- Email: [email protected]

Daniel Probst‡

this draft: 23:5:2004 (ﬁrst draft Summer 99)

Abstract We analyze the phenomenon of gradual increase of stakes in bilateral social interactions in the context of a repeated game with two-sided incomplete information. The underlying stage-game resembles properties of a ”Continuous Prisoner’s Dilemma” with a continuous action space plus the possibility of stealing the opponent’s investment. The continuous investment variable corresponds to cooperation and is interpreted as individual level of trust while the defective action is interpreted as break of trust which terminates the partnership. Players can be of two diﬀerent types/discount factors. We analyze the structure of eﬃcient equilibria and characterize the Pareto frontier of types’ payoﬀs. Eﬃcient equilibria are shown to display a typical pattern consisting of three phases. First, in the Communication Phase information about the respective opponent’s type is transmitted, after which in the Adjustment Phase stakes grow over time until in the Established Phase partners behave stationary as under complete information. The eﬃcient adjustment of stakes is governed by the impatience of the low type. Furthermore, we demonstrate that in some equilibria all players do strictly better than in a corresponding game with complete information. JEL Classification: C72, D82, D83 Keywords: trust, gradualism, repeated games, incomplete information

∗

We are grateful for insightful comments and advice from Christian Ewerhart, David Levine, Larry Samuelson, Joel Sobel and Joel Watson and two anonymous referees. The comments of seminar participants at the Universities of Berkeley, Bonn, Mannheim, San Diego, Stanford, UCLA and Wien as well as participants of the 2000 World Congress of the Game Theory Society, ESEM 2001 and the 2001 Meeting of the Verein f¨ ur Socialpolitik are appreciated. † Department of Economics, Goethe University, Frankfurt am Main, 60054 Frankfurt a.M., Germany; telephone: +49-(0)69-798-23858; e-Mail: [email protected] ‡ BASF, Germany e-Mail: [email protected]

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. . . What makes many agreements enforceable is only the recognition of future opportunities for agreement that will be eliminated if mutual trust is not created and maintained, and whose value outweighs the momentary gain from cheating in the present instance. . . . Neither (party) may be willing to trust the other’s prudence (or the other’s conﬁdence in the ﬁrst’s prudence, and so forth) on a large issue. But, if a number of preparatory bargains can be struck on a small scale, each may be willing to risk a small investment to create a tradition of trust. (Thomas Schelling, 1980, The Strategy of Conﬂict, p.45)

1

Introduction

How should business parties build up trust? There is a common and widely accepted perception that the mutual formation of trust is a key for successful business relationships and that it is a complex and genuinely dynamic process. Thomas Schelling suggests a number of preparatory bargains on a small scale to create a tradition of trust. In this article we analyze the structure of this dynamic formation process. More precisely, we study how many bargains on which scale should be struck under various conditions in order to successfully build up a relationship being able to perform on large issues. A salient part of any explanation how to build up trust is mutual uncertainty about the other partie’s true motives and characteristics. Our intuition tells us – as Schelling implicitly does – that foregone immediate larger business opportunities are the price to be paid in order to collect information and learn about prospective business partners. Hence, we are interested in more details of this information transmission process. How should we communicate how trustworthy we are? Clearly, networks and modern legal institutions oﬀer various possibilities to alleviate or circumvent the trust formation problem. However, we are convinced that a deeper understanding of a basic structure is paramount for the analysis of more realistic social settings. Therefore, we concentrate in this article on the pure bilateral problem of building up a relationship without any help by third parties or institutions. We analyze these questions in a repeated game framework with two-sided incomplete information. Suppose that in each period a party can either cooperate by investing a variable amount into a bilateral project or defect by stealing all tangible assets of his partner. If both parties cooperate, the joint project generates a proﬁt. On the other hand, if a party decides to defect, no return is generated. The defector steals the current period investment of his partner, and the relationship is terminated (simultaneous defection results in no transfer). We quantify trust in a relationship with the stakes that can get lost in case of betray. In this article we analyze the linear payoﬀ case which dissolves into two generic linear speciﬁcations for the cooperation proﬁt function1 . According to our motivation joint proﬁts should be maximized if both parties invest maximally and any partie’s individual payoﬀ should increase with the partner’s investment. Within a linear framework this leaves us with one further degree of freedom. Should individual payoﬀs decrease or increase with a partie’s own investment? The ﬁrst case is known as linear public goods 1

This paper reveals that the linear case is already ambitious. Although we believe that many of our results hold more generally non-linear speciﬁcations turn out to be much more challenging and are left to future research

2

game and we call the latter case linear or continuous trust game. Since we assess both cases as most relevant we study both of them. Both stage game structures share the common feature with the Prisoner’s Dilemma that the unique stage game equilibrium is non-cooperation. The continuous trust game, however, needs further motivation since it is less known and since it contains an additional and interesting twist that turns out to be very important for the structure of the results in the repeated game formulation. In contrast to the Prisoner’s Dilemma and the linear public goods game non-cooperation or quit Q is not a dominant strategy in the linear contribution game because the best response to a low cooperative investment of your partner may be to cooperate on the maximal investment level. In this case you can steal only very little but given your partner cooperates you can raise your cooperation proﬁts by investing more yourself2 . We believe this interdependence in which high investments can lead to high returns, but also result in high losses when a partner misbehaves, to be a central feature of many real-world economic and social situations: joint ventures, trading relationships, employeremployee relationships, social partnerships (co-authorships, marriage, friendships) and so forth. Our central and unifying result is that not only do there exist equilibria with gradually increasing investment levels. Most importantly, such equilibria whenever they exist they are uniquely eﬃcient and display a characteristic pattern. First, parties communicate their type, we call this Communication Phase. Then, both parties adjust their investments up to a ﬁnal stable level. We call this Adjustment or Enhancement Phase. Finally, partners behave stationary just as they should behave eﬃciently in the absence of uncertainty, which we label Established Phase. The phenomenon of gradually increasing stakes has entered the recent economic literature under labels such as “gradualism” or “starting small”. Most closely related to our approach and results are the articles of Watson (1996; 1999; 2002) as well as Ghosh and Ray (1996). Postponing a more detailed discussion of the related literature to Section 7, we note at this point that in contrast to these articles we embed our model in the standard setting of repeated games (albeit with incomplete information regarding discount factors). We therefore allow agents to select their possibly diﬀering investment levels endogenously in each period, which is not the case in other approaches. This possibility of asymmetric investment levels leads us to some surprising additional results regarding the information transmission process. Contrary to our ﬁrst intuition it is not generally true that incomplete information in this context necessarily worsens the expected payoﬀ of any type of agent. We show that for some parameters, there exist eﬃcient equilibria in which each possible type’s expected payoﬀ strictly dominates his payoﬀ in a benchmark setting of complete information.3 We proceed with a more detailed discussion of our main results. As already mentioned, we model the uncertainty about opponent’s motives by assuming double sided incomplete information regarding discount factors. Speciﬁcally, we assume the discount factors to be independently drawn by nature from a binary set 0 < δl < δh < 1 according to a commonly known distribution (p, 1 − p). Our preferred interpretation of this setup is as a prototypical model for general relationships in a society whereby two individuals are randomly matched to interact while each knows only his own type. In line with 2

Probably all co-authors are from time to time most familiar with this phenomenon In the benchmark setting assume that a type is randomly matched and observes the type of the opponent he is matched against. 3

3

this interpretation we focus on symmetric, pure strategy4 equilibria of the corresponding incomplete information game. We interpret equilibria as conventions of interaction in a society. Therefore, from a normative viewpoint, our prime interest will regard eﬃcient equilibria.5 We call an equilibrium interim eﬃcient if there exists no other equilibrium in which all types do as least as well and some type attains strictly higher expected payoﬀs. We also compare to the weaker notion ex-ante eﬃciency which we deﬁne regarding a player’s expected payoﬀ, i.e., before he observes his own type. As a benchmark we ﬁrst analyze the game with complete information, i.e. players observe their opponent’s type. Under the assumption that cooperative returns and losses are each linear in own investment and bounded, one can partition the discount parameter space into two non-trivial Regions I and II.6 In Region I only high type matches can potentially cooperate (low types are too impatient for cooperation to be a viable option in an equilibrium). Region II, in contrast, allows for equilibria in which mixed high-low matchings can cooperate too.7 Returning to the incomplete information case we observe that in the continuous trust game all interim eﬃcient Region I equilibria (with positive investment in some period) are of two diﬀerent types which we call Q−gradual equilibria and C−gradual equilibria. In contrast, we show that in the linear contribution game only Q−gradual equilibria exist. Both Q−gradual equilibria and C−gradual equilibria have a prominent feature in common: the behavior of a high type when matched against another high type. In this case the amount invested increases over time until it hits the maximal level where it stays thereafter. The rate of investment growth turns out to be mainly determined by the impatience of the low type. The low type is exactly indiﬀerent between his equilibrium behavior and imitating a high type in hope of cheating on a high type opponent in some later period. Q−gradual equilibria and C−gradual equilibria, however, diﬀer in the manner in which the separation of types occurs. In Q−gradual equilibria a low type quits in the ﬁrst period, while in a C−gradual equilibrium low types cooperate in the ﬁrst period at the highest level and then quit in the second. At ﬁrst sight, one may be irritated by the fact that, in an eﬃcient equilibrium, high type matches only increase investment gradually, even though they are sure of their opponent’s type along the equilibrium path because separation occurs in the ﬁrst period. We argue (see Remark 5) that such behavior is consistent with both renegotiation and reﬁnement considerations. Q−gradual equilibria reﬂect the intuitive eﬀect of incomplete information one would expect in such a setting: High types have an incentive to discover their opponent’s type in order to beneﬁt from cooperation with a high type opponent. Low types are enticed to reveal their identity by the oﬀer of a rent/transfer in the ﬁrst period, i.e., low types 4

If one were to either drop symmetry or allow mixed strategies, our results will still hold. We conjecture that this is also true for asymmetric, mixed strategy equilibria. 5 Alternatively, one could motivate our focus on eﬃcient equilibria as the standard procedure of characterization of the set of feasible equilibrium payoﬀs of the repeated game. 6 Trivial regions are those for which both discount factors are so low that no equilibria with positive investment exist or, alternatively, for which both discount factors are suﬃciently high such that all possible type combinations can cooperate at the highest level. Although a binary type space clearly simpliﬁes reality we think that this very qualitative structure of the parameter space is very intuitive and calls for a careful investigation. 7 In the continuous trust game in such equilibria low types invest more than high types (in each period). Low types do not steal as it is more worthwhile to obtain returns on the own large investment than to steal the opponent’s small amount. High types do not quit because they are patient.

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steal the ﬁrst period investment of a high type if they meet one. Furthermore there is a social loss due to the fact that high-high matches can only increase investment gradually to deter low types from posing as high types and stealing later on. Q−gradual equilibria can be indexed by the ﬁrst period investment a of a high type. While clearly a low type’s expected payoﬀ is strictly increasing8 in a, the high type’s expected payoﬀ turns out to be concave in a. This characterizes the Pareto frontier of Q−gradual equilibria in region I. Compared to expected (eﬃcient) complete information payoﬀs, the low type does strictly better9 , while the high type does strictly worse due to the low type’s stealing and the slow increase in high-high matchings. C−gradual equilibria only exist in the continuous trust game and are of a diﬀerent nature. In the ﬁrst period low types invest the maximal amount possible and quit in the second period. As before, high types start oﬀ with a small investment which gradually grows in case they are matched with a high type opponent. When they are matched with a low type they also quit in the second period. One may ask why low types do not quit and steal in the ﬁrst period? Low types face a tradeoﬀ in the ﬁrst period. If they cooperate they obtain a reasonably high payoﬀ, namely the cooperative return on the maximal possible investment. The amount they obtain from stealing depends on the type of their opponent. With probability p they meet a high type opponent from whom they can steal only little, whereas with probability 1 − p they can steal the largest amount possible from a low type opponent. We show that for a nontrivial parameter region this tradeoﬀ points in favor of cooperation, i.e., C−gradual equilibria exist. Given that in C−gradual equilibria a low type obtains the return from the maximal investment for one period, interim eﬃcient equilibria are obtained by maximizing the high type’s expected payoﬀ subject to low/high type non-deviation conditions. C−gradual equilibria are therefore generically unique. In contrast to both Q−gradual equilibria and eﬃcient complete information equilibria, low types obtain strictly positive payoﬀs in a C−gradual equilibrium against both low and high type opponents (they therefore do strictly better than in any complete information equilibrium). Similarly, high types also obtain positive payoﬀs against low type opponents. Compared to eﬃcient complete information equilibria, high types do strictly worse against high type opponents due to the ineﬃciency of increasing investment only gradually, while they do strictly better against low type opponents. We show that for some parameter regions the latter eﬀect compensates the former, i.e., there exist C−gradual equilibria which dominate all complete information equilibria with respect to interim eﬃciency. This surprising result indicates that the “high type pays low type information rent” story is not the complete picture. The possibility of diﬀering types investing diﬀering amounts in one period allow for the separation of types in a way which is beneﬁcial to all types, which is in contrast to results in the previous, related literature.10 While both Q− and C−gradual equilibria need not exist11 , at least one of them 8

With probability p a low type is matched with a high type from whom he can steal a. With complete information a low type’s payoﬀ is always 0 10 There are somewhat less related ﬁelds in which incomplete information has been shown to be beneﬁcial. In the reputation literature (see, for example, Kreps et al. (1982)), the addition of uncertainty can lead to previously infeasible cooperative outcomes. However, we would argue that our case is more salient in that we do not add uncertainty regarding an artiﬁcial, exogenously chosen type to a given economic setting. Instead we motivate a speciﬁc economic setting with given types, and then introduce incomplete information while keeping the type space ﬁxed. 11 This contrasts Watson (1996), where equilibria related to Q−gradual equilibria always exist because diﬀering types are assumed to invest the same amount, i.e., deviations to higher investment levels are 9

5

exists as soon as there is any equilibrium in which at least one type-matching invests a positive amount in some period. As Q−gradual equilibria can be shown to dominate C−gradual equilibria with respect to interim eﬃciency — and vice versa for a diﬀerent parameter region —, they together as well as the trivial “All Quit” equilibrium completely characterize the Pareto frontier of Region I equilibria. In Region II also low types can potentially cooperate with high types for prolonged periods of time. Only low-low matches cannot. We observe that both Q−gradual equilibria and C−gradual equilibria still remain equilibria in Region II. Since these equilibria do not take advantage of mixed matching cooperation they tend not to be eﬃcient. However, there are additional equilibria which do take advantage of mixed cooperation. We call them CII −equilibria. The structure of CII −equilibria is more involved for several reasons. First, in contrast to Region I we cannot exclude pooling phases. Further, since only low-low matchings leave the game after separation the following behavior is characterized by three potentially diﬀerent investment paths, those of a high type meeting another high type or a low type and that of a low type meeting a high type. These paths are interrelated by a complex variety of non-deviation conditions. Nevertheless, we can show that the previously discovered CAE−pattern of C ommunication Phase followed by Adjustment and E stablished Phases still prevails as eﬃcient mode of behavior. Since with linear payoﬀ functions all potentially binding non-deviation conditions are weak linear inequalities the Pareto frontier of the payoﬀ space in region II is simple to describe. The dimensionality of a detailed description of eﬃcient behavior in region II grows rapidly in the eﬃcient duration of the Adjustment Phase. To get a better intuition for this we further elaborate on the speciﬁc rapid enhancement equilibrium. This equilibrium with immediate adjustment (Adjustment Phase of length 0) is of particular interest since it is eﬃcient if it exists. Further, it only depends on two investment parameters, namely the separation investments, after which players switch to the stationary eﬃcient. Therefore it is the simplest CII −equilibrium and can be studied by graphical analysis. The paper is organized as follows: in Section 2 we present the formal model. Section 3 presents the benchmark case of complete information, characterizing the structure and payoﬀs of eﬃcient equilibria. Some preparatory results are presented in Section 4 before a complete characterization of eﬃcient Region I equilibria is given in Section 5. Section 6 characterizes Region II equilibria and provides a geometrical approach. Related literature is discussed in Section 7. Section 8 closes with the conclusions.

2

The Model

Stage game We ﬁrst deﬁne the stage game Γ := N := {1, 2}, (Ai), (vi ), where N is the set of players, Ai := [0, 1] ∪ {Q} is player i’s action set, and payoﬀ vi is deﬁned as fi (ai , aj ) if ai , aj ∈ [0, 1], −(1 + )ai if ai ∈ [0, 1], aj = Q, vi (ai , aj ) := aj if ai = Q, aj ∈ [0, 1], 0 if ai = aj = Q. where i, j ∈ N and i = j (which will always be assumed for indices i, j). The set of action proﬁles is denoted by A := A1 × A2 . not feasible and the separation of type can only occur by someone quitting.

6

As mentioned in the introduction the structure of the stage game Γ is closely related to the Prisoners’ Dilemma. However, we consider a continuous action space. By playing a cooperative action ai ∈ [0, 1] – denoted by C – each player individually chooses his level of trust measured by the potential loss if the partner defects12 . The project of the relationship only pays oﬀ fi (ai , aj ) when both players cooperate. Instead of investing, player i can quit – denoted by Q – and steal the other player’s stake aj . In order to allow for a social ineﬃciency of stealing, we assume that the victim j incurs losses of size (1 + )aj , which potentially exceed his investment (for > 0). We recapitulate with the following “collapsed” normal form matrix: C

Q

C

f1 (a1 , a2 ), f2 (a1 , a2 )

−(1 + )a1 , a1

Q

a2 , −(1 + )a2

0, 0

In this article we restrict attention to symmetric linear return functions f . While it seems intuitively plausible that many of our results also hold for concave return functions, such an analysis turns out to be technically extremely demanding13 . Within the linear formulation we consider two salient normalized speciﬁcations diﬀering by the individual incentive to raise the level of the relationship which encompass all relevant features of linear return functions. Case 1: fi (ai , aj ) = −λai + aj for i = 1, 2 and λ ∈ (0, 1). This is a normalized parametrization of the linear contribution game or linear public goods game. Here each player has the incentive not to contribute (free ride) while maximal contribution ai = aj = 1 is eﬃcient. Due to the normalization, contributing 0 is payoﬀequivalent to playing Q.14 Case 2: Let fi (ai , aj ) = (1 − λ)ai for i = 1, 2 and λ ∈ (0, 1). We call this the continuous trust game 15 . Contrasting the linear contribution game, in this speciﬁcation a player’s return only depends on his own stake when both players cooperate. We envisage a project which solely pays oﬀ when both players cooperate, and whereby returns are split proportionally to the investment levels. Here, a player would always like to invest if he were sure the opponent wouldn’t steal. An important diﬀerence to the standard Prisoners’ Dilemma is that the action Q is not strictly dominant. While (Q, Q) is the unique stage game equilibrium, maximal cooperation at level 1, and not Q, is a best response to low (i.e., below 1 − λ) levels of cooperation. This latter feature turns out to have surprising and interesting implications for the structure of information transmission under incomplete information. 12

Somewhat loosely, we use in this article ’investment’ and ’the level of the relationship or’ or what can be stolen synonymously for variable ai . 13 Our linear speciﬁcation can be seen as a piecewise linear approximation of a concave function since we assume an upper bound on investments which amounts to the assumption that additional investment from this upper bound does not result in additional returns. 14 In this case we will from here implicitly identify the strategies 0 and Q to avoid unsubstantial case diﬀerentiations. 15 An earlier version of this paper analyzed this speciﬁcation only.

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Repeated game We deﬁne the repeated game Γ∞ of incomplete information as follows: Players can be from a set of two diﬀerent types T := {l, h} — interpreted as low and high types. Types diﬀer in their discount factor δi ∈ {δl , δh }, where 0 < δl ≤ δh < 1. At the beginning of the game nature draws players’ types independently according to Pr(δh ) = p, Pr(δl ) = 1 − p. Unless indicated otherwise each player’s type is private information. For technical convenience we assume action Q or investment ai = 0 to end the game.16 Let ω 0 := ∅ and deﬁne nonterminal histories as ω t = (a0 , a1 , . . . , at−1 ) where at ∈ A and until then no player quits or invests 0, that is aτi = Q, 0 for all τ = 1, . . . , t − 1 and t i = 1, 2. Let Ωt := {ω t} and Ω := ∪∞ t=0 Ω . Terminal histories are either inﬁnite sequences of (stage game) outcomes in which Q was never played, or ﬁnite sequences of outcomes in which one (or both) players choose Q only in the last period. A strategy for player i is then given by si := Ω ∪ T → Ai . s := (s1 , s2 ) ∈ S := S1 × S2 . Type contingent strategies will be denoted si (h) and si (l). A tuple of strategies s induces an outcome o(s) consisting of four (one for each type-combination) terminal histories which we will also call outcome paths. Speciﬁcally o(s) := {ohh (s), ohl (s), olh (s), oll (s)} , where ohh (s) corresponds to the terminal history induced by reducing the strategies to s(h) = (si (h), sj (h)); and similarly the other expressions. For a history ω deﬁne by #ω the (possibly inﬁnite) index of the last period of ω; therefore #ohh (s) + 1 denotes the length of the outcome path/terminal history induced when both players play according to s = (si , sj ) reduced to the behavior of a pair of h-type. Furthermore index ohh (s) with t = 0, . . . , #ohh (s) such that othh (s) ∈ A. The preferences of player i over outcomes are given by utility ui (s) := p

2

#ohh (s)

δht vi (othh (s))

#ohl (s)

+ p(1 − p)

t=0

δht vi (othl (s))+

t=0 #olh (s)

+ p(1 − p)

δlt vi (otlh (s)) + p2

t=0

#oll (s)

δlt vi (otll (s))

t=0

Given that utility depends only on the outcome induced by strategies we will switch between the notation ui (s) and ui(o) where o is the outcome induced by s. We use Bayesian Nash equilibrium as our equilibrium concept. We call an outcome o an equilibrium outcome if there exist strategies s∗ and beliefs β ∗ such that (s∗ , β ∗ ) is a Bayesian Nash equilibrium with induced outcome o = o(s∗ ). Following the interpretation given in the introduction, namely a random matching problem in a population consisting of a proportion p of high types, we restrict attention to symmetric equilibria of Γ∞ . We therefore deﬁne type-contingent utilities for n ∈ T as follows: #onh #onl t t un (o(s)) := un (s) := p δn vn (onh ) + (1 − p) δnt vn (otnl ). t=0

t=0

Our prime interest regards equilibria which are undominated with respect to type contingent payoﬀs: Deﬁnition 1 An equilibrium (s∗ , β ∗ ) and its outcome o(s∗ ) are called interim eﬃcient or o) ≥ (weakly) interim undominated if there exists no equilibrium outcome o˜(˜ s) with uh (˜ 16

This is without loss of generality; all results would go through in the standard repeated game setting. By this assumption we just get rid of various cases that do not diﬀer in substance.

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uh (o) and ul (˜ o) ≥ ul (o) where at least one of the inequalities is strict. We will use “eﬃcient” and “undominated” synonymously. 2 A weaker concept which we will be using is: Deﬁnition 2 An equilibrium (s∗ , β ∗ ) and its outcome o(s∗ ) are called ex-ante eﬃcient o) + (1 − or ex-ante undominated if there exists no equilibrium outcome o˜(˜ s) with puh (˜ o) > puh (o) + (1 − p)ul (o). p)ul (˜ 2 Ex-ante and interim eﬃciency reﬂect player’s preferences over diﬀerent equilibria before and after learning their own type. Obviously, every interim ineﬃcient equilibrium is ex-ante ineﬃcient. Most results in this paper will refer to the more intuitive notion of interim eﬃciency.

3

Complete Information

As a benchmark for the inﬂuence of incomplete information, and to familiarize ourselves with the basic mechanics of our model, we start with a complete information, matching version of our game17 . Assume as before a population of players, of which proportion p have discount factor δh (and 1 − p have δl < δh ). Players are matched randomly in pairs, however, in this section, they observe their opponent’s type. The main goal is to characterize eﬃcient equilibria. We note that there are no general characterization results concerning repeated game equilibrium correspondences as a function of discount parameters. A ﬁrst simple observation for equilibria of the repeated game is: Lemma 1 In any ﬁnite equilibrium outcome path both players obtain payoﬀ 0. Any inﬁnite equilibrium outcome path where some player’s payoﬀ is not 0 consists of an inﬁnite number of periods with strictly positive investment. 2 Proof In the converse of both cases there exists a last period T where no player plays Q or invests 0. This cannot be an equilibrium since in this period players can always improve either by stealing or by investing more or less. As a consequence we can partition the λ, δl , δh parameter space into diﬀerent regions, corresponding to whether indeﬁnite cooperation (at a positive level) is viable or not. As we will see in the next section, this also determines the possible structure of equilibria in the incomplete information case. In any period t in an equilibrium it must hold that stealing is worse than continuing along the equilibrium path, i.e., fi (ati , atj ) + δi Vit+1 ≥ atj ,

(1a)

fj (ati , atj ) + δj Vjt+1 ≥ ati ,

(1b)

where Vit+1 and Vjt+1 denote the continuation payoﬀs along the corresponding outcome paths from period t + 1 on. A seemingly obvious but nontrivial observation is that eﬃcient outcome paths are stationary. 17

In contrast to the Folk Theorem literature in evolutionary game theory as for example Hirshleifer and Martinez-Coll (1988) we need to concentrate on the λ, δl , δh −parameter space where cooperation among the diﬀerent matchings is viable.

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Lemma 2 All eﬃcient outcome paths are stationary.

2

Proof Along eﬃcient equilibrium outcome paths (1a,1b) must either hold in equality or the investment bound of 1 is reached. Writing the non-deviation conditions for period t − 1 as: t−1 t t 2 t+1 fi (at−1 = at−1 i , aj ) + δi fi (ai , aj ) + δi Vi j , t−1 t t 2 t+1 = at−1 fj (at−1 i , aj ) + δj fj (ai , aj ) + δj Vj i .

Substituting the V t+1 from (1a,1b) we obtain: 1 t−1 t−1 ai − fi (at−1 i , aj ) δj 1 t−1 t−1 aj − fj (at−1 atj = i , aj ) δi ati =

(2a) (2b)

For the case of linear f , these equations deﬁne a linear dynamical system with ﬁxed point (0, 0). As we are restricting the system to the unit square we obtain either convergence to (0,0) or into a point on the boundary of the unit square. In a stationary equilibrium at level (ai , aj ), continuation payoﬀs in each period are given by (fi (ai , aj )/(1 − δi )), fj (ai , aj )/(1 − δj )). Inserting such continuation payoﬀs into (1a,1b) we obtain the following necessary equilibrium conditions: fi (ai , aj ) ≥ aj , 1 − δi fj (ai , aj ) ≥ ai . 1 − δj

(3a) (3b)

We deﬁne the set A(δi , δj ) as the set of points in the unit square for which inequalities (3a,3b) hold. Lemma 3 In the linear contribution game, the set of equilibrium payoﬀs for the repeated game with discount factors (δi , δj ) is equal to A(δi , δj ). For the continuous trust game all points in A(δi , δj ) for which ai ≥ 1−δi and aj ≥ 1−δj can be implemented as equilibria of the repeated game. 2 Proof The statement follows from standard results in the theory of repeated games. Any point in A(δi , δj ) can be implemented as an equilibrium payoﬀ18 of the repeated contribution game. Players could, for example, play well known “grim-trigger” strategies, sticking to the a point in A(δi , δj ) and punishing oﬀ-equilibrium behavior by quitting. The resulting equilibrium is also subgame perfect. For the continuous trust game, as mentioned in the model description, a deviation to 1 may also be a best response. The above inequalities ensure that such deviations are not beneﬁcial in an equilibrium with “grim-trigger” strategies. We can therefore proceed by analyzing the Pareto-frontier of A(δi , δj ) as a function of the discount factors. The next lemma shows that parameter λ describes two important regions of discount factors: 18

We gloss over the fact that one actually should renormalize the repeated game payoﬀs by a factor of 1 − δ to compare directly with stage game payoﬀs.

10

δ2

δ2

1

1

II I

λ

λ

II

II I II

I

I

δ1 1 λ (a) Contribution Game

λ (b) Continuous Game

0

0

1

δ1

Trust

Figure 1: In the lower left, white region no cooperation is possible, while in the upper right, white region all type combinations can cooperate. In Region I only high types amongst themselves have the option of cooperating, while in Region II all type combinations except low-low can do so. Lemma 4 If discount rates are such that λ > δh > δl then the unique equilibrium payoﬀ of the repeated game is (0, 0). Alternatively if λ ≤ δl < δh then the unique eﬃcient equilibrium has both players cooperating at the maximal level of 1 in every period. 2 Proof Note that if λ > δh > δl then A(δi , δj ) contains only (0, 0) (resp. is empty for the case of the continuous trust game in which case (Q, Q) is the unique equilibrium). To see the second point, note that (1, 1) ∈ A(δi , δj ) whenever λ < δi , δj . Obviously the more interesting case arises when δl < λ ≤ δh . This ensures that while low types cannot cooperate with one another, high types can. But how about mixed pairs consisting of a high and a low type? Restricting attention for the moment to δl < λ ≤ δh we deﬁne Region I to be the set of (δl , δh ) such that A(δl , δh ) is empty (i.e. mixed pairs cannot cooperate), while Region II is the set of (δl , δh ) such that A(δl , δh ) is nonempty. For the contribution game Region I is described by δl δh > λ2 and Region II by δl δh ≤ 2 λ . In the case of the continuous trust game Region I is described by (1 − δl )(1 − δh ) > (1 − λ)2 and Region II by (1 − δl )(1 − δh ) ≤ (1 − λ)2 These regions are graphed in Figure 1. Therefore, if mixed pairs cannot cooperate, i.e. in Region I, eﬃcient equilibria can be characterized as follows: Lemma 5 Assuming δl < λ ≤ δh and restricting attention to Region I, all eﬃcient equilibria have all type combinations except hh quitting in the ﬁrst period, while high types meeting high types cooperate at the maximal level 1 in every period. 2 It remains to analyze eﬃcient equilibria in Region II. We start with the contribution game: Lemma 6 Assuming δl < λ ≤ δh and restricting attention to Region II in the contribution game, eﬃcient equilibrium outcome paths are characterized as follows: • high types amongst themselves cooperate at the highest level in every period, • low types amongst themselves quit immediately, 11

• if a high type meets a low type, then the high type cooperates at level 1 forever, while the low type cooperates at some level in λ δl , . 2 δh λ Lemma 7 Assuming δl < λ ≤ δh and restricting attention to Region II in the continuous trust game, eﬃcient equilibrium outcome paths are characterized as follows: • high types amongst themselves cooperate at the highest level in every period, • low types amongst themselves quit immediately, • if a high type meets a low type, then the low type cooperates at level 1 forever, while the high type cooperates at level 1−λ . 2 1 − δl Lemmas 6 and 7 follow straightforwardly from Lemma 3 and the analysis of the Pareto frontier of A(δi , δj ). We note that for high-low matchings in the contribution game we obtain a Pareto frontier of equilibria in which the patient type invests maximally. In contrast in the continuous trust game we obtain a single point in which the low type player invests more than the high type and therefore obtains higher stage game payoﬀs in an eﬃcient equilibrium (however, the low type still has a strictly lower repeated game payoﬀ than the high type). To wrap up, we point again to Figure 1 which displays the partitioning of the discountparameter-space into regions with diﬀering eﬃcient equilibria. In Region I the payoﬀs attained in eﬃcient equilibria will play a role as a benchmark for the incomplete information setting. Recalling the population random matching setting we obtain uIl = 0, p(1 − λ) uIh = p . 1 − δh

4

(4) (5)

Incomplete Information: Preparatory Results

We now turn to incomplete information, i.e. nature selects types independently at the start of the game according to (p, 1 − p) (with p being the probability of a high type). Players observe their own types perfectly and obtain no information about their opponent. Our main goal is to characterize interim eﬃcient equilibrium payoﬀs and the corresponding equilibrium outcome paths. We focus on the interesting parameter region δl < λ ≤ δh (in the other regions eﬃcient equilibria induce the same outcome paths as in the complete information case). Remark 1 An important point to note is that given any equilibrium, setting all oﬀequilibrium behavior to Q retains the equilibrium property (recall that (Q, Q) is the only 12

equilibrium of the stage game). It therefore suﬃces to describe an equilibrium’s outcome path, implicitly assuming all oﬀ-equilibrium behavior to be Q, while taking beliefs to be consistent with behavior.19 In the following we will take this route and describe outcome paths when talking about (eﬃcient) equilibria.20 2 As usual we call an equilibrium with strategies s separating (with separation period T ) if otll (s) = otlh (s) = othl (s) = othh (s) for all t < T and oTll (s) = oTlh (s) = oThl (s) = oThh (s), i.e. both types l and h behave identically up to period T − 1 and diﬀer (are separated) in period T (recall we restrict attention to symmetric equilibria which allows this simplifying language).21 We will also call periods 0, . . . , T − 1 the pooling phase. An equilibrium with no separation will be called pooling equilibrium.22 For an outcome path to be an equilibrium one generally has to check numerous nondeviation conditions. To play Q oﬀ-equilibrium greatly simpliﬁes this task as only deviations in one period need to be veriﬁed. Note furthermore that playing either Q or 1 are best possible deviations in any period. Hence, these are the only deviations one has to check for since others are dominated by these. Therefore the set of potentially characterizing deviations is i) sticking to equilibrium outcome behavior for a (possibly zero) number of periods and then deviating to Q or 1, and ii) imitating the other type for a number of periods and then deviating to Q or 1. From i) it trivially follows that: Lemma 8 Consider a separating equilibrium with separation period T . Then all outcome paths restricted to periods after T are equilibrium outcome paths of the complete information game. 2 This tells us that the range of equilibrium behavior we can expect to observe may diﬀer in parameter regions I and II. We therefore discuss the structure in separate Sections 5 and 6. Before doing this, we proceed with observations which hold independently of the region. Equilibria in which all players obtain payoﬀ 0 are called 0-equilibria – i.e. players only invest 0 or play Q. First, we observe that there are no pooling equilibria exept 0-equilibria. Proposition 1 In any equilibrium — except 0-equilibria — separation occurs in ﬁnite time. 2 The intuition for this is that during a pooling phase in which all types invest the same strictly positive amount, the only way to keep a low type from quitting is by oﬀering him more to steal later on. However, as feasible investments are bounded from above, this is 19

As (Q, Q) is a subgame perfect equilibrium independently of discount factors, oﬀ-equilibrium beliefs will not play a role. 20 In Remark 5 we also discuss the appropriateness of thereby induced oﬀ equilibrium beliefs and possible renegotiation issues. 21 Note that separation will not necessarily be reﬂected in beliefs, if it occurs when one player plays Q (the game ends and there is no future period for the opponent to update his beliefs). 22 The outcome path of a pooling equilibrium can be of ﬁnite or inﬁnite length.

13

not feasible inﬁnitely often. The technicalities of the proof are relegated to page 32 in the Appendix. The next result shows that even if equilibria with a pooling phase are relevant in characterizing the Pareto frontier, there will always exist undominated equilibria with an empty pooling phase: Proposition 2 For any equilibrium E in which separation of types occurs in period T > 0 there exists another equilibrium E˜ in which separation of types occurs in period 0, and which is strictly preferred by the low type. 2 The proof is relegated to page 33 in the Appendix. As suggested by Lemma 8 we proceed with analyzing parameter Regions I an II in turn.

5

Region I

For this section we restrict the discount parameters to Region I, that is, after separation or in a complete information setting only cooperation amongst h types is viable. From Proposition 1 we know that separation of types occurs in ﬁnite time. This can occur in two distinct ways: i) the l type plays Q, or ii) both types cooperate at distinctive levels.23 For the low type to separate by a cooperative action can only be an equilibrium in the linear trust game since there his own investment is beneﬁcial to him. In the linear contribution game a low type could improve by quitting the game. As this diﬀerentiation will be important to our results, we deﬁne: Deﬁnition 3 An equilibrium in which separation occurs by the l type playing Q will be called Q−equilibrium or just Q-equilibrium. In contrast, an equilibrium in which separation occurs by both types playing distinct levels of cooperation will be called C−equilibrium or C-equilibrium. 2 The goal of this section is to give a complete characterization of interim eﬃcient (in the sense of deﬁnition 1) Q−or C−equilibria. The main results will be presented in the following subsections 5.1 and 5.2, respectively. It turns out that the structure of Q−equilibria is the same for the linear contribution game and the continuous trust game. However, C−equilibria do only exist for the continuous trust game since in the linear contribution game maximal investment is never a best response in the stage game. There is the salient unifying feature of gradualism or starting small in both corresponding sorts of eﬃcient equilibria: namely the path played after separation when two h types are matched. Both eﬃcient Q− and C−equilibria have qualitatively similar hh outcome paths, which can be described as follows: Deﬁnition 4 A gradual path α(a0 , a1 ) (for Region I) — indexed by a parameters a0 , a1 ∈ [0, 1] — is a sequence of actions:

2 τ (a1 ) λ λ λ α(a0 , a1 ) := a0 , a1 , a1 , . . . , a1 , 1, 1, . . . , (6) , a1 δl δl δl 23

Note that there cannot exist a separating equilibrium in which the h type separates by playing Q in T for the following reason: If h exits the game in T it follows from Lemma 8 that the low types play Q in the next period. Therefore, as l’s continuation payoﬀ is 0, it cannot be a best response for l to cooperate in T .

14

τ (a1 )

a 1

1

a1

a

1 a λ δl

2 λ δl

a0

t

0 0

1

2

3

4

5

6

7

Figure 2: An example of a gradual cooperation path. Starting oﬀ with investments a0 and a1 , in the following the investment level is increased by the factor δλl until period τ (a1 ), after which the maximal amount of 1 is invested thereafter. where

t max t ∈ {0, 1, . . .} | a1 λ < 1 for a1 < 1, δl 1 τ (a ) := 0 for a1 = 1.

(7) 2

While this might look somewhat complicated, the intuition is reasonably simple. Noting the graphic visualization of such a path in Figure 2 we subdivide a gradual cooperation path into three phases: Communication Phase: consists solely of period 0 in which amount a0 is invested. As we will see later, in this phase information about types is transmitted. Adjustment Phase: comprises periods 1 to τ (a1 ) (and therefore may be empty for a1 > δλl ). During periods 1 to τ (a1 ), the investment level is increased by a factor of δλl . The term τ (a1 ) is the maximal number of periods one can increase the investment by a factor of δλl before hitting the maximal stake 1. Established Phase: encompasses all periods from τ (a1 )+1 onwards where the maximal amount 1 is invested. The fraction δλl which determines the growth speed of the path in the Adjustment Phase will subsequently crop up frequently (as well as its reciprocal value). It has the following interpretation. Imagine for a moment type l playing along a gradual path (and his opponent too). Assume furthermore that we are in the Adjustment Phase and l is pondering quitting in t or t + 1 (both in the Adjustment Phase). λ/δl is the exact growth rate such that an l type is indiﬀerent between (i) stealing in t, and (ii) cooperating in t and stealing in t + 1, i.e., if at+1 = δλl at then: λ at = (1 − λ)at + δl at . δl

15

(8)

5.1 5.1.1

Separation by Quitting Characterization

We now use the deﬁnition of a gradual cooperation path given in the previous section to deﬁne a special class of Q−equilibria: Deﬁnition 5 A Q-gradual-equilibrium with starting value a — denoted by QGE(a) — is a Bayesian Nash equilibrium in which the outcome path of a pair of high types corresponds 24 to a gradual cooperation path, while low types quit in the ﬁrst period. The parameters are: a0 = a, a1 = δλl γa, and τa := τ (a1 ) = τ δλl γa where γ :=

pλ + (1 − p)(1 + ) . pλ

(9) 2

In order to present the central result of this section as early as possible we relegate a detailed description of the properties (existence etc.) of Q−gradual equilibria to Propositions 3 and the following. A point, however, which we cannot delay is the interpretation of γ. As will be shown later Q−gradual equilibria have the property that l is indiﬀerent between playing Q or imitating h for t − 1 periods and then playing Q (for all periods t ≤ τa ). If this statement is correct for t = 0 and t = 1, then the growth rate discussion preceding (8) shows that it is also true for t = 2, . . . , τa . The diﬀerence between period 1 and the following periods is: When l decides to deviate from playing Q in 0, to imitating h in 0 and playing Q in 1 there is a 1 − p chance that he will meet another l-type in which case he will loose (1 + )a0h . The γ parameter corrects for this feature, that is the indiﬀerence growth rate in the separation period is γ δλl while later on it is δλl . We now present the main result of this section: Theorem 1 Every interim eﬃcient Q−equilibrium is a Q−gradual equilibrium. Every Q−equilibrium is weakly dominated by a some QGE(a). 2 This implies that whenever Q−equilibria exist, a Q−gradual equilibrium exists also. A more detailed characterization of existence will follow in Proposition 3. For the proof of Theorem 1 we proceed in two steps: i) an interim eﬃcient Q−equilibrium after separation resembles a Q−gradual equilibrium (Lemmas 9 and 10), and ii) in all eﬃcient Q−equilibria separation occurs in the ﬁrst period (Lemma 11). For step i) we ﬁrst note that in an eﬃcient Q−equilibrium the path of play of an hh match must hit the top stake 1 at some stage (and stay there forever). Lemma 9 In any undominated Q−equilibrium there exists τ such that in the event of a hh match, both players invest the maximal amount of 1 in all periods after τ , i.e., othh = (1, 1) for all t > τ . 2 The proof can be found on page 33 in the Appendix. Now we can complete step i) by arguing that any equilibrium with a pooling phase is dominated by another equilibrium with the same pooling phase followed by behavior corresponding to a Q−gradual equilibrium. 24

We remind the reader that for convenience sake, all oﬀ-equilibrium behavior is set to Q and beliefs are consistent with actions (see Remark 1).

16

Lemma 10 Every Q−equilibrium E in which separation takes place in period T is weakly ˜ in which behavior starting from the separation period is dominated by an equilibrium E deﬁned by a QGE(a). 2 The proof can be found on page 34 in the Appendix. It remains to show that equilibria with pooling phases are dominated, i.e. step ii) above. Lemmas 9 and 10 leave us with equilibria which consist of a pooling phase followed by a gradual path. The following crucial lemma shows that unraveling the pooling phase, i.e. having the l quit earlier while keeping him indiﬀerent, is beneﬁcial to the h-type. Lemma 11 In any undominated Q−equilibrium, separation takes place in period 0.

2

The proof can be found on page 34 in the Appendix and concludes the proof of Theorem 1. 5.1.2

Existence

The expected payoﬀ of a low-type player in a QGE(a) is given by uQ l (a) := pa, whereas the high-type obtains:

uQ h (a)

t τa λ δhτa +1 := p(1 − λ) a + γa + δh δl 1 − δh t=1

(10)

− (1 − p)(1 + )a.

(11)

As mentioned previously, the relevant non-deviation conditions which have to be fulﬁlled in an equilibrium are i) sticking to equilibrium outcome behavior for a (possibly zero) number of periods and then deviating to Q or 1, and ii) imitating the opposite type for a number of periods and then deviating to Q or 1. Denote by a the maximum of the non-deviation conditions that bound initial invest¯ ment a from below (more precisely deﬁned in the proof of Proposition 3 in the Appendix) and let the solution of uQ h (a) − pa = 0 if there exists a solution in [0, 1], a ¯ := (12) 1 otherwise. Proposition 3 For any a ∈ [a, a ¯] there exists a gradual cooperation equilibrium with ¯ starting value a. 2 The proof is on page 35 in the Appendix. We proceed with a characterization of the Pareto frontier restricted to eﬃcient Q−equilibria. Deﬁne a∗ := max argmax uQ h (a).

(13)

a∈[a,¯ a] ¯

Proposition 4 Restricting attention to Q−equilibria, the Pareto frontier in the (ul ,uh ) space is a strongly decreasing, piecewise linear function described by the payoﬀs attained in a gradual cooperation equilibrium with a ∈ [a∗ , a ¯]. 2

17

Proof From the deﬁnition in (10), ul (a) is increasing in a. From the proof of Proposition 3 it follows that uh (a) is a concave, piecewise linear function which increases for a ∈ [0, a∗ ] ¯]. and decreases for a ∈ [a∗ , 1]. Therefore the Pareto frontier is described by a ∈ [a∗ , a It is furthermore possible to give a more precise characterization of the parameter region for which the optimal τ is small, i.e., when equilibria entail high type matches investing the maximal amount early on. Proposition 5 Let δl < (1 − λ)δh ,

(14)

and E be an interim eﬃcient Q-separating equilibrium. Then E is a Q−gradual equilibrium with τ = 0, (i.e., from period 1 onwards hh-matches play 1). 2 Proof We know from the characterization Theorem 1 that E is a Q−gradual equilibrium. It remains to show that τ = 0. Take a Q−equilibrium and let Vlt−1 be the best possible continuation payoﬀ of a low type in period t − 1, where t − 1 is after the separation period (i.e. assume l were to imitate h through periods 0 to t − 2 and let him play a best response thereafter). Assume l’s best response to be Q in period t, i.e., Vlt−1 = (1 − λ)at−1 + δl ath . This implies that (while keeping l’s incentive and Vlt−1 h constant) one can change investments at−1 and ath by h ∆ath = −

1 − λ t−1 ∆ah . δl

(15)

Such a change is strictly preferred by h if: (1 − λ)∆at−1 + δh (1 − λ)∆ath > 0. h

(16)

if δl > (1 − λ)δh Combining (15) with (16) we observe that h prefers a positive ∆at−1 h and a negative ∆at−1 if δ < (1 − λ)δ . Noting that a negative (positive) ∆at−1 implies l h h h t — via (15) — a positive (negative) ∆ah , we see that if (14) holds, h prefers to transfer cooperation returns from earlier periods (after separation) to later ones. This would not lead to paths of gradual increase of investment, but to U-shaped paths. However, we have proved in Theorem 1 that Q−gradual equilibria are undominated. The only Q−gradual equilibrium for which previously described Pareto-improvements are not feasible, are the ones with τ = 0. Remark 2 It is interesting to recapitulate the seemingly contradictory ideas in the preceding proof. We optimize h’s path while keeping l indiﬀerent to his equilibrium payoﬀ, i.e., his best possible deviation payoﬀ is equal to the equilibrium payoﬀ obtained by quitting in period 0. Under these conditions, the h faces following trade-oﬀ i) getting to the maximal investment level as early as possible versus ii) maximizing the earliest cooperation beneﬁts possible. In case i) we obtain a path resembling an U while for ii) we obtain a gradual path. Depending on whether (14) holds or not, h prefers i) to ii) or vice versa. This area is visualized in Figure 3. Having proved that gradual paths are a feature of undominated equilibria, a parameter region in which i) is preferred by h can only imply that such a change is not feasible. This is the case when both U-paths and gradual paths are degenerate and therefore coincide, that is, when τ = 0. 2 18

δ2 1 λ

λ

0

1

δ1

Figure 3: When δl < (1 − λ)δh — the darker colored parts of Region I —, then optimal QGE entail high types jumping immediately to the investment level of 1 in the second period.

5.2

Separation by Cooperation

We have already seen that for the low type to separate by a cooperative action cannot be an equilibrium for the linear contribution game. Consequently, for this case the Region I characterization is complete. In this subsection we consider the continuous trust game with fi (ai , aj ) = (1 − λ) ai for i = 1, 2. We shall investigate C−equilibria, i.e., equilibria in which separation occurs by diﬀerent levels of cooperation. It will be shown that they too resemble gradual cooperation paths. 5.2.1

Characterization

We start by noting an important property of C−equilibria which holds independently of eﬃciency: Lemma 12 In every C−equilibrium separation occurs in period 0. Furthermore in every C−equilibrium the l-type’s payoﬀ is uC l = 1 − λ.

(17) 2

Proof Assume separation occurs in T > 0. Since the l type separates by a cooperative action aTl > 0 and his continuation payoﬀ is 0, it must be that aTl = 1. Pooling in T − 1 implies alT −1 = ahT −1 = aT −1 . In an equilibrium it must hold that l has no incentive to play Q or 1 in period T − 1. This yields two conditions: (1 − λ)aT −1 + δl (1 − λ) ≥ aT −1 , (1 − λ)aT −1 + δl (1 − λ) ≥ 1 − λ. Both conditions together can only be satisﬁed at the same time if δl (1−λ) ≥ aT −1 ≥ 1 − δl . λ These inequalities imply δl ≥ λ which contradicts our assumption that λ > δl . As with respect to the second point, it follows straightforwardly that if a cooperative action is a best response in period 0 for the low type, and his continuation payoﬀ is zero, then he will cooperate at the maximum level of 1. His expected payoﬀ is therefore 1 − λ. Note that this contrasts Q−equilibria, for which equilibria with pooling phases exist but happen to be ineﬃcient. 19

Before proceeding with the characterization (and existence) of optimal C−equilibria we point out a few striking properties of equilibrium payoﬀs compared to the complete information case. Remark 3 Unsurprisingly, as in the case of Q−equilibria, the low type’s C−equilibrium I payoﬀ exceeds that of the complete information case, i.e., uC l > ul = 0. However, in contrast to Q−equilibria, a low type not only achieves a positive payoﬀ when matched with a high type, but also in a matching with another low type. Furthermore, again in contrast to Q−equilibria, a high type’s payoﬀ against a low type opponent is also higher than for complete information (although his payoﬀ against high types, as we will see, turns out to be lower). We will show and discuss later, that there exist parameters, such that in C−equilibria both high and low types do strictly better than in the corresponding complete information case. Straightforward intuition about the eﬀects of the introduction of incomplete information can be subsumed as follows. High types have an interest in discovering their opponent’s type. Therefore, they have to pay a rent to low types to extract this information. This eﬀect, which clearly shows up in the case of Q−equilibria, no longer holds for C−equilibria. The structure of the stage-game allows for information revelation in a way which is beneﬁcial to all types. 2 We now deﬁne C−gradual equilibria, of which we will show that they dominate other C−equilibria. Deﬁnition 6 A C-gradual-equilibrium — denoted by CGE(a0 , a1 ) — is a Bayesian Nash equilibrium in which the outcome path of high types corresponds to a gradual cooperation path α(a0 , a1 ), while low types play 1 in the ﬁrst period and quit thereafter (high types meeting low types also quit in the second period). 2 Theorem 2 Every interim eﬃcient C−equilibrium is a C−gradual-equilibrium.

2

Proof The theorem follows if we can show that, to be eﬃcient, the path of play of high types from period 1 onwards must resemble a gradual cooperation path. Given that the low type already obtains 1 − λ in equilibrium the only potentially binding deviation constraint for the low type is that he imitates the high type for a number of periods and then quits (for the case that he met a h-type opponent in 0). Denote such a constraint CIClt where t is the period he intends to quit when he meets a high type (implicitly assuming that he quits in 1 after having met another l-type).25 We ﬁrst note that the equivalent to Lemma 9 also holds for eﬃcient C−equilibria, i.e., in any undominated C−equilibrium there exists τ such that in the event of a hh match, both players invest the maximal amount of 1 in all periods after τ , i.e., othh = (1, 1) for all t > τ . Therefore , it remains to show that in an undominated equilibrium CIClt are binding for all t = 2, . . . , τ . Clearly CIClτ must be binding otherwise we could raise aτh , thereby increasing h’s expected payoﬀ. We distinguish between two parameter regions (known from Proposition 5) i) δl ≥ (1 − λ)δh and ii) δl < (1 − λ)δh . 25

When we say that CIClt is binding we mean that the low type’s 1 − λ equilibrium payoﬀ exactly equals the expected payoﬀ obtained by the low type when he imitates the high type for t − 1 periods and quits in t.

20

Case i): δl ≥ (1 − λ)δh . Assume that there exists a period t , 1 < t < τ such that CIClt does not bind. Assume furthermore, without loss of generality, that CIClt +1 binds. Raising ath and simultaneously lowering ath +1 , while ensuring

∆ath = −

δl ∆ath +1 , (1 − λ)

(18)

keeps CIClt +1 binding (i.e., does not induce the low type to deviate from his equilibrium behavior). Such a change is preferred by the h-type if:

(1 − λ)∆ath + δh (1 − λ)∆aht +1 ≥ 0.

(19)

which holds if δl ≥ (1 − λ)δh which is true by assumption. Case ii): δl < (1 − λ)δh . For this case we will take a slightly diﬀerent track. We ˜1 ), for which we argue that it deﬁne an equilibrium E˜ := CGE(˜ a0 , a dominates any other p−λ pδl 0 equilibrium — denoted by E. We set a ˜ = min p , 1 − (1−λ) and a˜1 = 1 (which implies that in E˜ hh matches always play (1, 1) except in period 0). For E˜ to be an equilibrium it must be that the low type has neither an incentive a) to quit in period 0, nor b) to imitate the high type in period 0 and then quit: 1 − λ ≥ p˜ a0 + 1 − p

⇔

1 − λ ≥ (1 − λ)˜ a0 + pδl

⇔

p−λ , p pδl a˜0 ≤ 1 − . 1−λ

a˜0 ≤

(20a) (20b)

˜ fulﬁlls these requirements, while equivalent conditions have to Note that by deﬁnition E ˜ will dominate any E for which a0 < a˜0 . As, due to (20a), hold E as well. Clearly E pδl a0 > a ˜0 cannot hold when a ˜0 = p−λ , the only case we need to check is a0 > a ˜0 = 1 − 1−λ , p ˜1 = 1. The deviation condition corresponding to (20b) for E is: while a1 < a (1 − λ) ≥ (1 − λ)a0 + pδl a1 ⇔ a0 ≤ 1 − If E were preferred to E˜ then: 0 (1 − λ) ah − 1 −

pδl (1 − λ)

pδl 1 a. 1−λ

(21)

− pδh (1 −

a1h )

> 0,

(22)

which, together with (21), reduces to

δl − δh p(1 − a ) 1−λ 1

> 0.

(23)

This, however, contradicts our assumption δl < (1 − λ)δh , which completes the proof. 5.2.2

Existence

In our deﬁnition of C−gradual equilibrium there remains a degree of freedom in the choice of a0 and a1 . The above conditions bound a0 , a1 and depending on parameter values C−gradual equilibrium — and therefore also C−equilibria — may not exist. However, when they exist, eﬃcient C−gradual equilibrium are generically unique. 21

Proposition 6 Eﬃcient C−gradual equilibrium (amongst the set of C− equilibria) are generically unique. 2 The proof can be found on page 37 in the Appendix. It therefore trivially follows: Corollary 1 Restricting attention to C−equilibria, the Pareto frontier in ul , uh space is a unique point. 2 The parameter values for which C−equilibria exist are those for which all non deviation conditions (42) through (46) given in the proof of Proposition 6 hold.

5.3

Eﬃcient Region I Equilibria

Having analyzed C−separation and Q−separation separately, we now combine both types. Combining Theorems 1 and 2 we obtain: Corollary 2 All region I equilibria which entail positive investment in some period by some type are weakly Pareto dominated (with respect to interim eﬃciency) by either a C−gradual-equilibrium or a Q−gradual-equilibrium. 2 The next proposition shows that, indeed, both types of equilibria are relevant in describing the Pareto frontier in Region I. Proposition 7 The Pareto frontier of Region I equilibria contains both C−gradualequilibrium and Q−gradual-equilibrium. 2 Proof The simplest proof is to note that for large enough Q−gradual-equilibrium no longer exist. C−gradual-equilibrium are then trivially better. On the other hand C−gradual-equilibrium do not exist for λ > p, which is not necessarily an impediment to the existence of Q−gradual-equilibrium. In Remark 3 on page 20 we commented on the existence of equilibria which dominate eﬃcient complete information equilibria: Proposition 8 There exist parameters such that an incomplete information equilibrium strictly dominates the eﬃcient complete information equilibrium with respect to interim eﬃciency. 2 1 19 Proof Let λ = 18 , p = 34 , δl = 20 , δh = 100 . The eﬃcient C−gradual-equilibrium entails 5 0 1 a = 6 , a = 1. Then, recalling (4) and (5), we obtain δh 1 C I uh − uh = (1 − λ) a + p − 1 − δh 1 − δh = (1 − λ)(a − p) 7 = > 0, 96

and I uC l − ul = 1 − λ =

22

7 > 0. 8

Remark 4 The central properties of the C−gradual-equilibrium underlying the previous proof where already discussed in Remark 3. While an l-type’s complete information payoﬀ is 0, he obtains a positive payoﬀ in a C−gradual-equilibrium both when matched with an l-type opponent or an h-type. This is also true for the h-type. However, compared with the complete information case the h-type faces a trade-oﬀ: a) against an l-opponent he earns more (cooperation at level a0 for one period), while b) against an h-opponent he has to wait longer until cooperation at the highest level of 1. As the example in the proof demonstrates, for some parameters this trade-oﬀ turns in favor of incomplete information. 2 Remark 5 As mentioned in the introduction, it may seem somewhat unconvincing that in both Q−gradual-equilibrium and C−gradual-equilibrium high type matches insist on gradually raising investment levels, although they are sure from period 1 onwards that their respective opponent is also a high type. If the focus is on eﬃciency, would not renegotiation in periods 1 and later break our equilibria? This is not the case. Indeed we constructed our equilibria by setting all oﬀ-equilibrium behavior to Q and having beliefs consistent with behavior (i.e., probability 1 on high types for later nodes), which may give rise to renegotiation concerns. However, even if we were to explicitly model renegotiation procedures, nothing would hinder us from setting beliefs such that all renegotiation attempts would be interpreted as coming from low types. Such type switches are not precluded by the speciﬁcation of Bayesian or sequential equilibrium, and the support restrictions of stronger reﬁnements like perfect sequential equilibrium are highly contentious.26 Furthermore, even alluding to heuristically motivated reﬁnements like the intuitive criterion would be to no avail. If any renegotiation would lead to a faster rise of investment than in our equilibria, low types would also have the strict incentive to pose as a high type and propose such renegotiation. Therefore, beliefs that renegotiation attempts stem from low types would be perfectly reasonable (and all renegotiation attempts would fail). 2

6

Region II

The remaining terra incognita of our parameter space is Region II. We learned in Lemmas 6,7 of Section 3 that in the complete information case these parameter constellations allow not only cooperation in hh-matches, but also in mixed hl-pairings. As in region I we expect that there exist equilibria where after separation behavior converges in ﬁnite time τ to the corresponding eﬃcient complete information behavior. Clearly, in that case separation has to be C−separation. The adjustment process after separation in period T in such a symmetric equilibrium is described by the 3 investment levels hh, hl, lh in each period T + 1, . . . , T + τ after separation, where hh and hl denote the investment paths of h after meeting another h or l and lh correspondingly denotes l’s investment path after meeting an h. Remember that l meeting another l cannot cooperate in region II and therefore must quit the game immediately after separation. Since we have 3 adjustment paths after separation (instead of one in region I) which are potentially interrelated by various equilibrium non-deviation conditions region II tends to be more complicated and less intuitive. Nevertheless, the salient feature of starting small or gradualism of eﬃcient outcome paths survives in a generalized way as is formulated precisely in the following deﬁnitions. 26

For a discussion of such issues see N¨ oldeke and van Damme (1990).

23

Deﬁnition 7 A gradual path or CAE-path27 α(τ, a∗ ) := (αhh , αll , αhl , αlh ) for region II with adjustment time τ is the 4-tuple of investment paths given by αhh := a0h , a1hh , a2hh , . . . , aτhh , 1, 1, . . . , αhl := a0h , a1hl , a2hl , . . . , aτhl , a∗hl , a∗hl , . . . , αlh := a0l , a1lh , a2lh , . . . , aτlh , a∗lh , a∗lh , . . . , αll := a0l , Q . As before we call t = 0 the Communication Phase described by C−separating investments a0h = a0l . The Adjustment Phase of length τ are the periods t = 1, . . . , τ described by ((a1hh , a1hl , a1lh ) , . . . , (aτhh , aτhl , aτlh )). Remaining periods t = τ + 1, τ + 2, . . . are called the Established Phase. Investments in the Established phase abbreviated by a∗ := (1, a∗hl , a∗lh ) are those deﬁned in Lemmas 6,7 of Section 3, i.e. by eﬃcient play in the corresponding complete information game28 . An equilibrium with a gradual or CAE-outcome path α(τ, a∗ ) := (αhh , αll , αhl , αlh ) is called CII −gradual equilibrium for region II with adjustment time τ . 2 The CII in the terminology CII −gradual equilibrium indicates the separation mode again by cooperation on diﬀerent levels and distinguishes these equilibria from the C−gradual equilibria and the Q−gradual equilibria which also may exist in region II but – as we will see– tend to be dominated by CII −gradual equilibria with their extended cooperation possibilities for mixed hl−matchings. The central result of this section is the following theorem stating that gradualism29 is also eﬃcient in region II. Theorem 3 Every interim eﬃcient equilibrium that is not a Q−gradual or a C−gradual equilibrium in region II resembles a CII −gradual equilibrium starting from the separation period. 2 Note that this is a weaker claim with respect to pooling phases compared to region I. We cannot exclude the existence of a pooling phase in region II as we did in region I. But with proposition 2 in section 4 we know that for every CII −gradual equilibrium with pooling phase there always exists an interim eﬃcient CII −gradual equilibrium without pooling phase that is strictly preferred by l. Since we know from proposition 1 in section 4 that separation occurs in ﬁnite time it remains to show for the proof that ﬁnitely many periods after separation eﬃcient behavior converges to the eﬃcient stationary behavior of the corresponding complete information game. This is done in the following lemma30 . Lemma 13 In any eﬃcient region II equilibrium that is not a Q−equilibrium or a C−equilibrium as in region I there exists τ such that in hh−matchings both players invest 1 and in all hl−matchings the l−type invests a∗lh and the h−type invests a∗hl in all periods after τ , i.e. othh = (1, 1) and othl = (a∗hl , a∗lh ) for all t > τ where a∗hl , a∗lh are those investment levels deﬁned by stationary eﬃcient play in complete information given in Lemmas 6,7 of Section 3. 2 27

CAE = Communication Phase, Adjustment Phase and Established Phase. Remember that while in the continuous trust game this was unique in the contribution game we have a continuum of such eﬃcient complete information equilibria. For this deﬁnition pick any of these and keep it ﬁxed within the Established Phase. 29 We interpret gradualism in region II in a less narrow sense than in region I because our deﬁnition of a region II gradual path is less explicit about the details of the corresponding Adjustment Phase. 30 Remember that lemma 9 for region 1 stated a similar result. 28

24

Proof Assume separation occurs in period T . Clearly, ll−matchings cannot cooperate after separation. For the hh−outcome path of an eﬃcient equilibrium there exists some τhh such that othh = (1, 1) for all t > τhh for the same reason as in region I (see proof of lemma 9). It remains to show that also the mixed outcome path at some point hits the ”upper bound” deﬁned by the eﬃcient stationary investment levels for the complete information case. First, note that #ohl = #ohh = ∞ since otherwise there must be a last period where someone invests after separation (which cannot be an equilibrium). An outcome path ohl with othl = (a∗hl , a∗lh ) for some t > T but not thereafter cannot be outcome path of an eﬃcient equilibrium since any h−investment above a∗hl or l−investment above a∗lh violates the equilibrium condition that nobody wants to steal and raising investment levels to (a∗hl , a∗lh ) after period t would raise equilibrium payoﬀs. The only remaining possibility would be that there is an eﬃcient equilibrium outcome path where either h or l do never hit their upper bound. The non-deviation conditions binding l’s path from above is that l imitating h should have no incentive to steal l’s investment for any t ≥ T . This condition must be binding inﬁnitely many times since otherwise equilibrium payoﬀs could be improved by raising investment after the last period where this condition was binding. Let us assume the condition binds in periods t1 , t2 with t2 > t1 > T . Denote the relevant segment of the low type path in the mixed matching by aTl , aTl +1 , . . . . This implies t2 −t1 λ t2 al ≥ atl 1 . δl But, as investment is bounded by 1 there must be a last such period which is a contradiction. Hence there must be period τl after which l invests a∗lh in the mixed matching. A similar argument deﬁnes τh for the h−path in the mixed matching. Now deﬁne τ = max {τhh , τl , τh } which completes the proof. We call τ ”eﬃcient adjustment time” because in an eﬃcient equilibrium in τ periods after separation players behave as eﬃcient players in the full information game – i.e. they play the stationary eﬃcient equilibrium. So far we did not provide a description of an eﬃcient Adjustment Phase. In order to further characterize region II-equilibrium outcome paths recall the set of relevant (potentially binding) equilibrium non-deviation conditions31 . • LQT:

l cannot improve by playing Q in period T .

• LQt˜:

l cannot improve by playing Q in periods T + 1 < t˜ ≤ τ .

l cannot improve by imitating h and play Q in periods T + 1 ≤ th , tl ≤ • ILQth tl : τ after meeting an h or l, respectively. • IH∞: h does not want to imitate l forever. Note that this condition together with δh > δl implies ul (ατ ) < uh (ατ ). • HQ0:

h cannot improve by playing Q in period 0 is dominanted by LQ0 since pa0h + (1 − p) a0l ≤ ul (ατ ) < uh (ατ )

h cannot improve and play Q in periods T + 1 ≤ tˆh , tˆl ≤ τ after • HQtˆh , tˆl : meeting an h or l, respectively. 31

Conditions for t > τ are satisﬁed by construction of τ .

25

• IHQtˆ:

h cannot improve by imitating l and play Q in periods T + 1 ≤ tˆ ≤ τ .

• All previous conditions with the deviation to play C (to invest 1) instead of playing Q. As we know from region I all these conditions are only relevant for the trust game and for existence since they deﬁne lower bounds for investments. As before, they have no role in the characterization of eﬃcient equilibrium outcome paths if these exist. Generally this rather large number of potentially binding non-deviation conditions exactly deﬁnes eﬃcient behavior in the Adjustment Phase. In order not to get lost between numerous case distinctions we proceed to describe eﬃcient behavior in region II geometrically. First, we provide a general description and second, to get a better intuition we study some special cases. Let a(τ, a∗ ) = (a0h , a0l , a1hh , a1hl , a1lh , . . . , aτhh , aτhl , aτlh ) ∈ A (τ, a∗ ) ⊂ [0, 1]2+3τ where A (τ, a∗ ) = {a(τ, a∗ ) |all non-deviation conditions are satisﬁed } ⊂ [0, 1]2+3τ is deﬁned as the set of investment parameters in the Communication and Adjustment Phases such that all non-deviation conditions are satisﬁed. CII −gradual equilibrium payoﬀs are given by

τ τ +1 δ uh (a(τ, a∗ )) = p (1 − λ) a0h + δht athh + h + 1 − δh t=1

τ τ +1 ∗ ∗ δ (θahl + ηalh ) δht θathl + ηatlh + h , + (1 − p) θa0h + ηa0l + 1 − δh t=1

τ τ +1 ∗ ∗ δ (θa + ηa ) lh hl δlt θatlh + ηathl + l + ul (a(τ, a∗ )) = p θa0l + ηa0h + 1 − δ l t=1 + (1 − p) (1 − λ) a0l ,

where θ = −λ, η = 1 for the linear contribution game and θ = 1 − λ, η = 0 for the continuous trust game. Both payoﬀs contain the two expessions reﬂecting continuation payoﬀs starting from the separation period conditional on meeting an h−type (with probability p) or an l−type (with probability 1 − p). For all matchings exept ll payoﬀs further reﬂect the three phases CAE of the relationship. Note that these payoﬀs are linear in all investment parameters. Hence, since all non-deviation conditions are linear weak inequalities A (τ ) is a closed convex linear 2 + 3τ −dimensional polygon and a subset of [0, 1]2+3τ ⊂ R2+3τ . Therefore the Pareto-frontier as in region I consists of subsets of the boundaries of these polygons A (τ ) for τ = 0, 1, . . .. More precisely, let Aˆ = A (τ, a∗ ) τ =0,1,...

then deﬁne P FII =

∗ ˆ there is no b ∈ A with un (b) > un (a(τ, a )) . a(τ, a∗ ) ∈ Aˆ for some type n = h, l and ≥ for both n = h, l

Clearly, P FII = ∅ if there exists some CII −equilibrium. While Q−gradual equilibria and C−gradual equilibria as deﬁned for region I do not take advantage of of hl−cooperation between high/low types, they still remain equilibria in Region II: 26

Lemma 14 Both QGE and CGE remain equilibria in Region II, if they exist.

2

Proof The non-deviation conditions for QGCE and CGCE do not depend on the Region II parameter restriction. Of course CII −equilibria are only eﬃcient if they are not dominated by some Q−gradual equilibrium or C−gradual equilibrium. In the remainder of this section we study the special case τ = 0 where players separate and then immediately switch to the eﬃcient complete information stationary mode of behavior. This case is of particular interest for two reasons. First, whenever this case exists CII −equilibria are eﬃcient and therefore substantial part of the general structure. Second, since this case can be described by just two parameters a0l , a0h we can use it for a graphical analysis which is helpful for a better understanding of region II in general. To avoid redundancies we will perform this anylysis of special cases only for the continuous trust game. Analogous considerations hold for the linear contribution game. Deﬁnition 8 A rapid enhancement equilibrium for Region II in the continuous trust game — parameterized by (al , ah ) — is an equilibrium in which an h-type’s actions are given by: (ah , 1, 1, . . .) when meeting an h-type, αh = 1−λ 1−λ , , . . .) when meeting an l-type, (ah , 1−δ l 1−δl while an l-type’s actions are given by: (al , 1, 1, . . .) when meeting an h-type, αl = when meeting an l-type. (al , Q)

2

Proposition 9 There exist parameters in Region II such that a rapid enhancement equilibrium exists and is eﬃcient. 2 Proof We show this by giving a numerical example. Let δh = .9, δl = .3, λ = .4 be in 9 and al = 1. Following 3 Region II and p = .4. As initial investment levels we set ah = 35 critical non-deviation-conditions have to be are satisﬁed: 1. LQ0 : l does not want to quit in t = 0: pδl ≥ pah + (1 − p)al (1 − λ) al + 1 − δl 2. LIQ1,1 : l does not like to pretend to be h and then quit in t = 1 for both opponent’s types: pδl δl (1 − λ) ≥ p ((1 − λ) ah + δl ) + (1 − p) (1 − λ) al + (1 − λ) al + 1 − δl (1 − δl ) 3. HIQ∞,∞ has no incentive to imitate l forever: (1 − p) (1 − λ) δh pδh ≥ (1 − λ) al + p+ (1 − λ) ah + 1 − δh (1 − δl ) 1 − δh 27

It is easy to check that all other non-deviation-conditions are either trivially satisﬁed or dominated by these. For example, h does not like to quit in t = 0 by combination of the ﬁrst and third conditions. An easy calculation shows that indeed for δh = .9, δl = 9 .3, λ = .4, p = .4, ah = 35 , al = 1 all conditions are satisﬁed. In order to show that this CII -equilibrium-outcome-path is eﬃcient we need to show that it is not dominated by a Q- or a C-equilibrium nor by another CII -equilibrium. First, with respect to the Q-equilibrium just check that for the given parameters δh (1 − p) (1 − λ) pδh CII > uQ uh = (1 − λ) ah + p+ > h. 1 − δh (1 − δl ) 1 − δh pδl (1−λ) II = uC . Second any C-equilibrium is always worse for l since uC l = 1 − λ < 1 − λ + 1−δl l Third, observe that for the given parameter values condition 1 is binding and does not depend on al since λ = p. Therefore al = 1 is maximal and a higher ah cannot be an equilibrium. Therefore the given outcome path is supported by an eﬃcient equilibrium which completes the proof.

As in the previous section, it is interesting to compare eﬃcient incomplete information equilibria with the complete information benchmark. In contrast to Region I we can no longer expect interim dominance of incomplete information equilibria in Region II. To see this, note that in an eﬃcient equilibrium in the complete information setting of the continuous trust game an h-type will cooperate at the highest possible level with both l-type and h-type opponents. Adding incomplete information can therefore only be detrimental to h-type. Nevertheless, low types can gain more as they loose by the introduction of incomplete information. This is due to the fact that in a C II -equilibrium separation works by cooperation on diﬀerent levels. Therefore, pairs of l-types can cooperate before and during the separation, which is not possible in the complete information case. Depending on parameters, one can show that this eﬀect can oﬀset the losses of h-types in the ex-ante sense. Proposition 10 There exist parameters in region II such that an eﬃcient C II -equilibrium strictly dominates the eﬃcient complete information equilibrium with respect to ex-ante eﬃciency. 2 Proof Consider the same numerical example δh = .9, δl = .3, λ = .4, p = .4, ah = 9 , al = 1 as in the proof of the previous Proposition 9. We calculate the according 35 equilibrium utility levels: pδl C II ul = (1 − λ) al + = 0.702857, 1 − δl δh (1 − p)(1 − λ) CII p+ uh = (1 − λ) ah + = 5.09143. 1 − δh (1 − δl ) Similarly for the eﬃcient complete information utilities we obtain: uII l = 0.342857, uII h = 5.48571.

28

II

II

C As we expect uC > uII < uII l (pairs of low types can cooperate one period) and uh h l (under complete information h cooperates with both types at the maximal level). Finally we check

UC U

II

II

II

= puC + (1 − p)uC h l =

puII h

+ (1 −

p)uII l

II

= 2.45829,

= 2.4,

II

and hence U C − U II = 0.0582857 > 0, which completes the proof.

For which parameters λ, p, δl , δh in region II exist CII −equilibria with τ = 0? A suﬃcient condition is A (0, a∗ ) = a0l , a0h |all non-deviation conditions are satisﬁed = ∅ It is easy to see that only the corresponding non-deviation conditions pδl 0 0 0 LQ0 : pah + (1 − p) al ≤ (1 − λ) al + 1 − δl p − λ 0 (1 − λ) δl al + , ⇔ a0h ≤ p 1 − δl pδl 0 0 ILQ11 : (1 − λ) ah + δl ≤ (1 − λ) al + 1 − δl pδ δ l l , − ⇔ a0h ≤ a0l + 1 − δl 1 − λ pδh pδh (1 − p) δh 1 − λ 0 0 + IH∞ : (1 − λ) al + ≤ (1 − λ) ah + 1 − δh 1 − δh 1 − δh 1 − δl (1 − p) δh 1 − λ ⇔ a0h ≥ a0l − 1 − δh 1 − δl are potentially binding. Looking at these conditions we obtain two cases: 1. If p ≥ λ then P FII ⊂ A (0, a∗ ) is the unique point p (1 − λ) [δl (1 − δh ) + δh (1 − p)] 0 al = min 1, λ (1 − δl ) (1 − δh ) (1−λ)[(p−λ)δh (1−p)+pδl (1−δh )] , λ(1−δl )(1−δh ) a0h = min (1−δl )(1−δh )−δh (1−p)(1−λ) (1−δl )(1−δh )

A possible constellation is illustrated in ﬁgure 1. 2. If p < λ then P FII (0) is a set of points which is the upper boundary of the set A (0, a∗ ). An example is illustrated in ﬁgure 2. This pattern extends to the other cases where τ > 0 in dimension 2 + 3τ . In other words, the Pareto frontier consists of subsets of the ”upper boundary” of the linear polygons A (τ, a∗ ) for τ = 0, 1, . . .which can be be unique points or a set of points depending on exogenous parameters λ, p, δl , δh in region II.

29

0

0

ah

ah 1

1

Q IL

LQ

11

0

PFII (0) 11 IH

4

A (0,a*) IH

4

A (0,a*)

IL Q

LQ 0

PFII (0)

0

1

1

0

0

al

(a) Case p ≥ λ

0

al

(b) Case p < λ

Figure 4: In ﬁgure (a) for p ≥ λ the Pareto frontier P FII (0) is a point. In ﬁgure (b) for p < λ the Pareto frontier P FII (0) is a set.

7

Related Literature

Numerous authors before and after Schelling wrote in a verbal style about trust and its necessity for doing business. Some of them even emphazised the worth of performing a rigorous analysis. To keep this section short, however, we only refer to closely related research with formal models or experiments and interpretations close to our approach. Gradual increases of stakes in partnership situations have been modeled in the context of random-matching games32 by Datta (1996) as well as Ghosh and Ray (1996) (see also Kranton (1996)). In their models, players from a large population are matched to play a potentially long term game (similar to ours). Players also have the option of terminating the relationship, in which case they drop back into the pool of unmatched players where they await a new matching. Due to the assumptions of large populations and the absence of information regarding player’s behavior in past matchings, contagion equilibria as in Kandori (1992) do not work. Instead, both papers show that a gradual increase of stakes can serve as an alternative disciplining device (for high types) by lowering the attractiveness of defecting and starting a new relationship.33 Datta works in a complete information setting (i.e., there are only high types) with an exogenous probability that matchings break up. Ghosh/Ray assume an incomplete information setting. Their low types players, however, are assumed to be myopic and therefore always quit in the ﬁrst period. Ghosh/Ray also apply a bilateral rationality constraint, which is similar in effect to our focusing on eﬃcient equilibria. Another paper which combines search with 32

The study of random-matching games was initiated by Rosenthal (1972). Kandori (1992), amongst others, extends the Folk Theorem to this context. 33 This is in contrast to our model, where the gradual buildup of investment serves as a disciplining device for the low types to keep them from posing as high types.

30

incomplete information (albeit one-sided) to generate gradualism is Rauch and Watson (2003). Lockwood and Thomas (2002) show in a complete information version of a repeated game with a general contribution game as a stage game, that the assumption of irreversibility of investments (in the sense that investments can only rise over time) also leads to gradualism. Sobel (1985) presents a loan model which is similar in ﬂavor to our setting in that stakes should change over time. The unique equilibrium has the lender starting oﬀ with small loans. The principle theoretical diﬀerence to our work is the one-sidedness of incomplete information. Closest to our work is a series of papers by Watson (1996; 1999; 2002) the ﬁrst of which contributed to inspire the writing of this paper. Watson’s ﬁrst article has been subsequently subdivided into the two latter articles. These are updated, generalized but substantially altered versions of the results in the ﬁrst unpublished article, which was in various respects closest to our framework. Watson’s work considers a game in which players at each instant of time only choose whether to continue or to quit. The players agree in advance upon a level function which describes the investment level as a function of time. Watson focuses both on cases in which players can commit to level functions or cases which are selected by a renegotiation condition. While we work in discrete time, Watson’s model is in continuous time. Watson is more general in his latter two papers by allowing for asymmetric probabilities to be a good type (including the symmetric case and the one sided incomplete information case). Furthermore, Watson considers mixed strategies (in his game strategies are simply stopping rules given by a probability distribution for quitting over time). We only consider symmetric type probabilities and only pure strategies. There are, however, two fundamental diﬀerences between our results versus Watson’s. Firstly, in Watson’s setup a gradual cooperation equilibrium regime exists even for very pessimistic players (i.e., low prior probabilities for being a good type). This is not so in our framework, as for very low starting values a high type (in a high-high matching) has an incentive to deviate to a high level after separation has taken place (which is not possible in Watson’s game, where deviations are restricted to quitting). Hence we obtain parameter cases where only the noncooperative equilibrium exists (everybody quits immediately). Secondly, in our framework asymmetric cooperation (players invest at two diﬀering levels) is supportable by an equilibrium (even with complete information). This generates a richer structure of eﬃcient equilibria than in Watson’s case. Conversely, Watson’s more general stage game and the possibility of asymmetric type probabilities generate other results which are beyond the scope of our current paper. Also germane is the literature on repeated games with incomplete information as presented for example in the seminal book by Aumann et al. (1995). Their book contains a systematic investigation of the nature of information revelation for repeated games with incomplete information. While most of the book is on repeated zero-sum games, the last chapter turns to non-zero sum games with one sided incomplete information. We are not aware of results (existence or characterization) on this level of generality for repeated non-zero sum games with two-sided asymmetric information. Since building and maintaining long-term relationships is an important issue if the players are governments of countries or institutions our framework could also be usefully applied on the impact of trust and reputation on investment levels in macro models. Noteworthy examples of this literature include Zak and Knack (2001), Marcet and Marimon 31

(1992) or Khan and Ravikumar (2002). Most of the experimental literature on trust and reciprocity as for example Berg et al. (1995), Smith (1998) or Houser and Kurzban (2002) deals with one-shot extensive-form games and concentrates on other issues as the explanatory power of self-interest versus altruism. We are not aware, however, of experimental studies within the narrow context of our article — i.e. repeated interaction with variable stakes and double-sided uncertainty.

8

Conclusions

We have presented a model of a repeated game between two players with double sided incomplete information regarding discount factors. Players invest variable stakes in a joint project. We interpret the stakes as levels of trust since this is what can be stolen and at the same time this measures the size of the damage if the partner steals. We explicitly allow for situations where these levels can diﬀer endogenously within a partnership. It turns out that this possibility to diﬀerentiate is a very eﬀective communication tool. For the benchmark case of complete information we partition the parameter space of discount factors into two non-trivial Regions in which either only high-high matches can potentially cooperate (Region I), or only low-low matches cannot cooperate (Region II). We characterize the structure and payoﬀs of interim and ex-ante eﬃcient incomplete information equilibria. Eﬃcient behavior in all such equilibria follows a typical pattern. First, in the Communication Phase players reveal their type. Then, in the Adjustment Phase players adjust the level of the relationship according to the relevant equilibrium non-deviation conditions which depend on the Region. In the Established Phase players behave as they do without mutual uncertainty. The Communication Phase can diﬀer in the way in which low types reveal themselves. On the one hand one obtains equilibria in which low types steal in the ﬁrst period, which can be interpreted as obtaining a rent from the high types in order to reveal information. On the other hand one obtains equilibria in which low types separate by playing a high investment level in the ﬁrst period and quit thereafter (in Region I) or continue only with high types (in Region II). Surprisingly, it can be shown that these equilibria may even dominate eﬃcient equilibria in a setting of complete information (irrespective of type). These eﬀects do not crop up in the immediately related literature, because it is usually either assumed that players cannot choose diﬀerent levels of investment endogenously in any period, or it is assumed that quitting is always the unique best response in the stage game (which need not be the case even if quit-quit is the unique stage game equilibrium).

Appendix Proof (of Proposition 1) Assume a pooling equilibrium with more than one period of play. Denote the actions of both types by at . We distinguish pooling with i) ﬁnite strictly positive cooperation, and ii) inﬁnite strictly positive cooperation. In the case of i) there must be a last pooling period where players invest strictly positively. This, however, cannot be an equilibrium since players can improve by playing Q. It remains to show that ii) implies a contradiction. We now have an inﬁnite pooling equilibrium (with strictly positive cooperation for an inﬁnite number of periods). Let a = sup at which is ﬁnite by the boundedness of the set of feasible investments and 32

strictly positive. In an equilibrium it must hold for the low type in any period t that continuing to cooperate entails higher payoﬀs than to quit, i.e., (1 − λ)at + δl V ≥ at , where V is the continuation payoﬀ of the low type from t + 1 onwards. V is bounded 1−λ a . Inserting this bound and rearranging we obtain: above by 1−δ l a a (24) δl t − λ + λδl 1 − t ≥ 0. a a By deﬁnition of a we can ﬁnd at arbitrarily close to a in which case a /at is arbitrarily close to 1 and the bracketed term can be made arbitrarily small. Then it suﬃces to note that δl < λ to see that there exists a period t such that (24) cannot hold. Proof (of Proposition 2) Assume without loss of generality that all oﬀ-equilibrium behavior of E is Q. Deﬁne E˜ as a copy of E except that all players skip period T − 1. By construction E˜ is an equilibrium too. Let Vlt (resp. V˜lt ) be the expected continuation ˜ equilibrium. As both equilibria induce the same payoﬀ of the l-type in the E (resp. E) behavior in periods 0, . . . , T − 2, the claim then follows if: V˜lT −1 − VlT −1 > 0.

(25)

Note that by construction V˜lT −1 = VlT and VlT −1 = (1 − λ)aT −1 + δl VlT . Substituting these equations into (25) and rearranging we obtain: (1 − δl )VlT > (1 − λ)aT −1 .

(26)

As E is an equilibrium, aT −1 must be such that a low type does not prefer stealing aT −1 to cooperation at level aT −1 plus the discounted continuation payoﬀ of period T , i.e. aT −1 ≤ δλl V T . Substituting this into (26) we obtain λ > δl , which holds by assumption. Proof (of Lemma 9) Assume separation occurs in period T . We ﬁrst note that #ohh = ∞. If this were not the case, i.e., if there is a period t > T in which ath = Q, then we and increasing subsequent could improve h’s equilibrium payoﬀs by setting ath = δλl at−1 h λ periods’ investments by δl until hitting 1 (and investment 1 thereafter). Therefore, let #ohh = ∞ and assume conversely that there is an eﬃcient equilibrium such that ath < 1 inﬁnitely often, i.e. for an inﬁnite set J ∗ := t ∈ {0, 1, 2, ...} ath < 1 of periods: #J ∗ = ∞. Then it must be the case that for inﬁnitely many of these periods t ∈ J ∗∗ ⊂ J ∗ (with #J ∗∗ = ∞) the l is indiﬀerent between (i) his equilibrium payoﬀ, and (ii) imitating h up to period t − 1 and playing Q in t . If this were not the case h’s payoﬀ could be strictly increased by raising ath in some period after the last such period. For any two such periods t , t ∈ J ∗∗ with t > t > T and for which an h imitating l-type is indiﬀerent between quitting in t or t , it must hold that: t −t λ t ah ≥ ath . (27) δl However, the investment level is bounded above by 1. Since δλl > 1 there must be a last such period which contradicts #J ∗∗ = ∞. This concludes the proof since there must be a last period τ where aτh < 1. 33

Proof (of Lemma 10) Separation takes place in period T . Following Lemma 9 there exists a ﬁrst period τ such that ath = 1 for all t > τ . Furthermore the l-type is indiﬀerent between his equilibrium payoﬀ (cooperating until T − 1 and quitting in T ) and imitating the h-type’s play and quitting in τ . We deﬁne a new equilibrium E˜ as a copy of E with the following changes: in period T to τ the h-type plays a gradual cooperation path such that a ˜τh = aτh . Note that this implies that a ˜th ≥ ath for all t ∈ [T, τ ]. We now have to ˜ As E and E˜ can diﬀer at most in periods prove that payoﬀs are weakly higher in E. between including)T and τ − 1, we restrict attention to subset of periods. Let τ(and τthis −1 t t ˜ τ −1 t t τ −1 t t ˜ −1 t t Vl := t=T δl ah , Vl := t=T δl a˜h , Vh := t=T δh ah , Vh := t=T δh a˜h . By construction, the l-type’s indiﬀerence implies: paTh = p ((1 − λ)Vl + δlτ aτh ) − (1 − p)(1 + )aTh , T τ τ ˜ p˜ ah = p (1 − λ)Vl + δl ah − (1 − p)(1 + )˜ aTh .

(28a) (28b)

˜τh , rearranging leads to: Subtracting (28a) from (28b) and noting that aτh = a a ˜Th − aTh =

p(1 − λ) V˜l − Vl . p + (1 − p)(1 + )

(29)

The high types payoﬀs from periods T, . . . , τ − 1 are given by: Uh = p(1 − λ)Vh − (1 − p)(1 + )aTh , aTh . U˜h = p(1 − λ)V˜h − (1 − p)(1 + )˜

(30a) (30b)

Subtracting (30a) from (30b), inserting (29) and dividing by p(1 − λ) we obtain: (1 − p)(1 + ) ˜ U˜h − Uh ˜ = Vh − Vh − Vl − Vl . p(1 − λ) p + (1 − p)(1 + )

(31)

We note that this expression is strictly positive due to following observations: The V ’s are are strictly monotonic in the discount factors, therefore, as a ˜th ≥ ath , it must hold that V˜h − Vh > V˜l − Vl > 0. Furthermore the fraction in the RHS of (31) is strictly below 1. ˜ (given that E is an The only deviation condition which could be violated in E ˜l > U˜h . Simple calculation shows equilibrium, which implies that Uh > Ul ) is that U ˜l − Ul . U˜h − Uh ≥ U Proof (of Lemma 11) Let E be an equilibrium in which separation takes place in ˜ which is strictly preferred by h and weakly T > 0. We will construct an equilibrium E preferred by l. Deﬁne E˜ equivalently to E except that the low type quits in period T − 1 instead of T and the high type invests a˜hT −1 and a ˜Th . Noting that ahT −1 = alT −1 , we deﬁne uh := (1 − λ) ahT −1 + pδh aTh − (1 − p)(1 + )δh aTh , (32a) ul := (1 − λ)ahT −1 + pδl aTh ,

(32b)

T −1 ahT −1 , ˜h + δh a ˜Th − (1 − p)(1 + )˜ u˜h := (1 − λ)p a

(33a)

and

u˜l :=

p˜ahT −1 .

(33b)

34

˜ We next set a˜hT −1 := δλl aTh and a˜Th := γ δλl a ˜hT −1 = γaTh . The non-deviation conditions of E for these parameter values are very similar to those of E but one period earlier (replace τ by τ − 1 and divide by δl or δh ). As E is an equilibrium, stealing in T − 1 is no strict improvement for the l-type, i.e., T −1 ah ≤ (1 − λ)ahT −1 + δl paTh , or: δl ahT −1 ≤ p aTh . (34) λ Inserting (34) plus the previous deﬁnitions into (32a), (32b), (33a), (33b) and rearranging we obtain: u˜h − uh (1 − p)(1 + )(δh − δl ) ≥ > 0, T ah λ u˜l − ul ≥ 0. aTh

Proof (of Proposition 3) To be able to treat the linear contribution game and the continuous trust game simultaneously we will use the notation fi (ai , aj ) = θai + ηaj . This implies θ = −λ, η = 1 for the linear contribution game and θ = 1 − λ, η = 0 for the continuous trust game. Existence of Q-gradual equilibrium is governed by whether the non-deviation conditions can be simultaneously fulﬁlled. We state the relevant conditions and start oﬀ the low type: • l cannot improve by imitating h and playing Q in period t > 0: As already noted in our discussion of gradual paths all such deviations are by construction weakly dominated by equilibrium behavior. • l cannot improve by investing 1 in period 0: pa ≥ p(θ + ηa) − (1 − p)(1 + ) ⇔ a ≥

θ 1−p − (1 + ). 1 − η p (1 − η)

• l cannot improve by imitating h and playing 1 in period 1: λ pa ≥ p (1 − λ)a + δl (θ + η γa) − (1 − p)(1 + )a ⇔ δl pθδl a≥ . p λ − ηγ + (1 − p)(1 + ) δl

(35)

(36) (37)

• l cannot improve by imitating h and playing 1 in period t > 1: one can show that if l wants to neither i) imitate h and play Q in period 1 (the ﬁrst point above), nor ii) imitate h and play 1 in period 1 (inequality (36)) then he does not want to imitate h and deviate to 1 in some later period. As we will be using this result later, we relegate the details of this claim to Lemma 15 later in this Appendix, page 37. We proceed with the high type: • h cannot improve by quitting in period 0: QICh0 (a) := uQ h (a) − pa ≥ 0.

(38)

Note, that this implies, that the h-type’s expected payoﬀ is always higher than an l-type’s. 35

• h cannot improve by quitting in a later period: After separation, the h-type’s continuation payoﬀs from cooperation are always strictly higher than feasible Qpayoﬀs.34 • h cannot improve by playing 1 in period 0: This is implied by combining (35) and (38). • h cannot improve by investing 1 in period 1: QICh1 (a) := uQ h (a) − p(θ + ηa) + (1 − p)(1 + ) ≥ 0.

(39)

h cannot improve by quitting in period 0: QICh0 (a) := uQ h (a) − pa ≥ 0. The monotonicity of continuation payoﬀs after period 1 guarantees that the corresponding non-deviation conditions for later periods do not bind. We now note that (35), (36) and (39) all bound a from below. We denote the maximum of these expressions with a. Now we show that QICh0 (a) is a continuous, piecewise linear function in a. Fur¯ thermore QICh0 (a) is weakly concave, attaining a strictly positive maximum on any set a ∈ [0, x] where x > 0. To see this note ﬁrst is that t max t ∈ {0, 1, ...} | γλ a λ < 1 for γλ a < 1, δl δl δl τa = τ (a1 ) = 0 for γλ a = 1. δl is a weakly decreasing, strictly positive step function with a countable number of jumps and lim+ a→0 τa = ∞. Payoﬀ uh (a) is therefore trivially linear for those a for which τa is continuous. For those a for which τa is discontinuous it suﬃces to verify that uh (a) is continuous. This proves the ﬁrst statement. To prove the second statement we ﬁrst note that QICh0 (0) = 0 and proceed in two steps: 1) δl = λδh and 2) δl = λδh . 1) δl = λδh : In this case, the LHS of condition (38) reduces to: δhτa +1 p(1 − λ) a + γaτa + − (1 − p)(1 + )a − pa. 1 − δh which we diﬀerentiate with respect to a (for those a where τa is constant) to obtain: p(1 − λ) + γτa − (1 − p)(1 + ) − p. + 0 Recalling that lim+ a→0 τa = ∞ we see that lima→0 ∂QICh (a)/∂a = ∞. For similar reasons the slope of successive linear segments of QICh0 (a) decreases in a. 2) δl = λδh : we rewrite the LHS of condition (38) as follows: τ δh λ δhτa +1 δh λ a 0 QICh (a) = p(1 − λ) a + aγ + 1− δl − δh λ δl 1 − δh − (1 − p)(1 + )a − pa. 34

This follows by construction of the Q−gradual equilibrium. Type l is indiﬀerent between playing the equilibrium and imitating h up to some t ≤ τ . Since h is strictly more patient he strictly prefers not to quit.

36

Diﬀerentiating with respect to a (for those a where τa is constant), substituting γ by its deﬁnition and rearranging we obtain: ∂QICh0 (a) = p(1 − λ) − (1 − p)(1 + ) − p ∂a τ δh λ a (pλ + (1 − p)(1 + ))δh 1− + p(δl − δh λ) δl

(40)

We analyze two subcases 2i) and 2ii) in turn: 2i) δl < λδh : As lim+ a→0 τa = ∞ the 0 ∂QIC (a)/∂a = ∞. For similar term on the second line of (40) goes to +∞, i.e. lim+ a→0 h reasons the slope of successive linear segments of uh (a) decreases in a. 2ii) δl > λδh : the expression with the τa exponent now converges to 0, and (40) can be written as follows: δh (pλ + (1 − p)(1 + )) −1 . δl − δh λ 0 This term is positive (as δh > δl ), therefore lim+ a→0 ∂QICh (a)/∂a > 0. As before, the slope of successive linear segments of uh (a) decreases in a. Combining these properties, it follows that QICh0 (a) is positive in a neighborhood of a = 0, and furthermore weakly concave. This implies that QICh0 (a) attains a strictly positive maximum on a ∈ [0, x] for any x > 0. Consequently uQ h (a) is a continuous, piecewise linear function which is diﬀerentiable almost everywhere. Furthermore, uQ h (a) is weakly concave, attaining a strictly positive maximum on any set a ∈ [0, x] where x > 0. This concludes the proof.

Lemma 15 Consider Region I and an equilibrium in which some gradual path is played amongst high types and separation takes place in period 0. If l wants neither i) to imitate h and play Q in 1, nor ii) to imitate h and play 1 in period 1, then he does not want to imitate h and deviate to 1 in some period t > 1. 2 Proof Assume without loss of generality that θ > 0 and l imitates h in 0 and he meets a high type opponent. The continuation payoﬀ of this branch is bounded below by θ + ηa1 which l gets when he plays 1. Could it be that he 1 to later periods? delays playing 1 1 1λ 1 l) This would be the case if θ + ηa ≤ (θ + η)a + δl θ + ηa δl or a ≥ θ(1−δ . However, θ+ηλ

l) is strictly larger than θ + ηa1 straightforeward calculation shows that to steal a1 ≥ θ(1−δ θ+ηλ since δl < λ. Therefore, if Q is not a best response in period 1, then to invest 1 in period 1 is better than doing so later on.

Proof (of Proposition 6) We proceed with a more detailed analysis of the nature of C-gradual equilibrium. An h-type’s payoﬀ is given by: τ (a1 ) τ (a1 )+1 δh λ t δ a0 + pa1 . uC +p h (41) h = (1 − λ) δ 1 − δ l h t=1 By the constructive nature of the proof of Theorem 2 it follows that if C-separating equilibria exist, then C-gradual equilibrium exist too. In any C-gradual equilibrium, the following non-deviation conditions must hold:

37

• l cannot improve with Q in period 0: 1 − λ ≥ pa0 + 1 − p ⇔ a0 ≤

p−λ . p

(42)

Note that this is an upper bound for a0 . It also implies that p ≥ λ. • l cannot improve by imitating h and playing Q in period 1: 1 − λ ≥ (1 − λ)a0 + pδl a1 ⇔ a0 ≤ 1 −

pδl 1 a. 1−λ

(43)

Recall that, by construction, corresponding non-deviation conditions must hold for all periods later than 1. • l cannot improve by imitating h and playing 1 in period 1: 1 − λ ≥ (1 − λ)(a0 + pδl ) ⇔ a0 ≤ 1 − pδl .

(44)

As in the case of Q-gradual equilibrium, the corresponding non-deviation conditions also do not bind for later periods (see again previous Lemma 15). • h cannot improve by playing 1 in period 0: uC h ≥ 1 − λ.

(45)

Note that together with (42) this implies that h does not want to play Q in period 0. • h cannot improve by playing 1 in period 1: 1 )+1 τ (a1 ) τ (a t δh λ δ . δh (1 − λ) ≤ (1 − λ) a1 + h δ 1 − δ l h t=1

(46)

Again as in the case of Q-gradual equilibrium the monotonicity of continuation payoﬀs after period 1 ensures that corresponding non-deviation conditions for later periods do not bind (both for Q-deviations as well as deviations to 1). Both (42) and (44) bound a0 from above. Call this bound a ¯0 . (46) bounds a1 from below, denote a1 . Therefore a0 ∈ [0, ¯a] and a1 ∈ [a1 , 1]. In an eﬃcient equilibrium — ¯ ¯ 0 1 0 1 given that uC l is independent of a and a — we wish to maximize a and a . Therefore, 0 1 either i) (43) holds with equality, or ii) a and a hit a boundary. In case of ii) the proposition trivially holds. In the case of i) we maximize uC h subject to equality of (43). given by equation (41) we obtain: Substituting into the deﬁnition of uC h t τ (a1 ) τ (a1 )+1 δh δh λ 1 pδl +p , (47) a + (1 − λ)p (1 − λ) 1 − 1−λ δ 1 − δ l h t=1 which we maximize with respect to a1 (keeping the relevant constraints in mind). Depending on whether the expression in the large brackets is strictly positive or negative (which is generically the case) we set a1 = 1 or a1 = a1 — with a0 determined by (43).35 ¯ 35

Obviously one then has to check, given optimal a0 , a1 values, that (45) holds.

38

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