Discriminatory Persuasion: How to Convince Voters Preliminary, Please do not circulate!

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1 Discriminatory Persuasion: How to Convince Voters Preliminary, Please do not circulate! Jimmy Chan Fei Li and Yun Wang September 4, 2015 Abstract We study a Bayesian persuasion game between a sender and a set of voters. A collective decision is made according to a K-majority rule. The sender can persuade each voter privately to influence her vote. In the optimal persuasion, (1) voters are provided discriminatory (private) and correlated information, (2) more than K voters may be persuaded so that there are multiple possible pivotal events for the sender s preferred action being chosen, and (3) some voters belong to multiple winning coalitions, and they are uncertain about the realized winning coalition given their private information. Keywords: Bayesian Persuasion, Voting, Selective Communication, Private Persuasion, Correlated Signal, Monotone Signal, Generalized Monotone Likelihood Ratio Property. JEL Classification Codes: D72, D83 School of Economics, Fudan University, School of Economics; jimmy.hing.chan@gmail.com Department of Economics, University of North Carolina Chapel Hill; lifei@ .unc.edu Wang Yanan Institute for Studies in Economics, Xiamen University; yunwang@xmu.edu.cn

2 1 Introduction An economic department is hiring a junior professor. The search committee ends up with one promising candidate. The hiring decision will be made by the department according to a majority rule. All faculty members have common interest: they want to hire a good candidate and to avoid a bad one. However, making a wise decision requires information. Due to the specialization in the profession, most faculties lack sufficient information to evaluate the type of the candidate. Suppose that faculties are Bayesian decision makers; they are willing to recruit the candidate only if he is good with a sufficiently large probability. Naturally, some faculties are easy to convince; while others are hard to convince. To convince the latter, it requires more evidence in favor of the candidate. Prior to the vote, the search committee can persuade voters by providing verifiable information. What is the optimal persuasion strategy? Does the committee want to persuade voters privately or publicly? If privately, whom to persuade? The above example is far from rare. To implement a new policy, a government communicates with multiple policymakers; An author of a research paper responses to multiple anonymous referees and journal editors in hope to convince them to accept the paper. In all these examples, the decision is made by a group, and a sender tries to convince a sufficiently large number of group members to support his preferred action. The aim of this paper is to understand the sender s optimal persuasion in such a scenario. We consider a Bayesian persuasion model à la Kamenica and Gentzkow (2011) with a novelty that the final decision is made by a group of N voters. The group has to decide between two actions: a and b. One of the action is chosen if it receives at least K votes. Voters prefer action a in state A and action b in state B. However, no voter knows the state. As they are all Bayesian, they will vote according to their beliefs about the state. A voter votes for a if only if her belief about the state being A is above a threshold (or cutoff belief). Voters have different thresholds. A voter is easier to convince to vote for a if her cutoff belief is lower. The sender wishes to choose a regardless of the state. The sender can influence voters beliefs by providing verifiable information. We endow the sender sufficient flexibility to choose the signal, which is a state-dependent joint distribution of voters individual realized signals. After observing the signal and their individual realized signal, voters apply Bayesian rule and update their beliefs privately. They then choose an action to vote and the collective decision is made according to K-majority rule. Since we assume the sender has perfect commitment power, we can apply the mechanism design approach to derive the sender s optimal persuasion signal. Namely, we examine the upper bound of the persuasion probability that the sender can achieve. As a benchmark, we first consider public persuasion where voters hold common posterior beliefs 1

3 after observing any realized signal. Since voters make decision according to their cutoff beliefs, whenever a hard-to-convince voter is convinced to vote for a, an easy-to-convince voter is convinced as well. Hence, the sender finds optimal to target at the median voter (the Kth easy-to-convince voter). As long as the median voter is convinced, a winning coalition consisting of K voters forms. However, excepts the median voter, the rest members in the winning coalition are over convinced. That is, their posteriors beliefs are strictly above their cutoff ones, which suggests that the sender wastes some information on these voters. Ideally, the sender can switch some wasted information from these easy-to-convince voters to convince additional hard-to-convince voters without affecting the posterior of the median voter. Two questions immediately raise. First, it is useful? After all, the sender only needs K votes to have the desire action. Having additional voters on board does not change the final outcome. Second, how to do it if it is beneficial? Apparently, switching information across voters leads to heterogenous posterior beliefs among voters. In any public persuasion, the mission is impossible. We then turn to study private persuasion in which each voter observes her own realized signal only. We demonstrate that the sender finds it beneficial to send private and correlated signals to voters. More interestingly, the sender prefers to persuade not only the K most easy-to-convince voters but also some more hard-to-convince ones. Voters hold heterogenous posterior beliefs and one of multiple winning coalitions may form. Notice that the advantage of private persuasion is not obvious because private persuasion does not necessarily cause heterogenous posterior beliefs among voters when voters can strategically aggregate spread information. Although we endow the sender sufficient flexibility of information controlling, we do not allow him to send non-monotone signal. Specifically, we assume that the signal must satisfy generalized monotone likelihood ratio property. 1 The condition is equivalent to monotone Bayesian updating. Intuitively, it implies that to convince additional voters to vote for certain action, the realized signal must lead to a larger posterior belief about the corresponding state. We make this assumption for two purposes. First, one can easily implement a non-monotone signal by (perhaps randomly) combining multiple independent simple experiments. A simple experiment is a state-dependent binary distribution and its outcome either favors state A or B. The sender controls the type-i error and type-ii error of each experiment, and each voter observes the outcomes of a set of such experiments. Multiple voters may observe the outcomes of a common set of experiments so that the signal can be positively correlated across voters. Second, the sender has the incentive to send a signal so that voters vote uninformatively with a sufficiently high probability. In such a case, no voter is pivotal with an arbitrarily large probability and the persuasion probability is one 1 In a single-receiver model, this condition becomes the standard monotone likelihood ratio property (Milgrom (1981)). 2

4 regardless of the preferences of voters. By focusing on monotone signals, one can rule out such a trivial case. Due to the strategic information aggregation among voters, the standard concavification methods (Aumann and Maschler (1995) and Kamenica and Gentzkow (2011)) does not apply. As a consequence, we work on the signal space directly. Thanks to the revelation principle, we can focus on state-dependent distribution over recommendation signals of voters. However, the number of variable is still remarkably large: there are CK N pivotal signal for each state.) However, as a voter cannot distinguish signals with identical recommendation to her, her incentive to follow the recommendation is driven by the posterior belief only conditional on the fact that she is pivotal. Motivated by this observation, we solve an auxiliary problem where the sender controls the probabilities of voters being pivotal and characterize the optimal solution. Then we show that the auxiliary problem and the original optimal persuasion problem are equivalent. The main result of the paper is the characterization of the optimal persuasion signal. To efficiently use information, the sender chooses a signal to create multiple pivotal events of action a. Some voters are pivotal for many wining coalitions. By observing her own realized signal, a voter is uncertain about the realized winning coalition. As a result, voters cannot efficiently aggregate spread information conditional on they are pivotal; and thus it gives rise to heterogenous posterior beliefs among voters. Since the sender can control the correlation of the signals that voters receive, he is able to manipulate the way that voters aggregate spread information and convince more voters with less informative signals to achieve a larger persuasion probability. Related Literature. Our paper relates to a growing literature on Bayesian persuasion. Kamenica and Gentzkow (2011) and Rayo and Segal (2010) study bayesian persuasion models between one sender and one receiver. Kolotilin, Li, Mylovanov, and Zapechelnyuk (2015) consider a one-senderone-receiver setting where the receiver has private information about his tastes. The sender is able to provide targeting signal based on the receiver s reported tastes. They refer such a model as private persuasion and model where the signal does not depend on the receiver s report as public persuasion. They show that two models are equivalent. Unlike their model, multiple receivers exist and interact directly in our model. As in our paper, Alonso and Camara (2014), Wang (2012), and Taneva (2014) also consider Bayesian persuasion between one sender and multiple receivers. However, Alonso and Camara (2014) assume that voters have heterogenous preferences on actions given the true state and focus on public persuasion; while we assume voters have the common preference for each state and focus on private persuasion. Wang (2012) studies both public and private persuasion in voting games, but in the private case of her model, the signals is assumed to be i.i.d. across voters, which results in the optimality of public persuasion. By allowing correlated signals, we show private persuasion 3

5 is strictly better than public ones. In a very general framework, Taneva (2014) studies information design by allowing arbitrarily correlated signals among receivers. On the contrary, we focus on signals satisfying generalized monotone likelihood ratio property to avoid a trivial case where the persuasion probability is one. Our paper also relates to the literature of communication with multiple audiences. Caillaud and Tirole (2007) also consider the strategy to convince a group by providing verifiable (hard) information. In their model, the sender chooses the optimal sequence to communicate with receivers. A receiver learns the state through two channels: (1) directly communicating with the sender, and (2) observing other receivers being convinced. They shed light on the advantage of selective communication to key group members and to engineer persuasion cascades in which receivers who are brought on board sway the opinion of others. By choosing the sequence of persuasion, the sender can manipulate receivers beliefs through the persuasion cascades effect, which has the same flavor as in our models. Farrell and Gibbons (1989) and Goltsman and Pavlov (2011) consider cheap talk models with multiple audiences. Unlike our paper, their focus is to examine how does the communication protocol affect the amount of information transmitted from the sender to receivers rather than released (to all parties). More broadly, our paper relates to the informative voting literature. There is a large body of literature considering information aggregation where voters (or committee members ) private information are exogenously given such as Austen-Smith and Banks (1996), Feddersen and Pesendorfer (1997), Feddersen and Pesendorfer (1998), Taylor and Yildirim (2010), Ekmekci and Lauermann (2015) and Li, Rosen, and Suen (2001). The literature has been enriched in many dimensons: Gerardi and Yariv (2007) compare different voting rules when voters are allowed to deliberate before casting their votes. Jackson and Tan (2013) allow voters consults experts before voting and examine how disclosure and voting vary with different voting rule and signal precision of experts. Li (2001), Persico (2004) and Cai (2009) assume that voters (or committee members) endogenously collect their information. The rest of the paper is organized as follows. In section 2, we present the model. In section 3, we provide some preliminary analysis of the optimal persuasion problem. Section 4 studies a three-voter case to illustrate the main idea of the paper. Section 5 consider the general case. Section 6 discuss the robustness of the main result and possible extensions. Section 7 concludes. 2 Model Voters. A group of N voters need to decide between one of the following two actions: x {a, b} 4

6 where b represents the status qua, and a represents the alternative (or risky) action. The collective decision is made according to a K-majority rule where K < N: action x is chosen if it receives at least K votes. We assume that the relevant state is binary: ω {A, B}. Voters are uncertain about the state and share a common prior belief µ 0 = Pr(ω = A) (0, 1). They want to match the state. Specifically, if the final decision is b, each voter obtains 0; otherwise, a voter s payoff is characterized by a von Neumann-Morgerstern utility function µ i (1 µ i )l i where µ i [0, 1] is her belief about the state being A and l i > 0: she obtains one unit benefit if ω = A; while she suffers l i unit loss if ω = B. The value of l i measures voter i s threshold of doubt of approving the risky action. While voters hold identical preference over actions conditional on the true state, their preference may be different for a given belief µ due to the differences in their threshold of doubt. In order to convince a voter with higher l i, one needs to provide more evidence to raise her belief about the state being A. We assume that l 1 < l 2 <... < l N, so l 1 -voter is the most easy-to-convince one; while l N is the most hard-to-convince one. We assume that µ 0 < (1 µ 0 )l K, so b is the default collective action without receiving further information. Persuasion Technology. A sender prefers the risky action regardless of the state. He chooses a signal to influence voters beliefs and therefore the collective decision. Following Kamenica and Gentzkow (2011), we assume the signal is costless and observed by voters. Formally, Definition 1. A signal consists of a set of finite realization space {S i },2,...N and a pair of probabilities {π( ω)} ω=a,b (,2,..N S i ) where S i denotes the realization space for voter i. A signal realization consists of N observations: s 1,...s N where s i S i, i = 1, 2,..., N. We assume that each voter can only observe her own signal realization rather than others. However, 5

7 as signals are allowed to be arbitrarily correlated across voters conditional on the state, voters may be able to infer others realized signals after receiving their owns. Naturally, a signal realization s = (s 1, s 2,...s N ) s.t. {s i S i },2,..N causes posterior beliefs: µ i (s i) = µ 0 s:s i =s π(s A) i µ 0 s:s i =s π(s A) + (1 µ 0 ) i s:s i =s π(s B) i for each voter i = 1, 2,...N. We allow signals to be arbitrarily correlated across voters given the realized state. In general, since {s i S i s i = s i} and {s i S i s j = s j} may not be identical for i j, voters do not necessarily hold common posterior beliefs. The sender s goal is to maximize the persuasion probability, that is, the probability of the collective decision being a. As the signal is observed by voters, one can focus on the revelation (recommendation, straightforward) signals where S i = {a, b}, i = 1, 2,..N, by applying the revelation principle: a voter only observes a recommendation for her action. 2 Furthermore, we focus on signals satisfying generalized monotone likelihood ratio property (GMLRP): (1) π(s s A)π(s B) π(s A)π(s s B) (GMLRP) for any realization s, s {a, b} N where s s = ( s 1, s 2,... s N ) s.t. s i = a unless s i = s i = b, i = 1, 2,...N. An immediate implication of the GMLRP assumption is that, for, s, s {a, b} N and any given prior µ 0, the posterior belief about the state being A caused by s s is greater than the one caused by s or s. Intuitively, to convince more voters to vote for a, it requires more good news favoring state A. Voting Strategies and Equilibrium. Given a recommendation signal {π( ω)} ω=a,b, one can define a voting game with private information. A pure strategy for voter i is a measurable function from the space of realized signals to the space of actions, i.e., σ i : {a, b} {a, b}. We say voter i votes uninformatively if σ i (a) = σ i (b). Namely, a voter votes for certain action regardless of her signal. In voting games, there always exist trivial equilibria where at least K voters vote uninformatively. In our model, the issue is more severe because the sender can send uninformative signal such as π(a, a,...a ω) = 1 for both ω = A and B. that voters vote for a only if she strictly prefer to do so. 3 To avoid a trivial case, we assume In addition, whenever there exist multiple Bayesian Nash equilibria, we select the one which maximizes the sender s payoff. Hence, to motivate voters to vote for certain action x, the sender may have to chose an informative signal to influence their beliefs. 2 The argument is a straightforward application of that in Proposition 1 of Kamenica and Gentzkow (2011), so it is omitted here. 3 This assumption can be motivated by small cost on voters to change the status quo. 6

8 Definition 2. Given a signal {π( ω)} ω=a,b, a voting equilibrium {σi },2,...N is an sender s optimal Bayesian Nash equilibrium of the voting game where σ i ( ) = a only if voter i strictly prefers a to b. Taking voters equilibrium strategies as given, the sender chooses a signal to maximize the probability that the risky action is chosen in the continuation voting game. We look at optimal signal in the closure of the set of feasible signals. 3 Optimal Persuasion Problem We begin by introducing some notations. Denote S x {a, b} N as the set of x-wining signals in which there are at least K voters realized signals are x where x = a, b. Let Sx S x be the set of pivotal x-wining realized signals in which there are exactly K voters realized signals are x., and S i,x = {s S s i = x} be the set of realized signals that voter i s realization is x {a, b}. Furthermore, denote Si,x as the set of pivotal x-wining signals for voter i in which 1. Si,x Sx, and 2. s i = x: voter i is a pivotal voter. 3.1 Incentive Compatible A voter prefers certain action only if the realized signal is sufficiently convincing to support the corresponding state. Suppose that voter i observes a recommendation s {a, b} N, she prefers action a if her posterior belief is large enough, i.e, µ 0 π(s A) µ 0 π(s A) + (1 µ 0 )π(s B) l i 1 + l i, or π(s A)/π(s B) l i (1 µ 0 )/µ 0 in terms of likelihood ratio. 4 Intuitively, a realized signal s is sufficiently convincing for action a if the true state is sufficiently unlikely to be B. In the extreme case, if π(s B) = 0, s is a perfect signal for state A; and thus a voter prefers action x for any l i. A similar logic applies to convince a voter to choose action b. In a strategic voting game, voter i only observes her own realized signal s i {a, b}. Since the signal π( ω) is observable, the voter can infer others realized signal s i {a, b} N 1. In addition, the voter s decision is relevant to the outcome only if she is pivotal. In such a scenario, to convince 4 Recall that we look at the closure of the set of feasible signals, thus the effective incentive compatible constraint takes the weak inequality. 7

9 voter i to vote for a, the sender needs to influence her belief to be higher than or equal to l i /(1+l i ). Formally, for each i such that one needs to ensure that µ i pivotal(a) = s S i,a π(s ω) > 0 µ 0 s S i,a π(s A) µ 0 s S π(s A) + (1 µ i,a 0 ) s S π(s B) i,a l i 1 + l i (IC-a) where s S i,a π(s ω) represents the probability of voter i being pivotal for action a in state ω. That is to say, to motivate voter i to support action a, one needs to convince her that her vote is relevant with a positive probability, and conditional on her being pivotal, the state is A with a sufficiently high probability. s S i,a Whenever it is well-defined, the likelihood ratio π(s A) measures the convincingness of s S i,a π(s B) the signal to voter i s vote for action a. The higher the likelihood ratio is, the more convincing the signal is to voter i. To ensure the collective decision is a, at least K voters incentive compatible conditions (IC-a) need to be satisfied. Similarly, if the realized recommendation signal is b, the voter follows the recommendation if and only if µ i pivotal(b) = 3.2 Simplifying the Problem µ 0 s S i,b π(s A) µ 0 s S π(s A) + (1 µ i,b 0 ) s S π(s B) i,b l i 1 + l i (IC-b) The sender s optimal persuasion problem is max µ 0 π(s A) + (1 µ 0 ) π(s B) π( ω) s S a s S a (P-0) s.t. π( A), π( B) (S), condition (GMLRP), condition (IC-a) for each voter i whenever s S i,a π(s ω) > 0 for ω = A or B (she is persuaded to vote for a with a positive probability) and condition (IC-b) for each voter i if s S i,b π(s ω) > 0 for ω = A or B (she is persuaded to vote for b with a positive probability). Notice that, for a given signal {π( ω)} ω=a,b, one only satisfy the incentive compatible constraints (IC-a) and (IC-b) of voters who are recommended to vote for the corresponding action with positive probability. Suppose that the sender never persuades voter i to vote for a. Then her posterior µ i pivotal (a) is not well-defined and her incentive compatible constraint (IC-a) is irrelevant. The following lemmas provide some characterizations for solutions of problem (P-0) which simplifies the analysis. 8

10 Lemma 1. In the optimal persuasion, π(s A) = 0, s S b so that s S a π(s A) = 1. Proof. By setting π(s A) = 0, s S b, incentive compatible constraint (IC-b) is trivially satisfied. Also, any signal s S b is a good news for action b so that the monotonicity condition for signals are still satisfied. As a result, one can set s S a π(s A) = 1 to raise the value of the objective function. Recall that the sender may recommend his less preferred action b in state B only to convince voters to follow the recommendation a. In state A, action b is Pareto inferior, so the sender finds suboptimal to recommend it. Clearly, Lemma 1 holds because the sender is able to choose (and commit to) a state-contingent signal. Thanks to Lemma 1, in any optimal persuasion, the persuasion probability is one in state A, and thus the sender s objective function effectively becomes s S a π(s B). As one can always renormalize l i = l i 1 µ 0 µ 0 without affecting the solution, in the reminder of the paper we assume that µ 0 = 0.5. Assumption 1. State A and B are ex ante equally likely, i.e., µ 0 = 0.5. The optimal persuasion problem can be further simplified by taking advantage of the GMLRP assumption. Lemma 2. In an optimal persuasion, π(s A) = π(s B) = 0, s S a /S a. Proof. In any incentive compatible persuasion signal, s S i,a π(s A) l i s S i,a π(s B) for each i = 1, 2,...N by constraint (IC-a). Hence, for each i, there exists at least one pivotal signal s S i,a such that π(s A) l i π(s B). By the monotonicity constraint (GMLRP), for any s S a /S a and there exists a pivotal s S i,a, π(s A)π(s B) π(s A)π(s B). Hence, by raising π(s ω) by π(s ω) for ω = A, B, the sender can be weakly relax voter i s incentive compatible constraint where s appears without affecting others. Lemma 2 further narrows down the set of recommendations used in an optimal persuasion signal: to convince the group to take the collective action a, the sender can focus on pivotal signals in S a. 5 The result directly results from the GMLRP assumption. To convince more voters, the signal must be more convincing. Hence, action a will be less likely recommended in state B, which is obviously undesirable from the sender s perspective. As the sender only needs to convince K voters support to choose action a, it is inefficient to use non-pivotal signal s S a /S a and convince more. 5 Notice that Lemma 2 has no implication on the use of non-pivotal signal for action b. 9

11 Furthermore, the results in Lemma 1 and Lemma 2 together implies that only pivotal recommendations s Sa are effectively adopted, and therefore the GMLRP are automatically satisfied in an optimal persuasion signal. Corollary 1. In an optimal persuasion signal where π(s A) = π(s B) = 0, s S a /Sa, constraint (GMLRP) is not binding. Proof. The result immediately comes from Lemma 1 and Lemma 2. As a consequence, one can focus on pivotal persuasion, and problem (P-0) becomes max π( ω) (S) s Sa π(s B) π(s B) l i π(s A). (P-1) (IC-a ) s S i,a s S i,a In problem (P-1), the incentive compatible condition (IC-a ) holds for each voter no matter she will be persuaded or not. For each voter whom the sender wants to persuade, s S π(s B), i,a s S π(s B) > 0, so condition (IC-a ) is equivalent to (IC-a). For those voters whom the i,a sender does not want to persuade, s S π(s B) = i,a s S π(s B) = 0, so condition (IC-a ) i,a trivially holds. Notice that this simplification works because we focus on pivotal persuasion. In the setting where (GMLRP) is not required, Lemma 2 fails, and the sender may find optimal to set π(s B) > 0 for s S a /S a. Then condition (IC-a ) is insufficient to ensure voters use unweakly dominated strategy to vote for a. For example, one can set an uninformative signal π(s ω) = 1 where s = (a, a,...a) for ω = A, B. As π(s ω) = 0 for s S i,a, ω = A, B, condition (IC-a ) trivially holds. However, none of N voters is pivotal; thus no voter is persuaded to vote for a. 4 A Three-Voter Case In this section, we study a simple example where N = 3, K = 2 to illustrate the main idea of the paper. Recall that we assume l 2 > 1 to avoid a trivial case. The sender s problem is max{π(aab B) + π(aba B) + π(baa B)} (2) π( B) 10

12 s.t. π(aab B)) + π(aba B) 1 l 1 [π(aab A) + π(aba A)] (3) π(aab B) + π(baa B) 1 l 2 [π(aab A) + π(baa A)] (4) π(aba B) + π(baa B) 1 l 3 [π(aba A) + π(baa A)] (5) First, we consider a simple way to persuade voters: sending a public signal so that all voters observe the common realized signal, so there is only one pivotal event. Whenever a hard-toconvince voter is willing to choose a, an easy-to-convince voter is automatically convinced as well. As a consequence, there are only four incentive compatible recommendations: {bbb, abb, aab, aaa}. Obviously, the sender finds optimal to persuade the two easy-to-convince voters only. The corresponding revelation mechanism consists of two recommendation signals: aab and bbb. Voter 1 and voter 2 vote for both a and b with positive probabilities; while voter 3 always votes for b. In the optimal public persuasion, voter 2 s IC constraint (4) is binding so that π(aab B) = π(aab A)/l 2 = 1/l 2. Since l 1 < l 2, voter 1 s IC constraint (3) must be slack. In addition, voter 3 s IC constraint (5) is trivially satisfied as she is never pivotal. In the optimal public persuasion, there is only one winning coalition consisting of voter 1 and 2, and there is only one pivotal recommendation signal for the risky action: aab. The risky action is chosen with probability µ 0 + (1 µ 0 )1/l 2. Proposition 1. Public persuasion is strictly suboptimal. Proof. We prove the result by finding another signal which strictly increase the probability that the risky action is chosen ( in state B. In the optimal public persuasion, (3) is slack and (4) is binding. Take ɛ = l 1 1 l 2 ). Let ˆπ(aab A) = 1 ɛ, ˆπ(baa A) = ɛ and ˆπ(aba B) = δ = min{(1 ɛ)/l 1, ɛ/l 3 } > 0. The new signal satisfies (3), (4), and (5), and the risky action is chosen with probability 1/l 2 + δ. So, public persuasion is suboptimal. The idea is to reallocate evidence across voters to manipulate their posterior beliefs conditional on they are pivotal one-by-one. Recall that to convince voter i to choose action a, one only needs to ensure that her posterior belief by seeing recommendation a is just above l i /(1 + l i ). As l i varies across voters, an efficient persuasion requires voters hold different posterior beliefs. 11

13 However, in the public persuasion, this is impossible since there is only one winning coalition consisting of voter 1 and 2. While voter 2 is just convinced to choose a, voter is over convinced, which means the sender wastes some convincingness on voter 1. Ideally, one would like to reallocate those redundant convincingness from voter 1 to voter 3 to create additional winning coalitions: one consists of voter 1 and 3 with a recommendation aba, and another one consists of voter 2 and 3 with a recommendation baa. If persuasion is private, such a reallocation of convincingness across voters is possible because multiple pivotal signals for action a can exist. By observing her own signal, a voter is uncertain about the pivotal signal and therefore the winning coalition that she is involved. Hence, there can exist pivotal event where a hard-to-convince voter is willing to choose a, but an easy-to-convince voter is not because the later does not know who else are in her winning coalition. For example, one can set π(aab ω), π(aba ω), π(bbb ω) > 0 for some ω, then there are three pivotal signals for action a, and each voter is involved in two winning coalition. The signal realization aba is incentive compatible because voter 2 is uncertain if voter 3 is choosing action a. As a result, voters posterior beliefs are not necessarily identical even they all observe the same recommendation. As discriminatory persuasion is feasible, the sender can reallocate redundant evidence from the easy-to-convince voter to the hard-to-convince voter to strictly increase persuasion probability. Proposition 2. When l 1 > l 3 (1/l 2 1) + 1, an optimal persuasion signal satisfies π(baa A) = l 3l 2 l 3 l 1 l 1 l 2 + l 2 l 3 ; π(baa B) = 0; π(aba A) = 0 ; π(aba B) = l 2 l 1 l 1 l 2 + l 2 l 3 ; π(aab A) = l 1l 3 + l 1 l 2 l 1 l 2 + l 2 l 3 ; π(aab B) = 1 l 2. l 3 +l 2 In state B, the persuasion probability is which is strictly decreasing in l l 2 (l 1 +l 3 ) i, i = 1, 2, 3, and s S b π(s B) > 0. When l 1 l 3 (1/l 2 1) + 1, the persuasion probability is one. Proof. See Appendix. Proposition 2 implies that the persuasion probability can achieve one as long as 0 < l 1 l 3 (1/l 2 1) + 1. Notice that when l 2, l 3 are sufficient large, l 3 (1/l 2 1) + 1 < 0, so the persuasion probability is less than one no matter how small l 1 is. Hence, the persuasion probability achieves one only if none of three voters are sufficiently hard-to-convince. More importantly, as l 2 > 1, the condition requires that l 1 < 1. Namely, voter 1 does not need further persuasion to vote for a. In fact, in the optimal persuasion mechanism, the sender will switch some evidence from voter 1 to 12

14 others so that she is just convinced by seeing a realization signal a conditional on she is pivotal: µ 1 pivotal (a) π(aab A) + π(aba A) = 1 µ 1 pivotal (a) π(aab B) + π(aba B) = l 1 but not convinced by seeing a realized signal b conditional on she is pivotal: µ 1 (b) = 0. 6 On the other hand, voter 2 and 3 will obtain more evidence so that they are also just convinced by seeing recommendation signal a respectively. When the persuasion probability is one, there is a continuum of signal maximizing the sender s payoff, so the optimal persuasion rule are not effectively sensitive to parameters. Hence, in the reminder of this section, we focus on the interesting case where the persuasion probability l 3+l 2 l 2 (l 1 +l 3 ) < 1. In such a case, persuasion probability in state B is strictly decreasing in each l i. This is quit intuitive. As l i increases, voter i becomes more cautious so that she needs more evidence to support the state being A. As s:s i =a π(s A) is bounded by 1, the sender has to reduce s:s i =a π(s B), which reduces persuasion probability. As we discussed before, it is critical that voters are uncertain about the winning coalition they are involved given each realized signal. In the three-voter case, there exists multiple possible winning coalitions only if voter 3 is involved, and voter 3 is involved if her voting is informative. Since lim l3 π(baa A) = (l 2 l 1 )/l 2, lim l3 π(aab A) = l 1 /l 2, voter 3 is always involved in the optimal persuasion no matter how hard-to-convince she is. Interestingly, lim l1 l 2 π(baa A) = lim l1 l 2 π(aba B) = 0, and lim l1 l 2 π(aab A) = 1. That is to say, the benefit of private persuasion disappear as voter 1 and voter 2 becomes equally easyto-convince. The reason behind is intuitive. As voter 3 is more hard-to-convince, it is optimal to convince her only if (1) voter 1 and voter 2 are both convinced, and (2) there is redundant evidence provided, so one of them are over convinced. When l 1 = l 2, one can just convince both of them by using public persuasion, so there is no room to improve by having voter 3 being involved without affecting the convincingness of others signals. Notice that, in the optimal persuasion, all voters IC constraints are binding because they do not know which pivotal event she is involved in, and π(aab A) π(aab B) (l 1, l 2 ); π(aba A) π(aba B) = 0; π(baa A) π(baa B) =. That is to say, if a voter can learn the pivotal event she is involved in, she may become not convinced or over convinced. For example, by seeing recommendation a, voter 2 cannot tell aab 6 Notice that both µ 1 pivotal (a) and µ1 pivotal (b) < µ 0. It is consistent with Bayes s rule because they are posterior beliefs conditional on voter 1 being pivotal. s µ1 non-pivotal (s) > µ 0 so that Bayesian plausibility (Kamenica and Gentzkow (2011)) holds. 13

15 from baa. In the first pivotal event, she is in the winning coalition with voter 1, and because π(aab A) π(aab B) < 1, she is not convinced; while in the second pivotal event, she is in the winning coalition with voter 3, as π(baa A) =, she is over convinced. Similarly, voter 3 is over convinced in pivotal π(baa B) event baa but not convinced in event aba. Consequently, although voters are willing to follow the sender s recommendation individually, they will refuse to do so if all recommendation signal are common knowledge; and thus, we conclude that, in the optimal persuasion, voting does not fully aggregate information in the sense of Feddersen and Pesendorfer (1997). Corollary 2. In the optimal persuasion, voting does not fully aggregate information. In the optimal persuasion, the sender deliberately create uncertainty about the pivotal events voters are involved in to further exploit voters willingness to approve the risky action. However, as voters will regret to follow the recommendation after knowing the pivotal event, the optimal persuasion is not voter-communication-proof. This observation should not bring any surprise. In the extreme case, if all voters can freely communicate their realized signals, private persuasion cannot do any better than public persuasion. 5 General Case In this section, we seek the optimal persuasion signal in a general setting with more than three voters. The main result is that the optimal persuasion follows a cutoff rule: the sender only tries to influence the beliefs of voters who are sufficiently easy-to-convince. Some of them will be convinced with positive probability; Voters who are hard-to-convince never vote for action a. Remarkably, to create multiple possible winning coalitions, more than K voters will be persuaded with positive probability. Proposition 3. Suppose that K < N. In the optimal persuasion, there is a positive integer i such that 1. s Si,a such that π(s A), π(s B) > 0 for i i, 2. π(s A), π(s B) = 0, s Si,a for i > i, and 3. K < i N, In the rest of this section, we prove the above result. The main challenge is that the number of choice variable is too large although it has been significantly reduced by Lemma 1 and Lemma 2. In principle, given N and K, there are 2CK N pivotal signals, and the number of variable increases geometrically as N grows. 14

16 Notice that in problem (P-1), voter i cannot distinguish different realized signal s Si,a. Her incentive to follow the recommendation a relies on π(s ω) only through the total probability about her being pivotal. A natural idea is to solve the optimal persuasion problem by choosing the probabilities that each voter being pivotal, then pin down the optimal signal π(s ω) for s Si,a. Motivated by this idea, consider the following problem: max Q B,α,β Q B (P-2) s.t. α i l i (Q B β i ), i (6) N α i K (7) β i = (N K)Q B (8) i α i [0, 1], β i [0, Q B ], i, and Q B [0, 1] (9) where Q B = s S a π(s B) represents the persuasion probability in state B, α i = s S i,a π(s A) is the probability that i is pivotal for action a in state A, and β i represents the probability that a is chosen but i is not pivotal in state B. Because of Lemma 2, in any optimal persuasion, Q B = β i s S π(s B). Hence, (6) is a reformulation of (IC-a ), (7) and (8) hold because each i,a pivotal signal is shared by K voters, and (9) are the feasibility constraint. Essentially, for each voter, we resemble her pivotal events together in problem (P-2). Obviously, for any {π(s ω)} s S i,a solves problem (P-1), the corresponding {α i, β i } i, Q B automatically solves problem (P-2). The following lemma shows that the opposite direction is also true. Lemma For any {α i } n with α i [0, 1], i and j α j = K, there exists a probability distribution π ( A) ({a, b} N ) such that s S a π (s A) = 1, and s S ia π (s A) = α i. 2. For any Q B [0, 1] and {β i } N with β i [0, Q B ], i and j β j = (N K)Q B, there exists a probability distribution π( B) ({a, b} N ) such that s π(s B) = Q B, and 1 b s S π(s B) = β i,b i. Proof. See Appendix. In consequence of Lemma 3, we can focus on problem (P-2) without loss of any generality. The following result generalizes the observation of Proposition 1 in the three-voter case. The intuition is similar, so it is removed here. Lemma 4. Public persuasion is strictly suboptimal under K-majority rule. 15

17 Proof. See Appendix. To solve problem (P-2), we first problem (P-3) by fixing Q B [0, 1] as a parameter and consider the following problem: U(Q B ) = min α,β N β i (P-3) s.t. l i Q B α i + l i β i, i (10) α K, α i [0, 1]; β i [0, Q B ], i (11) i Notice that problem (P-3) is not the dual of problem (P-2) unless constraint (8) is also satisfied. In the rest of this section, we first characterize the solution of problem (P-3) for a given Q, then we impose constraint (8) to find the solution of problem (P-2). In a solution of problem (P-3), (10) is binding for each i, otherwise, one can decrease α i without raising the value of the objective function; thus one can substitute out β i so that the sender s problem is to choose {α i },2,...N to maximize N (Q B α i /l i ) such that constraint (11). Lemma 5. For any Q B [0, 1], the solution of problem (P-3) satisfies where and 0 if i > i + 1, ˆα i (Q B ) = min{1, K i j=1 min{l jq B, 1}} if i = i + 1, min{l i Q B, 1} if i i. i = max{i N (12) i min{l j Q B, q} < K}, (13) j=1 The value function U(Q B ) is strictly increasing in Q B. Proof. See Appendix. ˆβ(Q B ) = Q B ˆα(Q B) l i. (14) Lemma 6. Suppose that the objective function of problem (P-2) takes value at Q B [0, 1] at optima. (ˆα(Q B ), ˆβ(Q B )) solves problem (P-2) and U(Q ) (N K)Q B where ˆα( ), ˆβ( ) are defined in (12) and (14). When Q B < 1, U(Q ) = (N K)Q B, and (ˆα(Q B ), ˆβ(Q B )) is the unique solution of problem (P-2). 16

18 Proof. See Appendix. Lemma 5 and Lemma 6 together imply that, in the optimal persuasion, βi = Q B as long as αi = 0. Hence, if voter i is never pivotal for action a in state A, she is never pivotal in state B. Furthermore, the optimal persuasion follows a cutoff rule. The sender only influences the beliefs of i most easy-to-convince voters. By Lemma 4, we also know that i > K so that there exist multiple possible winning coalitions. Hence, we immediately have the result in Proposition Comparative Statics We now analyze how the persuasion probability (in state B) are affected when (1) it requires more votes for the alternative action, and (2) voters become more hard-to-convince. As in the three-voter case, we are interested in the setting where the persuasion probability is less than one so that both persuasion probability and the optimal signal response to a slight change in parameters. The following lemma identifies a sufficient condition for such a scenario. Lemma 7. If l i > 1, i, Q B < 1. Proof. See Appendix. Lemma 7 says that when each voter needs to be persuaded to vote for a, the persuasion probability is bounded away from one. The condition is intuitive. If each voter needs to be persuaded, a convincing signal will recommend her action b with positive probability. Hence, the persuasion probability cannot be one. Proposition 4. When the persuasion probability is less than one, it is strictly decreasing in K and is strictly decreasing in l i, i s.t. αi 0. Proof. See Appendix. The intuition behind Proposition 4 is simple. As K increases, the sender needs to persuade more voters to vote for a in each winning coalition. As a result, additional hard-to-convince voters must be convinced, which costs a higher probability that b is chosen. As the choice of a nonpivotal voter is irrelevant to the collective decision, the persuasion probability is not affected by the change in the preference of such voters. On the other hand, when a pivotal voter becomes more hard-to-convince, the sender has to make the signal more convincing. Namely, he has to reduce the probability that a is recommended in state B, which causes the result. 17

19 6 Discussion Unanimous Rule. In the baseline model, we assume that K < N. What happens if the collective decision is made through the unanimous rule (K = N)? In such a case, there is a unique feasible pivotal event for action a where all voters vote for it. Hence, the seller is not able to create multiple pivotal event. Although the sender can privately communicate with each voter, they will strategic aggregate their private information conditional on being pivotal; and thus voters will hold common posteriors in the unique winning coalition. As a consequence, the advantage of private persuasion no longer exists. Non-Monotone Signals. Another critical assumption is that signals must satisfy generalized monotone likelihood ratio property. Thanks to this assumption, we can focus on pivotal persuasion (Lemma 2). One may wonder what happens if we open the box of non-pivotal persuasion? It turns out that the problem becomes trivial: the persuasion probability can be arbitrarily closed to one for regardless of voters preferences. To understand the result, imagine that the sender commit to the following signal. With probability 1 ɛ, the sender uses a manipulation signal that voter i = 1, 2,..., K, K + 1 are recommended to vote for a regardless of the state. With a complementary probability, the sender uses a pivotal signal as in Proposition 3 to persuade voter 1,2,...,i where i K. Voter i cannot distinguish the manipulation signal from a pivotal signal when her individual recommendation s i = a for i = 1, 2,..., K + 1. In the formal case, she is non-pivotal provided that other voters i i and i = 1, 2,...K + 1 vote for a; while the in the latter case, she weakly prefer to vote for a as the signal is sufficiently convincing. Hence, she is willing to follow the recommendation a as long as she is pivotal with positive probability. As a result, the persuasion probability is higher than 1 ɛ. By sending ɛ to zero, the persuasion probability achieve one. Persuasion in Large Elections. TBD Non-Correlated Signals. TBD 7 Conclusion TBD 18

20 A Appendix To prove Proposition 2, we start with the assumption that the persuasion probability is less than one in state B and characterize the corresponding solution, then we find the condition under which the persuasion probability is less than one in state B to complete the proof. The following two lemmas characterize the optimal solution of (2) by assuming the persuasion probability is less than one. Lemma 8. In the optimal solution, constraint (3-5) are binding. Proof. Suppose that (5) is slack, then either π(aba A) or π(baa A) > 0. In the formal case, one can reduce π(aba A) by ɛ, and increase π(aab A) by ɛ so that constraint (3)-(5) are all slack. Then one can increase the persuasion probability by raising π(aab B). In the latter case, a similar logic applies. Hence, in the optimal solution, (5) is binding. One can use the same logic to show that both constraint (3) and (4) are binding. Lemma 9. In the optimal solution, 1. π(baa B)π(baa A) = 0, and 2. π(aba B)π(aba A) = 0. Proof. First, suppose not and π(baa B), π(baa A) > 0. In this case, one can 1. decrease π(baa B) by ɛ and π(baa A) by l 1 ɛ, 2. increase π(aab B) by ɛ and π(aab A) by l 1 ɛ. When > 0 is sufficiently small, it leads to the persuasion probability unchanged and constraint (5) slack. By lemma 8, it is suboptimal, so π(baa B)π(baa A) = 0 is untrue in any optimal persuasion. Similarly, in the optimal persuasion, π(aba B), π(aba A) > 0 cannot hold. By Proposition 1, public persuasion is strictly suboptimal, so π(baa B) = π(aba B) = 0 is untrue, and π(baa A) = π(aba A) = 0 is untrue. As a result, Corollary 3. In the optimal solution, 1. either π(baa B) = π(aba A) = 0, or 2. π(aba A) = π(baa B) = 0. 19

21 Proof of Proposition 2. By Corollary 3, to search the optimal persuasion, one can focus on two cases. We start with the case where π(baa B) = π(aba A) = 0. Since constraint (3-5) are binding, one can solve the optimal solution of this case by choosing two variables: π(aab A) and π(baa A). The solution is given by π(baa A) = l 3l 2 l 3 l 1 l 1 l 2 + l 2 l 3 ; π(aab A) = l 1l 3 + l 1 l 2 l 1 l 2 + l 2 l 3 ; π(aba B) = l 2 l 1 l 1 l 2 + l 2 l 3 ; π(aab B) = 1 l 2 l and the persuasion probability in state B is 3 +l 2, which is positive for any {l l 2 (l 1 +l 3 ) i} i. In the case where π(aba A) = π(baa B) = 0, one can solve the optimal signal as well, but the persuasion probability in state B is 0. Hence, the solution in the first case admits the optimal signal of the original problem. Simple algebra implies that 1. the persuasion probability in state B is less than one if l 1 > l 3 (1/l 2 1) l 3 +l 2 l 2 (l 1 +l 3 ) is strictly decreasing in l i, i = 1, 2, 3. Hence, when l 3+l 2 1, the above signal is the optimal solution. In the case where l 3 +l 2 > l 2 (l 1 +l 3 ) l 2 (l 1 +l 3 ) 1, one can find (l 1, l 2, l 3) l 3 s.t. l i l i for i = 1, 2, 3 such that +l 2 = 1. In the model where voters l 2 (l 1 +l 3 ) preferences are characterized by {l i}, denote the optimal signal {π( ω) } ω=a,b and the persuasion probability is one. Obviously, {π( ω) } ω=a,b is also optimal in the original model with voters preference {l i }. Proof of Lemma 3. We prove part 1. The argument for part 2 is identical. Let h = C N K the number of pivotal signals. It is convenient to state the proposition in matrix form. Let s [1],..., s [h] be an order of the pivotal signals for voting a. Let θ = (1 α 1,..., 1 α n ). Let W be a N h matrix with W ij = { 1 if s [i] (j) = b, 0 if s [i] (j) = a. We need to show that there is a h-th vector π A = (π 1A,..., π ha ) such that i π ia = 1 and π ia [0, 1], and W π T A = θ T. Suppose by way of contradiction that no such π A exists. By Farka s lemma, there exists a n-th vector λ = (λ 1,..., λ N ) such that W T ij λ T 0; (15) θλ T < 0. (16) 20

22 Note that the row of Wij T that corresponds to the signal profile where players 1 to N K observe b begins with N K ones followed by K zeros. Thus, (15) implies that N K λ i 0. Since the player ordering is arbitrary, we can assume without loss of generality that λ i is ascending in i. Hence = min x i N K N λ i x i s.t. x i [0, 1] i, λ i 0. Since α i [0, 1] for all i, and N x i = N K, which contradicts (16). N λ i (1 α i ) 0, N x i = N K, Proof of Lemma 4. Suppose not. In the optimal persuasion, α i = Q A = 1, β i = 0, i = 1, 2,..., K and α i = 0, β i = Q B, i > K, there is only one pivotal event. As l 1 < l K, to satisfy voter K s IC, voter 1 s IC must be slack. Other pivotal voters i IC are satisfied because l i l K, i < K. For non-pivotal voters, their IC are also trivially satisfied. Also, it is obvious that the feasibility constraints for α and β are satisfied: and Q B < 1. K α i = K, N α i = 0, i=k Now consider the following procedure. K β i = 0, 1. increase α 1 byɛ so that voter 1 s IC is still slack. N β i = (N K)Q B 2. decrease α i by ɛ/k for i = 2, 3,...K + 1 so that N α i = K and voter i s IC are slack. 3. one can increase Q B by fixing β i, i K + 1 and increasing β i, i > K + 1 accordingly. i=k As a result, it is suboptimal to convince i = 1, 2,...K only. 21

23 Proof of Lemma 5. The Lagrangian function of (P-3) is n ( L = Q B α ) ( n ) n i + ρ α i k ψ i α i + l i n ξ i (α i min (l i Q B, 1)). (17) The Kuhn-Tucker conditions for α i are that L α i = 1 l i + ρ ψ i + ξ i = 0 i, and ρ, ψ i, and ξ i be positive (zero) if the corresponding constraints are binding (non-binding). Hence, for all i 0 if l i ρ > 1, α i = [0, min (l i Q B, 1)] if l i ρ = 1, min (l i Q B, 1) if l i ρ < 1. Let i = max{i N i j=1 min{l jq B, 1} < K} and set ρ = 1 li +1. Because l i is increasing, Hence, {α i },2,...N satisfies (12). ρl i < 1 if i i, ρl i > 1 if i > i + 1 Let U (Q B ) denote the solution to P3. Further define i = max{i N l i Q B 1}, and let î = min{i, i }. By the envelop theorem, we have du dq B = L Q B = N = N î î ( 1 l ) i lî ( ) 1 ρ l i l i > N î 0. (18) Proof of Lemma 6. Recall that l K > 1. When Q B = 1/l K, we can set { 0 if i > K, α i = l i l K if i K. so that the objective function takes the value (N K) 1 l K particular, N K l i α i = < K l K and all constraints are satisfied. In 22