Voting with Behavioral Heterogeneity

Transcription

1 Voting with Behavioral Heterogeneity Youzong Xu Setember 22, 2016 Abstract This aer studies collective decisions made by behaviorally heterogeneous voters with asymmetric information. Here behavioral heterogeneity models voters different levels of sohistication in handling information. Secifically, I consider a mandatory voting model with a grou of voters. Some voters take the information revealed by ivotality into account when making decisions call sohisticated voters, while other voters vote only according to their rivate information called naive voters. I show that behavioral heterogeneity can imose oosite influence on sohisticated voters behavior and information aggregation. When voter oulation is relatively small, the resence of naive voters can decrease the robability that sohisticated voters vote against the status quo but increase the robability that Tye I error occurs. 1 While when voter oulation is large, the resence of naive voters can increase the robability that sohisticated voters vote against the status quo but decrease the robability that Tye I error occurs. I also characterize the set of voting rules that aggregate information efficiently and rovide other results that are useful in choosing otimal voting rules and voter oulation for collective decisions when voters are behaviorally heterogeneous. I am very grateful to John Nachbar, Elizabeth Penn, and Marcus Berliant for their invaluable guidance and encouragement. I also thank Scott Baker, Randall Calvert, Anqi Li, John Patty, Werner Ploberger, Brian Rogers, Maher Said, Jonathan Weinstein, Yunfei Cao, Wei-Cheng Chen, Bo Li, and Junmin Liao for their helful comments. I areciate the financial suort from the Deartment of Economics, Washington University in St. Louis, and the Center for Research in Economics and Strategy CRES in the Olin Business School, Washington University in St. Louis. 1 Tye I error in this aer means that status quo is rejected by mistake when it should be maintained. 1

2 1 Introduction Most literature on voting theory, though some of which allows heterogeneity in voters references and quality of information, assumes that voters are behaviorally homogeneous, in the sense that voters show the same level of sohistication in using information. More recisely, the majority of this literature assumes that voters are sohisticated in using information in the sense that voters are fully rational and use both their rivate information and the information imlicit in other voters actions when making decisions. 2 Other literature assumes that voters make decisions only according their rivate information. Thus these voters adot a naiver method in using information when making decision. A artial excetion are aers in which a fraction of voters are artisans, whose reference over alternatives are indeendent of states and information. 3 But it is natural for us to think that in reality, individuals may actually handle information with different levels of sohistication. This intuition is confirmed by emirical research. For examle, Ladha, Miller and Oenheimer 1996 find evidence in their exeriments on voting behavior that not all voters are sohisticated in using information. Guarnaschelli, McKelvey, and Palfrey 2000 observe that some articiants in their exeriments on voting behavior always only follow their rivate information, while some other articiants act more sohisticatedly, taking into account both their rivate information and the information imlicit other subjects actions when making decisions. The henomenon that agents show different levels of sohistication in using information is observed in exeriments on other toics as well. For examle, in their exeriments on traders behaviors in financial markets, Ciriani and Guarino 2005 observe different levels of sohistication in using information in the articiants in the exeriments. Both of the intuition and the observations in related exeriments motivate me to introduce behavioral heterogeneity to model voters different levels of sohistication in using information. I consider a 2-state 2-alternative setting similar to Feddersen and Pesendorfer 1997, In this framework, voters have state-contingent references on alternatives but have different beliefs about the state of the world. But voters cannot observe the state of the world directly. Instead, each voter receives a noisy signal about the true state. Secifically, I consider a mandatory voting model in which a grou of voters need to 2 Some literature e.g., Eyster and Rabin 2005 allows voters to act artially rational, in the sense that voters do take the information imlicit in other voters actions into account, but they have incorrect beliefs about other voters equilibrium strategies. But this literature generally assumes that voters make mistakes in the same way and with the same robability, that is, voters are still behaviorally homogeneous. 3 Partisans behaviors reveal no information, which is the key difference between artisans and naive voters. Thus the resence of artisans imoses different influences on sohisticated voters behaviors and grou decisions than the resence of naive voters do. Here naive voters refer to the voters who only rely on their rivate information to make decisions. 2

3 choose between two alternatives, the status quo and a change. The two states are the original state and the new state. All voters refer the status quo if the true state is the original one and refers the change if the true state is the new one. But voters cannot observe the true state directly. Instead, there are a set of binary noisy signals. One of the two signals is the correct signal for the original state as it rovides more information that suorts the original state to be the true state. The other signal is the correct signal for the new state as it rovides more information that suorts the new state to be the true state. Each voter receives a noisy signal from this set. Voters are behavioral heterogeneous in the sense that there are two tyes of voters in this grou, sohisticated and naive. A naive voter only follows his rivate information when making decisions. More secifically, a naive voter votes the status quo if she receives the correct signal for the original state and votes the change if she receives the correct signal for the new state. Thus the action of a naive voter truthfully reveals her rivate information. A sohisticated voter, on the other hand, like voters in Austen-Smith and Banks 1996 and Feddersen and Pesendorfer 1997, 1998, is fully rational and uses both his rivate information and the information imlicit in other voters actions to make decision. 4 As Austen-Smith and Banks 1996 and Feddersen and Pesendorfer 1997 have shown, the truth-telling strategy of naive voters is generally not an equilibrium strategy for sohisticated, which means that a sohisticated voter s action generally does not truth-fully reflects his rivate information. That a sohisticated voter is fully rational means that he is a utility maximizer and votes only for the alternative that maximizes his exected utility. Note that a sohisticated voter s vote affects his own exected utility if and only if he is the ivotal voter, by which I mean that his choice alone determines the outcome of collective decision. That is, if the ivotal voter votes the status quo, then status quo is maintained; and if he votes the change, then the change is selected by the grou. For a sohisticated voter, call the event that he is ivotal ivotality. Thus a sohisticated voter can make decision conditioning himself on being ivotal. Since in an equilibrium, a sohisticated voter conditions himself on being ivotal with the belief that all other voters are following their equilibrium strategies, the information imlicit in other voters actions is equivalent to the information revealed by ivotality. Through the whole aer, I focus on symmetric Bayesian Nash equilibria in which 4 Sohisticated voters in this resent aer are called rational or strategic voters in other literature and naive voters are called sincere voters in some other literature. Since I focus on studying how different levels of sohistication in using information in voters affect voters behavior and grou decisions, I use the terms sohisticated and naive in this aer to emhasize this focus. 3

4 no sohisticated voter adots weakly dominated strategies. Symmetry here means that voters of the same tye take the same action if they receive the same signal. I refer a symmetric Bayesian Nash equilibrium in which no sohisticated voter adots weakly dominated strategies a voting equilibrium. Three main findings in this aer analyze sohisticated voters behavior in voting equilibria and information aggregation. The first one is that behavioral heterogeneity, or in other words, the resence of naive voters ensures the uniqueness of behavioral voting equilibrium. This is different than what we see in voting models in which all voters are sohisticated. In those models, there are always multile voting equilibria. Second, behavioral heterogeneity can imose oosite influence on sohisticated voters behavior and information aggregation. More secifically, on one hand, if the signal recision is relatively low and voter oulation is relatively small, then the resence of naive voters can decrease the robability that sohisticated voters vote against the status quo but increase the robability that Tye I error occurs. Here Tye I error means that status quo is rejected by mistake when it should be maintained. While on the other hand, if voter oulation is large, then regardless of the signal recision, the resence of naive voters can increase the robability that sohisticated voters vote against the status quo but decrease the robability that Tye I error occurs. Third, comared to the set of informationally efficient voting rules in the models with only sohisticated voters, 5 the set of informationally efficient voting rules in the model with behaviorally heterogeneous voters is smaller. That is to say, some voting rules that can aggregate information efficiently when all voters are sohisticated voters cannot aggregate efficiently aggregate information when voters are behaviorally heterogeneous. In other words, the resence of naive voters shrinks the set of informationally efficient voting rules. Besides these three main findings, I also rovide some results that are useful to grous with behaviorally heterogeneous voters that are looking for good voting rules and aroriate voter oulation for grou decisions. Note that my model can be considered as a generalization of the model with cursed voters in Eyster and Rabin Sohisticated voters act the same as uncursed voters and naive voters act the same as fully cursed voters. The difference between my model and their model is that, Eyster and Rabin 2005 assume voters have the same level of sohistication in handling information by assuming that all voters have the same degree of cursedness. 5 Say that a voting rule is informationally efficient if under this voting rule, the robabilities that mistakes occur diminish to zero as the voter oulation increases to infinity. 6 Cursed voters in Eyster and Rabin 2005 are rational voters who have incorrect belief about other voters equilibrium strategies. 4

5 This aer is organized as follows. Section 2 reviews some imortant results in the related literature. Section 3 is about the basic setu of the model with behavioral heterogeneity. Section 4 introduces all the related results in a traditional standard voting model in whic all votes are sohisticated as benchmarks. Section 5 focuses on how behavioral heterogeneity affects sohisticated voters behavior and robabilities that errors occur. Section 6 rovides results that are useful in choosing otimal voting rules and voter oulation for collective decisions when voters are behaviorally heterogeneous. Section 7 concludes and also discusses some otential extensions. All roofs are in the aendix. 2 Related Literature The roof of the Condorcet Jury Theorem and the extensive literature e.g., Condorcet 1785, Berg 1993, Ladha 1992, Young 1988, and etc. on this theorem assumes that every individual votes only according to their rivate information. This assumtion has been challenged by both theoretical research e.g., Austen-Smith and Banks 1996, Feddersen and Pesendorfer 1997, and etc. and emirical researche.g., Guarnaschelli, McKelvey, and Palfrey Austen-Smith and Banks 1996 and Feddersen and Pesendorfer 1997 oint out that such voting-only-according-to-rivate behavior cannot arise in any equilibrium in models in which all voters are sohisticated. Since then, most of the literature on voting theory has taken strategic voting as axiomatic, which assumes that all voters are sohisticated Feddersen and Pesendorfer 1998, Coughlan 2000, Kojima and Takagi 2010, and etc. In both of the two above frameworks, voters are behaviorally homogeneous, in the sense that they share the same level of sohistication in handling information. Exerimental observations, on the other hand, challenges the assumtion of behavioral homogeneity. For examle, Ladha, Miller and Oenheimer 1996 find evidence in their exeriments on voting behavior that not all voters are sohisticated in using information. Guarnaschelli, McKelvey, and Palfrey 2000 observe the existence of both sohisticated voters and naive voters in their exeriments on voting behavior. Behavioral heterogeneity is also observed in exeriments on other toics. For examle, Ciriani and Guarino 2005 observe different levels of sohistication in using information among the articiants in the exeriments on trading behavior in financial markets. Some literature assumes one tye of behavioral heterogeneity which involves sohisticated voters and artisans e.g. Feddersen and Pesendorfer This tye of behavioral heterogeneity has a fundamental difference from the one in my aer: artisans references 5

6 are indeendent of the true state, thus artisans actions neither deend on information nor reveal any information. On the oosite, naive voters actions solely deend on naive voters rivate information and then truthfully reveal naive voters rivate information. In other words, the resence of naive voters enriches information imlicit in voters actions, while the resence of artisans reduces the richness of information imlicit in voters actions. Thus the resence of artisans imoses an oosite influence on sohisticated voters behavior than the resence of naive votes does. See Feddersen and Pesendorfer 1996 for more details on the influence brought by the resence of artisans on sohisticated voters behavior. Note that naive voters in my aer do use some information to make decisions, but they do not fully utilize all available information, as they ignore the information imlicit in other voters actions. Another model in which voters fail to correctly utilize all information is Eyster and Rabin In Eyster and Rabin 2005, voters are sohisticated but cursed in the sense that, cursed voters do use both their rivate information and the information imlicit in other voters actions to make decisions, but they have incorrect beliefs about other voters equilibrium strategies. The extent to which a layer is cursed is arameterized by the robability χ 0, 1] χ is called the degree of cursedness she assigns to other layers laying their average distribution of actions irresective of tye rather than their tye-contingent strategy, to which she assigns robability 1 χ Eyster and Rabin Note that voters in Eyster and Rabin 2005 behave homogeneously as they are assumed to have the same degree of cursedness. Cursed voters with χ 0, 1] in Eyster and Rabin 2005 underareciate the information imlicit in voters actions, and the more cursed a voter is, the stronger the connection between his strategy and his rivate information is. 8 If χ = 1, a voter is fully cursed and his strategy is determined only by his rivate information. Then the more cursed a voter is, the more information about his own rivate signal is revealed by his action, and if a voter is fully cursed, his action truthfully reveals his rivate information. Thus a higher degree of cursedness among voters imlies that richer information imlicit in an equilibrium. But a voter with a higher degree of cursedness also imlies that this cursed voter more strongly underareciate the information imlicit in voters actions. Therefore cursedness imoses two oosite influences on cursed voters strategies. If all voters have the same degree of cursedness as in Eyster and Rabin 2005, the 7 Eyster and Rabin 2005 studies a general Bayesian game in which layers are cursed. They use voting game as an examle of the influence of cursedness on information revelation. See Section 5 in their aer for more details. 8 When χ = 0, a voters is uncursed and he correctly use all the information imlicit in voters actions as well as his rivate information. Therefore, an uncursed voters is the same as a sohisticated voter. 6

7 underutilization of information weakly dominates the other influence, which reduces cursed voters deendence on the information imlicit in voters actions. Comared to how voters behave in a voting model with only uncursed voters i.e., sohisticated voters as discussed in footnote 8, cursed voters vote the status quo with lower robabilities. My model can be considered as a generalization of the cursed voting game in Eyster and Rabin 2005 with voters who have different levels of cursedness. More recisely, naive voters can be considered as fully cursed voters whose degree of cursedness is χ h = 1, while sohisticated voters can be considered as uncursed voters whose degree of cursedness is χ l = 0. The main finding in my aer hold in a more general circumstance in which some cursed voters have a degree of cursedness χ h 0, 1 and the other ones have a degree of cursedness χ l, 0, χ h. This is because, in each voter s belief, the resence of cursed voters with a higher degree of cursedness χ h enriches the information imlicit in other voters actions, comared to the richness of information in a voting model with only cursed voters who share the same degree of cursedness χ l < χ h. Through changing the richness of information imlicity in other voters actions, the resence of more strongly cursed voters affects less strongly cursed voters behavior and then affects grou decisions and information aggregation. The exact influence on less strongly cursed voters behavior and grou decisions deends on signal recision and voter oulation. Feddersen and Pesendorfer 1997 and McLennan 1998 find that information can be aggregated more efficiently by grous with only sohisticated voters than by grous with only naive voters. The result in my model that the resence of naive voters shrinks the set of informationally efficient voting rules confirms their findings. 3 Basic Setu A grou consisting of N agents need to choose between two alternatives, the status quo and a change. I refer this grou as a grou and each agent as a voter. The grou decision is made by mandatory voting by all the N voters under a q-rule such that q 1 2, 1. That is, the change is selected if and only if at least qn voters vote for the change, otherwise the status quo is maintained. There are two states, the original state and a new state. All voters share the same state-contingent reference over these two alternatives, as all voters refer to maintain the status quo if the true state is the original one but refer to take the change if the true state is the new one. But voters cannot observe the true state directly, instead, each voter receives a rivate signal that rovides noisy information about the true state. Call the mistake that the change is selected when the true state is the original state the Tye I error and the mistake that the status quo is selected when the true state is 7

8 the new state the Tye I error. We use these terminology because if we discuss the grou decision in the context of hyothesis testing, we can write the grou s decision as whether the grou should accet or reject a null hyothesis that is the change should be taken. In this context of hyothesis testing, the Tye I error means the false rejection of the status quo a false accetance of the null hyothesis and Tye II error means the false reservation of the status quo the false rejection of the null hyothesis. For the ease of notations, let A denote the status quo and B denote the change, and let A denote the original state and B denote the new states. Without loss of generality, assume that the two states is chosen by nature with equal robability. After nature selects the true state, each voter receives a noisy signal s {α, β} such that Prα A = Prβ B = 1 2, 1. Thus α is the correct signal for state A and β is the correct signal for state B. I refer as the signal recision. Following these notations, the Tye I error means that B is selected in state A and the Tye II error means that A is selected in state B. All voters in this grou must articiate in voting and they vote simultaneously, using a q-rule where q 1 2, 1 to choose between the two alternatives. The change B is selected if and only if there are at least qn voters vote B, otherwise the grou stays with the status quo A. Without loss of generality, assume that qn is an integer. I also assume that 2q 1N 1 which means that when B is selected, the number of votes on B must be at least one more than the number of votes on A. I assume so because the number of votes on any alternative must be an integer. Voters are behaviorally heterogeneous in the sense that there are two tyes of voters, naive and sohisticated, who have different levels of sohistication in using information when making decisions. Let θ 0, 1 denote the rior robability that a voter is a naive voter and then 1 θ reresents the rior robability that a voter is sohisticated. Voters tyes are rivate information known by themselves, but θ is common knowledge. A naive voter always votes only according to her rivate signal. That is, a naive voter always votes A if she receives signal α and votes B if she receives signal β. The a naive voter s action truthfully reveals her rivate information. A sohisticated voter is an exected utility maximizer and votes for the alternative that maximizes his exected utility, using both his rivate signal and the information imlicit in other voters equilibrium actions to udate his belief on the true state. Thus sohisticated voter have a higher level of sohistication in using information than naive voters do. Let uo, S denote a sohisticated voter s utility if the true state is S {A, B} and the alternative O {A, B} is selected by the grou. I assume the following on uo, S: ua, A = ub, B = 0; and ub A = and ua B = 1, where 1 2, 1. 8

9 Here characterizes a sohisticated voter s threshold of reasonable doubt, that is to say, a sohisticated voter votes B only of he believes that the true state is B with robability greater than. 1 2, 1 means that between the two tyes of errors, all sohisticated voters dislike the Tye I error more than the Tye II error. Note that a sohisticated voter s vote affects his own utility if and only if he is the ivotal voter. A sohisticated voter is ivotal if his choice alone determines the outcome of collective decision, that is, if he votes A, then the outcome is A; and if he votes B, the outcome is B. For a sohisticated voter, call the event that he is ivotal ivotality. Given N voters articiating in voting, a voter is ivotal if and only if there are exactly qn 1 of the N 1 other voters voting B ignoring integer issues. 9 Since a sohisticated voter s action affects his own utility only if he is ivotal, he can make his decision conditioning on being ivotal. In an equilibrium, the information revealed by ivotality and the information imlicit in other voters equilibrium actions is equivalent: in an equilibrium, each sohisticated voter conditions himself on being ivotal, given that all other voters follow their equilibrium strategies. This imlies that there is information that a sohisticated voter can infer from other voters equilibrium actions if and only if the robability that this sohisticated voter is ivotal is strictly ositive. Since a naive voter sticks to the same strategy regardless of other voters actions, it is enough for us to use sohisticated voters common threshold of reasonable doubt, the recision of rivate signals,the voting rule q, the number of voters N, and the rior robability that a voter is naive to characterize the environment of grou decision we study in this aer. When voters are behaviorally heterogeneous, that is, when θ > 0, call {,, q, N, θ } as a behaviorally-heterogeneous voting environment. Here N is referred as grou size and N, θ as the grou structure. I use {,, q, N } to describe the voting environment in the standard voting model in which all voters are sohisticated. Here N stands for the grou size and N stands the grou structure. 10 Call {,, q, N } a fully-sohisticated voting environment. Say that a fully-sohisticated voting environment and a behaviorally-heterogeneous voting environment that share the same threshold of reasonable doubt for sohisticated voters, the same signal recision, the same voting rule, and the same grou size of voters, are dual. That is to say, for fixed,, q, and N, call {,, q, N } the dual fully-sohisticated voting environment of {,, q, N, θ } and call {,, q, N, θ } a θ-dual behaviorally-heterogeneous 9 Since I will be focusing on symmetric Nash equilibria, there is no need to identify articular voters in notation. 10 That all voters are sohisticated can be considered as a case in which θ = 0. To avoid confusion, I use N rather than N, 0 to reresent the grou structure when all voters are sohisticated. 9

10 voting environment of {,, q, N }. Since every naive voter sticks to the same strategy regardless of other voters actions, two equilibria differ only with resect to sohisticated voters equilibrium behavior. Then to study voters behavior in different voting environments, we only need to focus on sohisticated voters strategies in those voting environments. Through the whole aer, I focus on symmetric Bayesian Nash equilibria in which no sohisticated voter adots weakly dominated strategies. Call such symmetric Bayesian Nash equilibria voting equilibria. Particularly, I focus on voting equilibria that are informative. A voting equilibrium is informative if there is information revealed by ivotality in this equilibrium. Since there is information revealed by ivotality in a voting equilibrium only if ivotality occurs with ositive robability, we can also examine whether a voting equilibrium is informative or not by checking whether the robability that a single voter is ivotal is ositive in this voting equilibrium. There are two imortant reasons for me to focus on informative voting equilibrium. First, as discussed in later sections, the resence of naive voters affects sohisticated voters behavior through changing the richness of information revealed by ivotality. In a noninformative voting equilibrium if it exists, no single voter can be ivotal, which imlies that there is no information revealed by ivotality can be used by sohisticated voters and then sohisticated voters can only rely on their rivate information to make decision. In this case, the resence of naive voters imose no influence on sohisticated voters behavior. Second, as we will see in later section, the resence of naive voters ensures that there is a unique voting equilibrium in any behaviorally-heterogeneous voting environment. Moreover, this unique voting equilibrium is informative. This is different than what we see in fully-sohisticated voting environments. As shown by Feddersen and Pesendorfer 1998, there are at least two voting equilibria in a fully-sohisticated voting environment, and if grou size is large, there are three voting equilibria. Though there are multile voting equilibria in fully-sohisticated voting environments, luckily in a given fully-sohisticated voting environment, there is a unique informative voting equilibrium which is a natural benchmark for the study on how behavioral heterogeneity affects sohisticated voters behavior and grou decision. Besides sohisticated voters behavior, another imortant focus of the study on grou decisions is whether voters rivate information is aggregated well in grou decision. One of the most widely used criteria in judging how well information is aggregated is efficiency defined as following. 10

11 Definition 1 Efficient Information Aggregation Say that information is aggregated efficiently if for any sequence of informative voting equilibria corresonding to an increasing grou size, both the robability of Tye I error occurring and the robability of Tye II error occurring converge to zero. Say a q-rule is informationally efficient, or simly a efficient voting rule, if under this q-rule can aggregate information efficiently. Before roceeding further, a few notations should be clarified. In later sections, σ qθn s denotes the robability that a sohisticated voter votes B when he receives signal s {α, β} in a behaviorally-heterogeneous environment and σ qn s denotes the robability that a sohisticated voter votes B when he receives signal s in a fully-sohisticated model. γ qsn θ denotes the robability that a vote ends on B in state S {A, B} in the behaviorally heterogeneous model and γ qsn denote the robability that a vote ends on B in state S in the fully-sohisticated model. Here subscrit q stands for the voting rule, θ stands for the robability that a voter is a naive voters, N stands for the grou size, and S stands for the state. For given,, q, and θ, there are integers N q, N qθ, N M, N q, and N qθ such that N q N qθ N M < N q N qθ 1 The exlicit formulas for these integers can be found in Lemma 3 in the Aendix. 4 Benchmark: When All Voters Are Sohisticated In this section, I introduce the related results on sohisticated voters behavior and information aggregation in fully-sohisticated environments studied in Feddersen and Pesendorfer 1998, which will be used as benchmarks for my analysis of sohisticated voters behavior and information aggregation in behaviorally-heterogeneous environments. Proosition 1 Feddersen and Pesendorfer 1998 Consider a voting environment in which all voters are sohisticated, {,, q, N }. There is 11

12 a unique informative voting equilibrium in this voting equilibrium when < and N > N q or when and N 2. And in this unique informative voting equilibrium, 11 1 every sohisticated voter adots strategy 0, σqn β such that σqn β 0, 1 if < and N N q, N M ; 2 every sohisticated voter adots strategy 0, 1 if < and N [ ] N M, N q or if and N [ ] 2, N q ; 3 every sohisticated voter adots strategy σqn α, 1 such that σqn α 0, 1 if N > N q. For any N 2 there is a non-informative voting equilibrium in which all sohisticated voters vote A regardless what signals they receive; and for and N 1 1 q, there is another non-informative voting equilibrium in which all sohisticated voters vote B regardless what signals they receive. Note that a non-informative voting equilibrium can exist in a fully-sohisticated model because no sohisticated voter can be ivotal in this voting equilibrium, and then all sohisticated voters do not have motivation to deviate from their equilibrium strategies. More secifically, when all the other N 1 voters vote the same alternative regardless what signals they receive, a single voter cannot change the grou decision by changing his own vote. Thus he has no motivation to deviate from the strategy that he votes the same alternative as other sohisticated voter regardless what signal he receives. 11 Feddersen and Pesendorfer 1998 call these three informative voting equilibria resonsive in the sense that voters change their vote as a function of their rivate information with ositive robability in these voting equilibria. In Feddersen and Pesendorfer 1998, all voters are sohisticated, which means that a voter in their model changes his votes according to his rivate signal with ositive robability only if he knows that he has a ositive chance to be ivotal. In Feddersen and Pesendorfer 1998 the two non-resonsive voting equilibria, the one in which all sohisticated voters vote A regardless what signals they receive and the the one in which all sohisticated voters vote B regardless what signals they receive, exist because no single voter can be ivotal in these two equilibria. Thus in these two non-resonsive voting strategies in Feddesen and Pesendorfer 1998, there is no information revealed by ivotality for sohisticated voters to use. But in a behaviorally-heterogeneous voting environment, these two non-resonsive voting strategies can be equilibrium strategies for sohisticated voters not because ivotality cannot occur with ositive robabilities, but because the information revealed by ivotality is so rich that it overwhelms sohisticated voters rivate information. Thus sohisticated voters choose to follow the information revealed by ivotality, they act as if they are ignoring their rivate information. To avoid otential confusion and misunderstanding, I use informative and non-informative to describe voting equilibria in my aer rather than following the terminology in Feddersen and Pesendorfer

13 Proosition 2 Feddersen and Pesendorfer 1998 When all voters are sohisticated, any q-rule such that q 1 2, 1 aggregates information efficiently. Proosition 2 means that any q-rule is informationally efficient when all voters are sohisticated. 5 When Voters are Behaviorally Heterogeneous Now let s turn to voting environments in which voters are behaviorally heterogeneous, that is, the voting environments in which not only sohisticated voters but also naive voters articiate in grou decision. To study the influence on sohisticated voters behavior and grou decisions brought by the resence of naive voters in a behaviorally-heterogeneous voting environment, I choose the unique informative voting equilibrium in the dual fullysohisticated voting environment as the benchmark. I choose this secific benchmark because all voting equilibria in behaviorally-heterogenous voting environments are informative, thus using informative voting equilibria in fully-sohisticated voting environments are benchmarks is natural. There are three major influences on sohisticated voters behavior and grou decisions brought by behavioral heterogeneity, or more secifically, by the resence of naive voters. First, the resence of naive voters ensures the uniqueness of voting equilibrium in any behaviorally-heterogeneous voting environment and changes sohisticated voters behavior in informative voting behavior, comared to how sohisticated voters behave in the dual fully-sohisticated voting environment Theorem 1. Second, in the unique voting equilibrium in a behaviorally-heterogeneous voting environment, the resence of naive voters can imose oosite influences on sohisticated voters behavior and information aggregation, in the sense that the resence of naive voters can increase the robabilities with which sohisticated voters vote for against the status quo but increases decreases the robability that Tye I error false rejection of the status quo when it should be maintained occur at the same time Theorem 2. Third, the resence of naive voters also restrict on the ability of voting rules on aggregating information efficiently Theorem 3. 13

14 Theorem 1 In any behaviorally-heterogeneous voting environment, there is a unique voting equilibrium and in this unique voting equilibrium, sohisticated voters 1 vote against the status quo with lower robabilities if grou size is relatively small and signal recision is relatively low; but 2 vote against the status quo with greater robabilities if grou size is large, regardless of the level of signal recision, comared to how they behave in the unique informative voting equilibrium in the corresonding dual fully-sohisticated voting environment. The resence of naive voters ensures the uniqueness of voting equilibrium in any behaviorally-heterogeneous voting environment because it eliminates the existence of noninformative voting equilibria. Recall that among the multile voting equilibria in a fullysohisticated voting environment, there is a unique informative equilibrium while the other voting equilibria are non-informative. The non-informative voting equilibria exist in a fully-sohisticated voting environment because no voter is ivotal in those non-informative voting equilibria. While when naive voters also articiate in voting, the robability that a single voter is ivotal becomes strictly ositive, since naive voters always vote for different alternatives when they receive different signals also articiate in voting. Hence any voting equilibrium in a behaviorally-heterogeneous voting environment is informative, which imlies that the resence of naive voters eliminates non-informative voting equilibria in any behaviorally-heterogeneous voting environment. To facilitate illustration of the influence on how behavioral heterogeneity affects sohisticated voters behavior in the unique voting equilibrium in a behaviorally-heterogeneous voting environment, I introduce two roositions, Proosition 3 and Proosition 4, based on which we can derive Theorem 1 straightforward. In Proosition 3 and Proosition 4, σqθn s denotes the robability that a sohisticated voter votes B, or in other words, votes against the status quo A, when he receives signal s {α, β} in the unique voting equilibrium in a behaviorally-heterogeneous voting environment {,, q, N, θ }. σqn s denotes the robability that a sohisticated voter votes B when he receives signal s {α, β} in the unique informative voting equilibrium in the dual fully-sohisticated voting environment {,, q, N }. Before roceeding further, an imortant oint should be mention, that is, conditioning on being the ivotal voter, or ivotality for short, always reveals more information that 14

15 suorts state B to be the true state than the information that suorts state A to a sohisticated voter. To see this, first note that in any voting equilibrium, the robability that a vote on B is associated with a signal β is always greater than the robability that it is associated with a signal α. This is because, given the information imlicity in other voters actions, a voter always has stronger motivation to vote B when he receive signal β than when he receives signal α. Therefore the robability that a vote on B is made by a voter who receives signal β is greater than the robability that this vote is made by a voter who receives signal α. In the belief of a sohisticated voter who conditions himself on bing ivotal, there are exactly qn of the other N 1 voters voting B and the rest 1 qn voters voting A. Thus in this sohisticated voter s belief, there are exactly 2q 1N 1 more votes on B than on A. Since a vote on B is always associated with a signal β with a robability greater than the robability that it is associated with a signal α, ivotality reveals more information that suorts state B to be the true state than the information that suorts state A to this sohisticated voter. In the following analysis, let K qθn denote the exected numbers of signals β received by the other N 1 voters in the belief of a sohisticated voter who conditions himself on being the ivotal voter in a behaviorally-heterogeneous voting environment {,, q, N, θ }. Similarly, let K qn denote the exected numbers of signals β received by the other N 1 voters in the belief of a sohisticated voter who conditions himself on being the ivotal voter in the dual fully-sohisticated voting environment {,, q, N }. Note that in voting equilibria, K qθn and K qn may not equal qn, since if sohisticated voters adot mixed strategies in these voting equilibria, the robability that a vote on B is associated with a signal β is not one but less that one. But since the robability that a vote on B is associated with a signal β is always greater than the robability that it is associated with a signal α, in a given behaviorally-heterogeneous voting environment {,, q, N, θ }, the greater K qθn is, the stronger motivation a sohisticated voter has to vote B. Proosition 3 Given, q 1 2, 1, <, and N 2, there is a unique voting equilibrium for each θ. 1 When N N q, N M, there is a θn 0, 1 such that i if θ 0, θ N, the unique voting equilibrium is the one in which every sohisticated voter adots strategy σ qθn α, σ qθn β such that σ qθnα = 0 and σ qθn β 0, σ qn β ; 15

16 ii if θ [θ N, 1, the unique voting equilibrium is the one in which every sohisticated voter adots strategy σqθn α, σ qθn β = 0, 0. 2 When N [ ] N M, N q, given any θ 0, 1, the unique voting equilibrium is the one in which every sohisticated voter adots strategy σqθn α, σ qθn β = 0, 1. 3 When N > N q, there is a θ N 0, 1 such that i if θ 0, θ N, the unique voting equilibrium is the one in which every sohisticated voter adots strategy σqθn α, σ qθn β such that σ qθn α σqn α, 1 and σqθn β = 1; ii if θ [ θ N, 1, the unique voting equilibrium is the one in which every sohisticated voter adots strategy σqθn α, σ qθn β = 1, 1. Simle calculation shows that PrB β =, which means that when only using his rivate information, a sohisticated voter will not vote B regardless what signal he receives if <, as in this sohisticated voter s belief, the robability that the true state is B based only on that he receives signal β but not any other information is lower than his threshold of reasonable doubt. Thus in a voting environment with N voters, whether a sohisticated voter votes B with ositive robability when he receives some signal s {α, β} deends on the richness of information revealed by ivotality. To see how the resence affects the richness of information revealed by ivotality, let s start by looking at the benchmark model, the fully-sohisticated voting model first. Consider a fully-sohisticated voting environment {,, q, N } such that < and θ 0, θ N. By Proosition 1 we know that in the unique informative voting equilibrium, a sohisticated voter adots strategy σqn α, σ qn β such that σ qnα = 0 and σqn β 0, 1, which means that in this informative voting equilibrium, a sohisticated voter mixes between voting B and voting A if he receives signal β and votes A for sure if he receives signal α. Thus every vote on B is associated with a signal β with robability one, while some of the votes on A are also associated with signals β. Let KqN denote the exected numbers of signals β received by the other N 1 voters in the belief of a sohisticated voter who conditions himself on being ivotal in this voting equilibrium. Then KqN > qn since some of the 1 qn votes on A are made by sohisticated voters who receive signals β. Now consider the unique voting equilibrium in the θ-dual behaviorally-heterogeneous voting environment, {,, q, N, θ }, of the above fully-sohisticated voting environment. Let KqθN denote the exected numbers of signals β received by the other N 1 voters in 16

17 the belief of a sohisticated voter who conditions himself on being ivotal in this voting equilibrium. Then we must have KqθN < K qn. This is because a naive voter votes B if and only if he receives signal β, then at ivotality, less of the 1 qn votes on A are made by voters who receive signals β. Less signals β received means less information suorting state B is revealed by ivotality, which weakens sohisticated voters motivation in voting B regardless what signals they receive. Since a reresentative sohisticated voter already votes B with zero robability when he receives signal α, we can only see a decrease in the robability a reresentative sohisticated voter votes B when he receives signal β after naive voters join in grou decision, comared to how a reresentative sohisticated behaves in a fully-sohisticated voting environment. That is, σqθn β < σ qn β. Furthermore, since KqθN decreases with the roortion of naive voters in the total voter oulation increases, KqθN decreases with θ since θ, the rior robability that a voter is naive, is also the exected roortion of naive voters in the total oulation. Thus as θ increases, less information suorting state B revealed by ivotality, and then σqθn β, the robability that a sohisticated voter votes B when he receives signal β, also decreases. When θ is high enough, the richness of information suorting state B revealed by ivotality becomes so low that a reresentative sohisticated voter acts the same as when he can only rely on his rivate information to make decision. Recall that when <, a reresentative sohisticated voter who can only rely on his rivate information when making decisions always vote A regardless what signal he receives. Hence when θ is high enough, σqθn β = 0. The above discussions resent the intuition for Part 1 in Proosition 3. Different from what we see in voting environments with relatively small grou size, when grou size is large, the resence of naive voters increases the robability that a reresentative sohisticated voter votes B. To see the intuition behind this conclusion, let us look at the benchmark equilibrium, the unique informative voting equilibrium in a fully-sohisticated voting environment {,, q, N } such that < and θ 0, θ N. 12 As shown by Proosition 1, in this unique informative voting equilibrium, a reresentative sohisticated voter adots strategy σqn α, σ qn β such that σ qnα 0, 1 and σqn β = 1. Again, let KqN denote the exected numbers of signals β received by the other N 1 voters in the belief of a reresentative sohisticated voter who conditions himself on being ivotal in this voting equilibrium. Then KqN < qn since some of the qn votes on B are made by sohisticated voters who receive signals α while all the votes on A are made by sohisticated voters who receive signals α. Now consider the unique voting equilibrium in the θ-dual behaviorally-heterogeneous voting environment, {,, q, N, θ }, of the above fully-sohisticated voting environment. 12 Actually, this conclusion also holds when. See Proosition 4 for more details. 17

18 Let KqθN denote the exected numbers of signals β received by the other N 1 voters in the belief of a sohisticated voter who conditions himself on being ivotal in this voting equilibrium. Then we must have KqθN > K qn. This is because a naive voter votes B if and only if he receives signal β, then at ivotality, more of the qn votes on B are made by voters who receive signals β. More signals β received means more information suorting state B is revealed by ivotality, which strengthens sohisticated voters motivation in voting B regardless what signals they receive. Since a reresentative sohisticated voter already votes B with robability one when he receives signal β, we can only see an increase in the robability a reresentative sohisticated voter votes B when he receives signal α after naive voters join in grou decision, comared to how a reresentative sohisticated behaves in a fully-sohisticated voting environment. In other words, σqθn α > σ qn α. Furthermore, since KqθN increases with the roortion of naive voters in the total voter oulation increases, KqθN increases with θ since θ, the rior robability that a voter is naive, is also the exected roortion of naive voters in the total oulation. Thus as θ increases, more information suorting state B revealed by ivotality, and then σqθn α, the robability that a sohisticated voter votes B when he receives signal α, also decreases. When θ is high enough, ivotality reveals information suorting state B that is so rich that it overwhelms a reresentative sohisticated voter s rivate information, making sohisticated voters vote A regardless what signals they receive. Hence when θ is high enough, σqθn α = 1. This comletes the intuition for Part 3 in Proosition 3. The intuition for Part 2 in Proosition 3 is slightly different from above. Recall that in the unique informative voting equilibrium in a fully-sohisticated voting environment {,, q, N } such that < and θ [ ] N M, N q, all sohisticated voters adot the same strategy σqθn α, σ qθn β = 0, 1. In this informative voting equilibrium, sohisticated voters act exactly the same as naive voters, which means that the resence of naive voters does not affect the richness of information revealed by ivotality and then does not influence sohisticated voters behavior. Proosition 3 above shows the influence of the resence of naive voters on sohisticated voters behavior in voting environments in which signal recision is relatively low. Proosition 4 discuss such influence in behaviorally-heterogeneous voting environments in which signal recision is relatively high, that is, when. Proosition 4 Given, q 1 2, 1,, and N 2, there is a unique voting equilibrium for each θ. 1 When N N q, given any θ 0, 1, the unique voting equilibrium is the one in which 18

19 every sohisticated voter adots adots strategy σqθn α, σ qθn β = 0, 1. 2 When N > N q, there is a θ N 0, 1 such that i if θ 0, θ N, the unique voting equilibrium is the one in which every sohisticated voter adots strategy σqθn α, σ qθn β such that σ qθn α σqn α, 1 and σqθn β = 1; ii if θ [ θ N, 1, the unique voting equilibrium is the one in which every sohisticated voter adots strategy σqθn α, σ qθn β = 1, 1. Since PrB β =, when, a sohisticated voter who can only rely on his rivate information e.g., he is the only one in the grou, i.e. N = 1 will always vote B when he receives signal β. Recall that ivotality always reveals more information suorting state B than the information suorting state A, which means what when, in an informative voting equilibrium in a voting environment with N 2 voters, a sohisticated voter who has both his rivate information and the information revealed by ivotality always vote B when he receives signal β. This is why in voting equilibria in voting environments described in Proosition 4, we always have σqθn β = 1. The rest of the intuition for Proosition 4 is to the intuition for Part 2 and 3 of Proosition 3, so I omit it. Note that given the same,, q, and N, θ N in Proosition 4 is the same as θ N in Proosition 3. Proof of Theorem 1: Part 1 of Theorem 1 comes directly from Part 1 of Proosition 3. Since there are binary alternatives and A reresents the status quo, voting B means voting against the status quo. Thus that the resence of naive voters causes sohisticated voters to vote B with lower robabilities means that the resence of naive voters causes sohisticated voters to vote against the status quo with lower robabilities. Part 2 of Theorem 1 comes from Part 3 of Proosition 3 and Part 3 of Proosition 4. Again, since voting B just means voting against the status quo, that the resence of naive voters causes sohisticated voters to vote B with greater robabilities means that the resence of naive voters causes sohisticated voters to vote against the status quo with greater robabilities. Comare to the influence on sohisticated voters behavior brought by the resence of 19