- Email: [email protected]

⇤

† ˆ ODILON CAMARA

Marshall School of Business University of Southern California

June 3, 2014

Abstract In a symmetric information voting model, an individual (information controller) can influence voters’ choices by designing the information content of a public signal. We characterize the controller’s optimal signal. With a non-unanimous voting rule, she exploits voters’ heterogeneity by designing a signal with realizations targeting di↵erent winning-coalitions. Consequently, under simple-majority voting rule, a majority of voters might be strictly worse o↵ due to the controller’s influence. We characterize voters’ preferences over electoral rules, and provide conditions for a majority of voters to prefer a supermajority (or unanimity) voting rule, in order to induce the controller to supply a more informative signal. JEL classification: D72, D83. Keywords: Information control, persuasion, voting.

⇤

We thank the following audiences for their comments: Caltech, University of Southern California, Uni-

versity of Warwick, 2014 SWET Conference, 2014 MPSA Conference, Political Economy Workshop at the University of Bristol, and the Princeton Conference on Political Economy, in particular Dan Bernhardt and the discussants Tiberiu Dragu, Navin Kartik and Daniel Seidmann. †

USC FBE Dept, 3670 Trousdale Parkway Ste. 308, BRI-308 MC-0804, Los Angeles, CA 90089-

0804. [email protected] and [email protected]

1

Introduction Uncertainty gives rise to persuasion. — Anthony Downs (1957) Information is the cornerstone of democracy, as it allows voters to make better choices.

In many important cases, however, uninformed voters are not free to launch their own investigations, and must rely on the inquiries of others. For example, in most trials a juror may not choose which tests are performed during the investigation, or which questions are asked to a witness — jurors must rely on the prosecutor’s investigation and questions. In politics, the Legislative branch often must rely on information generated by investigative reports produced by the Executive. In firms, shareholders and the Board of Directors typically depend on reports commissioned by the CEO. If the individual choosing the questions and the voters have di↵erent preferences, then she may strategically design her investigation to persuade voters to choose her preferred alternative. In this paper we study how an individual (the “information controller”) influences the decision of voters by strategically designing a public signal, that is, by engaging in information control (e.g., Brocas and Carrillo 2007, Duggan and Martinelli 2011, Kamenica and Gentzkow 2011). We first study how the voting rule and the distribution of voters’ preferences a↵ect the controller’s choice of a signal. We show that with a non-unanimous voting rule and heterogenous voters, the controller can design a signal with realizations targeting di↵erent winning coalitions. That is, the signal exploits preference disagreement across voters. Consequently, under simple-majority, a majority of voters might be strictly worse o↵ due to the information supplied by the controller. To prevent this negative impact, voters may adopt a supermajority voting rule that induces the controller to supply a more informative signal. In our model, a group of uninformed voters must choose whether to keep the status quo (or default) policy, or to implement a proposed new policy. This can be interpreted as voters choosing between an incumbent politician and a challenger, shareholders choosing to approve or not a merger, members of a jury choosing between a guilty or not guilty verdict, or members of a legislature choosing to approve or not a new law. Prior to the election, 1

the information controller can sway voters’ decision by designing what voters can learn from a public signal, i.e. by specifying the statistical relation of the signal with the underlying state.1 After observing the signal and its realization, voters apply Bayes’ rule and reach a common posterior belief. They then choose an action (vote) and the electoral rule dictates the electoral outcome. We focus on the case of pure-persuasion, where the information controller wants to maximize the probability of approval of the proposal, independently of the state (in Section 5.2 we consider a controller with state-dependent payo↵s). We study two classes of institutional rules. The first is delegation, and serves as a benchmark. Under delegation, one member of the group acts as a dictator and chooses his preferred policy given his beliefs. The second is the class of k-voting rules, where a proposal replaces the status quo if it receives k or more votes. We focus on k -voting rules because they are important and prevalent in practice, and because they allow us to derive sharp equilibrium characterizations and comparisons. Since the controller’s signal is public and voters have no private information, there is no information aggregation problem. Hence, the strategic voting considerations related to the probability of being pivotal are absent in our model. We start by characterizing the optimal signal when the approval decision is delegated to voter i, who absent further information would reject the proposal. The controller maximizes the probability of approval by designing a signal such that, whenever voter i approves the proposal, he is just indi↵erent between approval and rejection. This implies that voter i does not benefit from this signal when he is the decision maker. In fact, voter i would prefer to delegate the approval decision to a “tougher” voter j to induce the controller to supply a more informative signal (where voter j is “tougher” than i if, for every belief, approval by j implies approval by i). We then characterize the optimal signal under a k-voting rule. We establish that a kvoting rule is payo↵-equivalent to delegating the decision to a particular “weak representative voter.” This result allows us to relate the optimal signal under the k-voting rule to that under delegation. Given a k voting rule, do voters benefit from the controller’s signal? We show 1

In our basic setup we consider an information controller who has no private information. In Section 5.1

we consider a controller who learns the state before choosing the signal.

2

that under a simple majority rule, the controller’s influence always makes a majority of voters weakly worse o↵. We then define conditions such that a majority of voters is strictly worse o↵. This happens when the controller’s optimal signal targets di↵erent winning coalitions, that is, when the controller exploits voters’ preference disagreement to increase the probability of approving the proposal. Interestingly, this can happen even when all voters would agree on their decision if they knew the true state. Anticipating the controller’s influence, which k-voting rule do voters prefer? Voting rules a↵ect outcomes not only by the amount of information voters have of the proposal’s relative merits, but also by the consensus required to approve the proposal. In particular, requiring a higher consensus (higher k-voting rule) may lead to an excessive rejection of the proposal. However, it may also induce the controller to provide a more informative signal. We study this trade-o↵ between control and information by posing two questions: (i) under delegation, what are each voter’s preferences over decision makers?, and (ii) when decisions are made collectively, how do preferences over decisions makers translate into preferences over k-voting rules? Consider a set of potential decision makers and a voter who can choose to whom to delegate the approval decision. The decision makers are totally ordered according to their “toughness”, and rank states in the same order as the voter, where “higher” states imply that the proposal delivers a higher net payo↵. For any such set, we show that the voter has single-peaked preferences over decision makers. Consider now a k-voting rule and an electorate who shares the same ranking of states. We extend the previous result to show that each voter has single-peaked preferences over k-voting rules. Moreover, a majority of voters always prefers a supermajority rule over simple majority. Finally, if voters also agree under full information, then every voter prefers unanimity over any other k-voting rule. That is, even heterogenous voters may agree on the optimal electoral rule. Our paper is related to the recent literature on information control. In Brocas and Carrillo (2007), a leader without private information sways the decision of a follower in her favor by deciding the timing at which a decision must be made. As information arrives sequentially, choosing the timing of the decision is equivalent to shaping (in a particular way) the information available to the follower. Duggan and Martinelli (2011) consider one media outlet that 3

can a↵ect electoral outcomes by choosing the “slant” of its news reports. The media in Duggan and Martinelli is constrained in the set of signals it can design (it must be a “slant”). In contrast, we consider an information controller who is unconstrained in her choice of signal. Our paper is most closely related to Kamenica and Gentzkow (2011) (KG henceforth). They develop the fundamental methodology to solve this class of unconstrained information control problems, when players have common priors. Alonso and Cˆamara (2014a) study information control when players have di↵erent prior beliefs. As in our paper, Michaeli (2014), Taneva (2014) and Wang (2013) focus on information control when there are multiple receivers.2 Our paper also relates to the broad literature on how institutional rules endogenously a↵ect the information available to voters. Following the work of Austen-Smith and Banks (1996) and Feddersen and Pesendorfer (1996, 1997, 1998), a large literature has focused on how voting rules a↵ect information aggregation via the strategic behavior of privatelyinformed voters. The literature has also studied the e↵ects of voting rules on the information available to voters when voters can deliberate (e.g., Austen-Smith and Feddersen 2005, and Gerardi and Yariv 2007), when voters can postpone a decision in order to wait for more information (e.g., Messner and Polborn 2012 and Lizzeri and Yariv 2013), and when voters can acquire costly information (e.g., Li 2001, Persico 2004, Martinelli 2006, and Gerardi and Yariv 2008). Recent papers focus on how voting rules a↵ect information provision by privately-informed experts. Jackson and Tan (2013) consider experts who can reveal verifiable information, while Schnakenberg (2014a,b) considers cheap-talk. In these papers experts are endowed with private information about the state, while in our paper the controller chooses the informational content of a public signal. Moreover, in Section 5.1 we contrast a 2

There are other recent papers that study the strategic design of a public signal. Gill and Sgroi (2008,

2012) consider a privately-informed principal who can subject herself to a test designed to provide public information about her type, and can optimally choose the test’s difficulty. Li and Li (2013) study a privatelyinformed candidate who can choose the accuracy of a costly public signal (campaign) about the qualifications of the politicians competing for office. Rayo and Segal (2010) study optimal advertising when a company can design how to reveal the attributes of its product, but it cannot distort this information. Kolotilin (2014) focus on how the receiver’s private information a↵ects the sender’s choice of a signal. In a somewhat di↵erent setting, Ivanov (2010) studies the benefit to a principal of limiting the information available to a privately informed agent when they both engage in strategic communication.

4

controller without private information and a controller who privately knows the state. The rest of the paper is organized as follows. Section 2 presents the basic model. Section 3 solves for the controller’s optimal signal. Section 4 studies voters’ preferences over institutional rules. Section 5 extends the basic model. Section 6 applies our results to relevant voting models. Section 7 concludes. All proofs are in the Appendix.

2 2.1

The Model General Setup

Policy and Decision Makers:

A group of n voters must choose one alternative from

a binary policy set X = {x0 , x1 }, where x0 is the status quo (or default) policy, and x1 is the proposal. This can be interpreted as voters choosing between an incumbent politician and a challenger, choosing to approve or not a ballot measure, members of a jury choosing between a guilty or not guilty verdict, or members of a legislature choosing to approve or not a new law. The collective decision is made following established institutional rules, which we discuss momentarily. Each voter i 2 I ⌘ {1, . . . , n} has preferences over policies that are characterized by a continuous von Neumann-Morgenstern utility function ui (x, ✓), ui : X ⇥ ⇥ ! R, with ⇥ a finite state space. State ✓ 2 ⇥ captures the realized value of payo↵ relevant variables, such as the relative competence of di↵erent politicians, or the productivity of di↵erent sectors of the economy. All players share a common prior belief p = {p✓ }✓2⇥ , which has full support in ⇥. Information Controller: One information controller C, who is not a member of the group, has preferences over policies characterized by a continuous von Neumann-Morgenstern utility function uC . We focus on the case of pure-persuasion were the controller’s preferences are independent of the realized state, uC (x) : X ! R (in Section 5.2 we consider a controller with state-dependent payo↵s). The controller can influence the decision of the group by designing a public signal that is correlated with the state. Before the group selects a policy, the controller chooses a signal ⇡, consisting of a finite realization space S and a family of likelihood functions over S, {⇡(·|✓)}✓2⇥ , with ⇡(·|✓) 2 5

(S). Signal ⇡ is “commonly

understood”: ⇡ is observed by all players who agree on the likelihood functions ⇡(·|✓), ✓ 2 ⇥ (see Alonso and Cˆamara 2014a for a discussion of this assumption). Players process information according to Bayes rule. Let q(s|⇡, p) be the updated posterior belief3 of every voter after observing ⇡ and its realization s. We make two important assumptions regarding the set of signals available to the controller. First, she can choose any signal that is correlated with the state. Thus, our setup provides an upper bound on the controller’s benefit from information control in settings with more restricted spaces of signals. In particular, he controller will not engage in designing a signal when she faces additional constraints if there is no value of information control in our unrestricted setup. Second, signals are costless to the controller. This is not a serious limitation if each signal is equally costly, and would not a↵ect the choice of signal if the controller decides to influence voters. However, the optimal signal may change if di↵erent signals have di↵erent costs. Gentzkow and Kamenica (2013) o↵er an initial exploration of persuasion with costly signals, where the cost of a signal is given by the expected relative entropy of the beliefs that it induces.

2.2

Institutional Rules

After observing the realization of the controller’s signal, the group chooses one policy x 2 X. The institutional rules governing the collective decision process are summarized by a mechanism

=(

1, . . . ,

outcome function h :

1

n , h),

⇥...⇥

which defines a strategy set n

for each member i and an

! X. Given belief q, mechanism

{ui }i2I define a Bayesian game G. Let

⇤

and utility functions

(q) ⌘ { i⇤ (q)}i2I be a Perfect Bayesian equilibrium

strategy profile played in this game. Together g(q) :

i

and

⇤

(q) implement a social choice function

(⇥) ! X, which defines the group’s equilibrium policy choice as a function of beliefs.

Therefore, for any signal ⇡ and realization s 2 S that yields belief q, the controller’s payo↵ is given by v(q) = uC (g(q)). 3

(1)

We use “posterior belief” to indicate the players’ belief about the state after the signal realization but

before voting. To simplify notation, we use q(s) or q as a shorthand for q(s|⇡, p).

6

Our main goal is to study how di↵erent institutional rules a↵ect the optimal choice of a signal and the equilibrium payo↵ of players. We focus on two classes of institutional rules: delegation, which serves as a benchmark, and k-voting rules, where a proposal replaces the status quo if it receives k or more votes. We focus on k-voting rules because they are important and prevalent in practice, and because they allow us to derive sharp equilibrium characterizations and comparisons. We now formally define these institutional rules. Delegation: Decision rights are fully delegated to a particular player d 2 I. Mechanism ={

1, . . . ,

n , h}

has

d

= X, where individual d chooses a policy

is implemented, h( 1 (q), . . . ,

n (q))

=

d (q)

and this policy

d (q).

In equilibrium, player d acts as a dictator and P chooses x 2 X that maximizes his expected payo↵, d⇤ (q) 2 arg maxx2X ⇥ q✓ ud (x, ✓). If there are multiple optimal policies, we assume he chooses the one preferred by the information controller. Delegation implements g(q) =

⇤ d (q),

and (1) becomes v(q) = uC ( d⇤ (q)). It follows

from Berge’s maximum theorem that v is upper-semicontinuous. k-voting rule: Proposal x1 is selected if and only if it receives at least k votes, where k 2 {1, . . . , n} is the established electoral rule. Mechanism where

i (q)

= 1 represents voting for proposal x1 , and

={ i (q)

we abstract from abstention. The outcome function h is 8 < x if P 1 i2I h( 1 (q), . . . , n (q)) = : x if P 0

i2I

1, . . . ,

n , h}

has

i

= {0, 1},

= 0 represents voting for x0 —

i (q)

k,

i (q)

< k.

Given a belief q, we apply the following two equilibrium selection criteria in case of multiple equilibria: 1. If policy x yields voter i a strictly higher expected payo↵ than policy x0 , then he votes for x; 2. If the two policies yield voter i the same expected payo↵, then he votes for the policy preferred by the information controller. The first criterion rules out uninteresting equilibria such as, when k < n, all voters vote for the status quo independently of expected payo↵s. Importantly, in our model voters have 7

no private information about the state, so there is no information aggregation problem. Hence, the strategic voting considerations related to the probability of being pivotal are not relevant in our setup. From the set of equilibria satisfying the first criterion, we select the subset of controller-preferred equibria, which guarantees that the controller’s expected payo↵ v is an upper semicontinous function of posterior beliefs. Let

⇤ i (q)

be the equilibrium

choice of voter i that satisfies the previous selection criteria. The social choice function is then g(q) = h( 1⇤ (q), . . . ,

⇤ n (q)).

As in Alonso and Cˆamara (2014a), we focus on language-invariant Perfect Bayesian equilibrium: a Perfect Bayesian equilibrium in which individual decisions depend on posterior beliefs, but not on the actual signal or realization — for every signals ⇡ and ⇡ 0 , and signal realizations s and s0 for which individual i has the same posterior belief q, he chooses the same equilibrium strategy

⇤ i (q).

Note that if game G has multiple equilibria, then the social

choice function g implicitly selects which equilibrium is played.

2.3

Information Controller’s Problem

For any signal ⇡ and realization s 2 S that yields posterior q, the social choice function g determines the implemented policy — the controller’s payo↵ v(q) is then defined by (1). The information controller selects a signal that maximizes E⇡ [v(q)]. Upper-semicontinuity of v both with delegation and with k-voting rules ensures the existence of an optimal signal (see KG). Moreover, choosing an optimal signal is equivalent to choosing a probability distribution over q, subject to the constraint E [q] = p. That is, V

= max E [v(q)],

s.t. E [q] = p.

For an arbitrary real-valued function f define fe as the concave closure of f , fe(q) = sup {w|(q, w) 2 co(f )} ,

where co(f ) is the convex hull of the graph of f . The following remarks follow immediately from KG: (R1) An optimal signal exists; (R2) If the approval decision is delegated to one voter, then there exists an optimal signal 8

with card(S) 2. With a k-voting rule, there exists an optimal signal with card(S) min{ (n

n! k)!k!

+ 1, card(⇥)};

(R3) The information controller’s expected utility under an optimal signal is V = ve(p);

(R4) The value of information control is V

v(p) = v˜(p)

(2) v(p).

For the remaining of the paper we focus on the case where uC (x0 ) < uC (x1 )4 . Without loss of generality, set uC (x1 ) = 1 and uC (x0 ) = 0. Therefore, the controller’s expected payo↵ V is simply the equilibrium approval probability under an optimal signal.

2.4

Definitions and Notation

We next present a series of definitions and notation that will be useful in our analysis. Notational Conventions: For vectors q, w 2 RJ , we denote by hq, wi the standard inner P product in RJ , i.e. hq, wi = Jj=1 qj wj , and we denote by qw the component-wise product of vectors q and w, i.e. (qw)j = qj wj .

Voter’s Type: Define the conditional net payo↵ for voter i when the state is ✓ as i ✓

⌘ ui (x1 , ✓)

ui (x0 , ✓).

= { ✓i }✓2⇥ captures the preferences of the voter, and we call i the type of voter P i. When voter i holds belief q, he votes for x1 if and only if ⇥ q✓ (ui (x1 , ✓) ui (x0 , ✓)) 0, The vector

i

that is, if and only if hq, i i i

and q,

0. Hence, equilibrium voting strategies

⇤ i

are fully defined by

8 < 1 if hq, i i 0, ⇤ i i (q) ⌘ a(q, ) = : 0 if hq, i i < 0.

Since a voter’s type defines his voting behavior, we use the term “voter ” to refer to a voter with type . 4

The remaining case uC (x0 ) > uC (x1 ) is equivalent to a proposal x ˆ1 = x0 and a status quo x ˆ0 = x1 , with

the corresponding relabeling of the collective decision process and social choice function.

9

Relevant Sets — Individual Voter: Consider a voter with type . Define the set of approval states D( ) = {✓ 2 ⇥|

✓

0} and the set of rejection states DC ( ) = ⇥ \ D( ).

Define the set of approval beliefs A( ) = {q 2 beliefs AC ( ) =

(⇥)| hq, i

(⇥) \ A( ). Under full information, voter

0} and the set of rejection approves x1 if and only if

✓ 2 D( ); while under uncertainty he approves x1 if and only if q 2 A( ). Finally, define the set of strong rejection beliefs R( ) = {q 2

(⇥)|✓ 2 D( ) ) q✓ = 0}, that is, the set of

beliefs that assign probability zero to every approval state. Relevant Sets — Electorate: Consider an electorate { 1 , . . . , Define the win set Wk = {q 2

(⇥)|

n X

a(q, i )

n

} and a k-voting rule.

k}.

i=1

That is, voters implement x1 if and only if q 2 Wk . Given the k-voting rule, there are n! (n k)!(k!)

possible minimal winning coalitions of k voters. The win set is then the union of

all possible minimal winning coalitions. Under unanimity rule k = n, the win set is the intersection of all approval sets, Wn = \i2I A( i ). If k = 1, then the win set is the union of

all approval sets, W1 = [i2I A( i ). Note that Wn is convex, but Wk might be a non-convex set when k < n. Given the electorate, define B as the collection of all coalitions of at least n

k + 1 voters, with typical element b 2 B. Define the set of strong rejection beliefs Rk = [b2B (\

2b R(

)) .

That is, Rk is the set of beliefs such that there exists a “blocking” coalition b, with voters 2 b assigning probability zero to every approval state. Finally, we use V ( ) and V (Wk ) to denote the equilibrium approval probability with delegation to voter

and with a k-voting rule with win set Wk .

Classes of Voters’ Types: It is useful to group voters according to their types. To this end, let z be a permutation z : ⇥ ! {1, . . . , card(⇥)} that strictly orders the states. Define the class of types Fz = { 2 Rcard(⇥) |

✓

>

✓0

() z(✓) > z(✓0 )}.

That is, class Fz includes all voter types who (strictly) rank states according to the conditional net payo↵

✓

in the order defined by z. We say that voter 10

i

“ranks states” according

to z if

i

2 Fz .

Ordering Voters: We introduce two orders on the space of voter types. First, we say that voter

is “tougher” than voter

0

if A( ) ⇢ A( 0 ). Second, we say that voter

(weakly) “harder-to-persuade” than voter

0

is

if V ( ) V ( 0 ). That is, under the optimal

signal, the equilibrium approval probability under an optimal signal with delegation to voter is (weakly) lower than with delegation to

0

.

Representative Voter: Fix a k-voting rule and an electorate { 1 , . . . ,

n

}. Voter

is a

“representative voter” if A( ) = Wk , that is, the proposal is approved with a k-voting rule if and only if it would be approved with delegation to voter . Voter is a “weak representative voter” if V ( ) = V (Wk ), that is, if the information controller’s equilibrium expected payo↵ is the same when she only has to convince voter

and when she has to convince at least k

voters.

3

Information Control

In this section we first solve for the controller’s optimal signal both with delegation and with a k-voting rule. We then study how the controller’s gain from designing the signal varies with the electoral rule and the distribution of voters’ preferences.

3.1

Information Control with Delegation

Suppose that the approval decision is delegated to voter . Which signal would the controller optimally supply? If p 2 A( ), then the controller provides a completely uninformative signal, as the voter approves the proposal in the absence of additional information. Now suppose p2 / A( ) and A( ) 6= ;. The characterization of the optimal signal obtains from two observations. First, the controller would never benefit from providing additional information after the realization of an optimal signal. Therefore, after observing a signal realization that induces rejection voter must assign zero probability to every approval state ✓ 2 D( ) (see also Proposition 4 in KG), and thus his posterior belief must lie in the strong rejection set R( ). Second, any signal realization that leads to approval must induce a posterior belief in A( ). 11

While there is a multiplicity of optimal signals, under delegation one can always construct an optimal signal with only two signal realizations. To do so, starting with an arbitrary optimal signal ⇡ ⇤ one can group all approval signal realizations into a single realization associated to a posterior that is the probability-weighted convex combination of all approval posteriors. One can likewise obtain a single signal realization by grouping all rejection signal realizations in a similar fashion. Importantly, A( ) and R( ) are both convex sets, so the posterior belief obtained from combining all approval realizations still leads the voter to approve, and combining all rejection realizations still leads the voter to reject, so that the probability of approval remains the same. We now provide a geometric interpretation of the controller’s optimal signal. Consider a signal ⇡ supported on {s , s+ }, where s+ induces approval posterior q + 2 A( ) and s

induces strong rejection posterior q 2 R( ). Holding constant q , s+ becomes more likely as the posterior q + becomes closer to the prior p. Conversely, holding constant q + , s becomes less likely as q

is further away from p. Consequently, the controller would like to resort

to both an approval belief q + in A( ) that is closest to the prior, and a rejection belief q

in R( ) that is farthest from the prior. The martingale property of Bayesian updating

requires, however, that q + , q and p must all be collinear. The following lemma shows that the optimal signal balances these two goals as it corresponds to lines through the prior that maximize the ratio of the distances described in (3). Lemma 1 Consider delegation to voter , with p 2 / A( ) and A( ) 6= ;. Let ⇡ ⇤ be any controller’s optimal signal supported on {s , s+ }, where voter only if s = s+ , with l⇤ = q(s+ )

approves the proposal if and

p. Let dl (p, A( )) and dl (p, R( )) be the (Euclidean)

distances from the prior to the sets A( ) and R( ) along the line l. Then dl⇤ (p, R( )) dl (p, R( )) = max , l dl⇤ (p, A( )) dl (p, A( ))

(3)

and Pr[Approval] =

dl⇤ (p, R( )) . dl⇤ (p, R( )) + dl⇤ (p, A( ))

The next Proposition shows that the solution to (3) can be understood as the optimal choice of a cuto↵ state. 12

Proposition 1 Consider delegation to voter , with p 2 / A( ) and A( ) 6= ;. Let ⇡ ⇤ be any controller’s optimal signal supported on {s , s+ }, where voter

approves the proposal if and

only if s = s+ . Letting ↵✓ = Pr [s+ |✓] , there exists ✓0 2 ⇥ such that 8 < 0 X ✓ < ✓0 ↵✓ = , and ↵✓ p✓ ✓ = 0. : 1 > 0 ✓

✓

(4)

✓2⇥

never gains by making decisions with the signal ⇡ ⇤ , the controller’s P expected utility under ⇡ ⇤ is V ( ) = ✓2⇥ ↵✓ p✓ . Moreover, while voter

To understand (4), first consider the voter’s ideal signal. Voter

would like to know

whether an approval state occurred; thus his best signal would induce s+ if ✓ 0, and s P if ✓ < 0. With this signal, the approval probability is {✓: ✓ 0} p✓ , and the voter’s net P value from approval is {✓: ✓ 0} p✓ ✓ . If ✓ > 0 for at least one ✓, then the controller can increase the probability of approval by distorting this signal in a way that rejection states with a small incremental loss (i.e. small | ✓ | ) still induce approval. The controller can do so until the voter’s net value from approval is identically zero, as indicated by (4). This also implies that the voter gains nothing from making decisions with ⇡ ⇤ , as he is indi↵erent between approval and rejection after observing s+ . Example 1: Consider three states, ⇥ = {✓1 , ✓2 , ✓3 }, and a single voter ✓2

<0<

✓3 .

such that

✓1

<

Figure 1 depicts on the simplex the approval set A( ) and the strong rejection

set R( ), which is the bottom line segment. The prior p is such that p3

✓3

+ p2

✓2

< 0.

Figure 1(a) shows the approval posterior q + 2 A( ) that is closest to the prior, and the strong rejection posterior q

2 R( ) that is furthest from the prior. However, there is no

binary signal that induces these posteriors, since q + , p and q

must be collinear. Figure

1(b) shows the posterior beliefs induced by an optimal signal satisfying condition (3). Using (4), ✓2 is the cuto↵ state: P r[s+ |✓3 ] = 1, P r[s+ |✓2 ] = approval probability is then V ( ) = 1 · p3 +

3.2

p3 p2 |

✓3 ✓2 |

p3 p2 |

· p2 . ⇤

✓3 ✓2 |

, P r[s+ |✓1 ] = 0. The maximum

Information Control with a k-Voting Rule

A basic insight of information control under delegation is the existence of a binary optimal signal: one realization leading to approval, and the other leading to rejection. This is possible 13

θ3

θ3 Approval%Set%

Approval%Set%

A(δ )

q

A(δ )

q+

+

dl* ( p, A(δ )) dl* ( p, R(δ ))

Prior%Belief%

θ2

p

q−

Strong%Rejec5on%Set%

R(δ )

θ1 θ 2

Prior%Belief%

p

Strong%Rejec5on%Set%

(a) Unfeasible Signal

q−

R(δ )

θ1

(b) Optimal Signal

Figure 1: Optimal Signal for Example 1 as the set of approval beliefs of any voter is convex. However, with a k-voting rule the set of approval beliefs Wk is, in general, not convex. Nevertheless, persuading voters with approval beliefs Wk is payo↵-equivalent for the controller to persuading voters with approval beliefs equal to the convex hull of Wk . To see this, note that any belief in co(Wk ) can be expressed as a convex combination of posterior beliefs each of which ensures approval. Therefore, if q 2 co(Wk ), then there is a signal that ensures approval with certainty. Lemma 2 shows that the same logic of Lemma 1 holds with a k-voting rule, replacing A( ) and R( ) with the sets co(Wk ) and Rk . Lemma 2 Fix a k-voting rule and electorate { 1 , . . . ,

n

}. The probability of approval under

an optimal signal is Pr[Approval] =

dl⇤ (p, Rk ) dl⇤ (p, Rk ) dl (p, Rk ) , with = max . (5) l dl⇤ (p, Rk ) + dl⇤ (p, co(Wk )) dl⇤ (p, co(Wk )) dl (p, co(Wk ))

We now contrast equilibrium payo↵s with a k-voting rule to equilibrium payo↵s with delegation to a voter . Voter

is a representative voter if A( ) = Wk : in this case, the

expected utility of all players with the k-voting rule is equivalent to delegating the decision to voter .5 In many situations, however, a representative voter does not exit. Nevertheless, 5

A representative voter exists for each k voting rule if all voters in the electorate are totally ordered

according to toughness. This is the case, for example, if there are only two states and voters have the same ranking of states.

14

the next proposition establishes that one can always construct a weak representative voter. Proposition 2 Fix a k-voting rule and electorate { 1 , . . . , sentative voter ⇤ ✓ (k)

6=

⇤ ✓ 0 (k)

⇤

(k), such that {q : hq,

⇤

n

}. There exists a weak repre-

(k)i = 0} is a supporting hyperplane of co(Wk ). If

for ✓ 6= ✓0 , then the expected utility of all players under a k-voting rule is the

same as under delegation to voter Typically voter

⇤

⇤

(k).

(k) is not part of the electorate. As {q : hq,

⇤

(k)i = 0} is a supporting

hyperplane of Wk , then Wk ⇢ A ( ⇤ (k)). That is, a k-voting rule is equivalent, from the controller’s perspective, to delegation to a “less tough” decision maker. Notwithstanding, ⇤

(k) is also the hardest-to-persuade among all voters with approval beliefs containing Wk .

To see this, let G(Wk ) = { : Wk ⇢ A( )} be the set of voters who are “less tough” than the k voting rule. The proof of the Proposition shows that Pr[Approval] =

inf

2G(Wk )

Pr[Approval( )] = Pr[Approval( ⇤ (k))].

Moreover, if the weak representative voter

⇤

(k) strictly ranks states, then the ex-ante

expected utility of all voters is the same with the k-voting rule as with delegation to voter ⇤

(k). This reflects the fact that if

⇤ ✓ (k)

6=

⇤ ✓ 0 (k)

for ✓ 6= ✓0 , then the optimal signal in

Proposition 1 is unique and provides the same expected utility to all voters as the outcome of a k-voting rule. An important di↵erence between delegation and k-voting rules is that, except for unanimity, the controller can exploit conflict between voters to increase the chance of approving the proposal. This is always the case if the optimal binary signal for

⇤

(k) would not in-

duce approval under a k-voting rule, i.e. letting q + ( ⇤ (k)) be the approval posterior belief associated to (4), then q + ( ⇤ (k)) 2 co(Wk ) but q + ( ⇤ (k)) 2 / Wk . Letting Q+ (⇡ ⇤ ) be the set of approval posterior beliefs induced by an optimal signal, we must have hq, q 2 Q+ (⇡ ⇤ ). Therefore, the representative voter

⇤

⇤

(k)i = 0 for

(k) describes precisely the direction of

voters conflict: if an approval realization doesn’t secure the support of one winning coalition, it must secure the support of another. Essentially, the controller exploits the change in the identity of the winning coalition (and hence the identity of the pivotal voter) with the signal realization by tailoring the signal to persuade winning coalitions with opposing interests on the hyperplane hq,

⇤

(k)i = 0. 15

In many important applications, voters have di↵erent preferences but agree on the ranking of the states — for example, state ✓ represents the “quality” of the proposal, as in the public good example in Section 6.1. The next lemma characterizes the weak representative voter from Proposition 2 when all voters agree on the ranking of the states. Lemma 3 Consider an electorate { 1 , . . . , i

n

} with all n

2 voters on the same class,

2 Fz for some permutation z. Given a k-voting rule, there exists a weak representative

voter

⇤

(k) in the same class as the electorate,

By establishing that

⇤

⇤

(k) 2 Fz .

(k) must be in the same class as the electorate, Lemma 3 allows us

to relate the optimal signal with a k-voting rule to the optimal signal with delegation. With delegation to

⇤

(k), there is an optimal signal supported on {s , s+ }, with a cuto↵ state ✓0

as described in Proposition 1. As voters rank states in the same order as

⇤

(k), they all

agree that s+ is “good news” about the proposal, while s is “bad news”. Although signal s+ is enough to persuade

⇤

(k) to approve, it might not secure k votes from the electorate.

As the next example shows, the controller then decomposes s+ into realizations targeting di↵erent winning coalitions. Example 2:

Consider 3 states, ⇥ = {✓1 , ✓2 , ✓3 }, and two voters,

A

and

B

, where prior

beliefs and net values are as follows: State

Prior

A ✓

B ✓

✓3

0.2

+1

+1

✓2

0.1

-0.5

-1.5

✓1

0.7

-6

-2

Without further information voters strictly prefer to reject the proposal. First suppose that the voting decision is delegated to voter

A

. Figure 2(a) depicts the posterior beliefs

induced by an optimal signal ⇡A⇤ , using Proposition 1.6 The equilibrium probability of approval is 0.325. Signal ⇡A⇤ is also optimal if the decision is delegated to

B

. In both cases,

the controller’s influence does not change voters’ expected payo↵. 6

Formally, S = {s , s+ }, P r(s+ |✓3 ) = P r(s+ |✓2 ) = 1, and P r(s+ |✓1 ) =

1 28 .

The possible posterior beliefs

1 4 8 are q = (1, 0, 0) and q + = ( 13 , 13 , 13 ). Probability of approval is P r(s = s+ ) =

0.325.

16

1 28

⇥ 0.7 + 1 ⇥ 0.1 + 1 ⇥ 0.2 =

θ3 Approval%Set%

A(δ ) Voter% Voter%

δ

W1

δA

Voter%

θ2

Strong%Rejec1on%Set%

p

R(δ A )

(a) Decision Delegated to

A

or

B

A(δ )

q +B

δA

Voter% Voter%

q +A Prior%Belief%

q− θ1 θ 2

*

δB

Voter%

q+ Prior%Belief%

Approval%Set%

Win%Set%

A

B

θ3

θ3

δ

q− θ1 θ 2

(b) Simple Majority

δ*

q*+ Prior%Belief%

R1

Voter%

δA

p

Strong%Rejec1on%Set%

B

Strong%Rejec1on%Set%

p q− θ1

R(δ * )

(c) Decision Delegated to

⇤

Figure 2: Optimal Signal for Example 2 Now consider simple majority (k = 1). Figure 2(b) depicts the posterior beliefs induced + 7 by an optimal signal ⇡1⇤ supported on S = {s , s+ A , sB }. Voter

s+ A occurs, while

B

A

approves if and only if

+ + approves if and only if s+ B occurs. Realizations sA and sB occur with

probabilities 0.15 and 0.225, respectively, hence the probability of approval is 0.375. To understand this optimal signal, we first analyze the weak representative voter simple majority. Voter

⇤

⇤

with

= ( 2, 0.5, 1) is represented in 2(c) by the dotted line, which

delineates the convex hull of the win set W1 . The optimal signal ⇡ ⇤⇤ with delegation to

⇤

induces posteriors q⇤+ and q .8 Note that belief q⇤+ is the weighted average of beliefs qA+ and qB+ . With posterior q⇤+ , voter

⇤

approves, but voters

A

and

B

strictly prefer to reject. This

implies that the controller’s influence strictly reduces the expected payo↵ of both voters and

B

A

.

We now use ⇡ ⇤⇤ to decompose ⇡1⇤ into two components: collective persuasion and targeted persuasion. The straight line connecting q and q⇤+ in Figure 2(c) is a direction of common interest: all voters agree that moving beliefs from q in the direction of q⇤+ represents “good news” about the proposal. The collective persuasion component of ⇡1⇤ is the belief change 7

+ + + + 3 1 Formally, S = {s , s+ A , sB }, P r(s |✓3 ) = 0, P r(sB |✓3 ) = 4 , P r(sA |✓3 ) = 4 , P r(s |✓2 ) = P r(sB |✓2 ) =

0, P r(s+ A |✓2 ) = 1, P r(s |✓1 ) = q

= (1, 0, 0),

+ qB

=

( 13 , 0, 23 ),

+ + 25 3 28 , P r(sB |✓1 ) = 28 , P r(sA |✓1 ) = 0. The + and qA = (0, 23 , 13 ). Probability of approval is

possible posterior beliefs are + P r(s = s+ B ) + P r(s = sA ) =

3 ( 28 ⇥ 0.7 + 0 ⇥ 0.1 + 34 ⇥ 0.2) + (0 ⇥ 0.7 + 1 ⇥ 0.1 + 14 ⇥ 0.2) = 0.225 + 0.15 = 0.375. 3 8 + + + Formally, S = {s , s+ ⇤ }, P r(s⇤ |✓3 ) = P r(s⇤ |✓2 ) = 1, and P r(s⇤ |✓1 ) = 28 . The possible posterior beliefs 3 4 8 are q = (1, 0, 0), q⇤+ = ( 15 , 15 , 15 ). Probability of approval is P r(s = s+ ⇤)=

17

3 28 ⇥0.7+1⇥0.1+1⇥0.2

= 0.375.

along this common interest direction. However, although belief q⇤+ is good news about the proposal, it is not enough to convince voters

A

and

B

. The controller then relies on targeted

persuasion. Starting from q⇤+ , signal ⇡1⇤ moves the belief to either qA+ or qB+ . The straight line connecting qA+ and qB+ in Figures 2(b) is a direction of opposing interest: moving beliefs from qA+ in the direction of qB+ represents “good news” about the proposal to voter “bad news” to voter

A

B

, but

. Importantly, the weak representative voter corresponds precisely to

this direction of opposing interest — as Figures 2(b) and (c) illustrate. From belief q⇤+ the controller ensures approval by exploiting the opposing interests of voters.

3.3

Value of information control

The next Corollary provides comparative statics of k-voting rules on the value of information control. Corollary 1 Consider an electorate { 1 , . . . ,

n

} and a k voting rule. Then

(i) Information control is not valuable if and only if the set Wk is empty or p 2 Wk . (ii) Information control is most valuable when p 2 / Wk and p 2 co(Wk ). (iii) Equilibrium probability of approval weakly decreases with k. (iv) The value of information control is a single-peaked function of k, possibly non-monotone. Parts (i) to (iii) follow immediately from Lemma 2. To understand (iv), suppose that the value of information control strictly decreases from rule k to rule k 0 = k + 1. This means that the value of information control was strictly positive under k. From (i) this implies p2 / Wk , yielding p 2 / Wkˆ for any kˆ > k. From (iii) we know that the probability of approval decreases in k, hence the value of information control must weakly decrease from rule k on.

3.4

Voter Heterogeneity and Information Control

Proposition 2 showed that the controller can, under non-unanimous voting rules, exploit voter heterogeneity by designing a signal that induces approval from di↵erent winning coalitions. In e↵ect, under a k-voting rule the controller designs approval signal realizations along directions of voter disagreement in such a way that there is always a coalition of at least k voters willing to approve the proposal. 18

A natural question then is: would the controller prefer to persuade a group of voters rather than an individual voter to whom the decision is delegated? To make this statement precise, suppose that voters are ordered according to how “hard” it is for the controller 0

to persuade them, i.e., if i < i0 then V ( i ) to persuade, while voter

n

V ( i ). Thus, voter

1

is the easiest voter

is the hardest. The following proposition provides a sufficient

condition for the controller to prefer a k voting rule to delegation to the k-th hardest voter. Proposition 3 Consider an electorate { 1 , . . . , i0

n

}, and index voters according to how hard

it is to persuade them individually, V ( ) V ( i ) for i < i0 . Then (i) For any voter

i

, V (Wn ) V ( i ) and V (W1 )

(ii) If voters rank states in the same order,

i

V ( i );

2 Fz , i 2 I, then V (Wn ) = V ( n ) and V ( k ).

V (Wk )

(6)

Part (i) captures the immediate observation that the controller can do no worse if she only requires one vote, regardless of the voter’s identity, rather than the vote of a given voter. Conversely, the controller cannot benefit from securing the approval of all voters simultaneously rather than the approval of a given voter. Part (ii) states that if voters are sufficiently aligned — i.e., all voters rank states in the same order — then the controller would prefer a decision process where he needs to persuade at least k voters, rather than persuading the k th hardest-to-persuade voter. That is, the controller benefits from some heterogeneity, but requires some alignment between voters. The intuition is that, when voters rank states in the same order, then the approval signal realization under an optimal signal to the k-th hardest-to-persuade voter also induces approval for any voter i < k. Therefore, V (Wk ) cannot fall below V ( k ). Finally, the controller su↵ers no loss from persuading a collection of voters under a unanimity rule rather than the hardest-to-persuade individual. That is, under unanimity (6) is satisfied with equality. Inequality (6) holds whenever voters agree on the ranking of states. If voters rank states di↵erently, then the reverse inequality to (6) may hold. The reason is that an optimal signal when facing the k-th hardest-to-persuade voter may not secure approval from all easier-topersuade voters i < k (see Example 5 in the online Appendix B). Interestingly, sometimes an

19

optimal signal does not target the easiest-to-persuade voter, even when voters agree under full information and rank states in the same order (see Example 6 in the online Appendix B).

4

Institutional Design

We start this section with a simple question: given a voting rule, do voters benefit from the signal chosen by the controller? We then study how di↵erent voting rules a↵ect the payo↵s of di↵erent voters. Importantly, voting rules a↵ect outcomes not only through the consensus required to approve a proposal, but also by the amount of information that voters endogenously receive. For example, under delegation the controller provides the sole decision maker with some information about the state, although the signal has no value for the latter (cf. Proposition 1). As a result, a single decision maker may prefer to delegate the choice to someone with di↵erent preferences than himself, but who will elicit more information about the benefits of the proposal.9 Voters face a similar trade-o↵ between control and information when evaluating di↵erent k-voting rules: a higher consensus (i.e. higher k) may lead to excessive rejection of the proposal, but may induce the controller to provide a more informative signal. We study this trade-o↵ by posing two questions: (i) under delegation, what are each voter’s preferences over decision makers?, and (ii) when decisions are made collectively, how do preferences over decision makers translate into preferences over k-voting rules?

4.1

Do voters benefit from the controller’s signal?

We start by comparing each voter’s ex ante expected payo↵ under two scenarios: their equilibrium payo↵ when the controller provides signal ⇡ ⇤ , and their equilibrium payo↵ if there was no controller providing a signal, and voters had to choose a policy solely on the basis of their prior beliefs. Clearly, if there is a single voter, then he cannot be made worse o↵ be the controller’s 9

In the context of organizations, Jensen and Meckling (1976) are among the first to point out that

delegation decisions are guided by a fine balance between the loss of control owing to conflict of interest, and the gain of information when delegating to experts (see also, Holmstrom (1982), Dessein (2002), Alonso and Matouschek (2008), and Armstrong and Vickers (2010)).

20

influence. In fact, the controller’s optimal signal does not change his expected payo↵. Similarly, if all voters in the electorate have the same type, then the expected payo↵ of all voters is the same with or without the controller, independently of the k-voting rule. This is not the case when voters have di↵erent preferences, as summarized by the next Corollary. Corollary 2 Fix a k-voting rule and consider the electorate { 1 , . . . ,

n

}. Compare voters’

ex ante expected payo↵ under the controller’s optimal signal ⇡ ⇤ and under no signal.

(i) If the voting rule is unanimity, then all voters are weakly better o↵ under the controller’s influence, independently of the prior belief; (ii) If k < n and p 2 Wk , then the controller’s influence does not a↵ect payo↵s; (iii) If k < n and p 2 / Wk , then at most k 1 voters are strictly better o↵ under the controller’s influence. Thus, at least n k +1 voters are weakly worse o↵ under the controller’s influence. These voters are strictly worse o↵ if there is no optimal signal with a binary realization space. In particular, with a simple majority voting rule, a majority of voters is weakly worse o↵ because of the controller’s influence. For any given k-rule, if p 2 Wk , then the controller’s optimal signal reveals no relevant information and voters approve the proposal. In this case, the controller’s influence has no impact on voters’ expected payo↵s — which concludes part (ii). Part (i) follows from the same logic when p 2 Wn , and the veto power of voters when p 2 / Wn : if the rule is unanimity, then in order to approve the proposal the controller must convince all voters at the same time. However, for any non-unanimous voting rule, the controller can exploit preference disagreement by choosing signal realizations that target di↵erent winning coalitions. Part (iii) highlights that it cannot be the case that k voters are strictly better o↵ by the controller’s influence. Otherwise, the controller could strictly increase the probability of approval by choosing a less informative signal that leaves the same k voters weakly better o↵, but at least one of them indi↵erent. Moreover, whenever the posterior belief q + 2 co(Wk ) obtained from

combining all approval realizations of ⇡ ⇤ is such that q + 2 / Wk , then n k+1 voters are strictly worse o↵. This is the case if there is no optimal signal with only two signal realizations, which implies that the controller must be targeting di↵erent winning coalitions.10 10

There is always an optimal signal with only one realization that leads to rejection — therefore, if every

21

Finally, with a simple majority rule, a majority of voters can be made strictly worse o↵ by the controller’s signal even when all voters agree under full information and rank states in the same order (see Example 4 in Section 6.1) .

4.2

Voter preferences over decision makers

Suppose that the approval decision is made by a single voter : the controller only needs to persuade this voter. Now suppose that voter decision. How would voter

can choose whom to delegate the approval

rank di↵erent decision makers? As mentioned earlier, voter

faces a well known trade-o↵ between the gain in information and a loss of control: delegating to someone with di↵erent preferences can lead to inferior decisions, but may induce the controller to provide a more valuable signal. To study this trade-o↵, we first characterize voter preferences over decision makers for a suitably-defined restricted domain. We then show that, in these domains, a voter can always resolve the previous trade-o↵ perfectly as a voter’s preferred decision maker would (i) induce from the information controller a most valuable signal for voter , and (ii) for that signal, there is no loss of control. The next proposition describes the preferences of a voter over decision makers that belong to the same class Fz , that is, rank states in the same order z. v

Proposition 4 Fix a permutation z and let

2Fz . Consider any totally ordered (according

to toughness) set of voters D ⇢Fz , and suppose that the approval decision is delegated to a voter in D prior to the controller supplying a signal ⇡. Then, (i) Voter

v

has single-peaked preferences over decision makers in D. That is, there exist

¯ 2 D such that for , of voter

0

v

2 D, voter

would (weakly) prefer to delegate to voter

if either A( ¯) ⇢ A( 0 ) ⇢ A( ) or A( ) ⇢ A( 0 ) ⇢ A( ¯).

(ii) If all voters in D agree with

v

under full information, then voter

preferences over decision makers in D. That is, for , to delegate to voter

0

instead of voter

if

(iii) The maximum expected utility of voter

0

0

2 D, voter

v

v

0

instead

has monotone

would (weakly) prefer

is tougher.

v

when delegating to any decision maker in R|⇥| ,

optimal signal must have at least three signal realizations, then the controller needs at least two di↵erent signals leading to approval, which implies that the controller must be targeting di↵erent winning coalitions.

22

is achieved by any voter

⇤

⇣

ˆ,

v

⌘

=ˆ

ˆ ( ˆ)1 2 Fz , where ˆ 2 Fz and

ˆ ( ˆ) =

X

✓2{✓:

v ✓

p✓ ˆ✓ .

(7)

0}

Parts (i) and (ii) of the proposition describe the preferences of voter

v

over decision

makers who share his ranking of states and are ordered according to toughness. This condition on alignment does not guarantee that there is no loss of control under delegation, as these decision makers may not have the same approval set as

v

. Part (i) shows that a voter

has single-peaked preferences over such decision makers. That is, the set inclusion ordering derived from toughness translates naturally to single-peaked preferences when one restricts attention to voters in the same class. Part (ii) shows that the voter’s preferences become monotone when the decision makers agree with

v

under full information.

These results follow from the basic structure of an optimal signal with delegation to a voter in Fz : the controller sets a threshold state and the optimal signal induces approval if a state with a higher net value occurs. Then, switching to a tougher decision maker implies a (weakly) higher threshold state and a (weakly) smaller set of approval states. Importantly, a tougher decision maker induces a signal that discriminates better between states of higher net value and states of lower net value for all voters in Fz . Therefore, switching to a marginally tougher decision maker benefits voter

v

whenever the current threshold state has a negative

net payo↵, but it proves detrimental whenever this net payo↵ is positive. If all decision makers agree with

v

under full information, then this net payo↵ is always negative.

Part(iii) identifies in Fz an ideal decision maker for voter

v

. If voter

v

could both

choose the signal ⇡ and decide whether to approve the proposal, then he only needs to learn whether the realized state corresponds to a positive net value. He can induce the controller ⇣ ⌘ ⇤ ˆ v to produce such a signal by delegating to a voter , = ˆ ˆ ( ˆ)1, with ˆ ( ˆ) given ⇣ ⌘ by (7). Note however that voter ⇤ ˆ, v and voter v disagree under full information: ⇣ ⌘ voter ⇤ ˆ, v would reject the proposal more often than v if they perfectly learned the state. Nevertheless, they fully agree on the decision given the controller’s optimal signal. In

this sense, the fact that the signal is not fully revealing eliminates the loss of control when ⇣ ⌘ ⇤ ˆ v delegating to a tougher voter. Therefore, by delegating to , voter v achieves the same expected value as if he both made decisions and controlled the signal himself. 23

4.3

Voter preferences over k-voting rules

How does each voter rank di↵erent voting rules? Recall that voting rules a↵ect voters’ payo↵s via two channels: the consensus required to approve a proposal, and the endogenous signal chosen by the controller. The following result follows from Corollary 2. Corollary 3 Consider an electorate { 1 , . . . ,

n

} with an odd number n

3 of voters. If p 2 /

W n+1 , then a majority of voters weakly prefer unanimity voting rule over simple majority.11 2

To draw stronger inferences about voters’ equilibrium payo↵ under di↵erent voting rules, we need to consider the nature of voters’ preference heterogeneity. The next lemma shows that, if voters belong to the same class, then each voter has single peaked preferences over k. Lemma 4 Consider an electorate { 1 , . . . , each voter

i

n

}, with

i

2 Fz , for some permutation z. Then

has single peaked preferences over k, in the sense that there exists k ⇤ ( i ) such

that his expected utility is non-decreasing in k for k < k ⇤ ( i ), and it is not increasing for k > k ⇤ ( i ). Proposition 2 shows that voters’ expected utilities with a k voting rule are the same as with delegation to the weak-representative voter

⇤

(k), as long as

⇤

(k) strictly ranks states.

Lemma 3 established that if all voters are in the same class, then the weak representative voter also belongs to that class. The intuition behind Lemma 4 is that since the weakrepresentative voter

⇤

(k) also belongs to the same class Fz , then a voting rule requiring a

higher consensus is equivalent to delegating to a tougher voter. As a result, the collection of representative voters

⇤

(k) describes a totally ordered set of voters in Fz , and Proposition

4(i) implies that each voter has single-peaked preferences over these decision makers, and hence, over k-voting rules.12 An important implication of Lemma 4 is that a majority of voters prefer a supermajority voting rule over a simple majority voting rule. 11

If p 2 W n+1 , then a majority of voters might prefer simple majority over unanimity when unanimity 2

makes approving the project too unlikely, e.g., if the win set Wn is empty. 12 If voters do not agree on the ranking of the states, then preferences might not be single-peaked even when voters agree under full information and are totally ordered according to toughness. See Corollary 4 and Example 7 in the online Appendix B.

24

Lemma 5 Consider and electorate { 1 , · · · , same class

i

n

} with an odd number n

3 of voters in the

2 Fz , and p 2 / W n+1 . Then a majority of voters: 2

(i) weakly prefer any supermajority voting rule k 0 >

n+1 2

over simple majority k =

n+1 ; 2

and

(ii) strictly prefer supermajority k 0 over simple majority if it leads to a lower (but positive) equilibrium probability of approval, 0 < V (Wk0 ) < V (W n+1 ). 2

The next Proposition provides sufficient conditions for all voters to have the same preferences over k-voting rules. Proposition 5 Suppose that all voters are in Fz and they agree under full information. Then every voter weakly prefers a (k +1)-voting rule to a k-voting rule, for k 2 {1, ..., n

1}.

This proposition implies that even heterogenous voters may have the same preferences over electoral rules. In fact, as long as there is agreement under full information and voters rank states in the same order, then they all prefer a unanimity rule to any other k-voting rule. Essentially, sufficient alignment among voters can induce perfect agreement over electoral rules if information is endogenous to the electoral rule. Indeed, while voters may disagree under uncertainty, if they agree under full information, then they also agree on the signal they would choose if they were in control of decisions and could design the signal themselves. The intuition is that the weak representative voters { ⇤ (1), . . . ,

⇤

(n)} (i) belong to

the same class, (ii) agree under full information, and (iii) are totally ordered according to toughness. Therefore, the conditions of Proposition 4(ii) apply and every voter has monotone preferences: as a higher k corresponds to a tougher weak-representative voter, voters prefer rules that require more consensus only because they induce the controller to supply a more valuable signal.

5 5.1

Extensions Controller knows the State

In our basic setup the controller has no private information. Suppose instead that the controller privately observes the true state ✓ before choosing signal ⇡. In this case, the 25

choice of ⇡ by the informed controller may itself convey information to voters. We ask two questions: does the controller benefit from her private information? and what is the signal that maximizes the expected payo↵ of the informed controller, when expectation over controller’s types is taking according to the prior p? In the online Appendix B we first apply the results from Alonso and Cˆamara (2014b) to show that the controller cannot benefit from privately observing the state. We then show that the maximum expected payo↵ is achieved in pooling equilibria where: (i) all controller’s types choose the same signal ⇡ ⇤ , and (ii) ⇡ ⇤ is also an optimal signal in the case of an uninformed controller. Together these two results imply that the equilibrium probability of approving the proposal is una↵ected by the controller privately learning the state.

5.2

Controller’s Payo↵ Depends on the State

In our basic setup we focus on the case of pure-persuasion. We now consider a controller with a state-dependent payo↵ uC (x, ✓) : X ⇥ ⇥ ! R. Let define the controller’s type

C

={

C ✓ }✓2⇥ .

C ✓

= uC (x1 , ✓)

uC (x0 , ✓) and C ✓

To simplify presentation, suppose

6= 0.

First suppose that the approval decision is delegated to voter . Proposition 6 in the online Appendix B generalizes Proposition 1. It shows that the optimal signal always induces an approval realization for states such that both controller and voter agree on approval, and it always induces a rejection realization for states such that both agree on rejection. In the set of states where there is disagreement, the optimal signal again defines a cuto↵ state. However, in the case of pure persuasion the cuto↵ state was defined by ordering the states solely according to the voter’s net payo↵

✓;

now the cuto↵ is defined by ordering the disagreement

states according to the absolute value of the ratio of players preferences,

C ✓ ✓

.

In many important cases the controller ranks states in the same order as the voter. For example, the controller receives the same payo↵ as the voter, plus some private benefit from approving the proposal (see also our application in Section 6.1). Proposition 7 in the online Appendix B shows that if the information controller and the electorate rank states in the same order, then each voter has single-peaked preferences over k-voting rules. Moreover, if voters also agree under full information, then the payo↵ of every voter is weakly increasing in k. To understand the result, consider delegation to the weak representative voter 26

⇤

(k). If

all players agree on the ranking of the states, then

⇤

(k) also agrees. The controller’s cuto↵

state is then defined according to this common ranking. A higher k-voting rule implies a tougher

⇤

(k) and a weakly higher cuto↵ state, without changing the ranking. Consequently,

all the results of Lemmas 4 and 5, and Proposition 5 continue to hold. Moreover, the proof of Proposition 7 shows that if all players rank states in the same order and the controller is sufficiently biased towards approval, then ⇡ ⇤ is an optimal signal for a controller with type C

if and only if ⇡ ⇤ is an optimal signal in the case of pure-persuasion.

5.3

Preference Shocks

In our basic setup the controller knows the preference profile of the electorate. However, in some instances voters are subject to idiosyncratic preference shocks, in which case the controller faces a probability distribution over preference profiles. To study the e↵ects of preference shocks, assume that voter i’s preferences are given by ui (x, ✓, µi ) and the conditional net payo↵ from approval with state ✓ and private shock µi is ui (x1 , ✓, µi )

ui (x0 , ✓, µi ) =

i ✓

µi .

Shocks µi are i.i.d. and jointly independent with ✓, with each shock distributed according to F (µ) with support in [ µ ¯, µ ¯]. The controller chooses the signal before shocks are realized. Proposition 8 in the online Appendix B shows that if preference shocks are small and high shocks are sufficiently likely, then with unanimity the controller behaves as if she is facing the toughest electorate. That is, the controller benefits from choosing a more informative signal that persuades even the electorate profile where each voter received the worst possible shock µ ¯. Under this optimal signal, voters have almost surely a strict preference between approval and rejection of the proposal, so that voters obtain a strictly positive gain with probability 1 when they approve the proposal. Therefore, unlike the case of non-probabilistic voting, with unanimity all voters can strictly benefit from the controller’s influence. For non-unanimous voting rules, the results from Corollary 2 carry over to cases with a small µ ¯ . In particular, with a simple majority voting rule, a majority of voters can be made strictly worse o↵ by the controller’s signal.

27

The same is true with delegation: the controller’s signal targets to persuade the voter with the worst possible shock. Hence, the voter can strictly benefit from the signal.

5.4

Heterogenous Prior Beliefs

In our base model, players share a common prior belief about the consequences of di↵erent policies. As argued by Alonso and Cˆamara (2014a), however, heterogeneous priors provide a powerful motive for persuasion, as a controller typically gains from shaping the learning of decision makers in the face of open disagreement. We can extend our main analysis to the case of heterogenous priors as follows. Suppose players hold di↵erent prior beliefs pl 2 int( (⇥)), with l 2 {C, 1, . . . , n}. Suppose that the controller’s signal is commonly understood in that all players agree on the conditional probabilities generating each realization. Then we can use the results from Alonso and Cˆamara (2014a) to characterize the controller’s optimal signal and her gain from information control. We now briefly discuss how heterogenous priors a↵ects our insights from Sections 3 and 4. With delegation to voter

and common priors, the controller’s optimal signal defines a

cuto↵ state where states are ordered solely according to the voter’s net payo↵

✓.

In the

case of heterogeneous priors, the optimal signal continues to define a cuto↵ state. However, the ordering of states might change depending on prior beliefs. Formally, the controller ranks states according to lowest

i i p✓ ✓ pC . ✓

i i p✓ ✓ pC ✓

and induces rejection only for the negative states with the

For example, the controller might now find it optimal to have a state ✓ with

a very negative

✓

inducing an approval signal simply because the controller assigns a very

high prior belief to ✓, while the voter believes that ✓ is very unlikely. In other words, the controller favors approval realizations for states with negative payo↵s whose likelihood he believes the voter underestimates. Proposition 5 showed that if voters share the same ranking of states and agree under full information, then they all have the same preferences over voting rules. In particular, unanimity is preferred to any other k-voting rule. This does not hold, however, if voters have heterogenous prior beliefs. Note that open disagreement does not per se induce disagreement over the public signal. Indeed, under the conditions of Proposition 5 all voters have the same preferences over the class of binary “approve-reject” signals that preserve the ranking 28

of states — i.e., signals with a cuto↵ state with higher ranked states always inducing the approval realization. The fact that Proposition 5 no longer holds with heterogenous priors owes to the fact that the controller’s signal no longer follows a cuto↵ on the ranking of states given by

i ✓,

but rather in the ranking according to

i i p✓ ✓ pC .

Nevertheless, if the two rankings

✓

coincide, then the results of Proposition 5 still hold with heterogenous priors.

6 6.1

Applications Voting on a Public Good

Consider a one-period k-voting model where an odd number n

3 of voters must choose

whether to approve (x = x1 ) or not (x = x0 ) the investment on a new public good, e.g., construction of a new highway overpass to improve traffic. If implemented, the cost c of the project is paid through a proportional tax t. Each voter i has a pre-tax income wi and the government budget must balance. For simplicity, suppose there are no other government expenditures. Hence, the status quo tax is t0 ⌘ 0, and it increases to t1 ⌘

P c

i2I

wi

if the

project is implemented. Voters’ payo↵ from the project depends on state ✓ 2 ⇥ ⇢ R. This represents the uncertainty about how the overpass will a↵ect the overall traffic flow. A voter-specific payo↵ yi : ⇥ ! R captures how each voter is a↵ected by traffic flow changes, depending on factors such as where the voter lives and works. Let yi be strictly increasing, so that a higher “quality” ✓ means a better traffic outcome. The utility function of each voter is then 8 < (1 ui (x, ✓) = :

t1 )wi + yi (✓) if x = x1 , wi

if x = x0 .

For each voter i compute the net payo↵ from approval i ✓

= (1

t1 )wi + yi (✓)

All voters belong to the same class Fz since

wi = yi (✓) i ✓

t1 w i .

(8)

strictly increases in ✓. Voter i with posterior

belief q votes to approve the project if and only if the expected payo↵ from the traffic outcome is greater than how much he has to pay in taxes to implement it, E[yi (✓)|q] 29

t1 w i .

Consider an information controller who has vested interests on the project — e.g., the controller is the Governor who proposed the project, but she needs voters to approve the ballot measure. Suppose that the Governor ranks states in the same order as voters. For example, her net payo↵ is proportional to the change in her “political capital,” which is increasing in the quality of the project. Moreover, suppose she receives additional private benefits (e.g., ego rents) from approving the project.13 Lemma 3 imply that for each k voting rule there is a weak representative voter

⇤

(k) 2

Fz , and from the point of view of all players the k-voting rule is payo↵-equivalent to delegating the approval decision to

⇤

(k). Moreover, the controller’s optimal signal ⇡ ⇤ defines a

cuto↵ quality ✓k⇤ such that the project is always rejected if the quality is below the cuto↵, ✓ < ✓k⇤ , and the project is approved with certainty if the quality is above the cuto↵, ✓ > ✓k⇤ . If it is optimal to target di↵erent winning coalitions, then ⇡ ⇤ contains multiple signal realizations that lead to approval. Cuto↵ ✓k⇤ weakly increases with k. Importantly, if the controller P is more biased towards approval than voters, that is, ✓2D( C ) p✓ ✓⇤ (k) < 0, then a signal is optimal for controller

C

if and only if it is optimal under the pure-persuasion benchmark

(see the proof of Proposition 7 in the online Appendix B). Next we present two examples based on this general setup. Example 3 considers voters with homogenous preferences for the public good but di↵erent incomes, which a↵ects their tax burden. It shows that the voter with the median income can benefit from delegating the approval decision to a richer voter. Example 4 considers voters with heterogeneous preferences. It shows that under a simple majority voting rule a majority of voters can be made strictly worse o↵ by the controller’s influence, even when voters have the same income, agree under full information, and rank states in the same order. Example 3:

Suppose voters have homogeneous quality preferences yi = y, i 2 I. Voter i

approves the project if and only if E[y(✓)|q]

t1 wi . Therefore, voters are totally ordered

— voters with higher income are both harder-to-persuade and tougher, wi < wj implies V ( i) Voter 13

V ( j ) and A( i ) k

A( j ). Let

k

be the voter with the k th lowest income.

is then a representative voter and a k-voting rule is equivalent to delegating the

Note that the controller’s ranking of the states does not change if her private benefit from approving the

project is either constant or strictly increasing with the project’s quality.

30

decision to him. Increasing the k-voting rule implies that the controller must target a richer voter. Suppose that the controller is more biased towards approval than the median voter m , P m ✓2D( C ) p✓ ✓ < 0. Lemma 5 implies that a majority of voters (the median and richer voters)

weakly prefer any supermajority voting rule over simple majority. Moreover, this preference relation is strict if voter

k

is strictly richer than the median voter and his approval set is not

empty. By delegating the approval decision to a richer voter, who pays more to implement the project, the electorate induces the controller to supply a more informative signal. This result does not require the median voter to agree with

k

under full information. ⇤

Example 4: Suppose yi = ✓ i , and consider three voters with

1

= 0.1,

2

= 0.5,

3

= 0.9.

Voters have the same income wi = 5. If implemented, the project costs 1.5, so the proposed tax t1 = 0.1 runs against the status quo t0 = 0. There are three possible quality levels for the project: it does not improve traffic (✓ = 0), it moderately improves traffic (✓ = 0.7), or it greatly improves traffic (✓ = 1.4), so that ⇥ = {0, 0.7, 1.4}. From (8) we have so

1

⇡ { 0.5, 0.46, 0.53},

2

⇡ { 0.5, 0.34, 0.68},

3

i ✓

=✓

i

0.5,

⇡ { 0.5, 0.23, 0.85}. Voters would like

to reject the project if it does not improve traffic, and approve if it has a moderate or great impact on traffic. Figure 3 depicts the prior belief p, the approval set of each voter, and the win set with simple majority. Note that there is no representative voter. The win set is not convex and the dotted lines delineate the convex hull of W2 . Consider a controller who prefers to approve the project in every state, which implies that she is more biased towards approval than voters. There is no optimal signal with only two signal realizations, but there is a ⇡ ⇤ with three signal realizations. One realization induces posterior q

and all voters

reject the project. Another induces posterior q1+ : voters 1 and 3 approve the project, while voter 2 strictly prefers to reject. The remaining realization induces posterior q2+ : voters 2 and 3 approve the project, while voter 1 strictly prefers to reject. Note that the weighted average of the two approval posterior beliefs is a belief on the dotted line connecting q1+ and q2+ . This average approval belief belongs to the convex hull of W2 , but it does not belong to W2 . Consequently, a majority of voters (voters 1 and 2) are made strictly worse o↵ by the controller’s influence. They strictly prefer the controller not to release the signal ⇡ ⇤ , so that voters keep their prior and vote to reject the proposal. Even though all voters

31

agree under full information and rank states in the same order, they sometimes disagree under uncertainty because of the di↵erences in the curvature of their utility functions. The information controller exploits this disagreement by designing a partially informative signal that targets di↵erent winning coalitions. Finally, all voters strictly prefer unanimity over simple majority, to induce the controller to provide a more informative signal. ⇤ θ=0.7$

θ=1.4$ Win$set$when$k=2$

Voter$3$ Voter$2$ Voter$1$

q1+−

q +2 p

θ=0$

q−

Figure 3: Win Set and Optimal Signal from Example 4

6.2

Spatial Model of Elections

Consider an election where a left-wing incumbent politician is running for re-election against an untried challenger from the opposing right-wing party. Let X = {L, R}, where L represents re-electing the incumbent and R electing the challenger. Each voter has a utility function ui (x, ✓) =

(yx

yi )2 , where yi captures the ideology of voter i and yx is the pol-

icy implemented by politician x. There is an odd number n

3 of voters with ideologies

symmetrically distributed around the median voter ymedian = 0. Voters know more about the incumbent than the challenger. Formally, voters know that the incumbent is committed to a policy yL < 0, but they are uncertain about the policy yR > 0 that the challenger would implement if elected. Voters’ prior belief is that yR = ✓ with probability p✓ , where ✓ 2 ⇥ ⌘ {✓1 , . . . , ✓M }, 0 ✓1 < . . . < ✓M , and ✓1 < |yL | < ✓M . Suppose that without further information the median voter strictly prefers the incumbent.

32

For each voter i the net payo↵ from electing the challenger is i ✓

=

(yR

yi )2 + (yL

yi ) 2 =

Voter i strictly prefers the challenger if (✓2

(✓2

yL2 ) + 2(✓

yL2 ) + 2(✓

and he strictly prefers the incumbent if yi <

✓+yL . 2

yL )yi .

yL ) > 0, that is, if yi >

(9) ✓+yL , 2

Therefore, the voter with the k-th

lowest ideology yi is the representative voter. Consider a simple majority rule, so that the median voter is decisive. Under full information, the median voter strictly prefers the challengers if yR < |yL |, and strictly prefers the incumbent if yR > |yL |. From the median voter’s perspective, the relevant information for his decision is: who is more moderate, the challenger or the incumbent? Let the information controller be a right-wing Interest Group (IG) with ideology yC > 0 and payo↵ uC (x, ✓) =

(yx

yC )2 . In this spatial model, the controller and voters rank

states in di↵erent orders.14 Nevertheless, in the online Appendix B we show that the op⇤ timal signal ⇡ ⇤ defines a cuto↵ state ✓R > |yL |: the challenger losses for sure if he is “too

⇤ ⇤ radical”, ✓R > ✓R , and he wins for sure if ✓R < ✓R . Importantly, we also show that there

exists an ideology cuto↵ y¯ > 0 such all radical IG’s with ideology yC > y¯ behave as in the pure-persuasion benchmark. That is, ⇡ ⇤ is an optimal signal for these policy-motivated IGs if and only if ⇡ ⇤ is an optimal signal for a purely office-motivated IG. Suppose that the IG is radical, yC > y¯. Proposition 1 implies that the IG’s influence does not a↵ect the payo↵ of the decisive median voter. However, voters to the left of the median are hurt by the IG’s influence, while voters to the right are better o↵. Although players do not rank states in the same order, the results from Lemma 5 continue to hold: a strict majority of voters (the median voter and all left-wing voters) prefer any supermajority voting rule over simple majority. Supermajority implies that a voter to the left of the median becomes decisive, which induces the IG to provide a more informative signal. If the IG is moderate, yC < y¯, then the IG’s preferences are sufficiently aligned with the median voter. In this case, the more informative signal provided by the IG strictly benefits the median voter. 14

Note that

@ ✓i @✓

= 2(yi

✓). Therefore, a strict majority of voters — voters with ideology yi < ✓1 — rank

all states in the same decreasing order. Voters yi > ✓M rank all states in the same increasing order.

33

6.3

Winners and Losers

We now study an application inspired by the model of Fernandez and Rodrik (1991), who highlight the role of individual-specific uncertainty when voters must decide whether or not to engage in an economic reform.15 There are three sectors in the economy, L, M and R. The population of workers, who are also voters, is distributed uniformly across the sectors. Voters must decide whether to implement an economic reform x1 (e.g., sign a trade agreement with other countries) that increases the productivity of one sector, but decreases the productivity of the other sectors. Players have a uniform prior believe over which sector ✓ 2 ⇥ = {L, M, R} will benefit from the reform. The reform increases the payo↵ of workers in sector ✓ by +1, and decreases the payo↵ of all other workers by

1.

Consider a simple majority voting rule. Without further information, each worker believes that he is more likely to be a loser than a winner. Therefore, the proposal delivers a negative expected payo↵ and all voters reject the proposal. With full information about the state, voters in the winning sector ✓ vote to approve, but voters in the two losing sectors form a majority and reject the proposal. Consider an information controller who wants to maximize the probability of approval. The controller can design a partially informative signal that guarantees the approval of the proposal. The optimal signal does not reveal the identity of the winning sector. Instead, it reveals the identity of one losing sector.16 Upon learning this information, the losing sector votes to reject, but the two other sectors vote to approve. They now believe that there is an equal chance of being a winner or a loser. With the controller’s influence and a simple majority rule, the proposal is approved independently of the state. Consequently, the controller’s strategic information provision strictly lowers the expected payo↵ of all voters. All voters would strictly prefer a unanimity voting rule to block the influence of the controller. With unanimity, the win set is empty 15 16

We are also grateful for suggestions by Navin Kartik. Formally, let s 2 S = {L, M, R}, P r[s|✓] = 0 if s = ✓, and P r[s|✓] = 0.5 if s 6= ✓. Therefore, upon

observing s, all players know that sector s is not the winner ✓, and the two remaining sectors are equally likely to be the winner.

34

and the reform cannot be implemented.

7

Conclusion

In important cases, acquiring information is infeasible or prohibitively expensive for individual voters. Voters must then rely on the information generated by certain individuals, who control the design of a public signal (e.g., jurors and prosecutor, voters and media, shareholders and CEO). Obviously, if the controller and voters share the same preferences, then the controller’s signal always benefits voters, as it allows them to make better decisions. However, this is not true if there is a conflict of interest between the controller and voters. We show that, with a simple majority rule, a majority of voters is always weakly worse o↵ by observing the information provided. In fact, all voters can be strictly worse o↵, even when they would agree on their decision if they knew the true state. This is so because the controller strategically designs a signal with realizations targeting di↵erent winning coalitions. To prevent this negative impact, voters may adopt a supermajority voting rule that induces the controller to supply a more informative signal. We also provide conditions for unanimity to be the rule preferred by all voters. We extend our analysis in a number of ways. We show that the controller cannot benefit from privately observing the state prior to choosing her signal. We study situations in which the controller also cares about the state and situations where voters are subject to idiosyncratic preference shocks. In these cases, a voter may now strictly benefit from the controller’s signal if he is the sole decision maker, although a majority of voters can still be worse o↵ under a simple majority rule. We also extend the analysis to allow voters to have heterogeneous prior beliefs, so that they openly disagree about the likelihood of the state. Importantly, even if they all share the same ranking over states and agree under full information, belief disagreement can translate into disagreement over the optimal electoral rule. Two interesting extensions are to allow for voters’ to privately acquire information and then deliberate prior to voting, and to allow voters to choose among multiple policy options. We see these extensions as promising and leave them for future work.

35

A

Appendix

Proof of Lemma 1: Under the assumptions, the set A( ) is non empty and the voter rejects the proposal if he has no additional information. Let ⇡ 0 be an arbitrary binary signal that induces posterior beliefs {q (⇡ 0 ), q + (⇡ 0 )} such that q (⇡ 0 ) 2 R( ) and q + (⇡ 0 ) 2 A( ) P with ✓2⇥ q✓+ (⇡ 0 ) ✓ = 0. Define the vector l as l = q + (⇡ 0 ) p. Then, Bayesian rationality implies that average posteriors must equal the prior so that

and q (⇡ 0 )

Therefore,

⌦ Pr[Approval] q + (⇡ 0 ) p and q + (⇡ 0 ) ⌦

q (⇡ 0 )

Pr[Approval] =

↵ p, l + (1

p are collinear so

⌦ Pr[Approval]) q (⇡ 0 )

↵ p, l = 0,

q + (⇡ 0 )

p

↵ p =

p, q + (⇡ 0 )

p

q (⇡ 0 )

.

hp q (⇡ 0 ), li k(q (⇡ 0 ) p)k = hq + (⇡ 0 ) p, li + hp q (⇡ 0 ), li k(q + (⇡ 0 ) p)k + k(q (⇡ 0 )

where, by construction, k(q + (⇡ 0 )

p)k = dl (p, A( )) and k(q (⇡ 0 )

p)k

,

p)k = dl (p, R( )). As

the optimal signal maximizes Pr[Approval], it must be that the optimal signal corresponds to a vector l⇤ that maximizes the ratio k(q (⇡ 0 )

p)k / k(q + (⇡ 0 )

⌅

p)k .

Proof of Proposition 1: The existence of an optimal binary signal is established in KG (Proposition 1, p. 2595). Let ⇡ be an optimal binary signal supported on S = {s , s+ } where the voter approves the proposal if and only if he observes s+ , and let ↵✓ = Pr[s+ |✓] so P that Pr[Approval] = ✓2⇥ ↵✓ p✓ . A Bayesian voter will approve after observing s+ if and only if

E[ |s+ ] =

X

q✓+

✓2⇥

✓

=

X ✓2⇥

↵ ✓ p✓ ✓ Pr[Approval]

0

Therefore, ↵ must solve the following linear program X

↵✓ p✓ = max

✓2⇥

Note that for any ✓0 such that

X ✓2⇥

✓0

↵✓0 p✓ ,

s.t. 0 ↵✓0 1,

X

↵✓0 p✓

✓

0.

(10)

✓2⇥

0 we must then have ↵✓0 = 1, as increasing ↵✓0 whenever

↵✓0 < 1 relaxes the approval constraint and increases the approval probability. Therefore, 36

suppose that by " (| ✓ |p✓ /|

✓

<

✓ 0 |p✓ 0 )

✓0

< 0 for ✓, ✓0 2 ⇥. If ↵✓ > 0 but ↵✓0 < 1, then increasing ↵✓0

while reducing ↵✓ by " leaves unchanged the approval constraint but in-

creases the probability of approval by "p✓ (| ✓ |/| Therefore, if ↵✓ > 0, then ↵✓0 = 1 for any Now suppose that voter

✓0

✓ 0 |)

>

"p✓ > 0, thus leading to a contradiction.

✓.

strictly ranks states so that

✓

6=

✓0

for ✓ 6= ✓0 . Then there can

only be one “cuto↵” state that satisfies (4), and given that the approval constraint is met with equality, the optimal binary signal must be unique. If p 2 A( ) then ↵✓ = 1, ✓ 2 ⇥, and the voter receives a completely uninformative signal. If p 2 / A( ) then the solution to (10) must have a binding approval constraint, implying

that the voter is indi↵erent between approval and rejection after observing s+ . That is, his ⌅

expected gain from making decisions with ⇡ is again zero.

Proof of Lemma 2: Follows immediately by replacing A( ) with co(Wk ), and R( ) with ⌅

Rk , in Lemma 1 and then applying the same reasoning as in the proof of Lemma 1.

Proof of Proposition 2: Let Wk be the win set under a k-voting rule and suppose that p2 / co(Wk ). Define G(Wk ) as G(Wk ) = { : q 2 Wk ) hq, i Note that each

0}

2 G(Wk ) corresponds to a “less tough” voter than the k-voting rule, in the

sense that any voter in G(Wk ) would approve the proposal if the electorate does so under a k-voting rule. Moreover, as G(Wk ) describes all hyperplanes that contain Wk , then we have (i) co(Wk ) = \ a

⇤

2G(Wk ) A(

), and (ii) V (Wk ) inf

2 G(Wk ) with V (co(Wk )) = V ( ⇤ ) so that

⇤

2G(Wk )

V ( ). We now show that there is

is a weak representative voter when the

win set is co(Wk ). As explained in the text, for any belief in co(Wk ) the controller can find a signal that induces approval with probability 1. Therefore V (Wk ) = V (co(Wk )), and

⇤

is

also a representative voter for Wk . Define the function f (↵) = inf

2G(Wk )

h↵, pi, 0 ↵ 1, which is concave as it is the

infimum of affine functions. The function f (↵) provides a representation of co(Wk ), since q = ↵p h↵,pi 17

2 co(Wk ) if and only if f (↵)

For every q 2

0.17 Let s+ be the event corresponding to approval of the

(⇥), the existence of a corresponding ↵ 1 is guaranteed by simply choosing ↵✓ =

37

proposal under an optimal signal, and let ↵✓⇤ = Pr[s+ |✓] so that Pr[Approval] =

P

✓2⇥

↵✓⇤ p✓ .

Since the expected approval posterior must be in co(Wk ), then we must have f (↵⇤ )

0.

Thus, the controller’s optimal signal must maximize Pr[Approval], i.e. X

↵✓⇤ p✓ = max ↵

✓2⇥

X ✓2⇥

↵✓ p✓ , s.t. 0 ↵✓ 1, f (↵)

0.

(11)

Program (11) is concave (as it maximizes a concave function over a convex set). Consider the Lagrangian L associated to (11) X

L =< ↵, p >

⌫✓ < ↵, 1✓ > +

✓

with ⌫✓ , µ✓ ,

X

µ✓ < ↵

1, 1✓ >

f (↵),

✓

0, and 1✓ is the unitary vector whose ✓-component equals 1. Suppose that

Wk is non-empty and has at least two di↵erent elements. This implies that Wk has a nonempty relative interior, so that the constraint qualification is satisfied and the Karush-KuhnTucker conditions are both necessary and sufficient for optimality (Boyd and Vandenberghe 2004). In particular, when p 2 / co(Wk ), ↵⇤ is an optimal solution if and only if there exist ⇤

, ⌫✓⇤ , µ⇤✓ > 0, ✓ 2 ⇥, such that ˜⌘

⇤

p

X

⌫✓⇤ < ↵⇤ , 1✓ > +

✓

X

µ⇤✓ < ↵⇤

1, 1✓ >2 @f (↵⇤ ) and f (↵⇤ ) = 0,

(12)

✓

where @f (↵⇤ ) is the set of subgradients of f at the point ↵⇤ , Define ⌧ ↵⇤ p ⇤ = ˜, , < ↵⇤ , p > ⇤

and consider the voter

=˜

⇤

1. By construction, h ⇤ , qi = 0 is a supporting hyperplane

of co(Wk ). Now consider the optimal signal ↵0 under delegation to voter X

↵✓0 p✓ = max ↵

✓2⇥

X ✓2⇥

↵✓ p✓ , s.t. 0 ↵✓ 1, h↵,

⇤

pi

⇤

which must satisfy

0.

Again, this is a concave program (in fact a linear program) with non-empty relative interior if co(Wk ) has a non-empty relative interior. Therefore ↵0 is optimal if and only if there exist ⇤

, ⌫✓⇤ , µ⇤✓ > 0, ✓ 2 ⇥, such that ⇤

p

X

⌫✓⇤ < ↵0 , 1✓ > +

✓

⇣

q✓ p✓

⌘

X ✓

/ max pq✓00 . ✓

38

µ⇤✓ < ↵0

1, 1✓ >= ˜

(13)

In particular, as ↵⇤ satisfies (12) it also satisfies (13) and thus provides an optimal signal when delegating to voter

⇤

. Therefore

that the representative voter

⇤

⇤

is a weak representative voter. Finally, suppose

(k) strictly ranks states. Then, following Proposition 1 the

binary optimal signal is unique and thus the expected utility of every player is the same under delegation to

⇤

⌅

or under a k-voting rule.

Proof of Lemma 3: We will prove the lemma by showing that if all voters in the electorate belong to Fz for some permutation z, then if {q 2

(⇥) : hq, vi = 0} is a supporting

hyperplane of co(Wk ) then v 2 Fz . Let Sk be the set of all k coalitions of voters with generic element s. Then \

2s A(

)

describes the win set associated with a unanimous decision when the electorate is restricted to the coalition s, and Wk = [s2Sk \ {q 2

(⇥) : hq, i

2s A(

). As \

2s A(

) is the finite intersection of half-spaces

0} , then any supporting hyperplane of \

be represented as a convex combination of { :

2s A(

) at q 2 int( (⇥)) can

2 s}. Moreover, co(s) ⇢ Fz , as ranking of

states is preserved under convex combinations. Thus any supporting hyperplane of \

2s A(

)

at an interior belief corresponds to a voter in Fz . Turning to co(Wk ), consider any point q 0 2 int( (⇥)) with q 0 2 @ (co(Wk )) and a

supporting hyperplane {q : hq, v(q 0 )i = 0} of co(Wk ) at q 0 . Since q 0 2 co(Wk ), Caratheodory’s theorem guarantees the existence of J card(⇥)+1 points, {q i }, i = {1, ..., J}, with q i 2 Wk

and q 0 2 co(q i , i = {1, ..., J}). We consider two possibilities: (i) at least one of the points q i is in int( (⇥)), (ii) any representation of q 0 as a convex combination of points {q i } with q i 2 Wk must correspond to points on the faces of the simplex

(⇥), i.e. for each

i = {1, ..., J 0 } there exists ✓i such that q✓i i = 0.

Consider first case (i) in which q i 2 int( (⇥)) for some i. Then {q : hq, v(q 0 )i = 0}

must also be a supporting hyperplane of Wk at q i . Furthermore, since q i 2 Wk there exists a k coalition s such that q i 2 \ hyperplane of \

2s A(

2s A(

) and thus {q : hq, v(q 0 )i = 0} is a supporting

). However, as all supporting hyperplanes to \

2s A(

) in int( (⇥))

can be associated to a voter in Fz , then v(q 0 ) 2 Fz .

Consider now case (ii) where every representation of q 0 as a convex combination of points

in Wk involves points that lie on (possibly di↵erent) faces of the simplex

39

(⇥), and let {q i },

i = {1, ..., J} be such a collection of points. As each q i 2 Wk , and {q : hq, v(q 0 )i = 0} is a supporting hyperplane of co(Wk ) then there exists a coalition si such that min and let

i

2si

hq i , i = 0,

be a voter at which this minimum is achieved. That is, coalition si would support

approval for belief q i with at least one voter being indi↵erent between approval and rejection. Now suppose by way of contradiction that v(q 0 ) 2 / Fz . This means that there exist

two states ✓ and ✓0 with ( the edge of beliefs {q : q = ↵1✓0 + (1

i 0 ✓ ) (v✓ 0 (q )

i ✓0

v✓ (q 0 )) < 0 for all i 2 {1, ..., J} . Now consider

(✓, ✓0 ) that put positive probability only on ✓ and ✓0 , i.e. ↵)1✓ } and let q˜ 2

(✓, ✓0 ) =

(✓, ✓0 ) be such that h˜ q , v(q 0 )i = 0. As v(q 0 ) is a

support hyperplane, we must have h˜ q , i i = 0 for some i = {1, ..., J}. The fact that v(q 0 ) rank ✓ and ✓0 di↵erently implies that either v✓0 (q 0 ) < 0 <

i ✓0

or v✓ (q 0 ) < 0 <

i

and

i ✓.

In

either case, it implies that there is one state that belongs to Wk (as it is approved by the coalition represented by ) but does not lead to approval by voter v(q 0 ). Therefore, v(q 0 ) ⌅

cannot be a supporting hyperplane of co(Wk ) and we reach a contradiction. Proof of Corollary 1: In the text. Proof of Proposition 3:

Part (i)- Follows immediately as any optimal signal under i

unanimity must induce approval of every voter, while an optimal signal for a voter

would

also induce approval if k = 1. Part (ii)- Note that if all

i

2 Fz , then Proposition 1 shows that the structure of the

optimal signal is the same for all voters: if ↵✓ ( i ) = Pr[approval|✓] represents the optimal signal under delegation to voter

i

, where ↵✓ ( i ) is given by (4), then ↵✓ ( i0 )

0

↵✓ ( i ) 0,

✓ 2 ⇥ if V ( i ) V ( i ). This implies that signal ↵( k ) would induce approval for any i < k such that V ( k ) V ( i ). Therefore, the optimal signal to persuade voter

k

has an

approval signal realization that would induce the approval vote of at least k voters. Therefore V (Wk )

⌅

V ( k ).

Proof of Corollary 2: In the text. Proof of Proposition 4: Without loss of generality, suppose that z(i) = i so that for 2 Fz we have

✓i

<

✓i+1 ,

i 2 {1, ..., card(⇥)

optimal signal when the decision is made by voter 40

1}. From Proposition 1, the controller’s 2 Fz is characterized by the approval

conditional probabilites ↵✓ ( ) = Pr[approval|✓] such that there exists i↵ ( ) with (i) ↵✓i ( ) = 0 P if i < i↵ ( ), (ii) ↵✓i ( ) = 1 if i > i↵ ( ), and (iii) ↵✓ ( )p✓ ✓ = 0. Also, for 2 Fz let i( ) = min {i :

✓i

0}. In words, if the realized state is ✓i then voter

proposal under full information as long as i by voter

only if i

would approve the

i( ), while the optimal signal induces approval

i↵ ( ). v

Part (i)- The increment in the expected utility of voter

under delegation to

rather

than choosing always the status quo is U = E[ui (x( ), ✓)] We now show that voter voters ,

0

v

⌦ E[ui (x0 , ✓)] = P (q + ( )) q + ( ),

v

↵

=

X

↵✓ ( )p✓ ✓v .

has single peaked preferences among voters in D. Select two

2 D with A( 0 ) ⇢ A( ). From Proposition 1, this implies that ↵✓ ( 0 )

✓ 2 ⇥. First, suppose that i↵ ( ), i↵ ( 0 ) < i( v ). Then, ↵✓i ( ) = ↵✓i ( 0 ) = 1 if i thus U ( 0)

X

U( ) =

(↵✓i ( 0 )

↵✓i ( )) p✓i

v ✓i

↵✓ ( ) 0,

i0 ( v ), and

0,

i

where the inequality follows from

v ✓i

< 0 if i < i0 ( v ). Second, suppose that i↵ ( ), i↵ ( 0 )

i( v ). Then, ↵✓i ( ) = ↵✓i ( 0 ) = 0 if i < i0 ( v ), and thus U ( 0)

U( ) = i

where the inequality follows from

v ✓i

X

(↵✓i ( 0 )

↵✓i ( )) p✓i

v ✓i

i0 ( v )

0,

i0 ( v ).

0 if i

Finally, divide voters in D into two groups D+ ( v ) = { 2 D : i↵ ( )

{ 2 D : i↵ ( ) < i( v )}. Then, for any ,

0

ers are given by their toughness, while if , are less tough. Therefore,

v

2 D ( v ), voter 0

v

2 D+ ( v ), voter

i( v )} and D ( v ) =

preferences over decision makv

prefers decision makers that

has single peaked preferences over voters in any totally ordered

chain (ordered according to toughness). Part (ii)- Let F (D)z be the set of voters that rank states according to z and who share the same set of approval states D. For any ,

0

2 F (D)z we have that i↵ ( ) < i( 0 ) (equality

is ruled out as voters strictly rank states). In words, if voters both agree on the ranking of states and on decisions under full information, then the controller would provide voter with a signal that always induces approval in states for which voter 41

0

would want to approve.

Therefore, Proposition 4 implies that all voters in F (D)z have monotone preferences over totally ordered chains in F (D)z . v

Part (iii)- The maximum expected gain to voter

(with respect to always selecting the

status quo) if he can design the signal himself is U⇤ =

X

i i0 (

p✓

v

(✓) .

v)

This corresponds to (i) a signal that reveals whether or not a state with a non-negative i0 ( v ) occurred, and (ii) the proposal is selected ⇣ ⌘ in that case. But, this is precisely the signal that the controller provides to a voter ⇤ ˆ, v , net value occurred, i.e. if a state ✓i with i

as to induce approval the controller would need to supply a signal such that E[

⇤

⇣

ˆ,

v

⌘

|s+ ] =

X

ˆ ( ˆ) =

✓

✓2⇥

which implies that ↵✓ = 1 only if v

q✓+

X⇣ ✓2⇥

v ✓

↵✓

1{✓:

v ✓

0}

⌘

p✓

✓

= 0,

0, which corresponds to the optimal signal to voter

⌅

.

Proof of Corollary 3: Corollary 2(i) implies that all voters weakly prefer unanimity with the controller’s influence than rejecting the proposal without further information; assumption p2 / W n+1 and Corollary 2(iii) imply that a majority of voters prefers to reject the proposal 2

without further information than having simple majority under the controller’s influence. ⌅

Corollary 3 then follows immediately.

Proof of Lemma 4: Lemma 3 implies that if all voters are in the same class Fz , then for each k there exists a representative voter

⇤

(k) 2 Fz and, furthermore, A ( ⇤ (k 0 )) ⇢

A ( ⇤ (k)) for k 0 > k. Therefore D = { ⇤ (k) : k 2 {1, ..., n}} forms a totally ordered chain,

and Proposition 4 implies that each voter in the electorate has single-peaked preferences in D. Finally, since

⇤

(k) 2 Fz , Proposition 2 implies that the expected utility of each voter

under a k-voting rule is the same as delegating to single peaked preferences over k.

⇤

(k). This implies that each voter has

⌅

Proof of Lemma 5: Consider the optimal binary signal targeting the weak representative voter

⇤

(Wk ). Let qk+ be the posterior belief after the approval signal. Under simple majority 42

n+1 , 2

rule there is a set M of voters, card(M )

D

+

such that for each 2 M we have q n+1 , 2

E

0. Hence the expected payo↵ of those voters under simple majority is weakly lower than their expected payo↵ from always rejecting the proposal. Moreover, since Wn 6= ;, under unanimity the payo↵ of all voters

2 M is weakly higher than their payo↵ from always

rejecting the proposal, since unanimity implies < qn+ ,

>

0. Therefore all voters in M

weakly prefer unanimity over simple majority. Using Lemma 4, single-peaked preferences over k implies that all voters in M weakly prefer k 0 over simple majority, concluding the proof of part (i). Part (ii) follows from 0 < V (Wk0 ) < V (W n+1 ) because it implies that the 2

0

optimal signal under k is not the same as the signal under simple majority. To see that the set M must exist, suppose by contradiction that it does not exist. Then D E + n+1 n+1 there are at least n + 1 = 2 voters such that q n+1 , > 0. Therefore, after 2 2

observing q +n+1 a majority of voters strictly prefer to approve the proposal, a contradiction 2

to the optimality of the signal.

⌅

Proof of Proposition 5: Proposition 4(ii) applied to Lemma 3 implies that all voters have monotone preferences over voters in D = { ⇤ (k) : k 2 {1, ..., n}} when they all agree under full information. Since

⇤

(k) 2 Fz , Proposition 2 implies that the expected utility of each

voter under a k-voting rule is the same as delegating to has monotone preferences over k.

⇤

(k). This implies that each voter

⌅

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