Using Cheap Talk to Polarize or Unify a Group of Decision Makers

Transcription

1 Using Cheap Talk to Polarize or Unify a Group of Decision Makers Daeyoung Jeong October 30, 015 Abstract This paper develops a model of strategic information transmission from an expert with informational superiority to a group of people who make a collective decision by voting on a proposal. An expert who observes the qualities of a proposal in a two-dimensional state space sends a public cheap talk message to a group of decision makers with limited information. We find that a simple cheap talk strategy of an expert with informational superiority can be surprisingly effective in persuading the collective decision makers by polarizing (or unifying) their beliefs and voting decisions: an expert s cheap talk message (1) polarizes (or unifies) the decision makers beliefs and () makes the decision makers vote collectively in the expert s interest without paying attention to their private information. We consider two different models based on the type of the expert: one with a highly biased expert who always wants a rejection and the other with a surplus maximizing expert who tries to maximize the total surplus of the group of decision makers. In an equilibrium, the highly biased expert polarizes decision makers opinions by sending a message which says the proposal is stronger in one dimension than in the other, whereas the surplus maximizing expert unifies them by sending a message which reveals the average qualities. We also observe that, when there is a significant informational gap between an expert and decision makers, decision makers vote in her interest by focusing only on the expert s message, even though they know the expert has her own bias. This implies that the equilibrium voting outcome would be irrelevant to the decision makers private information, so the voting procedure would fail in aggregating their private information. JEL Classification Numbers: D71, D7, D78, D8, D83 Keywords: Cheap Talk, Voting, Polarization The Ohio State University, 410 Arps Hall, 1945 North High Street, Columbus, Ohio, jeong.11@osu.edu. 1

2 1 Introduction In many economic situations, a group makes a collective decision, which determines the payoffs for all members. Moreover, decision makers may have some, but not all, information regarding proposal at hand. The unavailability of complete information may complicate reasonable decision making. It is quite common that collective decision making bodies hear testimony or advice from experts that could potentially be better informed about an issue at hand. For example, a policylobbying group, which has more precise information on the proposal, may give advice to a group of legislators. This well-informed agent, the expert, does not necessarily have the same preferences or incentives for outcomes as the deciding body. Hence the advice given has to be interpreted strategically. In this paper, we explore the information transmission from an expert (female) with informational superiority to a group of decision makers (male) with heterogeneous interests. Unlike the existing literature of cheap talk, which mainly focuses on credible information transmission of an expert to decision makers who make individual decisions, in our model the multiple decision makers vote on a proposal, which determines their payoffs. We show that, in a cheap talk model with collective decision makers, a simple cheap talk strategy of an expert with informational superiority can be surprisingly effective in persuading the collective decision makers by polarizing (or unifying) their beliefs and voting decisions: an expert s cheap talk message (1) polarizes (or unifies) the decision makers beliefs and () makes the decision makers vote collectively in the expert s interest without paying attention to their private information. More specifically, an expert who observes the qualities of a proposal in a two-dimensional state space sends a public cheap talk message to a group of decision makers with limited information. This model is inspired by a voting example of Chakraborty and Harbaugh (010). While their example has uninformed voters in two groups, within which the voters have identical preferences, we introduce a group of heterogeneous decision makers with private information. We consider two different models based on the type of the expert: one with a highly biased expert who always wants a rejection and the other with a surplus maximizing expert who tries to maximize the total surplus of the group of decision makers. In a cheap talk equilibrium, the highly biased expert polarizes decision makers opinions by

3 sending a message which says the proposal is stronger in one dimension than in the other. Given the expert s message, called the comparative cheap talk, a decision maker who cares more about the former dimension may be more favorable to the proposal, but a decision maker who cares more about the later dimension may be less favorable. 1 Among the heterogeneous voting bodies, the highly biased expert strategically makes an ally of a group decision makers, while giving up the support of the other group. On the other hand, the surplus maximizing expert unifies decision makers opinions by sending a message which reveals the average strength or qualities of the proposal in a cheap talk equilibrium. We call this messaging strategy the average cheap talk strategy. If the message says the average qualities of the proposal is sufficiently high, all decision makers believe the proposal is good enough for them to support, and vice versa. These polarized (or unified) beliefs of decision makers help an expert to get her way by resulting in polarized (or unified) voting behavior. We show that, under any super majority rule, either expert s manipulative cheap talk makes the decision makers focus only on the expert s message and vote in her interest, even though they know the expert has her own bias. When there is a significant informational gap between an expert and decision makers, the decision makers polarized (or unified) perspectives on the proposal based on the given expert s message cannot be significantly altered by their private information. As a result, either expert s manipulative cheap talk hinders decision makers from being responsive to their private information in the process of voting. This implies that the equilibrium voting outcome would be irrelevant to the decision makers private information, so the voting procedure would fail in aggregating their private information. We further show that, even when the decision makers information is arbitrarily accurate, either expert s cheap talk strategy specified above is still better than any other strategies which transmits more or less information in equilibrium. In other words, the more or less information transmission does not help the expert. Our main result is related to previous studies on polarization of agents beliefs. However, unlike the existing literature on polarization, which studies the effect of an exogenous public message on forming agents beliefs, in our model, a public message is strategically transmitted from an expert. 1 Chakraborty and Harbaugh (010) show that, mainly in a model with a single decision maker, a comparative message could be credible even when an expert is highly biased with state-independent preferences. 3

4 We show that the formation of decision makers beliefs in an equilibrium depends highly on the identity of the agent who sends a public message: different types of the experts, in an equilibrium, would induce different formations of the decision makers beliefs. 1.1 Related Literature Even though most of cheap talk literature consider one-dimensional state variable, there are some studies with multiple dimensions. Battaglini (00) studies multidimensional cheap talk with multiple senders and a single decision maker and shows that there exists a full revelation equilibrium. Chakraborty and Harbaugh (010) examine the credibility of a cheap talk by a very biased expert with state-independent preference. Their main result shows that a multidimensional space of state of nature guarantees the existence of an influential cheap talk equilibrium. Though they mainly focus on cheap talk models with a single decision maker, they also discuss, with a simple example, the possibility of the welfare loss of collective decision makers in a cheap talk equilibrium with a highly biased expert. They conclude that a cheap talk may benefit individual voters by letting them make more informed decisions, but in a final decision of voting body this benefit could be offset by strategic voting behaviors of the other voters, who have different preferences. We borrow the basic setting of theirs, and generalize it with various types of voters and experts. While they only consider two groups of voters who have identical preference within groups along with a highly biased expert, we consider a voting body of which each individual has a unique preference as well as two different types of experts. Moreover, we allow voters to possess not all but some private information about the state of the world. With theses features, we can explore how the optimal information transmission strategies varies with different types of experts and with different level of information gap between a sender and a group of receivers. We can also investigate the effect of cheap talk on information aggregation among voters. Our results are related to literature on polarization of opinion and/or belief. Literature on polarization studies polarized opinion and/or belief of people who observe the same public information, but have heterogeneous priors (Dixit and Weibull (007); Acemoglu et al. (Forthcoming)) or different private information (Kondor (01); Andreoni and Mylovanov (01)). While the literature studies the polarization of opinion among individuals who have private interests, we explore the polarization on voting decisions of collective decision makers who have heterogeneous interests 4

5 and private information. In addition to that, unlike the exogenously given public signal in the polarization literature, we assume in our model that the public message is strategically transmitted from the expert who does not necessarily have the same preferences or incentives for outcomes as the deciding body. In spite of theses differences, our results in polarization with a highly biased expert share the same features or intuitions with those from the polarization literature. For example, Andreoni and Mylovanov (01) conclude that, with theory and experiment, polarization of opinions could exist (and persist over time) when the dimensionality of uncertainty exceeds that of opinions. Similarly, we show that the highly biased expert polarizes the voting body on strategic voting decisions of binary choices by transmitting restrictive information of the two dimensional state of nature. Our paper is also related to literature on information aggregation. The primary question of information aggregation in voting literature is whether private information of individual voters could be aggregated through strategic voting behavior of a voting body. Feddersen and Pesendorfer (1997), for instance, shows that under the unidimensional state of nature, a voting with a large number of voters effectively aggregates information in the sense that the equilibrium outcome is full information equivalent. However, if there is uncertainty with higher dimension, information aggregation would be limited. We investigate, on the other hand, the degree of information aggregation when there exists an individual with relatively accurate information outside of the voting body, who may transmit some information of two dimensional state of nature to the voting body. We conclude that information aggregation among voters would be disturbed by manipulative information transmission of an expert, even though it allows each individual to make more informed decision. Higher dimension of uncertainty makes it available for an expert to send a credible message to voters and impede their informative actions. Several papers explore the possibility of persuasion of a biased informed agent on a voting body. Caillaud and Tirole (007) and Schnakenberg (015b) explore an expert s effort to persuade a voting body in the context of a mechanism design and a cheap talk, respectively. Both studies show that the outside expert could get her way by transmitting information to existing allies among a voting body, and leading persuasion cascades in which the allies convince the other members. However, interestingly, they seem to mainly show that an expert as a supporter of the proposal can increase the probability of approval by transmitting information to existing allies among a voting body. In 5

6 other words, they had paid more attention to the existence of equilibrium when there already exist the expert s allies among voters; relatively little is described of the expert s strategy to make allies among voters with various interests or preferences to get her own way in a voting outcome. On the other hand, our model with strategic information transmission illustrates the process of making allies among the heterogeneous voters, with two different types of experts. There are several working papers which study a cheap talk model with an informed decision maker. Lai (013) compare the models with informed decision maker to the basic uninformed receiver model. He assumes that a decision maker knows whether the state is high or low relative to a private threshold, and shows that the expert s message is more biased compare to the situation with uninformed receiver. Our structure of the binary private signal of a decision maker is similar to theirs except the fact that the private signal is not completely accurate. Ishida and Shimizu (013) assume that a sender as well as a receiver are incompletely informed. They mainly focus on the impact of the information gap between a sender, who has relatively accurate information, and a receiver. They show the less the information gap, the less the information transmitted. On the contrary, with collective decision makers, we show that the amount of information transmitted does not depend on the information gap between a sender and the group of decision makers. Other studies examine a cheap talk model with multiple decision makers. Board and Dragu (008) consider a cheap talk model with a general decision maker and a decision maker only with a veto power. They conclude that a veto threat makes the veto player worse off and the general decision maker better off. Yeung (013) investigates a cheap talk model with a binary state of nature and multiple free-riding decision makers. He shows that, when there is a severe negative externality, it is impossible for an expert to send a credible message to decision makers, because the expert has incentive to send exaggerated message to prevent free-riding behavior. Previous studies have also examined a cheap talk model with a voting. Most of the studies focus on deliberation among voters in the form of a cheap talk. Austen-Smith and Feddersen (00) explore the effect of deliberation among decision makers who have private information of the state of nature prior to a collective decision making via voting. By comparing the majority and unanimity rules, they show that the former induces more information transmission and fewer errors in decision making than the latter. With a outside sender s cheap talk to a voting body, we show that the optimal cheap talk strategy does not vary over voting rules. Rivas and Rodríguez- 6

7 Álvarez (013) examine the impact of a leadership in a deliberation among decision makers before a voting. They conclude that a very influential leader affect less on the equilibrium outcome than a moderately influential leader. The main difference of these previous papers and our study is that a sender in those papers is one of the voter, while a sender in our study is an expert who does not have a vote but has complete information about the true state. Schnakenberg (015a) considers a cheap talk from an outside expert, whose type is unknown to voters, to collective decision makers with no private information. He relates expert s manipulation opportunity to voting rules. He shows that there is a voter whose vote is always required to pass the proposal if and only if there is a cheap talk equilibrium in which all voters are worse off than in a babbling equilibrium. So, in his setting, the manipulative cheap talk does not work under the unanimity rule. On the contrary, in our setting where decision makers have private information and known expert s preferences, we show the existence of a manipulative cheap talk equilibrium in which all voters would be worse off than in a babbling equilibrium. The remainder of the paper is organized as follows. The next section illustrates the basic model. In Section 3, we examines a model with the highly biased expert. We show the existence of equilibrium, and compare the equilibrium behaviors under different voting rules and different degrees of the information gap. In Section 4, we introduce another type of the expert, the surplus maximizing expert, and conduct the equilibrium analysis. Section 5 concludes the paper. The Appendices contain proofs and some more technical materials not presented in the main body of the paper. Model There are an expert (female) and an odd number of decision makers (male) i N = {1,,..., n} where n 3. Decision makers (DMs) vote on a given proposal under a certain voting rule. 3 The space of the state of nature is two-dimensional, Θ [0, 1] [0, 1], the aspect 1 and, respectively, and the state of nature θ Θ is uniformly distributed, θ U[0, 1] [0, 1]. Each dimension represents a different aspect of the state of nature. We use the pronoun she for the expert and he for a decision makers. 3 q-majority Rule requires at least q/n of support to approve a proposal. When q = n, q-majority Rule is a unanimity rule, and when q = n+1, it is a simple majority rule. 7

8 Figure.1: Pipeline Example In order to effectively illustrate the important features of the model, consider, for example, a legislature who vote on a pipeline bill. They may ask two questions; is the project good for the environment and how many jobs will it create. Each legislator may put different weights on these two aspects; one may only care about either the environment or the jobs and another may care about both aspects equally. Figure.1 depicts the space of the state of nature and the preference of a legislator in the pipeline example. The vertical axis represents how good the project would be for the environment, and the horizontal axis represents how many jobs the project would create. The first figure from the left is for a legislator who only cares about the number of jobs, the middle one is for a legislator who only cares about the environment, and the right one if for a legislator who cares about both aspects equally. The vertical dot-dashed line on the first figure represents the cutoff of a legislator s preference over the state of nature: if the project lies on the right hand side of the line, then the legislator is favorable to the project, and vice versa. It may be impossible for the legislators to know exactly how the project will turn out, but they may have partial information on it. We assume, with own specialized knowledge or device, which is related to his private interest, each decision maker updates own individual belief on the probability that the proposal is good (or bad) for him. In the form of a conditionally independent private signal, he gets a high signal (a low signal) with a certain probability when the project lies on the right hand side (the left hand side) of the line. So, a legislature with a high private signal would believe the project is more likely to be favorable to him. In addition, there may be an outside lobbying firm, which has relatively accurate information about the project. In the model we assume the expert observes the exact quality of the project. 8

9 By sending a costless public message to the legislature, she tries to persuade the legislators to vote in her favor. The optimal messaging strategy may vary with the preference of the expert. In the basic model, a DM i s preference conditional on approval is characterized by the weighted average of the state of nature. u d γ i θ 1 + (1 γ i )θ 1 i =, if approved. 0, if rejected The parameter γ i, which is assumed to be publicly informed, captures the intensity of DM i s personal interest (or concern) on the aspect 1. γ i = 1 means DM i only cares about the aspect 1 (the first dimension), and γ i = 0 means DM i only cares about the aspect (the second dimension). A decision maker i has payoff γ i θ 1 + (1 γ i )θ 1 from approval and payoff 0 from rejection.4 To economize on notation, without loss of interesting generality, we assume γ i > γ j if and only if i > j. In analysis, to economize the notion, we define DM i s gross utility when the proposal has been approved given θ as follows. ũ i γ i θ 1 + (1 γ i )θ 1. An expert privately observes the state of nature, θ Θ = [0, 1] [0, 1]. Decision makers do not observe θ, but receives a private signal, s i S i = {h, l}, which may contains partial information about θ. For all θ P (s i = h γ i θ 1 + (1 γ i )θ 1 ) > 0 =p ( 1, 1) P (s i = l γ i θ 1 + (1 γ i )θ 1 ) > 0 =1 p. The value p, which is commonly known, represents an accuracy of DM s private information. The justification of this private information of a decision maker is as follows. Though a decision maker cannot observe the state of nature, he could still have some partial information about it by exploiting his own knowledge or device. For example, a legislator, who is specialized in the knowledge related 4 We borrow the model from Chakraborty and Harbaugh (010), and generalize it with n different decision makers. We normalize their utilities with the last term 1, which makes the ex-ante expected payoff be zero. 9

10 to his own interest, could review a bill by himself before he gets any further information from a lobbyist, and have a rough idea that whether the bill is favorable to himself or not. After observing a private signal, a decision maker forms an interim belief on the state of nature. To economize on notation, we define the region that DM i gets positive payoff with approval Di h {θ Θ γ i θ 1 + (1 γ i )θ 1 > 0} and Dl i θ \ Dh i. By Bayes rule, we get P P ( ) θ Di h s i = h = P ( θ Di h ) ( P si = h θ Di h ) P (s i = h) = P ( θ Di h ) ( P si = h θ Di h ) t S i P (θ Di t) P (s i = h θ Di t) = 1 p + 1 ( ) θ Di s l i = h =1 p. 1 p (1 p) = p Similarly, P ( θ Di l s i = l ) = p and P ( θ Di h s i = l ) = 1 p. Therefore, if p = 1, the signal conveys no information, but if p > 1, it conveys some accurate information. A strategy for Decision Maker i is a function v : M S i {0, 1} that associates the expert message and his private signal to a voting decision. A strategy for the expert is a function m : Θ M where M = {M k } K k=1 is a partition of Θ = [0, 1] [0, 1]. To economize the notation, we write v k i := (v i(m k, s h i ), v i(m k, s l i )), and denote Ŝi(v k i ) j is j be the set of events where DM i is pivotal under v k i = (vk j ) j i. DM i s utility function is a function u d i (v i v i, γ i, θ) = 1{ n i v i q}(γ i θ 1 + (1 γ i )θ 1 ) where v i = (v k i )K k=1, and the expert s utility function ue (v θ) also depends on the voting strategy profile v = (v i ) n i=1. Different types of experts would have different functional forms of payoff function u e (v θ). The game proceeds as follows: 1. Nature draws the sate θ U[0, 1] [0, 1].. The expert privately observes θ. 3. The expert sends a message m M to the group of decision makers. 4. A decision makers receives a private signal s i S i = {h, l}. 5. Each decision maker votes on the proposal, v i : M S i {0, 1}. 10

11 6. Under q-majority rule, agents receive payoffs. The solution concept employed in this paper is perfect Bayesian equilibrium. Definition 1 (Perfect Bayesian Equilibrium). A perfect Bayesian equilibrium consists of a pair of strategies (σ, v) and a set of beliefs µ, such that 1. for each θ, if m supp [σ( θ)], where s = (s i ) n i=1 and S = n i S i. m arg max u e (v(m, s), θ)f(s θ)ds m S. for each m and s i v i (m, s i ) arg max u v i Θ d i (v i θ, γ i )µ i (θ m, s i )dθ 3. for all m M and all s i S i σ(m θ)φ i (θ s i ), if Θ µ i (θ m, s i ) = Θ σ(m θ )φ i (θ s i )dθ σ (m) ψ i (θ s i ), if Θ σ (m) = where ψ i (θ s i ) is any distribution. We make the following assumptions: Assumption 1 (Two extreme decision makers). In the group of decision makers, there always exists extreme types of decision makers γ 1 = 1 and γ n = 0. This assumption is made for technical convenience. Assumption (Symmetric decision makers). In the group of decision makers, the decision makers are symmetric around n+1 -th decision maker. More precisely, γ k = 1, and γ n j+1 = 1 γ j for any j such that k > j > 1. 11

12 In order to focus solely on a meaningful information transmission which conveys some information about the state of nature and affects on the decision makers behavior, as in most of theoretical cheap talk literature, we restrict attention to informative and influential equilibrium. Definition (Informative equilibrium). An equilibrium is informative if any message m supp[σ( θ)] conveys any information of the state of nature, θ. For any θ, µ i (θ m, s) φ i (θ s) for all i with at least one strict inequality. Definition 3 (Influential equilibrium). An equilibrium is influential if there exists different messages m and m where m m induce different actions of decision makers, v i (m, s) v i (m, s) for some i and s. As in a general voting game, there always exists a trivial equilibrium with any type of experts in which no voter (or decision maker) is pivotal. In such trivial equilibrium, neither information which may be transmitted from an expert or private information decision maker may posses would rarely affect on equilibrium outcome from a voting game, since any unilateral deviation with a change in the belief of an agent would not affect on the voting outcome. Therefore, in order to sharpen predictions and rule out inappropriate equilibrium, we would like to focus mainly on non-trivial equilibrium in which there exists an individual decision maker with positive probability of being pivotal. Most of literature with simultaneous voting game restricts its focus on an equilibrium in which no weakly dominated strategy has been chosen. Although this refinement surely refine the set of equilibria with the classic implication of strategic dominance, it may have its own inherent limits or disadvantages. In particular, it may not rule out all trivial equilibria. In other words, there could still exist an equilibrium with no possibility of pivotal event even after requiring them not to chose weakly dominated strategy. Hence, in the current paper, we restrict our attention directly on the non-trivial equilibrium, which is formally defined as follows. Definition 4 (Non-trivial equilibrium). An equilibrium is non-trivial if the probability of being pivotal for some decision maker i is positive at all information set on the equilibrium path. 5 Therefore, unless otherwise stated, the term equilibrium in our analysis stands for non-trivial equilibrium. 5 Note that we require neither the existence of pivotal decision maker in all information set, nor the positive probability of being pivotal for all voters. 1

13 We would also like to restrict our attention on voters equilibrium strategies which is reasonably related to information the voters posses. In our Bayesian voting game, a voter s interim expected utility from the approved proposal would be not lower with a high private signal than with a low private signal. Therefore, it would be only reasonable that a voting strategy is monotonic in the voter s private signal, in the sense that a voter would be more likely (or not less likely) vote for the proposal when he gets a high signal. Therefore, we reduce our attention to an equilibrium in which each decision maker s voting strategy is monotonic in his private signal, which is formally defined as follows. Definition 5 (Monotonic Voting Strategy). A voting strategy v i is monotonic if, for a given M k m(θ), DM i votes for the proposal with high signal s i = h whenever he votes for the proposal with low signal s i = l. 6 v i (M k, h) v i (M k, l). For simplicity, we occasionally restrict our attention to an equilibrium with a monotonic voting profile, which is formally defined below. In an equilibrium with a monotonic voting profile, given the expert message M k, a voter votes for the proposal whenever a tougher voter votes for the proposal. We denote vi k := (v i(m k, s h i ), v i(m k, s l i )) and vk = (vi k)n i=1. Definition 6 (Monotonic Voting Profile). A voting strategy profile v k is monotonic if E[ũ i M k ] E[ũ j M k ] v k i v k j, for any i and j. 6 Note that this property should hold for an equilibrium strategy of any DM with positive probability of being pivotal. (Proposition 6 in Appendix proves it.) In addition to that, for both a pivotal and non-pivotal voter, the conditional expectation with low signal s l i cannot be higher than that with high signal s h i for a given message from the expert M k, E[ũ i M k, h] E[ũ i M k, l]. Therefore, it is fairly reasonable that we require monotonicity of a DM s voting strategy. We can also argue that non-monotonic voting could be eliminated by applying the concept of perturbed game in which each voter has small probability of being pivotal under any circumstances. 13

14 3 Babbling Equilibrium As in a general cheap talk model, there trivially exist many babbling equilibria. Among those, in this section we characterize the babbling equilibrium which is optimal for the highly biased expert. This equilibrium would be a benchmark for our basic model with the highly biased expert. We suppose Assumption 1 holds. First, consider unanimity, q = n. For any value of p > 1, there exists a babbling equilibrium equilibrium with (3.1) (1, 0), if i = 1, n v i (h, l) =. (1, 1), if 1 < i < n This equilibrium grants the highly biased expert the highest ex-ante expected payoff among symmetric babbling equilibria. In this equilibrium, only two DMs, i = 1 and n, votes depend on their private information. In other words, only DM i = 1 and n take informative actions, and all other voters always vote for the proposal. So, DMs as a group delegate their decision to the extreme voters, i = 1 and n. 7 Now, consider the simple majority rule, q = n+1. For any value of p > 1, there always exists a babbling equilibrium. Specifically, there is a pivotal babbling equilibrium equilibrium with (3.) i, v i (h, l) = (1, 0). In this babbling equilibrium, the ex-ante expected utility is exactly 1. 4 Highly Biased Expert In this section, we assume a heavily biased expert who always prefer the rejection. Definition 7 (Highly biased expert). A highly biased expert is an expert whose objective is to get rejection regardless of the state of 7 As mentioned above, there always exist multiple babbling equilbirium. In particualr, there could be some other equilibrium where a pair of decision makers other than i = 1 and n vote according to their signals. However, since DM 1 and n s interests are directly (or diagonally) opposite, it would be harder for them to agree on the decision by voting for the proposal than for any other pair of DMs. Therefore, from the highly biased expert s point of view, this equilibrium would be optimal. 14

15 Figure 4.1: Comparative Cheap Talk nature. The utility function of the highly biased expert is u e (v θ) = 1 1{ n v i q} u e (v). i As in pure persuasion game, the highly biased expert s preference is sate-independent. We define a particular type of a comparative cheap talk strategy, a -element partition cheap talk strategy, as follows. Definition 8 (-element Symmetric Partition Cheap Talk: Comparative Cheap Talk). m M +, θ M + (θ) = M, θ M where M + = {θ Θ θ 1 < θ } and M = {θ Θ θ 1 θ }. A message in this comparative cheap talk says the value of the state of nature in one dimension is higher than in the other dimension. By sending the message, the expert sacrifices the informational advantage in one dimension for the greater advantage in the other dimension. Consider the pipeline bill example again. By choosing this strategy, a lobbying firm reveals a simple truth about the quality of the pipeline project by saying, the project is better for the environment than it is for 15

16 jobs, (or vice versa). So, she sacrifices the advantage in the dimension of the environment for the advantage in the dimension of the number of jobs. As suggested in Chakraborty and Harbaugh (010), this trade-off may establish the credibility of the cheap talk with a multidimensional state of nature. Unlike the single decision maker model, in a model with collective decision making, any cheap talk strategy could be credible. In particular, along with a trivial equilibrium voting strategy where no voter is pivotal at any information set and the proposal is always rejected, any cheap talk strategy could be credible, since there is no incentive for the expert to lie. So, in this trivial equilibrium, an expert s cheap talk message may convey some credible information, but does not affect the equilibrium behavior. In other words, such equilibrium could be informative, but not influential. This justifies our direct attention on the non-trivial equilibrium, which allows us to focus on meaningful information transmission. However, even with a non-trivial equilibrium strategy, there generally exist the infinite number of credible cheap talk strategies. So, we start the analysis by focusing on the -element symmetric partition cheap talk strategy, of which the credibility has been already discussed by Chakraborty and Harbaugh (010). We first show that the highly biased expert could successfully transmit a part of information she has by choosing the strategy, and compare the expert s payoff from the strategy to the other strategies which convey more or less information. So, in analysis, we basically restrict our attention to the set of cheap talk strategies with a finer or coarser partitions than the -element symmetric partition. 4.1 Unanimity We now argue that the optimality of an equilibrium with the comparative cheap talk m (θ) for the expert with n-decision Makers under unanimity. Proposition 1 (The Existence of a rejection equilibrium). Suppose Assumption 1 holds. Consider the unanimity rule, q = n. 1. If p 5 6, there exists a rejection equilibrium with the comparative cheap talk m (θ) which is optimal for the highly biased expert. 16

17 . Specifically, if p 5 6, there exists an equilibrium with a cheap talk strategy m (θ) and a voting strategy {v +, v } such that v + 1 = (0, 0) and v+ i 1 = (1, 1) and vn = (0, 0) and v i n = (1, 1). 3. If p > 5 6, there is no rejection equilibrium with m (θ). 8 Proposition 1 shows that when the accuracy of the decision makers private signals is sufficiently low, there always exists an equilibrium that guarantees the rejection of the proposal with the comparative cheap talk message. 9 Then, how does the expert s message make the expert better off? The answer is polarization. In the rejection equilibrium, a cheap talk message M + (M ) ranks the decision makers in descending (increasing) order of γ i and make the highest ranked decision maker be least favorable toward the proposal. For example, by sending the message M + = {θ θ > θ 1 }, DM 1 with γ i = 1, who only cares about θ 1, would be least favorable toward the proposal, while DM n would be most favorable. Consider the pipeline bill example again. The highly biased expert reveals a simple truth about the quality of the pipeline project by saying, the project is better for the environment than it is for jobs (or vice versa). This message polarizes the legislators into groups with opposing beliefs; a group of legislators who care more about jobs believe the bill is good enough for them regardless of their private signals, and the other group who care more about environments would believe it is not. 10 By polarizing the legislature s beliefs, the highly biased expert could always induce her best outcome: the rejection of the proposal. In particular, under the unanimity rule, it is only natural that the former group would vote for the proposal regardless of their private signals. 11 At the same time, the legislators in the later group (except the DM 1) may want to vote against the proposal, but are delegating their decision to the toughest voter (DM 1) by voting for the proposal in spite of their negative belief on it. And DM 1 votes in the expert s favor, which is rejection. In summary, 8 This proposition is quite robust as it does not depend on any other assumptions but Assumption 1, the relaxation of which only affects on the cutoff of the probability. 9 Later, we show that under any super majority rule there always exists an equilibrium which results in the rejection of the proposal if the information gap between the expert and the voting body is sufficiently large. 10 The belief of a legislator with a moderate preference, γ 1, would depend on his private signal. 11 Given the fact that all voters strategies are monotonic, Definition 5, there is no incentive for them to vote against the proposal. 17

18 Figure 4.: Sincere Voting Strategy by exploiting the innate trade-off of the comparative message, the expert antagonizes one group of the decision makers, but turns the other of the group into allies and make them believe the proposal is not good enough to support. This polarized beliefs guarantees enough nay votes to reject the proposal. 1 To demonstrate the polarized opinions or decision, we use a sincere voting equilibrium where all decision makers vote according to their interim beliefs given their private signals and the expert s message. The condition specified in Proposition 1 is also a necessary condition for the existence of a sincere voting equilibrium. Figure 4. depicts decision makers sincere voting strategies: given a expert s message M +, decision makers who cares relatively more about θ 1 votes against the proposal and decision makers who care more about θ vote for the proposal. In the sincere voting equilibrium, we can observe more evenly polarized voting behavior as well as the polarized updated beliefs. Remark 1 (Sincere voting equilibrium). 1 With two intra-identical groups of voters, Chakraborty and Harbaugh (010) argue that the expert would be better off by attracting one group of voters at the expense of alienating the other. They call this a divisive effect of comparative cheap talk. So basically, they show that, in order to get her own way, the expert exploits the fact that voters has already been polarized in two groups. With more generalized model with n voters with diverse interests, we further show that the expert s cheap talk polarizes the voters into two groups with opposing opinions on the proposal, as well as cajoles one group into voting against the proposal. 18

19 There exists a non-trivial rejection equilibrium with the comparative cheap talk m (θ) if there exists an equilibrium with sincere voting strategies of all DMs given their private signals and the comparative cheap talk m (θ). Furthermore, the sincere voting equilibrium is outcome equivalent with the corresponding non-trivial rejection equilibrium. Another interesting observation from Proposition 1 is that DMs vote collectively in the expert s interest without paying attention to their private information. While a DM may differ his vote according to the expert s message M, he does not change his vote according to his private signal s i. That is, a DM s equilibrium voting behavior is not responsive to his own private signal. The expert s manipulative cheap talk message makes the DMs private information irrelevance to a final voting outcome: the voting outcome would never depend on the realization of the DMs private signals. This implies that the voting procedure would fail in aggregating their private information. However, if the accuracy of the DM s private signal p is sufficiently high, it may not be reasonable for DMs to ignore their private information. In particular, it is obvious that, when p = 1, a DM would vote for the proposal if and only if he gets a high signal: a sincere voting is a weakly dominant strategy. So this intuition says the higher p may lead to the more informative voting according to the DM s private signal. We do find relatively more aggregation of information with a higher value of p > 5 6. The following proposition demonstrates this under the expert optimal equilibrium when p > Proposition (Properties of the expert optimal equilibrium with p > 5 6 ). Suppose Assumption 1 holds. Consider the unanimity rule, q = n. In the expert optimal equilibrium with m (θ), two toughest decision makers for a given message M + or M vote according to their private signals, ṽ 1 + = ṽ+ = (1, 0) or ṽ n = ṽn 1 = (1, 0), and all others always vote for the proposal, v i> + = v i<n 1 = (1, 1). Now, with a relatively higher value of p > 5 6, two DMs vote according to their private signals. So, we can say that the voting outcome aggregates theses two DMs private information along with the information which is provided by the expert The complete characterization of the equilibrium is deferred to Proposition 7 in the Appendix. 14 Note that this number of responsive DMs to their private signals does not increase in n as well as p, so the proportion converges to zero as n approaches infinity. n 19

20 One may ask why a decision maker always votes for the proposal even when his private signal is very accurate p 1. We explain this behavior with delegation of decision making in voting. In the expert optimal equilibrium, a group of decision makers delegate their decision to the toughest decision maker(s) for a given message of the expert by voting for the proposal regardless of their private signals. In particular, under unanimity, all decision makers except DM 1 (and DM when p > 5 6 ) always vote for the proposal. For these supporters of the proposal, the decision of toughest decision maker(s) would be dependable or trustworthy. Delegation is frequently observed in any voting game especially with incompletely informed and (partially) ordered set of voters. Later in this paper, we show that the delegation behaviors are also observed under the majority rule (and any other super majority rule). Unlike the previous case with relatively low p, the probability of approval is positive when p 5 6 and the highly biased expert s ex-ante expected payoff is decreasing in p. That is, the expert s manipulating power of getting rejection with the comparative cheap talk is weakening as DMs is getting more accurate private signals. Moreover, the optimality of the equilibrium with m (θ) is now in question. But we can still show that the equilibrium with m (θ) is better for the expert than any other equilibrium with coarser or finer partition messaging strategy. 15 The following theorem demonstrates the optimality of the -element partition equilibrium for any value of p ( 1, 1). Theorem 1 (The highly biased expert optimal equilibrium). Suppose Assumption 1 holds. 1. For any p ( 1, 1), there exists a non-trivial equilibrium with the comparative cheap talk m (θ). 16. Moreover, the expert optimal equilibrium with m (θ) gives the expert weakly higher payoff than any other non-trivial equilibrium with a finer or coarser partition cheap talk. Theorem 1 shows that the highly biased expert cannot be better off by sending any more or less informative messages than the -element partition message m (θ) (the comparative cheap talk). As stated above, the only coarser partition is the entire set Θ = [0, 1] [0, 1], which is correspond 15 See Proposition 7 in Appendix. 16 See Proposition 1 and 7 for the full characterizations. 0

21 to the case under the babbling equilibrium, and we have already provided the intuition behind the dominance of the equilibrium with m (θ) in the expert s payoff over the babbling equilibrium earlier in this subsection. Then, why any finer partition cheap talk strategy does not help the expert? First, the finer partition, keeping the ranks of toughness of DMs as in the -element partition, just provides more precise information for them. So, there cannot be any further advantage from the schemed (or contrived) ordering, but could be a loss from providing too much information to the decision makers. Second, if there are too much of difference among the expected values of the state of nature given messages in a finer partition, so the expert s expected payoff from one message is higher than the other, it would not be credible since the expert would always prefer one message to another. Therefore, only possible equilibrium with a finer partition should have the same or similar level of expected values of the state of nature for a given message (or element in the partition), which would guarantee non-drastic differences in the expert s expected payoff and so the credibility of the messaging strategy. These expected values in a finer partition could not be too different from that in an element with -element symmetric partition, which is a weighted average of them by construction. So, the expert cannot be better off with a finer partition cheap talk strategy. Theorem 1 also shows that the cheap talk equilibrium exists even when p is arbitrarily close to 1. This means that the expert s cheap talk message would still be informative even when the decision makers have almost perfectly accurate private signals. So, for any value of p, in a cheap talk equilibrium characterized in Theorem 1, a DM makes more informative decision than in a babbling equilibrium, but be worse off in terms of ex-ante expected payoff. 4. Majority We now check the comparative cheap talk still works for the highly biased expert under majority. The following proposition characterizes the comparative cheap talk equilibrium under majority. Unlike under unanimity, we now assume the symmetry of the decision makers. (Assumption ) Proposition 3 (The existence of a comparative cheap talk equilibrium). Suppose Assumption holds. Consider the simple majority rule, q = n+1. For any n and p, 1. there is an equilibrium with the comparative cheap talk m (θ) which is optimal for the highly 1

22 biased expert.. Specifically, there exists an equilibrium with a cheap talk strategy m (θ) and a voting strategy {v +, v } such that (a) for M +, v i>q + = (1, 1), v+ i<q = (0, 0) and v+ i=q = (1, 0). (b) for M, v i>q + = (0, 0), v+ i<q = (1, 1) and v+ i=q = (1, 0). If we restrict our attention to the set of equilibrium where a DM s voting strategy is monotonic in the expected utility given any expert s message, the characterized equilibrium in the proposition above is weakly better for the highly biased expert than any other equilibrium along with any finer or coarser partition. Unlike the unanimity case, there is no rejection equilibrium even when the accuracy of DM s private signal is low under majority. Moreover, the ex-ante expected payoff of the expert in this cheap talk equilibrium is the same as in the babbling equilibrium. In other words, the highly biased expert loses the manipulating power of getting rejection under majority: the manipulating power of getting rejection is weaker under majority than under unanimity. This makes sense, since under unanimity she only needs one votes, but under majority she needs to persuade more than a half of the DMs. Though the cheap talk message may lose the power of getting rejection in expectation, however, its functions of polarizing DMs beliefs and limiting the information aggregation have still well (or even better) performed. In a babbling equilibrium under majority, all DMs information would be aggregated since all DMs voting strategies depend on their own private signals. On the other hand, in the cheap talk equilibrium characterized in Proposition 3, only one DM s voting strategy the median DM s voting strategy v q depends on his private signal. All DM other than the median DM q are polarized into two groups: one who always vote for the proposal and the other who always vote against it. So, all other DMs private information would be ignored, so not aggregated in the voting outcome. Furthermore, if there is arbitrarily small possibility that a DM is being misinterpreting a high private signal, so making errors by voting against the proposal more frequently than he is supposed to be, then the cheap talk equilibrium would be better for the highly biased expert than the

23 babbling equilibrium, since in the babbling equilibrium such an error committed by a DM could be compensated by the other DMs correct decisions. To sum up, under majority, though the manipulating power of getting rejection is weakened, the cheap talk strategy still has a significant manipulating power of polarizing DMs beliefs and preventing them from aggregating the private information. 5 Surplus maximizing expert In this section, we assume there is a surplus maximizing expert instead of a heavily biased expert. The definition of a surplus maximizing expert is as follows. Definition 9 (Surplus maximizing expert). A surplus maximizing expert is an expert whose objective is to maximize the sum of all decision makers utility. The utility function of the surplus maximizing expert is u e s = n i=1 u d i θ n 1 i=1 = γ i + θ n i=1 (1 γ i) n, if approved. 0, if rejected This expert s preference is coincide with the median DM s preference under Assumption. For this surplus maximizing expert, we are going to argue that the other -element partition cheap talk strategy, the average cheap talk strategy which is defined below, is the optimal. Definition 10 (Average Cheap Talk). t T +, θ T + (θ) = T, θ T where T + = {θ Θ θ 1 + θ > 1} and T = {θ Θ θ 1 + θ 1}. 3

24 Figure 5.1: Average Cheap Talk 5.1 Unanimity As in the model with a highly biased expert, we show that the surplus maximizing expert can always get his way in a cheap talk equilibrium. Proposition 4 (Surplus Maximizing Equilibrium). Suppose Assumption 1 holds. Consider the unanimity rule, q = n. For any value of p < 5 6, 1. there exists a surplus maximizing equilibrium with the average cheap talk t (θ).. Moreover, there exists an outcome equivalent sincere voting equilibrium in which the average cheap talk unifies the DMs; all DMs vote for the proposal given the message T + and vote against it given T. Proposition 4 shows that, when there is sufficient informational gap between the expert and DMs, the expert s average cheap talk now unifies the DMs beliefs and voting behavior. As depicted in Figure 5.1, in a cheap talk equilibrium, given the average cheap talk message T + (or T ), all DMs believe the proposal is good (or bad) for them. Because any negative private signal s i = l cannot overturn the DM i s belief on the proposal give the expert s cheap talk message T + (or T ), a DM votes only responsive to the expert s suggestions. Therefore, DMs votes in a cheap talk equilibrium are unified. 4