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Econometrica, Vol. 81, No. 3 (May, 2013), 1229–1247 PREFERENCE MONOTONICITY AND INFORMATION AGGREGATION IN ELECTIONS SOURAV BHATTACHARYA University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

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Econometrica, Vol. 81, No. 3 (May, 2013), 1229–1247

PREFERENCE MONOTONICITY AND INFORMATION AGGREGATION IN ELECTIONS BY SOURAV BHATTACHARYA1 If voter preferences depend on a noisy state variable, under what conditions do large elections deliver outcomes “as if” the state were common knowledge? While the existing literature models elections using the jury metaphor where a change in information regarding the state induces all voters to switch in favor of only one alternative, we allow for more general preferences where a change in information can induce a switch in favor of either alternative. We show that information is aggregated for any voting rule if, for a randomly chosen voter, the probability of switching in favor of one alternative is strictly greater than the probability of switching away from that alternative for any given change in belief over states. If the preference distribution violates this condition, there exist equilibria that produce outcomes different from the full information outcome with high probability for large classes of voting rules. In other words, unless preferences closely conform to the jury metaphor, information aggregation is not guaranteed to obtain. KEYWORDS: Voting, elections, information aggregation, preference monotonicity.

ARE LARGE ELECTIONS GUARANTEED to correctly identify the alternative preferred by the majority? Existing work invokes Condorcet Jury Theorem (CJT) to suggest an affirmative answer to the question. According to CJT, while individual members of the jury may not have precise knowledge of the defendant’s guilt, the aggregate outcome in a large jury under majority rule voting is correct almost surely. However, unlike in the jury setup, voters in the real world may have substantial diversity of preferences. The main message of our paper is that if there is a conflict of interest among voters such that a change in information affects different voter groups in opposite ways, there may be coordination problems, and thus elections are no longer guaranteed to deliver the majority-preferred outcome. When preference aggregation is also an issue, large elections may not aggregate information efficiently. In the canonical Condorcet jury model, there are two states of the world: the defendant is either guilty or innocent. Jury members get noisy but independent signals about the state. If each member votes his signal, the Law of Large Numbers implies that a majority of the members vote for the correct alternative almost surely if the jury size is large enough.2 Feddersen and Pesendorfer (1997) (henceforth FP) provided a game theoretic generalization of 1 I thank Daron Acemoglu, David Austen-Smith, Steven Callander, Joyee Deb, Tim Feddersen, Roger Myerson, three anonymous referees, and many participants in various seminars, and conferences for important inputs while writing this paper. All responsibility for any errors remaining in the paper is mine. 2 See Ladha (1992) for a statistical proof of CJT. Austen-Smith and Banks (1996) first pointed out that voting one’s signal (“sincere voting”) is not necessarily rational. McLennan (1998) showed that in the Condorcet setting, if there exists an outcome that aggregates information with sincere voting, then there exists a Nash equilibrium that does the same too.

© 2013 The Econometric Society

DOI: 10.3982/ECTA8311

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CJT. Their paper shows that for any threshold voting rule short of unanimity, large elections approximate the outcome that would have been obtained in the absence of uncertainty. In their model, the state is a continuous variable and, for every voter, the utility difference between alternatives is increasing in the state (the “common values” assumption). In terms of the jury metaphor, they considered the degree of guilt as the state, and while each member of the jury may have a different “threshold of doubt” at which his preferred alternative switches from acquit to convict, as the guilt increases, everyone’s utility from conviction (relative to acquittal) increases. Subsequent work on informational properties of elections (Feddersen and Pesendorfer (1999), Myerson (1998), Wit (1998), Meirowitz (2002), Bouton and Castanheira (2012), Krishna and Morgan (2012)) has considered either completely homogeneous preferences or retained the common values feature, that is, a change in state induces voters to switch only in one direction.3 However, there are realistic situations where a given change in the state of the world affects different groups of voters in opposite ways. For instance, suppose that there are two candidates in an election who are by and large centrists. In state A, candidate 1 is to the left of candidate 2; in state B, candidate 1 is to the right of candidate 2. Clearly, the extreme leftist voters prefer 1 in state A and 2 in state B, while the extreme rightist voters have the exact opposite preference in each state. As a second example, consider a city council voting on a proposal to fund a future public project with increased taxes. The project will not benefit every citizen uniformly and there is uncertainty about who is going to benefit. In state A, the main beneficiaries live in district 1, and in state B, they live in district 2. While the rich citizens in both districts vote against the proposal, the poor citizens in the two districts have opposed, state-contingent preferences: those in district 1 (2) will support (oppose) the proposal in state A and oppose (support) it in state B. As a final example, consider a country voting on a proposal to join the World Trade Organization (WTO) and suppose there is uncertainty about whether the comparative advantage for the country lies in agriculture or in industry. Under free trade, the sector with comparative advantage will be better off and the other sector will be worse off. Clearly, those engaged in agriculture and those in industry will have opposed preference in each state. In our paper, there are two alternatives (P and Q), two states (A and B), and two conditionally independent signals (a and b) much like the canonical CJT. We relax the assumption that all state-sensitive voters agree over what the best alternative is in each state. When individual voter utilities in each state are drawn from a more general distribution, the following statement is true (in an ex ante sense) for a randomly chosen voter holding a given belief over 3 Kim and Fey (2007) considered a setting with opposed rankings and showed that information aggregation can break down. Since they allowed abstention in their model, it is not clear whether the aggregation failure is driven by voter preferences or by the expanded voter strategy space.

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states: for an infinitesimally small increase in the belief he holds, he switches his ranking from P to Q with a certain probability and from Q to P with another probability. We denote these probabilities as “rates of switch” in favor of or against P. On the contrary, under the common values assumption, there is a positive probability of him switching in favor of only one alternative for any change in belief. We obtain a condition on the distribution of voter preferences that guarantees efficient information aggregation for any informative signal structure and any threshold voting rule short of unanimity. The condition is as follows: suppose that under full information about the state, P has more support in state A than in state B. Then, for every prior belief over states held by the randomly chosen voter, the likelihood of him switching from Q to P should be strictly higher than the likelihood of him switching from P to Q. We call this condition Strong Preference Monotonicity (SPM). For every preference distribution that does not satisfy SPM, there exist signal precisions (values of conditional probabilities) that induce equilibria with incorrect outcomes for any voting rule that requires P to win in state A and lose in state B under full information. In these “bad” equilibria, P either wins in each state or loses in each state almost surely. Since SPM is defined on the “rate of switch” in favor of or away from P (rather than on the mass of voters who support or oppose P), the condition is not robust to local perturbations of the distribution of preferences. In all the existing literature including FP, SPM is satisfied by the assumption of unidirectional shift. Our contribution is to point out that this extreme condition is necessary in a certain sense for large elections to aggregate information. If the preference distribution violates SPM, there exists a continuum of conditional probabilities over signals for which information is not aggregated efficiently. What can we say about the informational properties of elections given the probability distribution over signals along with preferences? Here, we show that elections are informationally efficient if and only if the following property (Weak Preference Monotonicity (WPM)) holds: for every prior belief held by the random voter, a change in signal from b to a leads to a higher likelihood of him shifting from Q to P rather than the other way around. A failure of this property for any belief leads to equilibria that induce wrong outcomes in at least one state. Unlike SPM, which is a condition on preferences alone, WPM is a joint property of preferences and signals. A preference distribution satisfies SPM if and only if it satisfies WPM for every signal distribution. Our result is driven by a single insight: the characteristics of the equilibrium outcome depend on the local properties of beliefs; in particular, they depend on the behavior of the expected vote shares around a given belief.4 To ensure that all equilibria induce the full information outcome, one must impose 4 Mandler (2012) also exploited this fact to show that information aggregation can break down in a common values model if there is a small uncertainty over signal precisions. We discuss the relationship with this paper at the end of Section 2.

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a global property on the vote shares that should hold true for all beliefs. SPM is such a property. The intuition for the necessity of SPM is best understood in the following way. For the voters, belief updating is a two-step process: first, the prior is updated, conditioning on the event of being pivotal (to what we call the induced prior); then the final posterior is formed based on the signals. Notice that in an equilibrium with symmetric strategies, all voters have the same induced prior. Consider a consequential rule, that is, a voting rule that, under full information, leads to P winning in state A and losing in state B. The failure of SPM implies that there is some belief where the switch rate in favor of P is higher than the switch rate away from P and there is some other belief where the opposite holds. By continuity, for some signal precisions, there exists a be for which the expected vote share for P is the same in each state. Now lief β  there is an equilibrium (for any voting rule) where each voter holds belief β conditioning on being pivotal, and it is rational to do so since this belief produces a high likelihood of a tie. In this equilibrium, the vote tally for P is the same in each state: therefore, if the policy P wins (loses) in one state, it wins (loses) in the other state too. Clearly, for a consequential rule, in one state a wrong alternative wins. 1. THE SETUP There is an electorate composed of n individuals who choose between P (policy) and Q (status quo). P wins if it gets more than a proportion θ ∈ (0 1) of the votes.5 A state of nature is S ∈ {A B}, and the commonly known prior probability of state A is Pr(A) = α ∈ (0 1). Given some information I we sometimes refer to the updated probability Pr(A|I) simply as the belief. We denote a generic value of the belief by t. The voter receives a private signal s ∈ {a b} that is drawn randomly from a conditionally independent distribution given by Pr(a|A) = qA and Pr(a|B) = qB . We make the usual assumption on informativeness of signals (i.e., 1 > qA > qB > 0). A specific pair of conditional probabilities {qA  qB } is called signal precisions. 1.1. Preferences Each voter’s utility from Q is normalized to 0. The utility of voter i from alternative P is denoted by the vector Ui = {Ui (A) Ui (B)} where Ui (S) is the utility from P in state S. Ui is drawn independently from some atomless distribution over a compact set in R2 ; the realized value of Ui (S) is denoted by ui (S). Voter i strictly prefers the alternative P (Q) in state S if ui (S) is strictly 5 To simplify the analysis, assume the tie breaking rule that if the policy receives exactly θ proportion of votes, the status quo wins.

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positive (negative).6 If ui (S) is weakly positive (negative) in each state, then voter i is committed to alternative P (Q).7 If ui (S) has opposite signs in the two states (i.e., if ui (A)ui (B) < 0), then his ranking over alternatives depends on the state. We refer to such voters as independents. A voter is committed to alternative X ∈ {P Q} with probability γX > 0 and is independent with a probability γI > 0.8 There are two kinds of independents: those who prefer policy P only in state A and those who prefer P only in state B. Call the former voters (u(A) > 0 u(B) < 0) the u-type and call the latter voters (u(A) < 0 u(B) > 0) the dtype. Notice that in the Condorcet setup, we have only one of these two types. For independents, the intensity of preference (i.e., the realized value ui (S)) does matter for their ranking under uncertainty. For an independent voter, deui (B) fine the quantity μi ≡ ui (B)−u ∈ (0 1) as a function of realized utilities. It is i (A) easy to see that a u-type (d-type) voter i holding belief t strictly prefers P (Q) if t > μi and Q (P) if t < μi . The only quantities relevant for an independent voter i’s behavior under uncertainty are (i) the cutoff value of belief μi ∈ (0 1) where he switches his ranking over alternatives, and (ii) the direction of switch xi ∈ {u d}. Instead of using the utility pair {ui (A) ui (B)} we describe an independent voter i’s preferences by the pair {μi  xi }. Conditional on a voter being independent, denote the joint distribution over the cutoff belief and the direction of switch by F(μ x). To be specific, F(μ x) is the probability that a voter has xi = x ∈ {u d} and μi ≤ μ ∈ [0 1].9 In particular, F(μ x) is a joint probability distribution and not the distribution of μ conditional on xi = x. Without loss of generality, we assume that F(1 u) > F(1 d) which implies that the probability that a random voter prefers P to Q in state A is higher than that in state B. Preferences and signals are private information. Since the partisan voters play very little role in our analysis, we focus on F and simply call it the distribution of preferences. The interpretation of F(μ x) is the following. Consider a random voter who holds belief t. The probability that he is u-type and prefers P to Q is F(t u); similarly, the probability that he is d-type and prefers P to Q is 1 − F(t d). Thus, the total probability that a random voter with belief t prefers P to Q is 1 + F(t u) − F(t d). The marginal density ∂F(tx) is denoted ∂t 6 For example, suppose there is a proposal for a tax-funded project where each individual i has to pay a proportion τ of his income yi as tax, and the monetary value of the project for i is ϕSi in ϕS

state S Then ui (S) = ϕSi − yi τ S ∈ {A B} Those with yii > 0 prefer the project in state S

7 Technically speaking, a voter u(A) = u(B) = 0 is committed to both states, but our assumptions ensure that such voters occur with zero probability. 8 The probabilities γP , γQ , and γI are derived from the atomless joint distribution of Ui as γP = Pr(Ui (A) ≥ 0 Ui (B) ≥ 0) γQ = Pr(Ui (A) ≤ 0 Ui (B) ≤ 0), and γI = Pr(Ui (A)Ui (B) < 0)

9 The distribution F over μ and x can be obtained from the original distribution over U as F(μ u) = Pr[μi ≤ μ|ui (A) > 0 ui (B) < 0] and F(μ d) = Pr[μi ≤ μ|ui (A) < 0 ui (B) > 0]. In particular, they must satisfy F(1 u) + F(1 d) = 1

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by fx (t) and is called the switch rate in favor of or away from P, as the case might be. As t increases, the probability that the random voter with belief t prefers P to Q changes by the “net switch rate” h(t) ≡ fu (t) − fd (t): loosely speaking, for any belief t there is a density fu (t) of voters who switch in favor of P and there is a density fd (t) of voters who switch away from P. We make the following assumptions on the distribution of preferences. ASSUMPTION A1: For all μ ∈ [0 1] the joint distribution F(μ x) has continuous and bounded marginal densities (switch rates) fx (μ) for each x ∈ {u d}. ASSUMPTION A2: The net switch rate satisfies h(μ) = 0 at μ ∈ {0 1}. Moreover, there is no open interval over which h(μ) = 0. 1.2. Strategies and Equilibrium—Definition Our equilibrium concept is symmetric Bayesian Nash equilibrium in weakly undominated strategies. All committed voters have weakly dominant actions: they vote for the alternative to which they are committed, irrespective of the signal. A pure strategy for an independent voter i is a function σi : [0 1] × {u d}×{a b} → {P Q}. We denote the pure strategy as a function σi (μ x s) ∈ {0 1} which assigns to every tuple (μ x s) a degenerate probability of voting for P. Formally, we define our equilibrium in the following way. DEFINITION 1—Equilibrium: The strategy profile where every independent voter i uses σi∗ (μ x s) is a symmetric Bayesian equilibrium in undominated strategies if there is a number β∗ ∈ [0 1] such that ⎧ 0 if β∗s < μ and x = u ⎪ ⎨ 1 if β∗s > μ and x = u σi∗ (μ x s) = ∗ ⎪ ⎩ 0 if βs∗ > μ and x = d 1 if βs < μ and x = d where, for s ∈ {a b}, β∗s is given by β∗a =

qA β∗  qA β∗ + qB (1 − β∗ )

β∗b =

(1 − qA )β∗  (1 − qA )β∗ + (1 − qB )(1 − β∗ )

and   ∗ = β∗

Pr A|piv σ−i Before proving existence, we explain the equilibrium definition. For a strategy profile σ fixing a voter i denote by σ−i the profile of strategies played by everybody else. Voter i best responds to a equilibrium profile σ−i by forming an updated belief using two pieces of information: (i) her own signal s and (ii) the

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distribution of signals among other voters given σ−i in the event of a tie. Denote by β the belief Pr(A|piv σ−i ) over states conditioning on pivotality given σ−i , but not taking into account the signal s. Since this belief is induced by the others’ strategies, we call it the induced prior belief. Also, suppose that βs is the posterior belief updated given the signal s and the induced prior β. The best response to σ−i is to vote for P if μ < βs and vote against P if μ > βs for the u-type and conversely for the d-type.10 Therefore, in equilibrium, every voter’s strategy is characterized by a pair of cutoffs {βa  βb } one for each signal. The equilibrium strategy profile σ ∗ satisfies the following condition: for each voter ∗ i, the induced prior obtained from the other’s profile σ−i is exactly equal to β∗ , ∗ ∗ that is, Pr(A|piv σ−i ) = β . In that case, all voters hold the induced prior β∗  and the strategy that involves cutoffs {β∗a  β∗b } obtained by Bayes rule from β∗ is a best response to all other independent voters using cutoffs {β∗a  β∗b }. 1.3. Existence If a randomly chosen voter holds belief t ∈ [0 1], then the probability that he votes for P is given by (1)



V (t) = γI F(t u) + F(1 d) − F(t d) + γP

The probability that a randomly chosen voter with induced prior β votes for P in state S is given by (2)

t(S β) = qS V (βa ) + (1 − qS )V (βb ) for S ∈ {A B} and

β ∈ [0 1]

where βs is the posterior belief based on prior β and signal s, and V (βs ) is obtained from equation (1). Note that t(S β∗ ) is the equilibrium expected vote share for P in state S. From now on we refer to the function t(S β) as the vote share function, stressing the dependence of expected vote shares on the shared induced prior belief β. It is important to note that the vote share function depends only on F and {qA  qB }. Nonatomicity of F in μ guarantees that V (·) is continuous and, as a consequence, so is t(S β) in β. Since 0 < γP ≤ V (t) ≤ 1 − γQ < 1 for all t ∈ [0 1] we have t(S β) ∈ (0 1) for all β ∈ [0 1]. To find β∗  fix i and consider a strategy profile σ−i (β) where the strategies of all voters except i are characterized by cutoffs {βa  βb } that arise from the prior β according to Bayes Rule. The probability of voter i’s vote being decisive in Those with μ = t are indifferent between voting for P or Q but the overall outcome does not depend on the behavior of indifferent voters as indifference is a zero probability event by the assumption of nonatomicity of F

10

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state S is given by equation (3), where t(S β) is calculated from equations (1) and (2),    n−1−[nθ] n−1 Pr piv|S σ−i (β) = (3) t(S β)[nθ] 1 − t(S β)  nθ

S ∈ {A B} where nθ is the largest integer weakly smaller than nθ. The fact that t(S β) ∈ (0 1) guarantees that Pr(piv|S σ−i (β)) ∈ (0 1) and, therefore, there is always a positive probability of being pivotal. Conditioning on being pivotal, the likelihood ratio over states is obtained from Bayes Rule as α Pr(piv|A σ−i (β)) Pr(A|piv σ−i (β)) =

Pr(B|piv σ−i (β)) 1 − α Pr(piv|B σ−i (β)) The belief βi induced by the strategy profile σ−i (β) is given by α Pr(piv|A σ(β)) βi Pr(A|piv σ−i (β)) = = 1 − βi 1 − Pr(A|piv σ−i (β)) 1 − α Pr(piv|B σ(β)) or H(β n θ) where 1 + H(β n θ) α Pr(piv|A σ−i (β)) H(β n θ) ≡ 1 − α Pr(piv|B σ−i (β))  t(A β)[nθ] (1 − t(A β))n−1−[nθ] α =

1−α t(B β)[nθ] (1 − t(B β))n−1−[nθ] βi =

By our definition of equilibrium, we must have βi = β. Thus, our equilibrium condition is simply (4)

β=

H(β n θ)

1 + H(β n θ)

Any solution to this equation is the equilibrium induced prior β∗ . Notice that we find the equilibrium as a fixed point in the space of beliefs rather than as a fixed point in the space of strategies. Continuity and boundedness of t(S β) guarantee that H(β n θ) is continuous and that it is bounded above 0 and below 1 for any given n and θ. The left hand side of (4), on the other hand, varies monotonically in (0 ∞) as β changes in (0 1). Therefore, a solution to equation (4) exists and it characterizes the equilibrium for any (n θ). We denote the equilibrium induced prior by βnθ . Moreover, we have βnθ ∈ (0 1) by the boundedness of H(β n θ).

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2. LARGE ELECTIONS AND INFORMATION AGGREGATION In the previous section, we characterized equilibria for an electorate with a finite size. In this section, we provide conditions on the distribution of preferences under which large elections are guaranteed to produce outcomes close to the full information outcome. To look at the voting outcome for a large electorate, we hold the preference distribution, signal precision, and voting rule fixed, and examine the limit of the equilibrium outcome of the sequence of games as the size of the electorate increases. Thus, we are looking at the limit of βnθ as n → ∞. Since the sequence belongs to a compact interval [0 1], a limit always exists, which we denote by β0θ . From condition (4), we obtain the following limit equilibrium condition, which identifies the limit of the induced prior belief held in a given sequence of equilibria for a voting rule θ: 

H(βnθ  n θ) 0 βθ = lim (5)

n→∞ 1 + H(βn  n θ) θ For large n the actual vote share for P in state S is close to the expected vote share t(S β0θ ) with a high probability. Thus, for large enough n P wins in state S with an arbitrarily high probability if t(S β0θ ) > θ and loses with an arbitrarily high probability if t(S β0θ ) < θ. We compare this (limit) outcome given θ with the full information outcome induced by θ to see whether information is indeed aggregated in the limit of the particular equilibrium sequence. To be able to do that, we first formally define (i) full information outcomes and (ii) the standard for information aggregation. 2.1. Full Information Equivalence To study information aggregation properties of an election, we first introduce a classification of voting rules according to the outcome under common knowledge of the state in a large election. Denote by VS the expected share of voters who prefer P over Q in state S. In a large election, if the state is known, the actual vote share of P is very close to VS . Since we assume that F(1 u) > F(1 d), equation (1) implies that VA > VB  that is, a larger fraction of voters support P in state A than in state B. For any voting rule θ ∈ (VB  VA ) the alternative P would win in state A and lose in state B if the state were known. Since the voting rules θ ∈ (VB  VA ) induce different outcomes in different states, we call them the consequential rules. On the other hand, the voting rules θ ∈ (0 VB ) are called P-trivial rules since P wins in both states; likewise, the voting rules θ ∈ (VA  1) are called Q-trivial rules. The yardstick of information aggregation used here is the full information equivalence (FIE) criterion used in FP. An equilibrium sequence of a game defined by preference F signal precisions {qA  qB } and voting rule θ is said to be full information equivalent if the limit outcome is the same as the full information outcome in each state with an arbitrarily high probability. The formal definition follows.

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DEFINITION 2—Full Information Equivalence: Fix the distribution of preferences F signal precision {qA  qB } and voting rule θ ∈ (0 1). An equilibrium sequence is said to satisfy full information equivalence if for any ε > 0, there is some m such that the equilibrium outcome in each state is the same as the outcome under full information with a probability greater than 1 − ε whenever the size of the electorate is greater than m. We show that information is aggregated in all equilibrium sequences (irrespective of the voting rule and signal precision) if a particular condition on the preference distribution F is satisfied, which we call Strong Preference Monotonicity (SPM). If F does not satisfy SPM, there exist signal precisions for which, given any consequential rule, the voting game has a sequence of equilibria that produces incorrect outcomes in the limit.11 Moreover, for any equilibrium sequence that fails to satisfy FIE, an alternative different from the correct one wins with a very high probability in at least one state. We first define and discuss the condition of SPM and then state our main result. DEFINITION 3—Strong Preference Monotonicity: A distribution of preferences F satisfies Strong Preference Monotonicity (SPM) if fu (μ) > fd (μ) for all μ ∈ [0 1] except possibly for a countable number of values of μ. SPM is a condition on the switch rates. According to this condition, the switch rate in favor of P is greater than the switch rate away from P for almost every belief. Another interpretation of this condition is the following: if a random voter holds belief t the probability that he prefers P to Q is strictly increasing in t.12 The “common values” assumption in FP implies, in the context of a continuous state space, that voters switch only in favor of policy P for any increase in state. Since the state space is binary in our case, we convexify the state space using the belief over state A. In a binary-state version of FP, common values would mean that fd (μ) = 0 for all μ ∈ [0 1]. The assumption of SPM is therefore only slightly weaker than the common values assumption. THEOREM 1: If the preference distribution F satisfies Strong Preference Monotonicity (SPM), every equilibrium sequence satisfies full information equivalence (FIE). If F does not satisfy SPM, there is some signal precision {qA  qB } such that for any consequential voting rule θ (except possibly for a single value of the rule), the sequence of games defined by F qA  qB , and θ has an equilibrium sequence that fails to satisfy FIE. In this equilibrium sequence, P either wins in both states 11 We sometimes refer to the sequence of voting games with F qA , qB , and θ (as n becomes large) simply as the voting game with F qA , qB , and θ (with no reference to n) when there is no reason for confusion. Similarly, we sometimes loosely refer to an equilibrium sequence as an equilibrium. 12 Notice that the probability of the random voter preferring P to Q is given by V (t) Since dV (t) = γI [fu (t) − fd (t)] it is easy to see that SPM is equivalent to V (t) being increasing for all t

dt

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with probability converging to 1 or loses in both states with probability converging to 1 as the size of the electorate grows. Given any consequential rule θ information is aggregated along a sequence if, for the limiting induced prior β0θ  we have t(A β0θ ) > θ > t(B β0θ ) that is, P wins in state A and loses in state B. Under SPM, we have t(A β) > t(B β) for all β and, in particular, the condition for FIE is always satisfied. On the other hand, if F fails to satisfy SPM, there exist signal precisions qA  qB and a sequence of equilibria for which t(A β0θ ) = t(B β0θ ) = t. In this equilibrium, P wins in both states if θ < t and loses in both states if θ > t. Two comments are in order regarding the content of the theorem. First, the theorem says that nonaggregating equilibria exist when F violates SPM. For the same parameters that lead to non-FIE equilibria, there may be other equilibria, some of which may satisfy FIE. However, the non-FIE equilibrium sequence we identify in the proof of the theorem (P either always loses or always wins) has a focality property: the equilibrium belief and strategies employed in the limit are independent of the particular voting rule in use. While there may exist equilibrium sequences satisfying FIE even when SPM is violated, the beliefs and strategies in such equilibria are very sensitive to the voting rule. Second, while the theorem only says that there exists some value of signal precision that leads to non-FIE equilibria for each F that violates SPM, outcomes in such equilibria are robust to small perturbations in signal precisions. From the construction used in the proof, it will be clear that, for any F that violates SPM, there is actually a continuum of such signal precisions for which these nonaggregating equilibria exist.13 We next provide the proof of the theorem step by step. 2.2. Proof of Theorem 1 As the first step of the proof, Lemma 1 identifies a specific relationship between the voting rule θ and the limit of the induced prior belief β0θ . This relationship is necessary to ensure that H(n βnθ  θ) does not explode to infinity or go to zero in the limit. LEMMA 1: Fix F qA  and qB . If the limit belief is β0θ in an equilibrium sequence, then θ   1−θ   θ   1−θ  = t B β0θ

1 − t A β0θ 1 − t B β0θ t A β0θ Suppose F violates SPM. The proof of Lemma 5 actually finds an ε such that for any ε ∈ (0 ε) there is an equilibrium in which FIE fails for any given consequential rule and signal precisions qA = 12 + ε and qB = 12 + ε

13

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PROOF: Whenever t(A β0θ ) = t(B β0θ ) the lemma holds trivially, which is the case if (but not only if) β0θ ∈ {0 1}. Now consider β0θ ∈ (0 1) and rewrite H(n βnθ  θ) =

n n θ n 1−θ 1−tA α n−m−1 (tA ) (1−tA ) m n n )θ (1−t n )1−θ 1−α 1−tB (tB B 1 n θ θ

[

]

[

] , where m =



θ

 tAn = t(A βnθ )

and tBn = t(B β ). Since m ≥ n −  we have m → ∞ as n → ∞. Also, since t(S β) ∈ (γP  1 − γQ ) and m − n ∈ [0 θ1 ] there is some 0 < t < t 1−t n

such that t ≤ [ 1−tAn ]n−m−1 ≤ t for all m and n. If there is some ε > 0 such (t n )θ (1−t n )1−θ

B

that (tAn )θ (1−tAn )1−θ > 1 + ε for all n large enough, then limn→∞ H(βnθ  n θ) > B B α ( 1−α )t[limm→∞ (1 + ε)m ] → ∞. Similarly, if there is some ε > 0 such that n )θ (1−t n )1−θ (tA A

< 1 − ε for all n large enough, then limn→∞ H(βnθ  n θ) = 0 To make sure that the right hand side (RHS) of equation (5) is bounded above 0 and below 1 the condition in the lemma must hold. Q.E.D. n )θ (1−t n )1−θ (tB B

For a given belief β, if t(A β) = t(B β) define a function θ∗ (β) as 1 − t(B β) 1 − t(A β) θ∗ (β) =

t(A β)(1 − t(B β)) log t(B β)(1 − t(A β)) log

(6)

COROLLARY 1: Fix F qA , and qB . If for some θ we have β0θ = β in some equilibrium sequence and t(A β) = t(B β), then it must be the case that θ = θ∗ (β). Moreover, θ∗ (β) lies strictly between t(A β) and t(B β). The corollary is easy to verify from Lemma 1.14 Moreover, this corollary suggests an interesting relationship between voting rules and equilibrium vote shares in large elections, which is empirically testable in principle.15 Fixing F , qA and qB  we basically fix the functions t(S ·) for S ∈ {A B}. Now, for any belief β with the property that t(A β) = t(B β), Corollary 1 identifies a unique candidate voting rule θ∗ (β) such that the sequence of games with F , qA  qB and θ∗ (β) would have a sequence of equilibria with induced prior converging to β. Lemma 2 identifies such candidate voting rules Θ(β) for all β ∈ [0 1]. LEMMA 2: Fix F qA , and qB . Define by Θ(β) the following correspondence: (i) If t(A β) = t(B β), then Θ(β) is the unique value θ∗ (β). 14 Corollary 1 makes use of the fact that if both z and θ lie in (0 1) the function zθ (1 − z)1−θ is single peaked in z

15 Suppose the equilibrium vote share for the alternative X ∈ {P Q} in state S ∈ {A B} in a large election with plurality rule θ is v(X|S) Lemma 1 suggests that if the election proθ duces different vote shares for the alternatives in the two states, then we must have 1−θ = v(P|A) v(Q|A) (log v(P|B) )/(log v(Q|B) )

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(ii) If t(A β) = t(B β) = t and β ∈ (0 1), then Θ(β) = (0 1). (iii) Θ(0) = (0 VB ] if fu (0) > fd (0) and [VB  1) if fu (0) < fd (0). (iv) Θ(1) = [VA  1) if fu (1) > fd (1) and (0 VA ] if fu (1) < fd (1). / Θ(β) then For a given β ∈ [0 1] consider any sequence βn → β. Now, if θ ∈ β as n → ∞. H(βn  n θ) is bounded away from 1−β PROOF: The proof for case (i) follows from Lemma 1. The statement of case (ii) of the lemma is vacuous. Next, consider case (iii) with the subcase that fu (0) > fd (0). In this case, tAn > tBn > VB as βn → 0. Notice that the the function z θ (1 − z)1−θ is single peaked in z and attains its maximum at z = θ. Now, if θ > VB  then for all large enough n we have θ > tAn > tBn  which would imply that (tAn )θ (1 − tAn )1−θ > (tBn )θ (1 − tBn )1−θ . In that case, we have α )t for all large n The subcase with fu (0) < fd (0) and case H(βn  n θ) > ( 1−α (iv) follow similar logic. Q.E.D. / Θ(β) no seFor a game defined by F qA  qB , and some voting rule θ ∈ quence βn → β satisfies the limit equilibrium condition (6). We want to show the converse, that is, for any β and any θ ∈ Θ(β) there exists a sequence of equilibria βnθ → β. However, that statement is true only with a few caveats. For the voting rule θ∗ (β) to support an equilibrium sequence βnθ → β we need to ensure that θ∗ (β) is not locally constant at β. When θ∗ (β) is not well defined, that is, for a belief such that t(A β) = t(B β) = t any voting rule θ ∈ Θ(β) \ {t} supports an equilibrium sequence βnθ → β if the vote share functions t(A β) and t(B β) “cross” at β. To state this formally, fixing F qA , and qB , we say that a belief β ∈ [0 1] is regular if any of the following ∗ three conditions holds: (i) β ∈ {0 1} or (ii) t(A β) = t(B β) and dθdβ(β) = 0 − dt(Aβ) = 0. or (iii) t(A β) = t(B β) and dt(Aβ) dβ dβ Lemma 3 is a partial characterization of the set of equilibria for large electorates. LEMMA 3: Fix F , qA , and qB . For any β ∈ [0 1] there exists a sequence of equilibria with induced prior beliefs βnθ converging to β only if θ ∈ Θ(β). If β is regular, there is a sequence of equilibria with induced prior βnθ converging to β if θ ∈ Θ(β) (except possibly for the case θ = t(A β) = t(B β)). The “only if” direction follows directly from Lemma 2. The proof for the “if” direction is as follows. H(βnθ) . The proof PROOF OF LEMMA 3: Define the function Gn (β θ) = 1+H(βnθ)   consists of showing that for any regular β if θ ∈ Θ(β), then there is a sequence  In this proof, we repeatedly of fixed points βnθ of Gn (β θ) such that βθn → β. xθ (1−x)1−θ use the following result: if Z(x y θ) = y θ (1−y)1−θ for some 1 > x > y > 0, then ∂Z(xyθ) > 0 (result A). ∂θ

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 such that t(A β)  = t(B β).  By Lemma 2, First, consider some regular β ∗    > t(B β)  Θ(β) = θ (β). We provide the proof for the case with t(A β)  dθ∗ (β) and dβ < 0; the other cases are similar. Since fx (·) is continuous and  − ε β  + ε) where (i) θ∗ (β) is bounded, there must be a neighborhood (β β  + ε) strictly decreasing and (ii) t(A β) > t(B β). Next, fix some β ∈ (β 1−t(Aβ) n−m−1 α m and write H(β n θ) as B(β)[g(x y θ)] , where B(β) = 1−α [ 1−t(Bβ) ] ,  x = t(A β) and y = t(B β). B(β) is bounded above and below. m = nθ

θ  > θ∗ (β) we must have g(x y θ∗ (β))  > g(x y θ∗ (β)) = 1, where Since θ∗ (β) the inequality follows from result A and the equality follows from definition of  m → ∞ implying that H(β n θ∗ (β))  → θ∗ (β). As n → ∞ [g(x y θ∗ (β))] ∗    ∞. Therefore, for any β ∈ (β β + ε) we must have Gn (β θ (β)) → 1 as  → 0 as  − ε β)  we must have Gn (β θ∗ (β)) n → ∞. Similarly, for β ∈ (β ∗  − ε β + ε) that solves n → ∞. By continuity, there must exist some βθn (β) ∈ (β ∗  Gn (β θ (β)) = β Therefore, for any ε > 0 small enough, there exists a se∗   such that for all n quence βθn (β) of fixed points of the function Gn (· θ∗ (β))  θ∗ (β)  large enough, |βn − β| < ε.  ∈ (0 1) such that t(A β)  = t(B β)  = t. Lemma 2 Next, consider a regular β  = (0 1) Notice that Gn (β  θ) = α for all (n θ). If β  = α suggests that Θ(β)  for all n and we are done. Now, suppose β > α consider a sequence βn = β  < α is analogous). By regularity, assume that t(A β)−t(B β) (the case with β  (again, the proof in the other case is similar). is strictly decreasing at β Consider any θ > t and choose ε small enough such that θ > θ∗ (β) for all  − ε β).  Since t(A β) > θ∗ (β) > t(B β) for all β ∈ (β  − ε β)  we β ∈ (β have Gn (β θ) → 1 in this interval by result A. Thus, for each n large enough,  − ε and negative at β = β.  Therefore, Gn (β θ) − β is positive at β = β  − ε β)  for large enough n. Thus, Gn (β θ) must have a fixed point βθn ∈ (β there exists a sequence of fixed points βθn of Gn (β θ) such that for any ε > 0  < ε for n large enough given any θ > t. To show the exsmall enough, |βθn − β|  for voting rules θ < t consider istence of a sequence of beliefs converging to β   the interval β ∈ (β β + ε) with ε appropriately chosen.  ∈ {0 1}. If β  = 0 and fu (0) > fd (0) then Finally, consider the cases with β = we must have t(A β) > t(B β) in some interval (0 ε). We also have α − β α > 0. By the above method, for any θ ∈ (0 VB ], we can show that for ε > 0 small enough, there exists a sequence of fixed points of Gn (β θ) in the interval  = 0. The other cases, that is, (i) β  = 0 and fu (0) < (0 ε) that converges to β   fd (0), (ii) β = 1 and fu (0) > fd (0) and (iii) β = 1 and fu (0) < fd (0) can be dealt with in the same way. Q.E.D. Lemma 3 allows us to pin down the set of equilibrium outcomes of large elections (given F θ and {qA  qB }) in the following way. Given some θ ∈ (0 1) find the set of beliefs B (θ) = {β : θ ∈ Θ(β)}. For any regular β ∈ B (θ) there is an equilibrium sequence with beliefs converging to β. We can determine the

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winning alternative in each state S by comparing the relevant vote share t(S β) with the voting rule θ. Notice that as long as θ is different from t(S β), we can pin down the outcome in a large election almost surely. Lemma 4 gives us the outcomes depending on the shape of the vote share functions. LEMMA 4: Fix F , qA  and qB . If VA > t(A β) > t(B β) > VB for every β ∈ (0 1) all equilibrium sequences satisfy FIE for each voting rule. If, for some regular β we have t(A β) < t(B β) then, given the voting rule θ∗ (β) there exists an equilibrium sequence for which P loses in state A and wins in state B with high probability. If, for any regular β ∈ (0 1) we have t(A β) = t(B β) = t then for every θ > t there is an equilibrium sequence in which P loses with high probability in each state and for every θ < t there is an equilibrium sequence in which P wins with high probability in each state. The proof is given in the Supplemental Material (Bhattacharya (2013)). Lemma 4 is a direct outcome of Lemmas 2 and 3. It shows that if the vote share in state A is greater than that in state B for every belief, then large elections approximate the full information outcome for all voting rules. If, on the other hand, there is some belief for which the vote share in state A is weakly less than that in state B and, additionally, if that belief satisfies the regularity property, then there is an equilibrium sequence that does not satisfy FIE for a consequential rule (since P either loses in state A or wins in state B). Lemma 5 concludes the proof of Theorem 1. LEMMA 5: Fix the distribution of preferences F . If F satisfies SPM, then for each {qA  qB } the vote share function satisfies VA > t(A β) > t(B β) > VB for every β ∈ (0 1). If F violates SPM, there exist signal precisions {qA  qB } such that the vote share function has the property that v(A β) = t(B β) for some regular β ∈ (0 1). See the Supplemental Material for the proof. The two direct implications of SPM are that (i) the vote share in state A is higher than that in state B for any belief and (ii) the vote share function is strictly increasing in β in each state. These implications lead to the conclusion that VA > t(A β) > t(B β) > VB  which, according to Lemma 4, is sufficient for full information equivalence. If SPM is not satisfied by some F , by continuity arguments, we can find signal precisions {qA  qB } that lead to vote share functions with the property that there is some regular β with t(A β) = t(B β) = t. By Lemma 4, the game defined by F , qA , qB , and any θ ∈ (VB  VA ) \ {t} has an equilibrium sequence that fails to satisfy FIE. Moreover, the proof of Lemma 5 shows that for each F that violates SPM, there exists a continuum of such signal precisions. The intuition for that is as follows. Suppose, for some F qA , and qB  there is some β at which vote share functions t(A ·) and t(B ·) “cross” each other. By continuity of the vote share functions in {qA  qB } if we perturb {qA  qB } a little bit, we will still have some β close to β at which t(A ·) and t(B ·) cross each other.

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2.3. Strong and Weak Preference Monotonicity SPM is a condition on the distribution of preferences alone. Theorem 1shows that there exist signal precisions that will lead to incorrect outcomes in some equilibria of large elections if preferences violate SPM. However, one may ask the following question: given a signal precision, what can we say about the information aggregation properties of a large election? The next proposition provides a joint condition on preferences and signals that guarantees information aggregation. Moreover, if the condition is violated, there exist equilibrium sequences that do not aggregate information. DEFINITION 4—Weak Preference Monotonicity: Together, a preference distribution F and signal precision {qA  qB } are said to satisfy Weak Preference Monotonicity (WPM) if, for all beliefs β ∈ (0 1) the following condition is satisfied: F(βa  u) − F(βb  u) > F(βa  d) − F(βb  d)

According to WPM, as the signal changes from b to a for every prior belief β ∈ (0 1) held by the randomly chosen voter, the probability of him switching from Q to P is greater than his probability of switching from P to Q. In other words, a given change in signal always induces a net shift (in probabilistic terms) in favor of the same alternative, irrespective of the prior belief. SPM is satisfied for some preference distribution F if and only if WPM holds for F and each {qA  qB } such that 0 < qB < qA < 1. To prove the proposition, we need an extra assumption that guarantees that for any admissible signal precision, beliefs that produce equal vote shares in each state will be regular, that is, t(A β) = t(B β) implies that β is regular as long as β ∈ (0 1). a ASSUMPTION A3: If, for any 1 ≥ a > b > 0 we have b h(μ) dμ = 0 then it cannot be the case that either (i) a(1−a) = h(b) or (ii) h(a) = h(b) = 0. b(1−b) h(a) PROPOSITION 1: Suppose Assumptions A1, A2, and A3 hold. If F and {qA  qB } satisfy WPM, then for each consequential rule, every equilibrium sequence satisfies FIE. If they violate WPM, then for each consequential rule (except possibly a single rule), there exists an equilibrium sequence that fails to satisfy FIE. In this equilibrium sequence, P either wins in both states with probability converging to 1 or loses in both states with probability converging to 1 as the size of the electorate grows. The proof is provided in the Supplemental Material. To see the intuition behind the proposition, notice that WPM is equivalent to t(A β) > t(B β) for all β ∈ (0 1) which, by Lemma 4, guarantees information aggregation for consequential rules. If WPM is violated, that is, if the

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same change in signal produces a net shift in favor of P for some belief β1 and a net shift away from P for some belief β2 , then, by continuity of the vote  that produces equal vote shares in each state. share function, there must exist β  Since regularity of β is guaranteed by A3, Lemma 4 tells us that information aggregation fails for almost every consequential rule. Mandler (2012) showed that information aggregation can fail even in a common values setup if there is aggregate uncertainty over the signal precision. In Mandler’s model, each voter prefers P in state A and Q in state B. The signal precision {qA  qB } itself is a random variable, and qS (the probability of signal a in each state S) is drawn from a commonly known distribution GS with full support over [0 1]. Notice that for realizations such that qA < qB  a change in signal a to b increases the relative likelihood of state A and for realizations qA > qB , a change in signal a to b decreases the relative likelihood of state A. Therefore, a change in signal from a to b makes every voter more or less likely to vote P depending on whether voters believe that the signal a occurs with a relatively higher frequency in state A or in state B. While WPM in our model is defined with respect to fixed signal precisions and varying preferences, essentially what we have in Mandler (2012) is a failure of WPM due to varying signal precisions. The broad lesson that arises from a comparison of these two papers is that, irrespective of the source of uncertainty, voting equilibria are driven by the local properties of the relationship between voter beliefs and expected vote shares, and unless a given change in the signal increases (or decreases) vote shares in favor of the same alternative for every belief, information may fail to aggregate. 3. CONCLUSION Our main result tells us that if we want to guarantee information aggregation for every signal precision, the preferences have to satisfy a strong condition, the SPM. According to SPM, the net rate of switch in favor of the alternative P must be strictly positive for almost all beliefs. Put differently, SPM requires that F(μ u) − F(μ d) be strictly increasing in μ. Notice that this is not a condition on the absolute mass of switchers, but on the net rate of switching. The condition is very fragile in the sense that we can always change the function F(· d) over any small open interval ζ ⊂ (0 1) of the support and set fd (μ) > fu (μ) over ζ and the condition is violated. This feature allows us to present a sense in which SPM is nongeneric and its failure is generic. Say that two distribution functions F and G are “ε-close” if the integral of |F − G| is less than some ε > 0. Now, for any distribution of social preference F and any ε there is some G that does not satisfy SPM. Thus, SPM is nongeneric. Moreover, for any F that does not satisfy SPM, we can find some small enough ε such that all functions that are ε-close to F fail to satisfy SPM. In this sense, the failure of SPM is generic.16 16

I thank an anonymous referee for suggesting this genericity interpretation.

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The bulk of the existing literature on CJT has assumed unidirectional switching, that is, fd (μ) = 0 for all μ which automatically implies SPM. The point of this paper is to demonstrate that the property of full information equivalence cannot be extended much beyond the jury setting. It is customary to mention here that we have not included “realistic features” of elections such as the possibility of abstention, signaling motivation of voters, unanimity rules, costly information, and so on.17 We have specifically excluded these features so as to demonstrate that the central source of aggregation failure is preference conflict and to indicate the possibility that such conflict may lead to inefficient outcomes in real world elections. REFERENCES AUSTEN-SMITH, D., AND J. BANKS (1996): “Information Aggregation, Rationality and the Condorcet Jury Theorem,” American Political Science Review, 90 (1), 34–45. [1229] BHATTACHARYA, S. (2013): “Supplement to ‘Preference Monotonicity and Information Aggregation in Elections’,” Econometrica Supplemental Material, 81, http://www.econometricsociety. org/ecta/Supmat/8311_proofs.pdf. [1243] BOUTON, L., AND M. CASTANHEIRA (2012): “One Person, Many Votes: Divided Majority and Information Aggregation,” Econometrica, 80, 43–87. [1230] FEDDERSEN, T., AND W. PESENDORFER (1997): “Voting Behavior and Information Aggregation in Large Elections With Private Information,” Econometrica, 65 (5), 1029–1058. [1229] (1998): “Convicting the Innocent: The Inferiority of Unanimous Jury Verdicts Under Strategic Voting,” American Political Science Review, 92 (1), 23–35. [1246] (1999): “Abstention in Elections With Asymmetric Information and Diverse Preferences,” American Political Science Review, 93 (2), 381–398. [1230] KIM, J., AND M. FEY (2007): “The Swing Voter’s Curse With Adversarial Preferences,” Journal of Economic Theory, 135, 236–252. [1230] KRISHNA, V., AND J. MORGAN (2012): “Voluntary Voting: Costs and Benefits,” Journal of Economic Theory, 147, 2083–2123. [1230] LADHA, K. (1992): “The Condorcet Jury Theorem, Free Speech and Correlated Votes,” American Journal of Political Science, 36, 617–634. [1229] MANDLER, M. (2012): “The Fragility of Information Aggregation in Large Elections,” Games and Economic Behavior, 74, 257–268. [1231,1245] MARTINELLI, C. (2006): “Would Rational Voters Acquire Costly Information?” Journal of Economic Theory, 129, 225–251. [1246] MCLENNAN, A. (1998): “Consequences of the Condorcet Jury Theorem for Beneficial Information Aggregation by Rational Agents,” American Political Science Review, 92 (2), 413–418. [1229] MEIROWITZ, A. (2002): “Informative Voting and Condorcet Jury Theorems With a Continuum of Types,” Social Choice and Welfare, 19, 219–236. [1230] MYERSON, R. B. (1998): “Extended Poisson Games and the Condorcet Jury Theorem,” Games and Economic Behaviour, 25, 111–131. [1230] PERSICO, N. (2004): “Committee Design With Endogenous Information,” Review of Economic Studies, 71 (1), 165–191. [1246] RAZIN, R. (2003): “Signaling and Election Motivations in a Voting Model With Common Values and Responsive Candidates,” Econometrica, 71 (4), 1083–1120. [1246] 17 For these other sources of aggregation failure, see Feddersen–Pesendorfer (1998), Razin (2003), Persico (2004), and Martinelli (2006).

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WIT, J. (1998): “Rational Choice and the Condorcet Jury Theorem,” Games and Economic Behavior, 22, 364–376. [1230]

Dept. of Economics, University of Pittsburgh, 4528 Posvar Hall, 230 South Bouquet St, Pittsburgh, PA 15260, U.S.A.; [email protected]. Manuscript received December, 2008; final revision received November, 2012.

Preference Monotonicity and Information Aggregation ...

{01} which assigns to every tuple (μ x s) a degenerate probability of voting for P. Formally, we define our equilibrium in the following way. DEFINITION 1—Equilibrium: The strategy profile where every independent voter i uses σ∗ i (μ x s) is a symmetric Bayesian equilibrium in undominated strategies if there is a number β∗ ...

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