

X

i=1

[wi − wˆ i ] ≥

i=1

[wi − wˆ i ] ⇒

i=1

[wi − wi ] − ε ≥

i=1

[wi − wi ] − ε,

168

but this contradicts the optimality of k ∗ (w, w) at (w, w). Similarly either k ∗ (w, ˆ w) ˆ = k ∗ (w, ˆ w) or k ∗ (w, ˆ w) ˆ < l. This means that we have four remaining potential cases to consider. (1) When k ∗ (w, w) ˆ = k ∗ (w, w), k ∗ (w, ˆ w) ˆ = k ∗ (w, ˆ w), it implies that W (w, w) ˆ = W (w, w) − ε and W (w, ˆ w) ˆ = W (w, ˆ w) − ε, so W (w, w) ˆ − W (w, ˆ w) ˆ = W (w, w) − W (w, ˆ w). ˆ < l, k ∗ (w, ˆ w) ˆ = k ∗ (w, ˆ w), we know that W (w, ˆ w) ˆ = W (w, ˆ w) − ε (2) When k ∗ (w, w) and k∗ (w,w)

k∗ (w,w)

X

W (w, w) ˆ ≥

[wi − wˆ i ] =

i=1

X

[wi − wi ] − ε = W (w, w) − ε.

i=1

Moreover, the inequality has to be strict, since otherwise k ∗ (w, w) ˆ would also be optimal in (w, w). Therefore W (w, w) ˆ − W (w, ˆ w) ˆ > W (w, w) − ε − W (w, ˆ w) + ε = W (w, w) − W (w, ˆ w). (3) Case k ∗ (w, w) ˆ = k(w, w), k ∗ (w, ˆ w) ˆ < l is not possible, since it would mean that k ∗ (w, w) ˆ = k(w, w) ≥ l > k ∗ (w, ˆ w) ˆ ≥ k ∗ (w, w). ˆ (4) Finally, when k ∗ (w, w) ˆ < l, k ∗ (w, ˆ w) ˆ < l, we can write [W (w, w) ˆ − W (w, ˆ w)] ˆ − [W (w, w) − W (w, ˆ w)]

=−

k∗ (w, ˆ w) ˆ

k∗ (w,w)

X

X

[wi − wi ] +

i=k∗ (w,w)+1 ˆ

k∗ (w,w) ˆ

[wˆi − wi ] +

i=k∗ (w, ˆ w)+1 ˆ

X

[wˆi − wi ] > 0,

i=k∗ (w,w)+1

Pk∗ (w, ˆ w) ˆ

− wi ] ≤ 0, since otherwise k ∗ (w, w) ˆ would not be Pk∗ (w,w) ˆ optimal at (w, w), ˆ (2) wˆi ≥ wi for each i, (3) i=k∗ (w,w)+1 [wˆi − wi ] > 0, since because (1)

[wi i=k∗ (w,w)+1 ˆ

otherwise k ∗ (w, ˆ w) would not be optimal at (w, ˆ w).

169

We have shown that the weak inequality holds for all possible arguments.

Lemma A.1.3. Let W (w, w) = maxk∈{0,...,m}

Pk

i=1 [wi

− wi ] for all w, w ∈ [0, 1]m such

that w1 ≥ w2 ≥ · · · ≥ wm . Let wn−j+1 ≥ · · · ≥ wn−j+1−m be generated so that P r(n = n ˆ ) = γnˆ , wk = w(vk:n ) when k > 0 and wk = 0 otherwise, and P r(vk:n ≤ vˆ) = Fk:n (v). Fix w ∈ (0, 1)m , l ∈ {1, . . . , m}, ε ∈ (0, 1 − wl ) and let wˆ be such that wˆ l = wl + ε and wˆ i = wi for all i = 6 l. Denote w = (wm , . . . , w1 ), w = (wn−j , . . . , wn−j+1−m ), wˆ = (wˆn−(j−1) , . . . , wˆn−(j−1)−m ). Then En,v [W (w, w) ˆ − W (w, ˆ w)] ˆ > En,v [W (w, w) − W (w, ˆ w)]. Proof Lemma A.1.2 gives weak inequality for any realization of (n, v). Moreover, the proof shows that whenever k ∗ (w, w) ˆ < l ≤ k ∗ (w, w), the inequality is strict. Therefore it suffices to construct a positive measure set of wi ’s, where k ∗ (w, w) ˆ < l ≤ k ∗ (w, w). We construct the set as follows: (i) wn−j+1 ≥ · · · ≥ wn−j+1−(l−1) > max{wm , . . . , wl+1 } (ii) wn−j+1−l ∈ (wl , wl + ε) (iii) wn−j+1−m ≤ · · · ≤ wn−j+1−l−1 < min{w1 , . . . , wl−1 } A combination of wi ’s satisfying these restrictions occur with strictly positive probability. By construction, • [wn−j+1−i − wm+1−i ] > 0 for all i ≤ l and [wn−j+1−i − wm+1−i ] < 0 for all i > l, therefore k ∗ (w, w) = l. • [wn−j+1−i − wˆ m+1−i ] > 0 for all i < l and [wn−j+1−i − wˆ m+1−i ] < 0 for all i ≥ l, therefore k ∗ (w, w) ˆ = l − 1 < l.

170

Therefore under all these realizations the inequality is indeed strict.

Proof of Corollary 1.3.3 We apply Theorem 1.3.1 to get w and the general structure of the optimal mechanism. Suppose w1 ≤ · · · ≤ wm . The claim is that (n + 1 − (m − z) − i)th highest customer with virtual surplus wn+1−(m−z)−i is offered delayed service if and only if wn+1−(m−z)−i > wi . To see this, let k ∗ = arg max

k X [wn+1−(m−z)−i − wi ],

k∈{0,...,m} i=1

kˆ = max k ∈ {0, . . . , m} : [wn+1−(m−z)−k − wk ] > 0 . For each i ≤ kˆ we have wn+1−(m−z)−i ≥ wn+1−(m−z)−k and wi ≤ wk , so [wn+1−(m−z)−i − wi ] ≥ [wn+1−(m−z)−k − wk ] ≥ 0. For each i > kˆ we have by definition that [wn+1−(m−z)−i − ˆ wi ] ≤ 0. Therefore we must have k ∗ = k.

A.2. Waiting list Proof of Theorem 1.3.5 By assumptions, allocation assigns each buyer i either instant ˆ i , so that this buyer service, refusal of service, or promise of delayed service at some set D ˆ i . The waiting list assumption says that will be served next period if and only if (n, v) ∈ D ˆ = (D ˆ 1, . . . , D ˆ N ) must be fixed before observing (n, v). Feasibility requires that vector D the total number of buyers served in the current period must be at most m, and promises ˆ i } ≤ m for each (n, v). Remember that of delayed service are such that #{i : (n, v) ∈ D ˆ = (D ˆ 1, . . . , D ˆ N ), and we denote state by D = (D1 , . . . , DN ), the vector of contracts by D the corresponding probabilities d = (d1 , . . . , dN ), dˆ = (dˆ1 , . . . , dˆN ) st di = P r(Di ) and ˆ i ). dˆi = P r(D

171

Let π(D) denote the maximized revenue at state D and with some abuse of notation let π(d) be the maximized revenue for given probabilities d. That is, π(d) = maxD:P r(Di )≥di π(D). We assume that the functions π(d) are well-defined and will verify it later. After the standard manipulations we can express the revenue recursively as " (A.2)

π(d) = max En,v (1 − δ)

n X

# ˆ , φi wi + δπ(d)

i=1

where φi = 1 if i is assigned instant service, φi = 0 if refused service, and φi = β dˆi when promised delayed service with probability dˆi . We can make some immediate observations. It is clear that giving instant service or promising delayed service to buyers with negative virtual surpluses is dominated by not serving them. Moreover, it is optimal to serve higher-valued buyers before lower valued buyers. That is, if buyer i is served, buyers j and j 0 are offered delayed service with ˆ j and D ˆ j 0 such that dˆj > dˆj 0 , and r is refused service, then it must be that respective sets D wi ≥ wj ≥ wj 0 ≥ wr , since otherwise the seller could strictly increase revenue by swapping promises given to these buyers. Finally, the seller will never assign positive probability of delayed service to more than m buyers. Then we could take the lowest-valued buyer who receives positive probability of service and reassign his probability to higher-valued buyers without changing the feasibility or continuation states, but increasing the flow revenue. The final immediate observation is that the monotonicity of promises also holds in ˆ1 ⊃ · · · ⊃ D ˆ m . This is a terms of sets. In particular, contracts are ordered so that2 D priority ranking—the keeper of the first contract will always be served before the keeper 2More

ˆj \ D ˆ j−1 ) = 0 for all j. precisely this relation must hold with probability one, that is P r(D

172

of the second contract. If the ordering were not true for some j, the seller could replace 0 ˆ j−1 and D ˆ j by new promises D ˆ j−1 ˆ j−1 ∪ D ˆ j and D ˆ j0 = Dj ˆ −1∪D ˆj promises for D =D

while keeping other contracts unchanged. The new contracts do not violate feasibility but extract more revenue from the current buyers. ˆ are already optimally Step 1: allocation rule. Suppose first that the contracts D designed. Fix realization (n, v) and the corresponding ordered virtual surplus vector w and denote the number of slots unavailable by z = #{i : (n, v) ∈ Di }. By the observations above, it is clear that m − z highest-valuing buyers with strictly positive surplus will receive instant service and buyers with negative virtual surplus are never served. If there are more than m − z buyers with strictly positive virtual surplus, k highest of them will receive ˆ 1, . . . , D ˆ k . The remaining question is the choice of optimal k ∈ {0, . . . , m}. contracts D Let ∆1 be the difference between revenue when no promises are given compared to ˆ 1 , that is, ∆1 = π(0, . . . , 0) − π(D ˆ 1 , 0, . . . , 0). revenue when one buyer received promise D More generally, let the revenue effect of having assigned the contract k by ˆ 1, . . . , D ˆ k−1 , 0, . . . , 0) − π(D ˆ 1, . . . , D ˆ k , 0, . . . , 0). ∆k = π(D ˆ k in the allocation Again, it will serve as the opportunity cost of allocating kth contract D problem. Define wk =

δ ∆k . 1−δ β dˆk

173

Let π j (n, w) denote the maximum revenue when j instant service slots are unavailable at realization (n, w). Then we can express it as m−j j

π (n, w) = (1 − δ)

X

max{0, wn+1−i } + δπ(0, . . . , 0)

i=1

+ (1 − δ)β

max k∈{0,...,m}

k X

dˆi [wn+1−(m−j)−i − wi ].

i=1

Again, the sum of the first two terms is the maximal static revenue that the seller can extract without giving any promises of delayed service. The last term is the extra revenue extracted through delay and always non-negative. Step 2: fulfilling promises optimally. The next question is how to choose particular ˆ Using the fact that D ˆ if for fixed probabilities of service d. ˆ1 ⊃ · · · ⊃ D ˆ m we contracts D can express the expected revenue from a given dˆ as ˆ = π(d)

max

ˆ r(D ˆ i )≥dˆi D:P

n h io c 0 1 m ˆ ˆ ˆ ˆ En,v 1[D1 ]π (n, w) + 1[D1 \ D2 ]π (n, w) + · · · + 1[Dm ]π (n, w) .

With some straightforward manipulations, we can rewrite it as

(A.3)

ˆ = En,v [π 0 (n, v)] − π(d)

m X j=1

min

ˆ j :P r(D ˆ j )≥dˆj D

h i ˆ j ][π j−1 (n, w) − π j (n, w)] . En,v 1[D

Note that the problem in (A.3) is separable in j, since both the objective and the ˆ j and dˆj . Now, the difference in revenues, π j−1 (n, w)−π j (n, w), constraint only depend on D is equal to 0 whenever there is less than m − j new buyers with strictly positive virtual valuations, and is strictly increasing otherwise. That is, it is strictly positive and strictly increasing if and only if wn−(m−j) > 0.

174

This means that we have two cases to consider. When (n, v) such that wn−(m−j) ≤ 0, then π j−1 (n, w) − π j (n, w) = 0. Therefore if dˆj < P r(wn−(m−j) ≤ 0) = En [Fn−(m−j) (p∗ )], then the problem is simple to solve, any subset of {(n, v) : wn−(m−j) ≤ 0} works. However, it is not optimal, since by increasing the probability dˆj to En [Fn−(m−j) (p∗ )] the seller can increase the revenue while not violating the feasibility and not decreasing the continuation value. Therefore the optimum must be in the second case where dˆj ≥ En [Fn−(m−j) (p∗ )]. In ˆ j has to include some points where wn−(m−j) > 0 so that this case the contract set D π j−1 (n, w) − π j (n, w) is positive. Since then π j−1 (n, w) − π j (n, w) is strictly increasing ˆ j includes exactly measure dˆj of points where in wn−(m−j) , we get that each contract D π j−1 (n, w) − π j (n, w) is the smallest. ˆ j by the upper bound to the distortion This means that we can characterize contract D π j−1 (n, w) − π j (n, w). We will denote the maximum distortion level by bj . Then ˆ j = {(n, v) : π j−1 (n, w) − π j (n, w) ≤ bj } D ˆ j ) = dˆj . for uniquely determined maximum distortion level bj such that P r(D Step 3: optimal probabilities of delayed service. Since bj is strictly increasing with dˆj in the relevant region, there is a one-to-one mapping between dˆj ∈ [En [Fn−(m−j) (p∗ )], 1] and bj ∈ [0, 1 − δ]. The choice of optimal contracts takes the form n h io ˆ 1c ]π 0 (n, w) + 1[D ˆ1 \ D ˆ 2 ]π 1 (n, w) + · · · + 1[D ˆ m ]π m (n, w) , max En,v 1[D bj

175

subject to 0 ≤ bj ≤ 1 − δ and bj ≤ c (commitment constraint). Notice that here is when the commitment level affects the decision problem. So far we analyzed allocation rule for a given contract and choice of contracts for a given probability (which was not necessarily possible under partial commitment). Now the choice of probability of delay or equivalently the maximum distortion bj must be such that when contract j is assigned, then under any realization (n, v) the maximum benefit from not fulfilling contract j is less than c, which is exactly what bj ≤ c guarantees. We will now define two new distribution functions. First, Gbj (bj ) = P r({(n, v) : π j−1 (n, w) − π j (n, w) ≤ bj }). This allows us to compute the probability of delayed service from its corresponding maximum distortion bj by dˆj = Gbj (bj ) as well as the average distortion from promising the optimal contract corresponding to the probability dˆj by Rb ∆j = 0 j bj dGbj (bj ). The second new distribution will be the distribution of the virtual surplus of an average ˆ j . Let’s define person who gets the contract D " gwj (wˆj ) = En 1[D1c ]fn+1−(m−j) (wˆj )

+

m−1 X

# 1[Di \ Di+1 ]fn+1−(m−j)+i (wˆj ) + 1[Dm ]fn+1−(m−j)+m (wˆj )

i=1

and Gwj (wˆj ) =

R wˆj 0

gwj (wˆj0 )dwˆj0 . Remember that wk = w(0) whenever k ≤ 0. The

distribution Gwj is now a mixture of distributions of virtual surpluses w(vn+1−(m−j):n ), . . . , w(vn+1−(m−j)+m:n ) with partition {D1c , D1 \ D2 , . . . , Dm−1 \ Dm , Dm }.

176

We can now compute the marginal effect of bj to the revenue π i (n, w). For all (n, w) and all i, 0

∀wn+1−(m−i)−j ≤ wj , ∂π i (n, w) = ∂bj (1 − δ)βgb (bj )wn+1−(m−i)−j − δbj gb (bj ) ∀wn+1−(m−i)−j > w . j j j Therefore the first order condition from the expected revenue at state D with respect to bj is 0 = En,v

∂π 0 1[D1c ] ∂bj

∂π 1 ∂π m ∂π m + 1[D1 \ D2 ] + · · · + 1[Dm−1 \ Dm ] + 1[Dm ] , ∂bj ∂bj ∂bj

which we can rewrite as

(A.4)

δbj = (1 − δ)β

Z

1

wˆj wj

dGwj (wˆj ) = EGwj wˆj |wˆj > wj . 1 − Gwj (wj )

As in the illustrative example, we get that the maximum distortion from jth contract— adjusted with discount factor differences and de-normalized—must be equal to the average virtual surplus of a customer whom the seller expects to allocate this contract. Step 4: characterization of optimal contracts. By definition of wj we can now write

(A.5)

wj =

δ

R bj 0

bj dGbj (bj )

(1 − δ)βGbj (bj )

=

δEGbj [bj |bj ≤ bj ] (1 − δ)β

.

For any j ∈ {1, . . . , m} have now two equations that relate wj and bj , equations (A.5) and (A.4). Both describe a strictly increasing continuous relationships between bj and wj . Let us first look at (A.4). At the lower bound wj = 0 we get that 0] ∈ (0, 1) and near the upper bound wj = 1 we have

δbj (1−δ)β

δbj (1−δ)β

= EGwj [wˆj |wˆj >

= EGwj [wˆj |wˆj ≥ 1] = 1.

177

Equation (A.5) gives that at the lower bound bj = 0 we have wj = δEGb [bj ] j δ the upper bound bj = 1 − δ we have wj = (1−δ)β ∈ 0, β .

δEGb [bj |bj ≤0] j

(1−δ)β

= 0 and at

Therefore we have either that there exists interior solution (wj , bj ), so that the probability dˆj < 1 or that the first order condition (A.4) always holds as strict equality and therefore the optimal bj = 1 − δ which means dˆj = 1 or bj = c which means dˆj < 1 because promising delayed service with certainty is impossible due to partial commitment. Step 5: existence of π(d). To complete the analysis, we need to argue that the mechanism description given above gives well-defined functions π(d) which we initially assumed to ˆ exist. The characterization above is a mapping from a continuation value function π(d) to current expected revenue π(d). We denote the mapping by T π. We will first use Blackwell’s sufficient conditions to show that the mapping T is a contraction with speed of convergence δ. Then we can apply the contraction mapping theorem to show that π(d) exists. From the recursive formulation (A.2) it is clear that since

P

i

φi wi ∈ [0, N ], mapping

ˆ to bounded functions. If we have two functions, π, π T maps bounded functions π(d) ˆ such ˆ ≥ π(d) ˆ for all d, ˆ then T π ˆ ≥ Tπ ˆ for all d, ˆ since at each state with any that π ˆ (d) ˆ (d) ˆ (d) promises the continuation revenue is increased and the flow revenue unchanged. Therefore the optimal promises cannot lead to lower revenue. This guarantees monotonicity. To ˆ = π(d) ˆ + a. Then each π j (n, w) will simply have an extra verify discounting, suppose π ˆ (d) δa at the end. In the dynamic part of the problem δa terms cancel in π j−1 (n, w) − π j (n, w) ˆ = En,v [π 0 (n, w) + δa] = T π(d) ˆ + δa. and therefore T π ˆ (d)

178

A.3. Fully optimal mechanism Proof of Theorem 1.3.6 We can make the same immediate observations as in the proof of Theorem 1.3.5: (1) all buyers with negative virtual surplus are never served; (2) the allocation is monotone in values in the sense that the highest types get instant service and the next highest will get delayed service; (3) only up to m of the new buyers are offered delayed service; (4) the optimal promises are ordered both in probabilities dˆ1 ≥ · · · ≥ dˆm ˆ1 ⊃ · · · ⊃ D ˆ m ; and (5) the recursive definition of the revenue function and also in sets3 D is again " (A.6)

π(d) = max En,v (1 − δ)

n X

# ˆ , φi wi + δπ(d)

i=1

where φi = 1 if i receives instant service, φi = 0 if is refused service, and φi = β dˆi if is promised delayed service with probability dˆi . Step 1: allocation rule. Let π j (n, w) denote the maximum revenue that the seller can achieve when reserving exactly j current service slots for buyers from the previous period (for example j = #{i : (n, v) ∈ Di }). That is, the seller can serve only up to m − j of the new buyers instantly and gives optimal promises for next period service to m buyers who were not served instantly. Because of monotonicity in the order of the promises, the buyers with virtual surpluses wn , . . . , wn+1−(m−j) receive instant service (if their virtual surpluses are positive) and buyers with virtual surpluses wn−(m−j) , . . . , wn+1−(m−j)−m receive the

ˆj \ D ˆ j−1 ) = 0 for all j ∈ the exception of zero-measure sets. The precise statement is: P r(D {2, . . . , m}.

3With

179

corresponding promises such that dˆ1 ≥ · · · ≥ dˆm . Then π j (n, w) can be computed as m−j j

π (n, w) = max(1 − δ) dˆ

X

max{0, wn+1−i } + (1 − δ)β

i=1

m X

ˆ dˆi wn+1−(m−j)−i + δπ(d)

i=1

such that dˆj ∈ [0, 1] for each j ∈ {1, . . . , m} and the commitment constraint is satisfied. Differentiation with respect to the probability of delay dˆi gives (A.7)

ˆ dπ j (n, w) ∂π(d) = (1 − δ)βwn+1−(m−j)−i + δ . ddˆ ∂ dˆ i

i

The m first order conditions define vector of optimal probabilities dˆ as a function of virtual surpluses wn−(m−j) , . . . , wn+1−(m−j)−m , which we will discuss in detail later when we have determined the properties of

ˆ ∂ π(d) . ∂ dˆ i

Step 2: fulfilling promises optimally. We have determined the optimal probabilities ˆ Next we will find the precise optimal promises in terms of sets D, ˆ where of service, d. these promises are fulfilled. We argued above that the promises are ordered in the sense that higher types are ordered higher probability of service. Analogously to the waiting list mechanism case we can write

(A.8)

ˆ = En,v [π 0 (n, w)] − π(d)

m X j=1

min

ˆ r(D ˆ j )≥dˆj D:P

En,v

h

i j−1 j ˆ 1[Dj ][π (n, w) − π (n, w)] .

Since both the objective and the constraints are separable in j, we can again optimize ˆ j such that π j−1 (n, w) − π j (n, w) is the point-wise. That is, to choose each contract D smallest, conditional on P r(dˆj ) ≥ dˆj . Again, when wn−(m−j) ≤ 0, then all the buyers starting from n − (m − j) will be refused service, so π j−1 (n, w) − π j (n, w) = 0, whereas when wn−(m−j) > 0, this difference is strictly

180

positive. Therefore the binding promise constraint is always binding in the optimum, ˆ j ) ≥ En [Fn−(m−j) (p∗ )]. Finally, note that π j−1 (n, w) − π j (n, w) is still strictly dˆj = P r(D increasing in wn−(m−j) . ˆ j = {(n, v) : π j−1 (n, w) − π j (n, w) ≤ bj } for Now, for each j, the optimal contract is D ˆ j ) = dˆj . uniquely defined bj such that P r(D Step 3: optimal probabilities of delayed service. We can again change the optimization variable, instead of choosing the optimal probability of delayed service dˆj we can choose the optimal maximal distortion bj . Using this we get that

ˆ ∂ π(d) ∂ dˆj

= −bj , which is independent

of dˆi for i 6= j. Moreover, if dˆj = 1 (seller is promised service with certainty), then we know that ˆ j must include all pairs (n, w), therefore bj = 1 − δ and so D

∂ π(·,1,·) ∂ dˆ

= −(1 − δ). If

j

dˆj ≤ En [Fn−(m−j) (p∗ )], the constraint is not binding and therefore

∂ π(·,dˆj ,·) ∂ dˆ

= 0, whereas if

j

dˆj > En [Fn−(m−j) (p∗ )] we have that bj > 0 and so

∂π ∂ dˆj

= −bj < 0.

ˆj ( D ˆ j 0 and therefore Whenever En [Fn−(m−j) (p∗ )] < dˆj < dˆj 0 we must have that D bj < bj 0 . Therefore

∂ π(·,dˆj ,·) ∂ dˆ j

= −bj > −bj 0 =

∂π(·,dˆj 0 ,·) , ∂ dˆ 0 j

therefore

∂2π ∂ dˆ2j

> 0.

We can now study the first order condition (A.7). Suppose for a moment that commitment level c is high. Let’s consider the upper corner solution first. Suppose dˆj = 1. This is optimal only if (1 − δ)βwn+1−(m−z)−j ≥ δ(1 − δ), which is equivalent to wn+1−(m−z)−j ≥ βδ . Now, let’s take the lower corner solution. We know that

∂π ∂ dˆj

= 0 for all

dˆj ≤ En [Fn−(m−j) (p∗ )], so dˆj ≤ En [Fn−(m−j) (p∗ )] implies that (1 − δ)βwn+1−(m−z)−j ≤ 0, which means that wn+1−(m−z)−j ≤ 0. This is what we already argued above—whenever the buyer to whom the seller is supposed to promise delayed service (according to her position in the ordered values) has negative virtual surplus, it will be optimal to promise service

181

with zero probability. Finally, interior solutions, dˆj ∈ En [Fn−(m−j) (p∗ )], 1 have to satisfy (1 − δ)βwn+1−(m−z)−j = δ ∂∂dˆπ = δbj . Therefore dˆj is strictly increasing in wn+1−(m−z)−j for j

the interior solutions. If commitment level is not high, it gives a tighter upper bound, bj ≤ c. The lower bound and the interior points are not affected. We can summarize the choice of bj for a buyer4 i who gets the promise by a function bj = b(wi ), where

b(wi ) =

0

∀wi ≤ 0,

(1−δ)β δ wi ∀0 < wi < (1−δ)β min{1 − δ, c}, δ min{1 − δ, c} ∀wi ≥ δ min{1 − δ, c}. (1−δ)β

Therefore we showed that the optimal probability of delayed service dˆj (wi ) is equal to 0 when δ wi ≤ 0, equal to the upper bound Gbj (min{1 − δ, c}) ≤ 1 when wi ≥ (1−δ)β min{1 − δ, c} and is strictly increasing function Gbj (1−δ)β wi ∈ (0, 1) in the interior. δ

Step 4: existence of π(d). The proof that functions π(d) are well defined uses the Contraction mapping theorem and is analogous to the proof in the waiting list case.

4This

is to simplify the notation, in particular the contract j goes to ith highest valuing buyer with i = n + 1 − (m − z) − j.

182

APPENDIX B

Proofs of Results in Chapter 2 Derivation of Equation 2.3 A marginal-signal passenger is indifferent between expected value from ticket E[V

r˜+1

(θ∗ )] − p and expected value from not having a ticket E[V r˜(θ∗ )],

where the number of other ticket holders r˜ has binomial distribution with parameters (n − 1, F (ΘT )). That gives n−1 X n−1 r+1 ∗ p= F (ΘT )r (1 − F (ΘT ))n−1−r [V (θ ) − V r (θ∗ )]. r r=0 s˜

Expected total revenue is π = nF (ΘT ) p + E[˜ sΠ + (n − s˜)Πs˜], where s˜ has binomial distribution with parameters (n, F (ΘT )). Therefore (B.1)

n X n s π = n(1 − F (θ ))p + F (ΘT )s (1 − F (ΘT ))n−s [sΠ + (n − s)Πs ]. s s=0 ∗

From the equation for p we get that nF (ΘT )p = nF (ΘT )E[V

r˜+1

] − nF (ΘT )E[V r˜].

Rewriting positive and negative parts of this gives

nF (ΘT )E[V

r˜+1

] = nF (ΘT )

n−1 X r=0

=

n X s=0

(n − 1)! r+1 F (ΘT )r (1 − F (ΘT ))n−1−r V s!(n − 1 − r)!

n! s F (ΘT )s (1 − F (ΘT ))n−s sV . s!(n − s)!

183

The negative term is ∗

∗

s

n(1 − F (θ ))E[V ] = n(1 − F (θ ))

n−1 X s=0

(n − 1)! (1 − F (θ∗ ))s F (θ∗ )n−1−s V s s!(n − 1 − s)!

n

1 − F (θ∗ ) X n! = (n − s)(1 − F (θ∗ ))s F (θ∗ )n−s V s . ∗ F (θ ) s=0 s!(n − s)! Inserting nF (ΘT ) E[V gives Equation 2.3.

r˜+1

] and nF (ΘT ) E[V r˜] to Equation B.1 and collecting similar terms

Proof of Proposition 2.4.3 For reported values consistent with equilibrium ticket purchasing, the allocation is determined by comparison of the virtual surpluses. For other reports, we specify the allocations as follows. First, for any s and for any ticket holder j reporting vj above the support of G (·|ΘT ), and for any v−j , qj (vj , v−j ; s) = 1 while qi (vj , v−j ; s) = 0 for all i 6= j. Second, for any s and for any ticket holder j reporting vj below the support of G (·|ΘT ), and for any v−j , the allocation is the same as for passenger j report equal to the minimum of the support of G (·|ΘT ). Third, for any s and for any non-ticket holder k reporting vk above the support of G (·|Θ\ΘT ), and for any v−k , the allocation is the same as in case k reports the maximum of the support of G (·|Θ\ΘT ). Finally, for any s and for any non-ticket holder k reporting vk below the support of G (·|Θ\ΘT ), and for any v−k , qk (vk , v−k ; s) = 0 and qi (vk , v−k ; s) = 0 for all i 6= k. Using Part (i), the allocations defined above satisfy, for any s, any passenger i (whether or not he has purchased a ticket), and any v−i consistent with equilibrium ticket purchasing, qi (·, v−i ; s) is non-decreasing over [v, v¯]. This guarantees that the period-1 mechanism, defined by the above allocations and the transfers Equation 2.10, is ex-post

184

incentive compatible irrespective of whether the passenger follows the equilibrium strategy in purchasing (or not purchasing) a ticket. It remains to check that passengers prefer to purchase tickets if and only if their signal is in ΘT . The expected payoff to a passenger j with θj ≥ θ∗ , following his equilibrium strategy of purchasing a ticket and then reporting his value optimally, is V¯ (θ ) − p + ∗

Z

θj

θ∗

∂zj (y, ε˜j ) E qj (z (y, ε˜j ) , v ˜−j ; r˜ + 1) dy, ∂θj

where r˜ is the number of tickets purchased by other passengers playing equilibrium strategies). Given that V (θ∗ ) = V¯ (θ∗ ) − p, the payoff is given by exactly the same formula also if he does not purchase a ticket, except replacing the allocation qj (z (y, ε˜j ) , v ˜−j ; r˜ + 1) with that obtained when not purchasing a ticket. By Part (ii) of the proposition and construction of the extended allocation, the probability of allocation for a ticketed passenger is never lower than for an unticketed one given the same realization of values (noting that values are always reported truthfully in equilibrium by incentive compatibility of the period-1 mechanism). Therefore, the passenger’s expected payoff is higher when he purchases a ticket than when he does not (an argument along the same lines shows that for a passenger with a signal below θ∗ , his expected payoff from not purchasing a ticket is higher than from purchasing one).

Proof of Proposition 2.4.4

185

(1) Fully informative signals mean that vi = θi . Therefore for all vj ≥ θ∗ the virtual surplus expression for ticketholders from Theorem 2.4.2 takes the form

VS(vj ) = vj −

1−

F (vj )−F (θ∗ ) 1−F (θ∗ ) f (vj ) 1−F (θ∗ )

= vj −

1 − F (vj ) . f (vj )

Similarly, for all vk < θ∗ , the virtual surplus for non-ticketholders takes the form VS(vk ) = vk −

1−

F (vk ) F (θ∗ )

1−F (θ∗ ) F (θ∗ ) f (vk ) F (θ∗ )

+

(1 − 0)

= vk −

1 − F (vk ) . f (vk )

Notice that the virtual surplus expressions and therefore the maximized revenue are independent of θ∗ . Since the virtual surplus expressions are the same for ticketholders and non-ticketholders, the ticket price is always 0 and therefore ticket sales do not affect allocation and revenue. (2) Fully informative signals mean that vi is independent of θi , so g(vi |·) = g(vi ) and G(vi |·) = G(vi ). Therefore VS(vj ) = vj −

VS(vk ) = vk −

1 − G(vk ) +

G(vj ) − G(vj ) = vj , g(vj )

1−F (θ∗ ) (1 F (θ∗ )

g(vk )

− G(vk ))

= vk −

1 − G(vk ) 1 . g(vk ) F (θ∗ )

It remains to show that the optimal mechanism sets θ∗ = 0, so that F (θ∗ ) = F (∅). Suppose that this is not the case, so that θ∗ > 0. For each s ∈ {0, . . . , n} number of tickets sold, let the allocation rule of s-mechanism be q(v, s) and expected revenue π s . Notice that since VS(vi ) ≥ VS(vi ) for each vi , we must have that π s is increasing in the number of tickets sold, s. Since buyers who arrive in the first

186

period have the same information about their true valuation, they must all be indifferent between buying a ticket and not buying a ticket in the first period. Suppose we fix allocation rule q(v; S) in the second period and the ticket price, but instead of selling tickets to fraction 1 − F (θ∗ ) passengers, sell tickets to all buyers who arrive at date 0 (fraction 1 − F (∅)). Since we did not change any transfers nor allocation in the second period and the buyers were indifferent in the first period, the incentive constraints still hold and the expected revenue from any realized number of tickets sold, s, is unchanged. But it is easy to see that expected revenue is now increased, since we are selling more tickets.

Proof of Proposition 2.4.5 Consider the following strategy profile. Ticketed passengers with value vj bid vj . Unticketed passengers with value vk bid vj (vk ). To show that this is a dominant-strategy equilibrium, first consider a ticketed passenger with value vj when the profile of other bids is b−j . Among those bids that exceed r, let bn−j denote the nth highest bid, possibly equal to r in case the number of such bids is smaller than n. If j’s bid bj exceeds bn−j then j will be seated and receive no transfer. If he bids less than bn−j , he will volunteer his seat and receive compensation in the amount of bn = bn−j . Passenger j wishes to volunteer his seat at this price if and only if vj ≤ bn−j . Thus by bidding according to the specified strategy profile, i.e. bj = vj , he volunteers his seat exactly when it is optimal to do so. Next consider an unticketed passenger with value vk . Let bn−k denote the nth highest bid among the other passengers whose bid exceeds r, again with bn−k equaling r if there are

187

fewer than n. If k’s bid bk exceeds bn−k then k will win a seat and pay vk (bn−k ). If he bids less than bn−k , he will lose and pay nothing. It is in the interest of passenger k to fly and pay vk (bn−k ) if and only if vk ≥ vk (bn−k ). By bidding bk = vj (vk ) as dictated by the strategy profile, he flies if and only if vj (vk ) ≥ bn−k , i.e. if and only if vk ≥ vk (bn−k ), i.e. exactly when it is in his interest to do so. Finally to show that this dominant-strategy equilibrium implements the optimal allocation of seats it is enough to show that the ranking of bids is the same as the ranking of virtual utilities. This is immediate for any pair of ticketed passengers. They bid their values and, due to the regular case, virtual utilities are monotone in values. Unticketed passengers bid vj (vk ) which in the regular case is a monotone function of vk , which is in turn monotonic with respect to virtual utilities. Now consider a ticketed passenger j with value vj and an unticketed passenger k with value vk . Passenger k outbids j if and only if vj (vk ) ≥ vj . By construction, vj (vk ) = vj if and only if the two passengers’ virtual utilities are the same. And due to the regular case, the inequality holds if and only if k’s virtual utility is larger. Finally, a bid exceeds the reserve price if and only if vj ≥ r in the case of a ticketed passenger and vj (vk ) ≥ r. In both cases the condition is equivalent to non-negative virtual utility.

Proof of Proposition 2.5.1 As for the restricted mechanisms studied above, we define a mechanism on an extended message space to allow a passenger who is in the market at date 0 but fails to participate at that date to still report his value truthfully at date 1. This is notationally convenient, although the additional out-of-equiilibrium messages can be dispensed with. To define the allocations for these messages, let the probability

188

passenger i receives the good if reporting for the first time at date 1 a value below the support of G (·|∅) be zero (any feasible allocation can be chosen for the other passengers). Let the probability of allocation in case the first report at date 1 is above the support of G (·|∅) be the same as if the report were equal to the maximum of this support. With an allocation rule in hand for the extended message space, now consider a candidate for the transfers which implement it. First, note that

W

ΩM

1 − G (˜ vi |0) ˜ (θi ) = E qi ∅, θ−i , v˜i , v ˜−i dy|θ˜i = ∅ g (˜ vi |∅) Z θi ∂z (y, ε˜i ) ˜ ˜−i dy qi y, θ−i , z (y, ε˜i ) , v E + ∂θi 0

and that h i ˜−i W ΩM (θi ) = E W ΩM θi , θ˜−i , z (θi , ε˜i ) , v ΩM θi , θ˜−i , z (θi , ε) , v ˜−i W = E R |θ˜i = θi , v˜i ˜−i dy + z(θi ,ε) qi θi , θ˜−i , y, v where ε is the minimum of the support of H. We will construct transfers which guarantee equilibrium payoffs, for all θi ∈ [0, 1] , all θ−i and v−i , W ΩM (θi , θ−i , z (θi , ε) , v−i ) 1 − G (˜ vi |0) ˜ ˜ =E qi ∅, θ−i , v˜i , v ˜−i |θi = ∅ g (˜ vi |∅) Z θi ∂z (y, ε˜i ) ˜ qi y, θ−i , z (y, ε˜i ) , v + E ˜−i dy ∂θi 0 Z v˜i qi θi , θ˜−i , y, v −E ˜−i dy|θ˜i = θi . z(θi ,ε)

189

For θi = ∅, we let W ΩM (θi , θ−i , z (θi , ε) , v−i ) = 0. In any incentive compatible mechanism, payoffs are determined for all signals and values, by this constant and the allocation. In particular, given

W

ΩM

(θ, vi , v−i ) = W

ΩM

Z

vi

(θ, z (θi , ε) , v−i ) +

qi (θ, y, v−i ) dy. z(θi ,ε)

Note that the payoffs to be earned in equilibrium are pinned down solely by the allocation rule, the fact that the airline is optimizing and that the date 0 participation constraint binds at θi = 0. With these intended payoffs in hand, we can define transfers simply by the identity ti (θ, v) = qi (θ, v) θi − W ΩM (θ, v). The mechanism described above is ex-post incentive compatible at date 1 since each passenger’s probability of receiving the seat is non-decreasing in his value for it, conditional on the reports of the other passengers. Incentive compatibility of truthful reporting of the signal at date zero, given that the passenger participates at that date, also follows from the monotonicity of the allocation in both signals and values (see Pavan, Segal and Toikka (2011), for instance) and from the first-order stochastic dominance property of the stochastic process. What is left to check is that all passengers with signals in [0, 1] prefer to participate at date 0 rather than delaying participation to date 1. This follows from essentially the same argument as in the previous result, after noting that, for each reported value of a passenger i (holding fixed the reports of the other passengers) the probability of that passenger receiving the good, for each possible value vi , is lower if he delays participation until date 1.

190

APPENDIX C

Proofs of Results in Chapter 3 C.1. Symmetric Stationary Subgame Perfect Nash Equilibrium We will now introduce formally the equilibrium concept used in this paper, Symmetric Stationary Subgame Perfect Nash Equilibrium (SSSPNE). Denote the vector of bids at round t by bt = (bt0 , . . . , btN ), where bti ∈ {0, 1} is 1 if player i submitted a bid at period t. Denote the leader after1 round t by lt ∈ {0, . . . , N }. The information that each player has when making a choice at time t, or history at t, is ht = (b0 , l0 , b1 , l1 , . . . , bt−1 , lt−1 ). The game sets some restrictions to the possible histories, in particular to become a leader, one must submit a bid, so btlt = 1, and the leader cannot submit a bid, btlt−1 = 0, and ht is defined only if none of the previous bid vectors bτ is zero vector. Denote the set of all S t possible t-stage histories by Ht , and the set of all possible histories, H = ∞ t=0 H . In this game, a pure strategy of player i is bi : H → {0, 1}, where bi (ht ) = 1 means that player submits a bid at ht and 0 that the player passes. The strategies2 of the players are σi : H → [0, 1], such that σi (ht ) is the probability that player i submits a bid at history ht . Note that by the rules of the game, at histories ht where lt = i, player i is the leader and can only pass.

1That

is, the non-leader that submitted a bid at t and became the leader by random draw. game has perfect recall, so by Kuhn’s theorem any mixed strategy profile can be replaced by an equivalent behavioral. Since it makes notation simpler, whenever we are talking about strategies in the text, we mean behavioral strategies.

2The

191

Def: A strategy profile σ is Symmetric if for all t ∈ {0, 1, . . . }, for all i, ˆi ∈ {0, . . . , N }, ˆ t = (ˆbτ , ˆlτ )τ =0,...,t−1 ∈ Ht satisfies and for all ht = (bτ , lτ )τ =0,...,t−1 ∈ Ht , if h

ˆbτ = j

bτj bτi b τ ˆi

∀j ∈ / {i, ˆi}, j = ˆi,

ˆlτ =

j = i,

l τ i ˆi

lτ ∈ / {i, ˆi}, lτ = ˆi,

∀τ = {0, . . . , t − 1},

lτ = i,

ˆ t ) = σˆ(ht ). then σi (h i The Symmetry assumption simply states that when we switch the identities of two players, then nothing changes. This means that we could also call it Anonymity assumption. Intuitively, the assumption means that given that other N opponents make exactly the same choices and the uncertainty has realized the same way, different players would behave identically. Let function Li be the indicator function that tells whether player i is leader after history ht or not, Li (ht ) = 1[i = lt ],

∀i ∈ {0, . . . , N }, ∀ht ∈ H.

Let S be the set of states in the game and S : H → S the function mapping histories to states. In particular, we define these as (i) If ε = 0, then S = {N + 1, N }, and

S(ht ) =

N + 1 ht = ∅, N

ht 6= ∅.

192

The reason: in infinite game the price does not increase, so the only thing players will condition their behavior is the number of active bidders, which is N + 1 in the beginning and N at any round after 0. (ii) If ε > 0, then S = {0, 1, . . . }, and t

S(h ) =

t−1 X N X

bτi .

τ =0 i=0

That is, the total number of bids made so far or equivalently, the normalized price pt . Note that we do not have to explicitly consider two cases with two different numbers of players, since at ht = ∅ we have S(ht ) = 0 and at any other history S(ht ) > 0. Def: A strategy profile σ is Stationary if for all i ∈ {0, . . . , N }, and for all pairs of t ˆ tˆ = (ˆbτ , ˆlτ ) ˆ tˆ histories ht = (bτ , lτ )τ =0,...,t−1 ∈ H, h τ =0,...,tˆ−1 ∈ H such that Li (h ) = Li (h ),

ˆ tˆ), we have σi (ht ) = σi (h ˆ tˆ). and S(ht ) = S(h Stationarity assumption means that the time and particular order of bids are irrelevant. The only two things that affect player’s action are current state and the fact whether she is a leader or not. Def: SPNE strategy profile σ is Symmetric Stationary Subgame Perfect Nash Equilibrium SSSPNE if it is Symmetric and Stationary.

Lemma C.1.1. A strategy profile σ is Symmetric and Stationary if and only if it can be represented by q : S → [0, 1], where q(s) is the probability bidder i bids at state s ∈ S for each non-leader i ∈ {0, . . . , N }.

193

Proof Since q is only defined on states S and equally for all non-leaders, it is obvious that it is a strategy profile that satisfies Symmetry and Stationarity, so sufficiency is trivially satisfied. For necessity, take any strategy profile σ = (σ0 , . . . , σN ), where σi : H → [0, 1], that satisfies Symmetry and Stationarity. Construct functions q0 , . . . , qN , where qi : S → [0, 1] by setting t

qi (S(h )) =

0

∀ht : Li (ht ) = 1,

∀ht ∈ H.

σi (ht ) ∀ht : Li (ht ) = 0, Our construction of S and Stationarity ensure that qi is well-defined function. We claim that adding Symmetry means that we get qi (s) = q(s) for all i and s ∈ S. To see this, fix any i and ht such that s = S(ht ) and Li (ht ) = 0. By construction, qi (s) = qi (S(ht )) = σi (ht ). ˆ t that Now, fix any other non-leader, ˆi, so that Ltˆ(ht ) = 0. Construct another history h ˆ t ) = s (obvious is otherwise identical to ht , but such that i and ˆi are swapped. Then S(h ˆ t ) = 0. By Symmetry we have σi (ht ) = σˆ(h ˆ t ). Therefore for both cases) and Lˆi (h i ˆ t )) = σˆ(h ˆ t ) = σi (ht ) = qi (s). qˆi (s) = qˆi (S(h i

So, if strategy profile satisfies Stationarity and Symmetry, we can greatly simplify its representation. We can replace σ by q that is just defined for all s ∈ S instead of full set of histories H. In the following two lemmas we show that at least in the cases considered in this paper the solution method is also simplified by these assumptions, since any SSSPNE

194

can be found simply by solving for stage-game Nash equilibria for each state s ∈ S taking into account the solutions to other states and the implied continuation value functions. Lemma C.1.2. With ε > 0, a strategy profile σ is SSSPNE if and only if it can be represented by q : S → [0, 1] where q(s) is the Nash equilibrium in the stage-game at state s, taking into account the continuation values implied by transitions S. Proof Necessity: If σ is SSSPNE, then by Lemma C.1.1 it can be represented by q and since it is a SPNE, there cannot be profitable one-stage deviations. Sufficiency: By Corollary 3.5.3 any auction with ε > 0 ends not later than p˜ + N . So, although our game is (by the rules) infinite, it is equivalent in the sense of payoffs and equilibria with a game which is otherwise identical to our initial auction, but where after time p˜ + N the current leader gets the object at the current price. This is finite game and checking one-stage deviations is sufficient condition for SPNE.

Lemma C.1.3. With ε = 0, a strategy profile σ is SSSPNE if and only if it can be represented by q : S → [0, 1] where q(s) is the Nash equilibrium in the stage-game at state s, taking into account the continuation values implied by transitions S. Proof Necessity is identical to Lemma C.1.2. Sufficiency:3 Suppose q is Nash equilibrium in the stage-game equilibrium at each state s. To shorten the notation we will use the following notation: qˆ0 = q(N + 1), qˆ = q(N ), vˆ0 is the continuation value of the game at state N + 1, vˆ is the continuation value of a non-leader and vˆ∗ is the continuation value of a leader at state N . By Theorem 3.4.1 we get qˆ ∈ (0, 1), defined by (1 − qˆ)N ΨN (ˆ q ) = vc , 3Note

that since the game does not satisfy continuity at infinity, checking one-stage deviations may not be sufficient for SPNE.

195

qˆ0 < 1, vˆ = 0, and vˆ∗ = (1 − qˆ)N v. We need to show that there are no profitable unilateral multi-stage deviations from the proposed equilibrium strategy profile. Take any history ht 6= ∅ and individual i who is not the leader at ht . Let σi be the strategy that ensure the highest expected value to player i at history ht . Denote continuation value using σi at history hτ by V (hτ ) for all hτ following ht . To shorten the notation, denote Vˆ = V (ht ). Suppose there exists profitable deviation at ht . Then σi must also be profitable deviation and therefore Vˆ > vˆ = 0. Some of the histories ht+1 following ht and i playing σi (ht ) are such that i is a nonleader. In these situations all the other players use the same mixed strategy in all the continuation paths, so all payoff-relevant details are the same as at ht . This means that at such histories ht+1 , it must be V (ht+1 ) = Vˆ . It cannot be higher, since Vˆ is maximum, and it can’t be lower, since i could improve V (ht ) by changing strategy starting from this ht+1 . Other histories ht+1 following following ht , σi (ht ) are the ones where i is the leader. Being the leader at ht+1 , two things can happen to i’s payoff. First, game may end at ht+1 and player i gets v. This happens with probability (1 − qˆ)N as argued above. Secondly, i can become a non-leader at history ht+1 following ht+1 . For the same reason as above, V (ht+2 ) = Vˆ for all such histories. Therefore in histories ht+1 where i is the leader, V (ht+1 ) = (1 − qˆ)N v + (1 − (1 − qˆ)N )Vˆ .

196

The expected value at ht is the expectation over all the continuation values V (ht+1 ) following mixed action σi (ht ) minus the expected bid cost. So, we can write Vˆ = V (ht ) =

X

P (ht+1 |ht , σi (ht ))V (ht+1 ) − cσi (ht )

ht+1 |ht ,σi (ht )

Using the values V (ht+1 ) derived above and the fact that conditional on submitting a bid, the probability of becoming the leader at t + 1 is ΨN (ˆ q ). So, the probability of become the leader is σi (ht )ΨN (ˆ q ), which gives us Vˆ = σi (ht )ΨN (ˆ q )[(1 − qˆ)N v + (1 − (1 − qˆ)N )Vˆ ] + [1 − σi (ht )ΨN (ˆ q )]Vˆ − cσi (ht ) = cσi (ht ) + σi (ht )ΨN (ˆ q )[1 − (1 − qˆ)N − 1]Vˆ + Vˆ − cσi (ht ) ⇐⇒ σi (ht )c

Vˆ = 0. v

By assumptions c > 0, Vˆ > 0, and therefore σi (ht ) = 0. What we got is that by not bidding at ht and at any following ht+1 and so on the player can ensure strictly positive expected payoff Vˆ , which is impossible since the only way to get positive value is to be a leader and for this necessary condition is to bid. So there cannot be profitable deviations at any ht 6= ∅. We showed that at any history that follows h0 , always playing qˆ ensures highest possible payoffs. Therefore at round 0 if there is profitable deviation, it must be one-stage deviation. But this is not possible, since we assumed that qˆ0 is Nash equilibrium in the stage-game, taking into account the continuation values from qˆ in the following periods.

197

C.2. Properties of ΨN (q) The following Lemma helps us to characterize the set of equilibria and its properties in case when ε = 0. The interpretation of ΨN (q) and the intuition of the three properties are discussed in the text. Let

ΨN (q) =

N −1 X K=0

N −1 K 1 q (1 − q)N −1−K . K K +1

Lemma C.2.1. Let N ≥ 2. Then

(i) ΨN (q) is strictly decreasing in q ∈ (0, 1). (ii) limq→0 ΨN (q) = 1, limq→1 ΨN (q) =

1 . N

(iii) ΨN (q) > ΨN +1 (q) for all q ∈ (0, 1).

Proof

(i) ΨN (q) is a differentiable function of q, so it is sufficient to show that

dΨN (q) dq

(0, 1). Differentiation and reordering of terms gives N −1 (N − 1)!K dΨN (q) X = q K−1 (1 − q)N −(K+1) dq (N − 1 − K)!(K + 1)! K=1

−

N −2 X

(N − 1)!(N − (K + 1)) K q (1 − q)N −(K+1)−1 ]. (N − 1 − K)!(K + 1)! K=0 N −1 X

(N − 1)!q K−1 (1 − q)N −(K+1) K = −1 . (N − 1 − K)!K! K +1 K=1 K K+1

< 1, so all terms in the sum are strictly negative for any q ∈ (0, 1).

< 0, ∀q ∈

198

(ii) As q → 0, all terms of ΨN (q) where q is in positive power disappear, so only the one corresponding to K = 0 survives. Thus (N − 1)! q 0 (1 − q)N −1 = 1. q→0 (N − 1)!0! 1

lim ΨN (q) = lim

q→0

Similarly, as q → 1, all terms where (1 − q) is in positive power disappear, so only the one where K = N − 1 survives and we get

lim ΨN (q) =

q→1

1 (N − 1)! 1 = . 0!(N − 1)! N − 1 + 1 N

(iii) Want to show that ∆N (q) = ΨN (q) − ΨN +1 (q) > 0, ∀q ∈ (0, 1), where ∆N (q) =

N −1 X

(N − 1)!q K (1 − q)N −1−K qN (qN − K) − . (N − K)!(K + 1)! N +1 K=0

We prove it by first transforming the sum in a way that we get expectation of linear function over Binomial distribution with parameters (q, N + 1), since we know that expectation of the variable itself is q(N + 1) and expectation of constant is constant. The expression that remains after this manipulation depends only on q and N and is easy to analyze directly. First, change of variables in the sum, L = K + 1 N X (N − 1)!q L−1 (1 − q)N −L qN ∆N (q) = (qN + 1 − L) − (N + 1 − L)!L! N +1 L=1 N X 1 (N + 1)!q L (1 − q)N +1−L qN = (qN + 1 − L) − . N (N + 1)q(1 − q) L=1 (N + 1 − L)!L! N +1

199

Finally, we need to add and subtract terms with L = 0 and L = N + 1 N +1 X 1 (N + 1)!q L (1 − q)N +1−L ∆N (q) = (qN + 1 − L) N (N + 1)q(1 − q) L=0 (N + 1 − L)!L!

−

(1 − q)N +1 (qN + 1) −q N +1 (1 − q)N qN − − . N (N + 1)q(1 − q) N (N + 1)q(1 − q) N + 1

Using the properties of Binomial distribution and rewriting gives

∆N (q) =

qN + 1 − q(N + 1) (1 − q)N +1 (qN + 1) − N (N + 1)q(1 − q) N (N + 1)q(1 − q) =

1 − (1 − q)N (qN + 1) . N (N + 1)q

Therefore, to show that ∆N (q) > 0 for all q ∈ (0, 1), it is sufficient to show that 1 − (1 − q)N (qN + 1) > 0. Note that when q = 0 this expression is equal to 0, and it is strictly increasing in q d [1 − (1 − q)N (qN + 1)] = qN (N + 1)(1 − q)N −1 > 0, ∀q ∈ (0, 1). dq C.3. A penny auction with multiple equilibria Let N + 1 = 3, v = 9.1, c = 2, ε > 0. In this case, there are three SSSPNE, in Tables C.1, C.2, and C.3 (which differ by actions at p = 2).

200

Table C.1. Equilibrium with q(2) = 1 p q(p) v ∗ (p) v(p) Q(p) Qpp 0 0.509 0 0.1183 1 0 8.1 0 0.3681 0.4175 2 1 0 0 0 0 3 0.6996 0.5504 0 0.0119 0.0135 4 0 5.1 0 0.4371 0.4958 5 0.4287 1.3381 0 0.0211 0.0239 6 0.0645 2.7129 0 0.0277 0.0314 7 0 2.1 0 0.0157 0.0178 8 0 1.1 0 0.0001 0.0001 9 0 0.1 0 0 0 Table C.2. Equilibrium with 0 < q(2) < 1 p 0 1 2 3 4 5 6 7 8 9

q(p) 0.5266 0 0.7249 0.6996 0 0.4287 0.0645 0 0 0

v ∗ (p) 8.1 0.5371 0.5504 5.1 1.3381 2.7129 2.1 1.1 0.1

v(p) 0 0 0 0 0 0 0 0 0 0

Q(p) 0.1061 0.354 0.0298 0.0273 0.3344 0.0484 0.0636 0.036 0.0003 0

Q(p) 0.3961 0.0333 0.0306 0.3741 0.0542 0.0711 0.0403 0.0003 0

201

Table C.3. Equilibrium with q(2) = 0 p 0 1 2 3 4 5 6 7 8 9

q(p) 0 0.7473 0 0.6996 0 0.4287 0.0645 0 0 0

v ∗ (p) 0.5174 7.1 0.5504 5.1 1.3381 2.7129 2.1 1.1 0.1

v(p) Q(p) Q(p) 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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