THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS CARL HEESE. University of Bonn STEPHAN LAUERMANN. University of Bonn

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1 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS CARL HEESE University of Bonn STEPHAN LAUERMANN University of Bonn Abstract. This paper studies the Bayes correlated equilibria of large majority elections in a general environment with heterogeneous, private preferences. Voters have private signals and a version of the Condorcet Jury Theorem holds when voters do not receive additional information. We show that any state-contingent outcome can be implemented in Bayes-Nash equilibrium by some expansion of the given private signal structure. We interpret the result in terms of persuasion and show that persuasion does not require detailed knowledge of the distribution of voters preferences and one expansion of the private signal structure can be used uniformly across environments. Another interpretation is that an outside analyst who only knows that voters receive a minimal amount of private information cannot make a robust prediction on the election outcome. It is very hard to make predictions, especially about the future. Danish proverb Date: June 6, 208. We are grateful for helpful discussions with Ricardo Alonso, Dirk Bergemann, Mehmet Ekmekci, Erik Eyster, Daniel Krähmer, Gilat Levy, and Ronny Razin, as well as comments from audiences at the LSE, Bonn, the European Winter Meeting of the Econometric Society 207 and the CRC TR 224 Conference in Offenbach 208. This work was supported by a grant from the European Research Council (ERC 6385) and the CRC TR224. This draft is preliminary and incomplete. Comments and suggestions are welcome.

2 2 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS Elections are ubiquitous instruments of collective choice. This paper studies the Bayes correlated equilibria of standard majority elections. Correlated information might arise for example through communication of voters or through persuasion by a manipulator. We treat the persuasion application most prominently: An interested party has information that is valuable for voters and tries to affect voters choices by the strategic release of this information. Examples of interested parties holding and strategically releasing relevant information for voters are numerous. Consider the vote on a reform. The advantages of the reform are unknown to the public, and an informed politician can decide how to release information. Or consider the election of a CEO at an annual shareholder meeting. The Board of directors provides information on the candidates with the shareholder meeting brochure, through conversations, and presentations. This paper revisits the general voting setting by Feddersen and Pesendorfer [997]. There are two possible policies (outcomes), A and B. Voters preferences over policies are heterogenous and depend on an unknown state, α or β, in a fairly general way: Some voters may prefer A in state α, some prefer A in state β, and some may prefer A independently of the state (while others always prefer B). Preferences are drawn independently and identically across voters. Their preferences are each voters private information. The election determines the outcome by a simple majority rule. Voters have a common prior over the states. We explore the possibility and limits of persuasion, (Kamenica and Gentzkow [20]): Prior to the election, a manipulator commits to an information structure, which is a joint distribution over states and signal realizations that are privately observed by the voters. We ask: Can the manipulator ensure that a majority supports his favorite policy in a large election by choosing an appropriate signal? In this setting, Feddersen and Pesendorfer [997] have shown that, within a broad class of monotone preferences and conditionally i.i.d. signals, equilibrium outcomes of large elections are equivalent to the outcome with publicly known states ( information aggregation ). This may suggest that elections are robust. Our main result (Theorem 2) shows that, nevertheless, within the same class of monotone preferences for any possible statecontingent policy there exists an expansion of the conditionally i.i.d. signal structure and a natural equilibrium that ensures that the policy is supported by a majority with probability close to one. In particular, the supported policy can be the opposite of the outcome with publicly known states, for every state. For clarity, at first, we assume that all information of voters comes from a manipulator. Specifically, the main result for this baseline model roughly shows that the manipulator can persuade a large electorate to elect any See Bergemann and Morris [207] for a detailed analysis of how the study of Bayes correlated equilibria relates to existing work, including that on communication in games (Myerson [99]), and Bayesian persuasion (Kamenica and Gentzkow [20])

3 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 3 state-contingent policy with probability close to if there is one belief about the likelihood that the state is α such that a voter with randomly drawn preferences prefers A with probability larger than /2 given this belief and another belief such that the probability of preferring A given this belief is smaller than /2 (Theorem ). Denote these beliefs by p A and p B, respectively. In particular this condition guarantees that there is a belief r at which a random voter prefers A with probability of exactly 50% since we assume prefereces to be continuous in beliefs. Clearly, some such condition is necessary for persuasion to be effective: For example, if for all beliefs, each voter prefers A with probability less than /2, then, whatever the induced beliefs, in a large election the expected share of voters supporting A will be less than /2. We show that the condition is sufficient. For example, when the manipulator s goal is to get A elected in both states we construct a signal structure as follows. Roughly speaking, with high probability ε the voters receive conditionally independent draws of a binary signal, a or b, with a being relatively more likely in state α and b relatively more likely in state β. With monotone preferences and ε = 0, this would generally allow for information aggregation in equilibrium as in Feddersen and Pesendorfer [997]. However, with probability ε > 0, the manipulator induces an additional state-of-confusion: In this additional state, almost all voters will receive a common signal z while only few voters receive signals a or b. Conditional on observing z, a voter knows that most other voters have also observed z. The consequence is that, in contrast to the usual calculus of strategic voting, there is no further information about others signals contained in the event of being pivotal. This is the critical observation and it implies that voters behave essentially sincerely conditional on z. By choosing the relative probability of z in the two states appropriately, the posterior conditional on z will be r, meaning, each voter prefers A with probability /2 and, hence, the election is close to being tied. Thus, even from the viewpoint of the few voters observing signals a or b, conditional on the election being tied, it is likely that the other voters received the common signal z. By appropriately choosing the probabilities of a and b in the state-of-confusion, the posterior conditional on the state-of-confusion and conditional on a or b is p A. Hence, in the standard state, when there are only signals a and b, a large majority supports A. The main idea of the construction is that one can first characterize equilibrium for voters receiving a z signal and then use that to extend the construction to the other voters. For persuasion it is sufficient to have the ability to block the usual calculus of strategic voting in a state with probability ɛ by sending a common signal z to almost all voters to prehibit information aggregation in all states. We argue that persuasion is robust in various dimensions. First, the played equilibrium is simple and insures voters against errors. Specifically, the equilibrium profile is almost identical to voting sincerely given one s

4 4 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS signal, conditional on the state-of-confusion. One may argue that this behavior is simple. In particular, voters just need interpret their own signal conditional on that state; they do not need to make any further inference about other voters signals using the equilibrium strategy profile or have to know the preference distribution of the electorate. Furthermore, as will be explained in detail later, sincere behavior is safe in the sense of being an ε best response conditional on being pivotal for a neighborhood around the actual environment. Thus, even if a voter s belief about the environment and the equilibrium is slightly wrong, the cost of this error is small (conditional on being pivotal). Second, the played equilibrium is attracting. In particular, its basin of attraction for the best response dynamic is essentially the full set of strategy profiles, except for the one (essentially unique) strategy profile that corresponds to the one type of other equilibrium: If we start with any strategy profile that is close to but not exactly equal to that type of equilibrium and if we consider the voters best response to it and the voters best response to this best response, then the resulting strategy profile is arbitrarily close to the manipulated equilibrium when the number of voters is large (Proposition 5). Third, the sender does not need to know the exact parameters of the game (meaning, the distribution of the private preferences and the prior) when choosing the signal structure. One may interpret this by saying that the signal structure satisfies a version of the Wilson Doctorine of not requiring excessive knowledge by the principal: Fix the signal structure and some parameter of the game. There will be an open set of parameters containing the fixed one such that for every parameter from this set there is a manipulated equilibrium that implements the senders preferred outcome. By way of contrast, as discussed momentarily, existing work assumes that the manipulator knows the exact preference of each individual voter and this knowledge is indeed used. In the second part of the paper, we consider the setting in which voters already have access to information of the form studied in Feddersen and Pesendorfer [997]. Thus, if the manipulator adds no further information, the outcome would be as with publicly known states. We show that, by adding additional information, the manipulator can still persuade the voters effectively to elect any state-contingent policy (Theorem 2). In this setting, the manipulator does not have the ability to block information in a small added state. However, the main idea of the construction of the baseline model works here, too. We can first characterise equilibrium for voters receiving z. In the added state (where almost all voters receive z), the game converges to a game with only the exogeneous binary signal. It is known that the equilibrium limit of such a game is uniquely determined. In particular, this pins down the behavior in the added state. We extend the construction to the other voters, which received a or b.

5 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 5 The main result has second interpretation: If an outside observer only knows that voters have the preferences as in Feddersen and Pesendorfer [997] and access to information that is at least as fine as theirs, then it is not certain that information is aggregated in equilibrium. Moreover, no robust prediction is possible if the observer cannot exclude that voters might receive additional information (Corollary ). In an extension, we discuss the possibility of persuasion with public signals. Suppose that preferences are monotone, voters have no private signals about the state and hold a prior at which a majority votes B. Suppose, for example, that the manipulator s goal is to get A elected in both states. Revealing that the state is α clearly increases the probability of the outcome A. However, it is not possible to induce a posterior distribution such that a majority supports A for all posteriors. For public signals, Bayes consistency implies that the expected posterior is equal to the initial prior. So persuasion is only partial (Proposition 2). 2 When voters receive the exogeneous private signals as in Feddersen and Pesendorfer [997] and preferences are monotone, persuasion is not possible with public signals: When adding a public signal to the setting, this is equivalent to a shift in the common prior. However, we know that information is aggregated for all possible nondegenerate priors (Theorem 0). A degenerate posterior can only be induced by revealing a state, but this only helps information aggregation. The paper is related to work on information design in general (see Bergemann and Morris [207] for a survey paper) and especially to persuasion with multiple receivers (e.g., Mathevet et al. [206]) and to persuasion of a receiver with private information about its preferences (Kolotilin et al. [205]) and with private signals about the state (Guo and Shmaya [207]). Persuasion in the context of elections has been studied in a number of papers under various restrictions. Alonso and Câmara [205] study persuasion through a public signal that is observed by all voters simultaneously. Consequently, voters do not condition on being pivotal. We allow for private signals. In many settings, it is natural that signals are not commonly observed. Bardhi and Guo [206a] study persuasion with unanimity rule. With unanimity, every voter needs to be persuaded and hence the problem is more similar to a single-receiver persuasion problem. Wang [203] studies private persuasion by conditionally independent signals. This rules out the type of persuasion through a state-of-confusion that we consider. We believe that correlation of signals is feasible in many natural applications. Chan et al. [206] study persuasion with known and monotone preferences through private signals. Our work shares with theirs the observation that the voters conditioning on being pivotal allows relaxing the Bayesian consistency requirement. However, in our work, persuasion is achieved differently, 2 However, when preferences are non-monotone, complete persuasion might be possible, for n.

6 6 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS namely, through a state-of-confusion. Moreover, we allow for general preference heterogeneity and, in particular, voters preferences are their private information. The latter means that the type of targeted persuasion that is studied in the related work is not feasible here. When the preferences of individual voters are known, signals can be individualized so that they make a specific individual just indifferent. Methodologically, with known preferences, a revelation principle argument implies that individual signals are binary without loss of generality. 3 A more detailed discussion of the related literature is in Section 7 and in the conclusion. In Section 7 we also discuss in depth the existing work on failures of information aggregation, especially Mandler [202], Feddersen and Pesendorfer [997] (their extension to aggregate uncertainty about preferences), and Bhattacharya [203]. The rest of the paper is organized as follows: In Section we present the model. In Section 2 we discuss a binary-state version of Feddersen and Pesendorfer [997] as in Bhattacharya [203]. In Section 3 we show that persuasion is essentially limitless when the information designer is monopolistic (Theorem ) and illustrate the robustness of the manipulated equilibrium. In particular, any state-contingent policy can be the outcome of a Bayes correlated equilibrium when the electorate is large (Corollary ). In Section 4 prove the main result of this paper by showing that persuasion is essentially limitless even when a manipulator can only add information to arbitrarily precise exogeneous private signals (Theorem 2). In particular, in a voting game as in Feddersen and Pesendorfer [997] any state-contingent policy can be the outcome of a Bayes correlated equilibrium. Section 5 discusses other equilibria (Proposition ), feasibility and evidence for the strategic voter paradigm. In Section 6 we discuss the possibility of persuasion with public signals (Proposition 2) and when voters do not have heterogeneous types, but common values (Proposition 3). In Section 7 we discuss the paper s contribution to the existing literature and compare our results especially to other results on voter persuasion and other reported failures of information aggregation. The conclusion discusses the relation to the literature on auctions with general information structures. 3 Furthermore, given that there is a deterministic relation between signals and induced votes, the signal structure can be chosen such that the signal vote A is pivotal in different profiles from the signal vote B. For example, with voters, if the induced signal profile is 6 vote A and 5 vote B signals, then only voters with a vote A signal are pivotal. By judiciously choosing distributions over such profiles across states, being pivotal with a vote A signal implies that the state is α with certainty and being pivotal with a vote B signal implies that the state is β with certainty. In our setting, the interpretation of being pivotal is independent of one s signal.

7 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 7 Model There are 2n+ voters, two possible election outcomes A and B, and two states of the world ω {α, β} = Ω. Voters hold a common prior. The prior probability of α is p 0 (0, ), and the probability of β is p 0. Voters have heterogeneous preferences. The preferences are private information. A preference type is a pair t = (t α, t β ) [, ] 2, with t ω the utility of A in ω. We normalise the utility of B to zero. 4 Preference types are independently and identically distributed according to a commonly known distribution G that has a strictly positive, continuous density. An information structure π is a finite set of signals S and a joint distribution of signal profiles and states. We also denote by π only the joint distribution. We assume that π ω is exchangeable with respect to the voters for all ω Ω. 5 We consider information structures π that are the combination 6 of two information structures π and π 2 that are independent conditional on ω and have signal sets S and S 2 respectively. This means that S = S S 2 and that π ω = (π ) ω (π 2 ) ω. We write π = π π 2. The information structure π sends binary signals {u, d} that are independently, and identically distributed conditional on the state of the world ω Ω. We make the following assumption on informativeness of signals, > Pr(u α) 2 Pr(u β) > 0. Hence, u is weakly indicative of α, and d weakly indicative of β. 7 A symmetric strategy of the voters is a function of the signal s and the type t, and denoted by σ : S [, ] 2 [0, ] where σ(s, t) is the probability of type t to vote A after s. Aggregate Preferences. For a given strategy σ, denote by piv the event in which, from the viewpoint of a given voter, n of the other 2n voters vote for A and n for B. If she votes A, the outcome is A, if she votes B, the outcome is B. In any other event, the outcome is independent of her vote. Thus, a strategy is optimal if and only if it is optimal conditional on piv. Given σ, a voter of type t who received s weakly prefers to vote A if 4 Otherwise, we can view tω as the difference of the utilities of A and of B in ω. 5 The joint distribution F of discrete random variables Y,..., Y 2n+ is called exchangeable if Pr F (y = z,..., y 2n+ = z 2n+) = Pr F (y h() = z h(),..., y h(2n+) = z h(2n+) ) for all realisations (z,..., z 2n+) and all permutation h of {,..., 2n + }. 6 See Bergemann and Morris [206] for a more general definition. 7 Note that we allow for uninformative signals from π, that is Pr(u α) = Pr(u β), as well as uninformative signals from π 2.

8 8 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS and only if 8 () Pr(α s, piv; σ, π) t α + ( Pr(α s, piv; σ, π)) t β 0. A central object of our analysis is the aggregate preference function (2) φ(p) := Pr(t p t α + ( p) t β > 0), which maps a common belief p to the probability that a random type t prefers A under p. 9 Note that φ is continuously differentiable, since G has a continuous density. Equilibrium. Any information structure π induces a Bayesian game of voters, denoted by Γ(π). We analyse the symmetric Bayes-Nash-equilibria of Γ(π) in weakly undominated, pure strategies and call them (voting) equilibria. Voter types t 0 (A-partisans) have the weakly dominant strategy to vote for A. Voter types t 0 (B-partisans) have the weakly dominant strategy to vote for B. The restriction to undominated equilibria rules out trivial equilibria: the distribution G puts strictly positive probability on voter types t >> 0 and t << 0 by the assumption that it has a strictly positive density. Hence, there exists ɛ > 0 such that for all s S, and any undominated strategy σ, (3) ɛ < Pr(t : σ(s, t) = ) < ɛ. Since we only consider undominated strategies σ, for any ω Ω and s S, the posterior Pr(ω s, piv; σ; π) is defined by Bayes rule. The restriction to equilibria in pure strategies is without loss, because, by and the continuity of G, a voter has a unique strict best response with probability. A strategy σ is a cutoff-strategy if for all s S there exists p s [0, ] such that σ(s, t) = t α p s + t β ( p s ) 0. Any best reponse is a cutoff strategy with cutoffs p s = Pr(α s, piv; σ, π) by. We use the terminology of Bergemann and Morris [206] who define Bayes correlated equilibrium, and who show Theorem (Bergemann and Morris [206]). A decision rule θ : (S [, ] 2 ) 2n+ Ω {A, B} 2n+ is a Bayes Correlated equilibrium of Γ(π ) if and only if for some expansion π of π, there is a Bayes Nash equilibrium σ of Γ(π) that induces θ. 0 8 Throughout the paper we ensure that posteriors are well-defined through Bayes formula. For example, for any σ, any π and any s S, the posterior Pr(α s, piv; σ, π) is defined through Pr(α s,piv;σ,π) = p 0 Pr(s α;π) Pr(piv α,s;σ,π). Pr(α s,piv;σ,π) p 0 Pr(s α;π) Pr(piv α,s;σ,π) 9 Sometimes it will be convenient to work with belief ratios instead of beliefs. We define φ(y) := φ( y p ) for any y R which is equivalent to φ( ) = φ(p) for any p (0, ). +y p The function φ maps belief belief ratios y to the probability that a random type t prefers A under y. 0 Note that not only expansions π = π π 2 where π 2 and π are independent conditional on the state are considered for this statement.

9 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 9 t β t α t β = p p t α Figure. For any given belief p = Pr(α) (0, ) the curve of indifferent types is t β = p p t α. Hence, our analysis can be understood as studying the Bayes correlated equilibria of Γ(π ). From the viewpoint of an outside observer who only knows that voters receive at least information π, these are the Bayes-Nash equilibria that can arise. Therefore, we call π the exogeneous private signals of the voters. Information Aggregation. The full information outcome in ω Ω is the outcome which is prefered by a random voter with probability weakly larger than 2 conditional on ω. The literature on information aggregation in elections concerns with the question if strategy sequences σ n imply the full information outcome when n grows to infinity. (4) Monotone Preferences. Typically, we make the following assumption φ(p) is strictly increasing in p, φ(0) < 2 < φ(). Otherwise, we say this explicitly. When Assumption 4 holds, we say that preferences are monotone. Note that when φ is strictly increasing, the assumption φ(0) < 2 < φ() excludes the trivial cases: If φ(p) < 2 for all p [0, ], in any equilibrium sequence σ n, the probability that B gets elected, converges to. If φ(p) > 2 for all p [0, ], in any equilibrium sequence σ n, the probability that A gets elected, converges to. If φ(0) < 2 < φ() holds, the full information outcome is A in α and B in β. Convergence. Convergence of strategies means pointwise convergence (up to measure 0). A sequence of cutoff-strategies σ n with cutoffs (p s,n ) s S converges to a cutoff strategy σ with cutoffs (p s ) s S if and only if p s,n ps

10 0 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS for all s S. When we speak of distances between two cutoff strategies, we mean the Euclidean distance. When we discuss limits of statistics of sequences of strategies, we implicitly refer to a converging subsequence such that the limit is well-defined. Remark. The collection of posteriors conditional on piv and s, namely (Pr(α s, piv; σ, π)) s S, is a sufficient statistic for the unique best response (recall ). The possibility of writing equilibria in terms of posteriors is what makes our model easily amenable to the Bayesian persuasion literature. Remark 2. Imagine that a sender, who we do not model, commits to π 2. By analysing the equilibria of the Bayesian game of voters induced by π = π π 2, we implicitly analyse the scope of voter persuasion with exogeneous private signals π. Remark 3. Given the general preference distribution G, the model nests almost common values. Besides, it does not only include the case in which the full information outcome is A in α and B in β, but also all cases in which the full information outcome does not match the state. 2 Benchmark: Condorcet Jury theorem In this section, we consider the case when the signals of π 2 are uninformative. For the simplicity, we therefore omit π 2 and its signals from the notation. We write π = π, and assume that (5) Pr(u α) > Pr(u β). Under these assumptions, the model in this paper describes a binary-state version of Feddersen and Pesendorfer [997]. Each voter either receives u which is indicative of α, or d which is indicative of β. The sincere strategy ˆσ is the strategy that acts upon the posteriors conditional on the signal s only. Definition. The sincere strategy ˆσ is the pure strategy given by ˆσ n (s, t) = t α Pr(α s; π) + t β ( Pr(α s; π)) 0. Sincere Voting. When voters vote sincerely and the prior is sufficiently extreme, sincere voting does not necessarily aggregate information: For example, if p 0 is sufficiently low such that φ(pr(α u)) < 2, a random voter votes A with probability smaller 2 after any signal. The law of large numbers implies that B is elected with probability converging to. However, if priors are not too extreme, then, if voters vote sincerely and signals are relatively precise, the full information outcome is elected with probability converging to. For example, suppose that the prior is sufficiently close to φ ( 2 ) such that φ(pr(α u)) > 2 and φ(pr(α d)) < 2. If in addition signals are sufficiently precise, we have Pr(ˆσ(s, t) = α) > 2 > Pr(ˆσ(s, t) = β). The law of large numbers implies that the full information outcome is elected

11 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS with probability converging to. This instance of the Condorcet Jury Theorem (Condorcet [793]) is illustrated in the following figure on the right hand. φ(p) φ(p) 2 Pr(ˆσ(s, t) = α) 2 Pr(ˆσ(s, t) = β) Pr(α d) p 0 Pr(α u) Under ˆσ, information aggregation can fail with sufficiently extreme priors. p Pr(α d) p 0 Pr(α u) Under ˆσ, information can be aggregated with intermediate priors and sufficiently precise signals. p Strategic Voting. In the exact setting of this section, Bhattacharya [203] has replicated a result by Feddersen and Pesendorfer [997], namely that the Condorcet Jury Theorem extends to strategic voting. Moreover, under strategic voting information is aggregated even with extreme priors. Theorem 0. (Bhattacharya [203]) If π = π and > Pr(u α) > Pr(u β) > 0 and when preferences are monotone (that is 4 holds), then for any sequence of equilibria σ n, Proof. In the Appendix. lim Pr(A is elected α; σ n) =, lim Pr(B is elected β; σ n) =. This theorem is a special case of Theorem in Bhattacharya [203]. The notation for the function φ( ) is h( ) in Bhattacharya [203]. We provide the proof of this special case for the convenience of the reader.

12 2 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS lim Pr(σ n (s, t) = α) 2 lim Pr(σ n (s, t) = β) φ(p) Pr(α b, piv) Pr(α a, piv) p Figure 2. Condorcet Jury Theorem in Bhattacharya [203]: In any equilibrium sequence, lim Pr(σ n (s, t) = α) 2 = lim 2 Pr(σ n(s, t) = β) holds. 3 The Bayes Correlated Equilibria without Exogeneous Private Signals In this Section 3, we consider the case when the signals from π are uninformative, that is Pr(u α) = Pr(u β) holds. For the simplicity of notation, we omit π and its signals from the notation. We write π 2 = π. In this section we drop the assumption of monotone preferences (4). To make the analysis interesting, we assume that there exist beliefs p A, p B [0, ] such that 2 (6) φ(p A ) > 2 > φ(p B). 3. A Class of Information Structures We consider a class of information structures π n (q, l, r) with S = {a, b, z}. We illustrate π n (q, l, r) by two diagrams: 2 If φ(p) < for all p [0, ], the election outcome is B with probability converging 2 to, for n, for any information structure that does not condition on types, and for an equilibrium sequence. Conversely, if φ(p) > for all p [0, ],the election outcome is 2 A with probability converging to for n, for any information structure that does not condition on types, and for an equilibrium sequence.

13 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 3 ɛ α a 2n β 0 a 0 b b α ɛ 2 a β n 2 a ɛ α 2 ɛ 2 ɛ 3 z 2n β 2 2 n 2 z ɛ 3 The distribution of signals of π n (q, l, r) conditional on α. b n 2 The distribution of signals of π n (q, l, r) conditional on β. b q q We set ɛ = p 0 r p 0 r 2n, ɛ 2 = r r and ɛ n 2 3 = r l r l. First, nature draws the state ω {α, β} according to p 0. Then, a substate ω j is n 2 drawn with j {, 2}. Conditional on ω j voters receive independently and identically distributed signals s {a, b, z}. The probabilities by which the sub-states ω j are drawn and the probabilities by which the signals are sent to voters conditional on ω j are indicated along the arrows. The parameters q, l, r have an easy interpretation: By definition, the posteriors conditional on the signal and conditional on Ω 2 := {α 2, β 2 } 3 satisfy (7) (8) (9) Pr(α a, Ω 2 ; q, l, r) = q for all n N, lim 2; q, l, r) = r, Pr(α b, Ω 2 ; q, l, r) = l for all n N. Definition 2. For any (q, r) [0, ] 3, the game Γ n (q, l, r) is the game of n voters induced by π 2 = π n (q, l, r). Remark 4. In Ω 2, voters receive an almost public signal z. Intuitively, in Ω 2, in contrast to the usual calculus of strategic voting, there is almost no further information about others signals contained in the event of being pivotal: Information aggregation is blocked. Without Ω 2, signal a perfectly reveals the state, and signal b perfectly reveals the state. By concentrating the analysis on π n (q, l, r) we restrict a potential sender to blocking information aggregation in states of vanishing probability. 3 Similarly, we define Ω := {α, β } and we denote the generic element of Ω i by ω i.

14 4 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS Remark 5. The following story shall illustrate π n (q, l, r): Imagine a politician who places ads (signals) on a webpage. Most of the time in α, ad a appears with probability to a visitor, most of the time in β ad b appears with probability. However, in state α, with small probability ɛ, ad a and ad b appear with probability ɛ 2 and ɛ 3 respectively, and ad z appears with probability ɛ 2 ɛ 3. In state β, with small probability 2n, ad a and ad b appear each with probability, and ad z appears with probability 2. n 2 n 2 This ad placement strategy implements π n (q, l, r) when the arrival of voters to the webpage is independently, and identically distributed. 3.2 Result Without Exogeneous Private Signals Definition 3. The Ω 2 -sincere strategy ˆσ Ω2 (q, l, r) is the pure strategy under which a voter votes A if and only if 4 t α Pr(α s, Ω 2 ; q, l, r) + t β ( Pr(α s, Ω 2 ; q, l, r)) 0. We also say that ˆσ Ω2 is sincere voting conditional on Ω 2 or conditional sincere voting. Recall that we assumed existence of p A, p B for which 6 holds. Since φ is continuous,the intermediate value theorem implies that there exists r for which (0) φ( r) = 2. The following theorem shows that any state-contingent outcome can be implemented in an equilibrium sequence that converges to Ω 2 -sincere voting. Theorem. When 6 holds: For any state-contingent outcome x α {A, B} and x β {A, B}, the Ω 2 -sincere strategy ˆσ Ω2 (p xα, p xβ, r) is the limit of some equilibrium sequence σ n in the games Γ n (p xα, p xβ, r), and it holds that lim Pr(x ω is elected ω; σ n, π n (p xα, p xβ, r)) = for all ω Ω. Intuitively, the condition 6 describes two aspects of G: No majority of A- or B-partisans. The voters that prefer to vote B regardless of their belief do not represent a majority, for n. These are the types with t << 0. The same holds for A-partisans. Asymmetry of Information-Sensitive Types. There must be an asymmetry between the voter types who prefer A only in state α, that is, those for which t α > 0 and t β < 0, and the voter types who prefer A only in state β, that is, those for which t α < 0 and t β > 0. If both groups of voter types are equally likely, and the density of G is symmetric, meaning that it takes the same values at (t α, t β ) and at ( t α, t β ) for all (t α, t β ) with t α > 0 and t β < 0, then the function φ(p) = Pr G (p t α + ( p) t β > 0) is constant in 4 Recall the interpretation of the parameters q and r in terms of the posteriors Pr(α s, Ω 2; q, r) in 8 and 7.

15 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 5 p. Then the condition cannot be fulfilled. Information Design without Elicitation.? distinguish three cases of information design: when the designer is omniscient, when receivers have private information and an information designer may be able to elicit and condition on the private information (information design with elicitation) or he may be unable to do so (information design without elicitation). Information design without elicitation has the strongest constraints and hence relates to the smallest set of Bayes correlated equilibria. Note that the information structures π n (q, l, r) do not condition on the private information of voters. Theorem implies that all state-contingent outcomes can be implemented in a Bayes correlated equilibrium of a large election just by information design without elicitation. Corollary. For any state-contingent outcome x α {A, B} and x β {A, B}, there exists a sequence of Bayes correlated equilibria θ n of Γ(π ) such that under θ n the probability that x ω gets elected in ω converges to for all ω Ω. Weakened Bayes Consistency Constraints: Note that under ˆσ Ω2 ( q, r), for n, agents act upon the posterior q with probability converging to, independently of the prior p 0. The Bayes consistency constraints for persuasion of multiple voters vanish completely for n. This is in stark contrast to persuasion of a single receiver where posteriors have the martingale property (Kamenica and Gentzkow [20]). In Section 3.3 we give a proof of Theorem 2 for the case in which x ω = A for all ω Ω. For this we restrict to q = l, such that ɛ 2 = ɛ 3, and we write π n (q, r) for π n (q, l, r) and ˆσ Ω2 (q, r) for ˆσ Ω2 (q, l, r). We discuss the robustness of the equilibrium construction for this case in Section 3.4. A proof for the general case is done in the Appendix for the setup with exogeneous private signals π. 3.3 Proof Recall that by, the posteriors (Pr(α s, piv; σ n, q, r)) s {a,b,z} are a sufficient statistic for the best response. Hence, we can study equilibria in terms of belief triples. We show that there exists an equilibrium sequence that converges to lim (Pr(α s, Ω 2 ; σ n, q, r)) s {a,b,z} = (q, q, r). In the states Ω 2, almost all voters receive the common signal z, for any q, r. Conditional on observing z, a voter knows that either α 2 or β 2 holds and that most other voters have also observed z. In fact, the probability that all voters received z in α 2 and β 2 converges to for n : e.g. lim Pr(all voters received z β 2 ) = lim ( n 2 ) 2n+ = e 2 lim n =. Intuitively, in contrast to the usual calculus of strategic voting, there is no

16 6 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS further information about others signals contained in the event of being pivotal. Formally, we establish Lemma, (i) in the Appendix and apply it to x n = Pr(σ n (s, t) = α 2 ; σ n, q, r), and y n := Pr(σ n (s, t) = β 2 ; σ n, q, r). We record Lemma 2. For any sequence (σ n ) n N of strategies, it holds that lim Pr(α piv, z; σ n, q, r) = lim Pr(α Ω 2, z; σ n, q, r) = r. Proof. In the Appendix. Lemma 2 implies that after signal z agents behave sincerely for n. This means that we can control the behavior of agents getting z perfectly - and in particular make the election arbitrarily close to being tied in ω 2 for any ω 2 {α 2, β 2 } by choosing r appropriately. Since r satisfies 0, under the belief r a random voter prefers A with probability 2. So, given π n(q, r) and any equilibrium sequence σ n, () lim Pr(σ n(z, t) = σ n ; q, r) = Pr(t α r + t β ( r) > 0) = φ( r) = 2. Remark 6. We call the states Ω 2 the states-of-confusion, because information aggregation is not possible in Ω 2, since voters receive an almost public signal z. Also, by, the election outcome is purposefully highly uncertain in Ω 2. Lemma 3. If (2) holds, then max ω 2 Ω 2 lim Pr(σ n(s, t) = ω 2 ; q, r) 2 < min ω Ω lim Pr(σ n(s, t) = ω ; q, r) 2, lim Pr(α s, piv; σ n, q, r) = Pr(α s, Ω 2 ; q, r) = q for s {a, b}. Hence, and the unique best response to σ n in the games Γ n (q, r) converges to ˆσ Ω2 (q, r) for n. Proof. In the Appendix. Lemma 3 shows that if the limit of the expected margin of victory in the states Ω is strictly larger than the limit of the expected margin of victory in the states Ω 2 (call this the margin-of-victory condition ), the unique best response converges to Ω 2 -sincere voting ˆσ Ω2 (q, r) for n. Intuitively, when the margin-of-victory-condition holds, conditional on being tied, the states Ω 2 are infinitely more likely than the states Ω for n. Hence,

17 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 7 being pivotal contains the information that the states ω do not hold, but no information beyond that, by Lemma 2. This is precisely the information that Ω 2 -sincere voters condition on. Hence, the best reply converges to ˆσ Ω2. Pr(ω More precisely, the margin-of-victory condition implies lim s,piv;σ n,q,r) Pr(ω 2 s,piv;σn,q,r) = Pr(ω lim s;q,r) Pr(piv ω ;σ n,q,r) Pr(ω 2 s;q,r) Pr(piv ω 2 ;σ = 0 for s {a, b} and any ω n,q,r) Ω, and any ω 2 Ω 2. This can be seen as follows: The probability of the election being tied is decreasing exponentially faster in states ω than in states ω 2. Conditional on the signal s {a, b}, the states Ω 2 are less likely than the states Ω. However, note that the ratios Pr(ω s;q,r) Pr(ω 2 s;q,r) are only increasing at a rate proportional to n 3 for s {a, b}. So, the exponentially decreasing terms Pr(piv ω ;σ n,q,r) Pr(piv ω 2 ;σ n,q,r) dominate. So, the posteriors conditional on being pivotal and conditional on s {a, b} vanish on ω. Being pivotal contains the information that the states Ω do not hold for n. Equilibrium Construction. By Lemma 3 we can control the limit behaviour of agents getting s {a, b} by choosing q = 8 Pr(α s, Ω 2 ; q, r) appropriately. By 6, there exists a commmon belief q := p A (0, ) such that under q a majority of voters prefers to vote for A, (3) φ( q) > 2. In the games Γ n ( q, r), under the Ω 2 -sincere strategy ˆσ Ω2 ( q, r) and for n, a strict majority of agents votes A after getting s {a, b}: lim Pr(ˆσ Ω2 (s, t) = s; q, r) Definition = Pr G (t α q + t β ( q) > 0) = φ( q) > 3 2 for s {a, b}. By, under ˆσ Ω2 ( q, r), the limit of the expected margin of victory in the states Ω 2 is zero. So, ˆσ Ω2 ( q, r) satisfies the margin-of-victory condition 2 of Lemma 3. So when agents vote Ω 2 -sincerely or use a cutoff strategy close-by 5 to ˆσ Ω2 ( q, r), the best reply converges to ˆσ Ω2 ( q, r). A fixed-point argument, Lemma 4, closes the loop of best replies. Hence, there exists a sequence of equilibria that converge to ˆσ Ω2 ( q, r). This shows the claim of Theorem for the case x ω = A for all ω {α, β}, because under any strategy close-by to ˆσ Ω2 ( q, r), A gets elected with certainty, for n. We illustrate the equilibrium construction: 5 By close-by we mean that the Euclidean distance of the cutoffs is small.

18 8 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS φ(p) 2 p 0 r q p Under ˆσ Ω2 ( q, r) citizens vote A with probability φ( r) = 2 after z; citizens vote A with probability φ( q) > 2 after a and b. This supports the belief that conditional on being pivotal the states Ω do not hold, and justifies that voting according to Pr(α Ω 2, z; q, r) = r after z and Pr(α Ω 2, s; q, r) = q after s {a, b} is optimal, for n (see Lemma 3). Remark 7. (Belief Trap Ω 2.) The states Ω 2 function as a belief trap. When voters believe that being pivotal contains (only) the information that Ω 2 holds, and best respond to this belief by voting Ω 2 -sincerely, behaviour can be arbitrarily manipulated by choice of r and q (compare 7-9). As lons as q and r are chosen such that Ω 2 -sincere voting satisfies the margin-of-victory condition 3, voters are trapped into believing that Ω 2 holds conditional on being pivotal. Remark 8. (Stability of ˆσ Ω2 ( q, r)) There exists ɛ > 0 such that the margin-ofvictory condition 2 holds for all σ B ɛ (ˆσ Ω2 ( q, r)). Consequently, lim BR(σ) = ˆσ Ω2 ( q, r) for all σ B ɛ (ˆσ Ω2 ( q, r)) by Lemma 3. Hence, there exists n N such that any cutoff strategy σ B ɛ (ˆσ Ω2 (q, r)) is ɛ-close to its best reponse BR(σ) for all n n. Formally, this means that the cutoffs of σ and BR(σ) are ɛ-close. So, after any signal s, any type that makes different choices under σ and BR(σ) must be ɛ-close to the indifferent type (the cutoff of BR(σ)); consequently the type s loss is smaller than ɛ conditional on being

19 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 9 pivotal. We say that σ is a conditional ɛ-equilibrium. 6 The equilibrium limit ˆσ Ω2 ( q, r) is stable or safe in the sense that all strategies in a neighbourhood of ˆσ Ω2 ( q, r) are conditional ɛ-equilibria for n n. Revenue Equivalence. Note that the proof of Theorem does not rely on voting by simple majority rule. The result extends to any non-unanimous majority rule if we replace the assumption 6 by the assumption that there exist beliefs p A, p B [0, ] with φ(p A ) > 2 > φ(p B). when preferences are monotone this assumption is satisfied for all τ (0, ). We consider Theorem therefore as a revenue equivalence theorem. Revenue equivalence of non-unanimous voting rules has also been observed by Feddersen and Pesendorfer [997] (cf. Theorem 0) and in the context of deliberative voting by Gerardi and Yariv [2007]. Computational Example. We specify the preferences, the prior and the information structure by the assumptions that Pr(t α > 0, t β < 0) = 7, that Pr( t β t α+t β p) = p for all p [0, ] 8 which implicitly defines G, and that p 0 = 4. Further we set r = 2, and q = 3 4. In the Appendix we show that under these primitives an equilibrium σ n close to conditional sincere voting exists for n 200. Additionally, in this equilibrium A is elected with a probability of more than 99 percent. To do so, we show that under the specified primitives the best reponse is a self-map on the set of strategies σ satisfying Pr(σ(s, t) = s) 0.7 for s {a, b}, and Pr(σ(z, t) = ) [0.45, 0.54] for n 200. This yields an equilibrium in which voters with an a-or b-signal vote A with a probability of at least 70%. 3.4 Robustness This section further analysises the games Γ n (p A, r) of Theorem. We denote with q = p A as before. Conditional Sincere Voting is Simple. The voting strategy is simple to operationalise: If we want to tell a voter to behave conditionally sincere, then this will only require the voter to calculate his personal beliefs. It 6 The classical notion of ɛ-equilibrium (see e.g. Radner [980]) is void for the voting games analysed, since the probability of being pivotal converges to 0 for n. Therefore any strategy is an ɛ-equilibrium for n large enough. 7 Note that this is slightly inconsistent with the assumption that G has a strictly positive density, but made for the simplicity of presentation. 8 Note that this assumption is equivalent to saying that φ(p) = p for all p (0, ). One distribution G on [0, ] [, 0] that induces such a uniform distribution of thresholds of doubt is given by the density + ( t β t g(t α, t β ) = α ) 2 (2 t α > t + ( t β β t α ) 2 dt) + ( tα t β ) 2 (2 t α > t + ( t β β t α ) 2 dt) if if t β t α t β, 2 t β t α t β. 2

20 20 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS would not require knowledge of G or the strategies of others. It is simple to rationalise: Conditional sincere voting is an equilibrium (limit) by the simple logic that it is optimal to condition on the states Ω 2 if the expected margin of victory is smaller in Ω 2 than in Ω (cf. Lemma 3). If all voters actually condition on Ω 2 and vote Ω 2 -sincerely, the underlying assumption on the order of the margin of victories is indeed true, intuitively, because voters receive an almost public signal in Ω 2 which induces a close election outcome by construction. Best Response Robustness. Recall that we use a stability argument in the equilibrium construction of the proof of Theorem (we use Lemma 4, recall also Remark 8): If we start ɛ-close to ˆσ Ω2 ( q, r) with r and q satisfying and 3 respectively, then for n large enough, the best response is ɛ-close to ˆσ Ω2 ( q, r). So, in the games Γ n ( q, r), conditional sincere voting has a nontrivial basin of attraction with respect to the best response dynamics for n sufficiently large. In fact, something stronger is true. For any ɛ > 0, if n is sufficiently large, there is a set of strategy profiles that has measure ɛ 9 such that the following is true: Take any strategy profile σ from this set and calculate the best response to it, σ = BR(σ ). Then, the best response to σ, will be ɛ-close to ˆσ Ω2 ( q, r). Thus, for an arbitrarily large set of strategy profiles, the best response dynamics will be arbitrarily close to ˆσ Ω2 ( q, r) already after two iterations. Denote by BR(σ) the best response to a strategy σ, and by BR 2 the twice iterated best response, BR 2 (σ) = BR(BR(σ)). Further for any ɛ > 0, and any n N define Σ 2 (ɛ, n) := {σ : σ cutoff strategy for which BR 2 (σ) ˆσ Ω2 ( q, r) < ɛ}. Lemma 5. (Global Basin of Attraction) 20 When φ(0) < 2 < φ(): For any ɛ > 0, the measure of Σ2 (ɛ, n) in the space of cutoff-strategies [0, ] 3 converges to, for n. In particular, there exists n(ɛ) N such that all cutoff strategies σ for which (4) Pr(σ(s, t) = α ) 2 Pr(σ(s, t) = β ) 2 > n 4 and (5) min Pr(σ(s, t) = ω ) ω 2 min Pr(σ(s, t) = ω 2) ω 2 > n 4 hold, are elements of Σ 2 (ɛ, n) for n n(ɛ). Proof. In the Appendix. 9 When viewed as cutoffs, cutoff strategies are points in [0, ] 3, and we mean that the set of cutoffs has Lebesgue measure of at least ɛ. 20 The result holds more generally. If we consider any random (not necessarily cutoff) strategy as starting point, for any ɛ > 0 the probability that the twice iterated best response lies in an ɛ-neighbourhood of conditional sincere voting converges to, for n.

21 THE BAYES CORRELATED EQUILIBRIA OF LARGE ELECTIONS 2 Lemma 5 implies that the best response dynamics converges to ˆσ Ω2 (q, r) for almost any starting point, when n is large. Sketch of Proof. Whenever the margins of victory are larger in Ω than in Ω 2, than the best response converges to conditional sincere voting ˆσ Ω2 (q, r) for n by Lemma 3. In the Appendix we show that a difference of n 4 between the probability that a random voter votes A in α and the probability that a random voter votes A in β - as in 4 and 5 - is sufficient for this result. Conversely, whenever the margins of victory are sufficiently smaller in Ω than in Ω 2, for n, being pivotal contains the information that states Ω 2 do not hold. By the same reasoning, if the difference of the margins of victory in α and β is sufficiently large, after signals a and b being pivotal contains the information that either α does not hold or β does not hold for n. In any case under the best response voter behaviour in Ω is almost as if it is known that a specific state holds. When φ(0) < 2 < φ(), a strict majority of voters prefers A in α and B in β with probability converging to for n. Since in any equilibrium the limit of the expected marginof-victory is zero in Ω 2, the best response satisfies the margin-of-victory condition 2 for n sufficiently large. Consequently, the twice iterated best response converges to ˆσ Ω2 (q, r) by Lemma 3. The Instability of Other Equilibria. Suppose that there exists an equilibrium sequence σ n in Γ n ( q, r) with lim σ n = σ ˆσ Ω2 ( q, r). Hence, Lemma 3 implies that under σ n the limit of the margin-of-victory in α is weakly smaller than in α 2 or β 2 or the limit of the margin-of-victory in β is weakly smaller than in α 2 or β 2 ; otherwise the best response to σ n converges to ˆσ Ω2. Now, Lemma 2 and imply that the margin of victory under σ n converges to zero in α 2 and β 2, hence the same must hold in either α or β. 2 Lemma 5 now implies that for arbitrarily large n, arbitrarily small changes to σ n such that 4 and 5 hold suffice to enter the basin of attraction of ˆσ Ω2 ( q, r) under the best response dynamics. Hence, all equilibrium sequences that do not converge to conditional sincere voting ˆσ Ω2 ( q, r) are instable. Conditional ɛ-equilibrium BR 2 (σ). By Remark 8 any strategy σ in an ɛ-neighbourhood of conditional sincere voting ˆσ Ω2 ( q, r) is ɛ-close to its best reponse BR(σ) for n sufficiently large; σ is a conditional ɛ-equilibrium. So, Proposition 5 implies that for a set of cutoff strategies σ of measure ɛ, the twice iterated best response BR 2 (σ) is a conditional ɛ-equilibrium when 2 We derive this property and other necessary properties of equilibrium sequences σn with lim σ n ˆσ Ω2 ( q, r) in Section 5.. There, we also show that such equilibrium sequences typically exist.