Voting Rules and Informed Decision Makers
Cheap Talk and Collective Decision-Making: Voting Rules and Informed Decision Makers Daeyoung Jeong∗ April 15, 2015
Abstract We investigate a cheap talk model with a collective decision making. In our model multiple decision makers vote on a proposal which determines their payoffs, and an expert tries to persuade them to choose the outcome she prefers. We allow decision makers to possess not all but some of information regarding the state of nature, which determines the gross utility of them based on the voting outcome. Two different types of experts has been considered: a heavily biased expert who always wants a rejection and a surplus maximizing expert who tries to maximize the total surplus of the group of decision makers. We show that experts can transmit credible and influential information to voters by using their respective optimal cheap talk strategies and try to prevent voters from taking informative actions. This limited information aggregation induced by each type of experts results in either polarization or unification of the voters: the highly biased expert polarizes the voters to achieve her aim, while the surplus maximizing expert unifies them. Furthermore, with the highly biased expert, polarization among voters has been observed under any voting rules, but voters tend to be more conservative under a voting rule that requires less votes in favor.
JEL Classification Numbers: Keywords: ∗
The Ohio State University, 410 Arps Hall, 1945 North High Street, Columbus, Ohio, [email protected].
1
Contents 1 Introduction 1.1
3
Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Model
8
3 Highly Biased Expert 3.1
3.2
4
12
Unanimity with n-Decision Makers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.1
Babbling Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2
2-element Partition Equilibrium
. . . . . . . . . . . . . . . . . . . . . . . . . 15
Majority with n-Decision Makers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.1
2-element Partition Equilibrium
. . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3
Existence of Rejection Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4
Continuum of voters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Surplus maximizing expert 4.1
29
Unanimity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Conclusion
31
A Appendix
32
A.1 HBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.1.1 Unanimity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.1.2 Majority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 B Appendix: Technical Appendix
34
2
1
Introduction
In many economic situations, a group makes a collective decision, which determines the payoffs for all members. For example, the Supreme Court justices decide whether a bill is constitutional, and a board of directors or a group of policy makers have to decide whether to approve a proposal. Each decision maker may have different criteria for choices, bills or proposals, and this variety of criteria may cause some conflicts over their collective decision. 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 policy-lobbying group or an affiliated office of a private company does not usually have the right to decide even though they have more precise information on the proposals than the decision makers. This wellinformed 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. The primary question of the “cheap talk” literature is whether an expert with informational superiority could send credible messages, cheap talk, to a single decision maker, even when messages are unverifiable and there is no cost for lying. In my basic model, on the other hand, multiple decision makers vote on a proposal which determines their payoffs, and an expert tries to persuade them to choose the outcome she prefers. This feature of collective decision making in a cheap talk model allows me to explore several interesting research questions. First, this model investigates the expert’s optimal information transmission behavior, which may vary with the voting rule as well as the preference of the expert. I find that, under unanimity rule, the expert who always prefers rejecting the proposal can polarize the decision makers and increase the chances of rejection by sending a simple but credible message. I also show that the same messaging strategy may not help the expert with extreme preferences much under the simple majority rule. Different types of experts, such as an expert who tries to maximize the total welfare of the group of decision makers could also be considered. For example, in a jury trial, a prosecutor could be the welfare-maximizing expert, while a defense attorney is an expert with extreme preferences for acquittal. By comparing their different communication strategies that may
3
polarize or unify the opinions of the decision makers, I can provide economic implications for the persuasion of the prosecutor and defense attorney or of the lobbying behavior of different interest groups. Second, this study also explores whether information transmission helps voters make better decisions. Relevant policy implications can be derived from understanding the general setting I present with collective decision making that cannot be derived from the single decision maker case. Finally, I study the impact of the degree of the information dominance of the expert over the group of decision makers. The amount of information transmitted and the agents’ profits may depend on the gap between the amounts of information the expert and the group of decision makers initially have. Therefore, by investigating the impact of the informational gap between the agents, I am able to qualify the importance of the initial information distribution over economic agents when decisions are collective.
1.1
Related Literature
Even though most of cheap talk literature consider one-dimensional state variable, there are some studies with multiple dimensions. Battaglini (2002) studies multidimensional cheap talk with multiple senders and shows that there exists a full revelation equilibrium. Chakraborty and Harbaugh (2010) examines 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 equilibrium which influential cheap talk of very biased expert on decision making. Though they mainly focus on cheap talk models with a single sender, they also discuss, with a simple example, the possibility of the welfare loss of voters in a cheap talk equilibrium with a highly biased expert. They conclude that 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. Though the basic setting of two dimensional state of nature is the same as theirs, we consider a voting body of which each individual has a unique preference as well as two different types of experts, while they consider only a highly biased expert and two groups of voters who have identical preference within groups. Moreover, we allow voters to possess not all but some private information about the state of the world. With this feature, we can investigate the effect of cheap talk on information aggregation 4
among voters. 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) shows that under the unidimensional state of nature, voting with a large number of voters effectively aggregates information among voters 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 to voters under two dimensional state of nature. 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. This result provides a bridge between the study of information aggregation and polarization. Literature on polarization studies polarized opinion of people who observe the same information. For example, Andreoni and Mylovanov (2012) develop a theoretical model and conduct an experiment to investigate polarization of opinions and its persistence when people would be provided more information over time. They conclude that polarization of opinions could exist and persist over time when the dimensionality of uncertainty or information, as in our model, exceeds that of opinions. While each individual makes her own decision in their model, people make collective decision in our setting. In spite of this difference, our result with a highly biased expert is consistent with theirs. We show that polarization of voters’ opinions is the result of limited information aggregation among the voters, which induced by a highly biased expert’s strategic information transmission. In addition to that, we conclude that this polarization highly depends on the style or the types of additional information, by comparing the result of polarization with a highly biased expert to that of unification with a surplus maximizing expert. Several papers explore the possibility of persuasion of a biased informed agent on a voting body. Caillaud and Tirole (2007) and Schnakenberg (2015) explore an expert’s effort to persuade a voting body in the context of a mechanism design and a cheap talk, respectively. However, interestingly, they seem to merely show that an expert as a supporter of the proposal can increase the probability 5
of approval by transmitting information to existing allies among a voting body. In 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. There are several working, not yet published, papers which study a cheap talk model with an informed decision maker. Lai (2013) assumes that a decision maker knows whether the state is high or low relative to a private threshold. He shows that the expert’s message is more biased compare to the situation with uninformed receiver. Ishida and Shimizu (2013) investigate a cheap talk with incompletely informed agents. Unlike general cheap talk models, they 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 information transmitted. de Barreda (2013) assumes that a decision maker receives a signal which is affiliated with the state of nature. She shows the existence of CS type (partition) equilibrium and compare with basic uninformed receiver situation. Previous studies has also examined a cheap talk model with multiple decision makers. Board and Dragu (2008) 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 (2013) 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. Other studies examine a cheap talk model with voting. Austen-Smith and Feddersen (2002) explore the effect of deliberation prior to a collective decision making via voting. The deliberation is a type of cheap talk among decision makers who have private information of the state of nature. They compare majority and unanimity rules, and conclude that the former induces more information transmission and fewer errors in decision making than the latter. Guo (2013) analyzes an informed voter’s incentive of information transmission. In her model, there are three voters: a left biased, a right biased, and an unbiased voter. She conclude that a biased voter has more incentive to transmit her private information to the others than an unbiased voter. Rivas and Rodr´ıguez´ Alvarez (2013) examine the impact of a leadership in a deliberation among decision makers before 6
a voting. They conclude that a very influential leader affect more 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. 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.
7
2
Model
There are an expert (female) and an odd number of decision makers (male) i ∈ N = {1, 2, ..., n} where n ≥ 3. Decision makers (DMs) vote on a given proposal. We rename the voting rules from unanimity to simple majority rule as follows. Definition 1 (r-Majority Rule). r-Majority Rule requires at least r/n of support to approve a proposal. When r = n, l-Majority Rule is a unanimity rule, and when r =
n+1 2 ,
it is a simple
majority rule. The space of the state of nature is two-dimensional, θ = (θ1 , θ2 ) ∈ Θ ≡ [0, 1] × [0, 1], the aspect 1 and 2, respectively. Each dimension represents a different aspect of the state of nature. A DM i’s preference conditional on approval is characterized by the weighted average of the state of nature.
udi =
γi θ1 + (1 − γi )θ2 − 1 , 2
0,
if approved if rejected
γi captures the intensity of DM i’s personal interest (or concern) on the aspect 1. γi = 1 means DM i only cares about the aspect 1 (the first dimension), and γi = 0 means DM i only cares about the aspect 2 (the second dimension). A decision maker i has payoff γi θ1 + (1 − γi )θ2 −
1 2
from approval and payoff 0 from rejection.1 To economize on notation, without loss of interesting generality, we assume γi > γj if and only if i > j. We assume that a decision maker’s preference parameter γi is publicly informed. 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, si ∈ Si = {h, l}, which may contains partial information about θ. For all θ 1 P si = h|γi θ1 + (1 − γi )θ2 − > 0 =p ∈ 12 , 1 2 1 P si = l|γi θ1 + (1 − γi )θ2 − > 0 =1 − p 2 p, which is commonly known, represents an accuracy of DM’s private information. To economize 1
Chakraborty and Harbaugh (2010)
8
on notation, we define
Dih ≡ {θ ∈ Θ|γi θ1 + (1 − γi )θ2 −
1 2
> 0}
and
Dil ≡ {θ ∈ Θ|γi θ1 + (1 − γi )θ2 −
1 2
≤ 0} = Θ/Dih .
By Bayes’ rule, we get
P θ∈
Dih |si
P θ ∈ Dih P si = h|θ ∈ Dih =h = P (si = h) P θ ∈ Dih P si = h|θ ∈ Dih =P t t t∈Si P (θ ∈ Di ) P (si = h|θ ∈ Di )
=1
2p
+
1 2p 1 2 (1
− p)
=p
P θ ∈ Dil |si = h =1 − p. Similarly, P θ ∈ Dil |si = l = p and P θ ∈ Dih |si = l = 1 − p. Therefore, if p = 12 , the signal does not convey any information, but if p > 12 , it conveys some information. The game proceeds as follows: 1. Nature draws the sate θ ∼ U [0, 1] × [0, 1]. 2. 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 si ∈ Si = {h, l}. 5. Each decision maker votes on the proposal, vi ∈ {0, 1}. 6. Under r-Majority rule, agents receive payoffs as follows. P • udi (vi |v−i , γi , θ) = 1{ ni vi ≥ r}(γi θ1 + (1 − γi )θ2 − 12 ) • ue (v|θ) where v = (vi )ni=1 = (v1 , v2 , ....vn ) 9
Different types of experts would have different functional forms of payoff function ue (v|θ). In the basic model, we assume a heavily biased expert who always prefer the rejection. Definition 2 (Highly biased expert). A highly biased expert is an expert whose objective is to get rejection regardless of the state of nature. The utility function of the highly biased expert is ue (v|θ) = 1 − 1{
n X
vi ≥ r}.
i
Thus, the expert’s preference does not depend on the state of nature. The solution concept employed in this paper is perfect Bayesian equilibrium. Definition 3 (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)], Z
∗
ue (v(m, s), θ)f (s|θ)ds
m ∈ arg max m
where s = (si )ni=1 and S =
Qn i
S
Si .
2. for each m and si Z
∗
vi (m , si ) ∈ arg max vi
udi (vi |θ, γi )µi (θ|m∗ , si )dθ
Θ
3. for all m ∈ M and all si ∈ Si
µi (θ|m, si ) =
R
Θ
σ(m|θ)φi (θ|si ) , σ(m|θ0 )φi (θ0 |si )dθ0
ψi (θ|si ),
if Θσ (m) 6= ∅ if Θσ (m) = ∅
where ψi (θ|si ) is any distribution. Assumption 1 (Tie breaking rule). If DM is indifferent between approval and rejection of the proposal, he is more likely to vote for it. 10
To simplify the analysis, we occasionally assume “symmetry” in a group of decision makers. Assumption 2 (Symmetric decision makers). In the group of decision makers, the decision makers are symmetric around
n+1 2
≡ r-th decision maker, and there are always extreme types γn = 1 and
γ1 = 0. More precisely, γk = 21 , γn = 1, γ1 = 0, and γn−j+1 = 1 − γj for any j such that k > j > 1. 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 4 (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, µi (θ|m, s) ≥ φi (θ|s) for all i with at least one strict inequality. Definition 5 (Influential equilibrium). An equilibrium is influential if there exists different messages m and m0 where m 6= m0 induce different actions of decision makers, vi (m, s) 6= vi (m0 , s) for some i and s. Definition 6 (Influential in outcomes). An equilibrium is influential in outcomes if there exists different messages m and m0 where m 6= m0 induce different outcomes of decision makers, i.e., P P 6 1{ ni vi (m0 , s) ≥ r} for some i and s. 1{ ni vi (m, s) ≥ r} = 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 order to sharpen predictions and results, we only focus on non-trivial equilibrium in which there exists an individual decision maker with positive probability of being pivotal at some information set.2 Definition 7 (Non-trivial equilibrium). An equilibrium is non-trivial if the probability of being pivotal for some decision maker i is positive at some information set. In order to economize the notion, we may abuse the term “equilibrium” for “non-trivial equilibrium”. So, in our analysis, if there is no other comment, an “equilibrium” means a “non-trivial equilibrium.” 2
Note that we do not require the existence of pivotal decision maker in all information set.
11
3
Highly Biased Expert
To economize the notion,
1 u ˜i ≡ γi θ1 + (1 − γi )θ2 − . 2 With any message M k , strategy profile v k and signal si , k k E(ui (vi (si ) = 1)|M k , v−i , si ) − E(ui (vi (si ) = 0)|M k , v−i , si ) k k =P r(Sˆi (v−i )|M k , si )E(˜ ui |M k , Sˆi (v−i ), si ),
k )⊆× k where Sˆi (v−i j6=i Sj is the set of events where DM i is pivotal under vi . So, if
k E(˜ ui |M k , Sˆi (v−i ), si ) > 0,
then v k (si ) = 1 is a strictly (weakly, with equality) dominant strategy.
3.1
Unanimity with n-Decision Makers
Proposition 1. [A property of a non-trivial partition equilibrium] Consider the unanimity rule. If there exists a non-trivial equilibrium with a partition {M k }K k=1 and a corresponding strategy profile {v k }K k=1 , then for any decision maker i who is pivotal with positive probability given a message M k and the corresponding strategy profile v k , vik (shi ) ≥ vik (sli ). Proof of Lemma 1. Since DM i is pivotal with positive probability under a message M k and the corresponding strategy k ) 6= ∅. So, it is clear that profile v k , Sˆi (v−i
k k E[˜ ui |M k , Sˆi (v−i ), shi ] ≥ E[˜ ui |M k , Sˆi (v−i ), sli ],
which proves the lemma. Corollary 1.1. Consider unanimity, r = n, if there exists a non-trivial equilibrium {(M k , v k )}K k=1 12
with a positive probability of approval, then for any i and k, vik (shi ) = 1 ≥ vik (sli ). 3.1.1
Babbling Equilibrium
Proposition 2. [The Existence of Babbling Equilibrium] Consider unanimity, r = n. For any value of p >
1 2,
there always exists a babbling equilibrium.
Specifically, there is a pivotal babbling equilibrium equilibrium with
− vi (s+ i , si ) =
(1, 0),
if i = 1, n
.
(1, 1), if 1 < i < n Moreover, this equilibrium grants expert the highest ex-ante (and interim) 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. Since these two DMs’ 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 pair of DMs. Therefore, from the expert’s point of view, this equilibrium would be optimal. Lemma 1. Consider unanimity r = n. In a babbling equilibrium with v, for any i, vih = 1. Proof of Lemma 1. For any v−i which makes DM i pivotal, E[˜ ui |Sˆi (v−i ), shi ] ≥ E[˜ ui |shi ] > 0
Lemma 2. Consider unanimity, r = n. There always exist a babbling equilibrium with v = (1, 0, 1, 1, ...1, 1, 1, 0). Proof of Lemma 2.
13
From Lemma 1, we know vih = 1 for any i. For any s1 and sn E[˜ u1 |Sˆ1 (v−1 ), s1 ] = E[˜ u1 |s1 , sn ] = E[˜ u1 |s1 ],
and E[˜ un |Sˆn (v−n ), sn ] = E[˜ un |s1 , sn ] = E[˜ un |sn ].
So,
E[˜ u1 |sn , sl1 ] = E[˜ u1 |sl1 ] < 0,
and
E[˜ un |s1 , sln ] = E[˜ un |sln ] < 0. For any i such that 1 < i ≤ 21 , E[˜ ui |Sˆi (v−i ), sli ] = E[˜ ui |sh1 , shn , sli ] =
Similarly, for any i such that
1 2
(1 − γi )(1 − 7γi )(2p − 1) >0 6(1 − 2(1 − γi )p − 5γi )
< i < n, we can show that
E[˜ ui |Sˆi (v−i ), sli ] = E[˜ ui |sh1 , shn , sli ] > 0.
Lemma 3. Consider unanimity, r = n. There is no symmetric babbling equilibrium with more than 2 decision makers with vi = (1, 0). Proof. Suppose there exists a symmetric equilibrium with vˆi = vˆn−i+1 = (1, 0) and vˆj = vˆn−j+1 = (1, 0), which is different from the voting strategy from Lemma 2, v = (1, 0, 1, 1, ...1, 1, 1, 0). By
14
construction, E[˜ un−i+1 |Sˆn−i+1 (ˆ v−(n−i+1) ), sln−i+1 ] > E[˜ un−i+1 |Sˆn−i+1 (v−(n−i+1) ), sln−i+1 ],
and from Lemma 2, we know E[˜ un−i+1 |Sˆn−i+1 (ˆ v−(n−i+1) ), sln−i+1 ] > E[˜ un−i+1 |Sˆn−i+1 (v−(n−i+1) ), sln−i+1 ] > 0.
Therefore, vˆn−i+1 should be (1, 1). So, contradiction. Lemma 4. Consider unanimity, r = n. If there exists a symmetric pivotal babbling equilibrium with vi>1 = vn−i+1 = (1, 0) and vj6=i = (1, 1), then the expert’s payoff is less than 34 . Proof.
3.1.2
2-element Partition Equilibrium
As in pure persuasion game, the highly biased expert’s preference is sate-independent. Then the rationale behind the partition equilibrium in Crawford and Sobel (1982), which makes the cheap talk credible, does not work in our situation. Instead, as in Chakraborty and Harbaugh (2010) where also have mutidimensional information structure, we argue that, the 2-element symmetric partition cheap talk could be credible even if the expert is highly biased, since it has a intrinsic trade-off for the expert. Definition 8. A partition M = {M + , M − }, is a 2-element symmetric partition when
M + = {θ ∈ Θ|θ1 < θ2 }
and M − = {θ ∈ Θ|θ1 ≥ θ2 }.
A message in the 2-element symmetric partition cheap tale says the intensity of a state of nature in one dimension is (relatively) higher than in the other dimension. So, by sending that, the expert
15
sacrifice the informational advantage in one dimension for the greater advantage in the other. This trade-off establishes the credibility of the cheap talk strategy. This argument also provides the intuition behind the impossibility of information transmission in one dimension. If there is only one dimension, unlike in the multidimension case, any generic partition strategy would not be credible. Since one element in a partition generally carries (weakly) higher expected values of the state of nature than the other, it would never be credible. We now argue that the optimality of the 2-element partition equilibrium for the expert. Lemma 5. For uninformed decision makers, p =
1 2,
the 2-element symmetric partition is the
optimal messaging strategy for the expert. The interim expected payoff of the expert depends on the interim probability of being approved. Under the unanimity,3
(3.1)
P r(approved|θ, Mk , vk ) i Yh = P r(shi |θ)vik (shi ) + P r(sli |θ)vik (sli ) i
i Yh k k = P r(shi |θ)1(E[˜ ui |M k , Sˆi (v−i ), shi ] > 0) + P r(sli |θ)1(E[˜ ui |M k , Sˆi (v−i ), sli ] > 0) i
≡
Y
qi (θ, Mk , vk ).
i
Notice that the expressions in the indicator functions are independent on the state of nature θ. On the other hand, all the terms of conditional probabilities, P r(shi |θ) and P r(sli |θ), do depend on θ. In addition to that, the former depends on Mk and vk,−i , but the later does not. In any equilibrium with {(Mk , vk )}K k=1 , Equation (3.1) should be constant over Mk for any given θ ∈ Θ. Lemma 6. Consider the unanimity rule. If there exists a rejection equilibrium with a (weakly) finer partition + − k K than 2-element partition, {M k }K k=1 ≤ {M , M }, and the corresponding strategy profile {v }k=1 , + − then there exists a rejection equilibrium with {M k }K v k }K k=1 ≤ {M , M } and {˜ k=1 such that 3 Without loss of interesting generality, we ignore the case in which vik (si ) = 1 when E[udi (θ)|Mk , vk , si ] > 0, since we focus on the expert optimal equilibrium.
16
1. for any k with M k ⊂ M + , v˜1k = (0, 0) and v˜i6k=1 = (1, 1) and 2. for any k with M k ⊂ M − , v˜nk = (0, 0) and v˜i6k=n = (1, 1). Proof of Lemma 6. Since it is a rejection equilibrium, vjk should be (0, 0) for some j. So, k E[˜ uj |M k , shj ] ≤E[˜ uj |M k , Sˆj (v−j ), shj ] ≤ 0.
Consider any pair of signal (shi , shj ). For DM i and j such that i < j, we know E[˜ ui |M k , shi ] × p(M k , shi ) =p(Di+
∩M
k
)p(shi |Di+
k
Z
∩M ) + k ˆ θ∈D i ∩M
u ˜i dθˆ
Z + + + + k h k + p( Dj \ Di ∩ M )p(si | Dj \ Di ∩ M )
+ + k ˆ θ∈(D j \Di )∩M
+ p(M k \ Dj+ )p(shi |M k \ Dj+ )
Z k \D + ˆ θ∈M j
u ˜i dθˆ
u ˜i dθˆ
and
Eθˆ[˜ uj |M k , shj ] × p(M k , shj ) =p(Di+ ∩ M k )p(shj |Di+ ∩ M k )
Z + k ˆ θ∈D i ∩M
u ˜j dθˆ
Z + p( Dj+ \ Di+ ∩ M k )p(shj | Dj+ \ Di+ ∩ M k )
+ + k ˆ θ∈(D j \Di )∩M
k
+ p(M \
Dj+ )p(shj |M k
\
Dj+ )
Z k \D + ˆ θ∈M j
u ˜j dθˆ
ˆ u ˜j dθ.
The right hand sides of both equations have three terms. For any M k ⊂ M + , each term, from the first to the third, is no greater in the former equation than in the later equation. Therefore,
E[˜ uj |M k , shj ] ≤ 0
17
implies k E[˜ ui |M k , Sˆi (˜ v−i ), shi ] = E[˜ ui |M k , shi ] ≤ 0.
So, for any M k ⊂ M + , there could be an equilibrium with v˜k such that v˜1k = (0, 0) and v˜ik = (1, 1) for any i > 1 is an equilibrium strategy profile. Similarly, we can show that, for any M k ⊂ M − , there could be an equilibrium with v˜k such that v˜nk = (0, 0) and v˜ik = (1, 1) for any i < n is an equilibrium strategy profile. Proposition 3 (The Existence of rejection equilibrium). Consider the unanimity rule. 1. If p ≤
5 6,
there exists a rejection equilibrium with 2-element partition {M + , M − } which is
optimal for the highly biased expert. 2. Specifically, if p ≤ 65 , there exist a equilibrium with a partition {M + , M − } and a corresponding strategy {v + , v − } such that • v˜1+ = (0, 0) and v˜i6+=1 = (1, 1) and • v˜n− = (0, 0) and v˜i6−=n = (1, 1). 3. If p > 56 , there is no rejection equilibrium with 2-element partition {M + , M − }. Proof of Proposition 3. To prove this proposition, we use Lemma 6. Check, for some p, if k E[˜ ui |M k , Sˆi (˜ v−1 ), sh1 ] =
6p − 5 ≤ 0. 6(3 − 2p)
So, there exists a rejection pivotal equilibrium with 2-element partition if and only if p ≤ 56 . Proposition 4 (The Existence of an approval equilibrium). Consider the unanimity rule. 1. If p > 56 , there is a 2-element partition equilibrium with positive probability of approval. 18
2. Specifically, if p > 56 , there is a 2-element partition equilibrium with {M + , M − } + (a) for M + , v˜1+ = v˜2+ = (1, 0) and v˜i>2 = (1, 1) and − − (b) for M − , v˜n− = v˜n−1 = (1, 0) and v˜i γ3 ≥
1 2
+ − k E[˜ u3 |M + , Sˆi (˜ v−3 ), s− u3 |M + , s+ 3 ] = E[˜ 1 , s2 , s3 ]
=
γ32 ((3 − 2p)p − 1) − γ22 (p(γ3 (γ3 (6p − 7) + 2) + 2p − 3) + 1) + 2γ3 (p − 1)(2p − 1)γ2 ≥ 0. 6γ2 (γ3 (p(2p − 3)(γ2 + 1) + 1) + ((3 − 2p)p − 1)γ2 )
So, there cannot be more than 3 zeros in an equilibrium strategy v. Moreover, any other 2-element symmetric partition equilibrium with two zeros in v is worse for the expert than the
19
equilibrium characterized in Proposition 4. Hence, there is no 2-element symmetric partition equilibrium with other voting strategy that grants the expert a higher payoff than with v˜. Dominance over the finer partition equilibrium. First of all, there cannot be any rejection equilibrium with a finer partition. (∵ if there exists a rejection equilibrium with a finer partition, from Lemma 6, there should be a rejection equilibrium with 2-element partition.) Second, for any partition equilibrium with positive probability of approval, M k ∩ D1 6= ∅, and there should be same number of zeros. (If not for any θ ∈ D1 , one is better than the other, so the message is not credible.) Now, suppose there is any other pivotal equilibrium with a finer partition and a voting strategy v 0 . If v 0 = v˜, then the payoff is the same with the characterized equilibrium. Suppose v 0 6= v˜. If there is one zero, then the payoff cannot be higher than the 2-element partition equilibrium. (The best equilibrium with one zero is v = (1, 0) which gives the payoff less than the 2-element partition equilibrium.) Similar for the case with two zeros. For the last step, we show that there cannot be more than two voters with vik = (1, 0) in v k + − for any element M k in a finer partition {M k }K k=1 ≤ {M , M }. To have more than two zeros, we
should have k E[˜ ul |M k , Sˆl (v−l ), s− l ] 56 , the expert optimal 2-element partition equilibrium is better for the expert than the optimal symmetric babbling equilibrium.
20
Proof. In the equilibrium specified in Proposition 4, 1 3 1 2 p + (1 − p) 3(1 − p) − (1 − 5p) ≥ . E[u (v|θ)] = 1 − 4 γ2 4 e
Proposition 5 (Optimality of the 2-element symmetric partition equilibrium). For any p ∈ [ 12 , 1], the 2-element symmetric partition equilibrium gives the expert weakly higher payoff than any other pivotal equilibrium with a finer or coarser partition. This means that the more precise information transmission compare to the 2-element partition equilibrium does not help the expert. First, the finer partition, keeping the order of toughness of DMs, just provides more precise information for them. So, there cannot be a further advantage from the schemed (or contrived) ordering, but could be a loss from excessive information transmission. Second, if the element in any finer partition could be ordered in the expected values of them, it would not be credible since the expert could be better off with one signal than with another. Therefore, only equilibrium with finer partition which is credible should have the same or similar level of expected values. This expected value (or these expected values) would not be so different with that in an element with 2-element symmetric partition. So, the expert cannot be better off with a finer partition. In a 2-element partition equilibrium, the DMs would be ordered in toughness. And in the expert optimal equilibrium, they delegate their decision to the voter (or voters) with toughest criterion (or criteria).
21
3.2 3.2.1
Majority with n-Decision Makers 2-element Partition Equilibrium
Proposition 6 (The existence of 2-element partition equilibrium). Consider the simple majority rule, r =
n+1 2 .
For any n and p,
1. there is a 2-element symmetric partition equilibrium. 2. Specifically, there is a 2-element partition equilibrium with {M + , M − } + + + (a) for M + , vi>r = (1, 1), vir = (0, 0), vi γr = 21 . Moreover, + ˆ Eθˆ[ui (v = 1|θ)|M , sr = l, si = h] < E[u(v = 1)|M + , sr = l, si = h].
Therefore, any i < r dose not have incentive to deviate. For any i > r, i is pivotal only when sl = h. 1 + ˆ Eθˆ[ui (v = 1|θ)|M , sr = h, si = l] = (1 − 2γi ) > 0 6 , since γi < γr = 21 . Moreover, + ˆ Eθˆ[ui (v = 1|θ)|M , sr = h, si = h] > E[u(v = 1)|M + , sr = h, si = l].
Therefore, any i > r dose not have incentive to deviate.
22
Voter i = r is always pivotal, and it is easy to show that he has no incentive to deviate for any p > 21 . 2p − 1 + ˆ ≥ 0. Eθˆ[ul (v = 1|θ)|M , sr = h] = 6
+ ˆ Eθˆ[ul (v = 1|θ)|M , sr = l] =
1 − 2p ≤ 0. 6
Part 3. One zero other than 1/2? Lower than 1/2 implies lower payoff Higher than 1/2. to have higher payoff, there should be at least (n − i) − (r − 1) (0, 0) Impossible, since from r, it should be (1, 0) or (1, 1). maximum (n − i) − r (0, 0) Two zeros? If both γi > γj > 21 , g 2 p 2 − 8r2 − 1 + 2gr(2p(r − 1) + 1) + (2p − 1)r2 21 , g 2 p 2 − 8r2 − 1 + 2gr(2p(r − 1) + 1) + (2p − 1)r2 n−(r−1) = (1, 1) and − − (b) for M − , v˜i≥r = (0, 0) and v˜in−(r−1) = (1, 1). For any i ≤ n − (r − 1), who is pivotal,
we need to show E[˜ ui |M + , Sˆi (˜ v−i ), s+ i ] < 0, which also implies E[˜ ui |Sˆi (˜ v−i ), s− i ] < 0. We know E[˜ ui |M + , Sˆi (˜ v−i ), s+ ui |M + , s+ i ] = E[˜ i ]=
4γi (−2γi (p − 1) + p − 1) − 2p + 1 , 6(4γi (p − 1) − 2p + 1)
and this is decreasing in γi , in other words, increasing in i. So it is enough to show that E[˜ un−(r−1) |M + , Sˆn−(r−1) (˜ v−(n−(r−1)) ), s+ n−(r−1) ] < 0, which is guaranteed by the condition p < p¯(r; n). For M − , it is symmetric, so we prove the existence. Now, we want to show there is no rejection equilibrium if p ≥ p¯(r; n). We prove by contradiction. Suppose p ≥ p¯(r; n). Assume there is a rejection equilibrium with a voting strategy v + and v − . To have any rejection pivotal equilibrium with 2-element partition, there should be n − r voters who always vote no, i.e. vi = (0, 0), and the others should have (1, 0) or (0, 0). Let’s denote j the DM with the lowest value of γ among DM’s who always vote no for M + . Then, the index j should be
24
greater than n − (r − 1). In order to be an equilibrium strategy, we should have E[˜ uj |M + , Sˆj (v−j ), s+ j ] < 0.
However, we know E[˜ un−(r−1) |M + , s+ uj |M + , s+ uj |M + , Sˆj (v−j ), s+ j ] ≤ E[˜ j ], n−(r−1) ] ≤ E[˜ and
E[˜ un−(r−1) |M + , s+ n−(r−1) ] ≥ 0, since p ≥ p¯(r; n). Therefore, vj+ = (0, 0) cannot be an equilibrium strategy for DM j. Contradiction.
Remark 1. 2.
∂ p¯(r;n) ∂n
1. > 0 if
3. p¯(r = n; n) =
∂ p¯(r;n) ∂r n+1 2 5 6
> 0.
< r < n.
and p¯(r =
n+1 2 ; n)
= 12 .
pˆ(r; n) 2 2 2 2 2 2 2 γn−(r−2) γn−(r−1) −4γn−r +1)−2γn−(r−1) γn−r +γn−r −γn−(r−1) γn−r +2γn−(r−1) γn−r γn−(r−2) (8γn−r = 2 2 2 2 2 2 2 2γn−(r−2) 4γn−(r−1) +1 γn−r +γn−(r−1) −2(γn−(r−1) +1)γn−(r−1) γn−r −2γn−(r−1) γn−r +4γn−(r−1) γn−r γn−(r−2)
Proposition 7 (Existence of Approval Equilibrium). Assume there are n decision makers. Consider r-majority rule where
n+1 2
≤ r ≤ n.
1. If p¯(r; n) ≤ p < pˆ(r; n), there exists a 2-element partition approval equilibrium. 2. Specifically, if p¯(r; n) ≤ p < pˆ(r; n), there is a 2-element partition equilibrium with {M + , M − }
+ + + + (a) for M + , v˜in−(r−2) = (1, 1)
and 25
− − − − (b) for M − , v˜i>r = (0, 0), v˜i=r = v˜i=r−1 = (0, 0) and v˜i
1 2,
n+1 2
< r ≤ n. For any value
there always exists a babbling equilibrium. Specifically, there is a babbling equilibrium
equilibrium with
− vi (s+ i , si ) =
(1, 1), if r < i < n − (r − 1) (1, 0),
.
o.w.
Moreover, this equilibrium grants expert the highest ex-ante (and interim) expected payoff among symmetric babbling equilibria.
26
3.4
Continuum of voters
In a babbling equilibrium, all voters take informative action. p portion of voters vote for it. So, if p ≥ r, always approved, and if p < r, not. Proposition 9. Let’s assume there are continuum of voters with γi ∈ [0, 1] where i ∈ [0, 1]. Consider the r-majority rule which requires at least r ∈ ( 21 , 1] portion of support to approve a proposal. Then, for any value of p ∈ ( 12 , 1], there is a 2-element symmetric partition equilibrium with (0, 0), γi ∈ (¯ γ (p), 1] vi+ = (1, 0), γi ∈ [γ(p), γ¯ (p)] , and (1, 1), γi ∈ [0, γ(p))
(0, 0), γi ∈ [0, γ(p)) vi− = (1, 0), γi ∈ [γ(p), γ¯ (p)] (1, 1), γi ∈ (¯ γ (p), 1]
where
γ(p) =
q 3 − 1 (3p−1) , p < 4 4 (1−p)
5 6
p≥
5 6
0,
, and
γ¯ (p) = 1 − γ.
Proportion of approval φ(p) is
φ(p) =
h q q i 1 (3 − 2p) − (3p−1) + 2p (3p−1) , p < 4 (1−p) (1−p)
5 6
p≥
5 6
p,
ue (p, r) =
1, φ(p) < r 0, φ(p) ≥ r
Remark 2. 1.
dE(ue ) dp
≤0
2.
dE(ue ) dr
≥0
3. E(ue ) > E(ue |Babbling) if φ(p) < r ≤ p
27
.
4. E(ue ) = E(ue |Babbling) if r ≤ φ(p) or r > p
28
4
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
ues =
=
n X
udi
i=1 θ1 Pn γi + θ2 Pn (1 − γi ) − n , i=1 i=1 2
if approved
0,
With three decision makers with γ1 = 1, γ2 =
ues =
=
1 2
if rejected and γ3 = 0, the utility function turns out to be
θ1 3 + θ2 3 − 3 , 2 2 2 0, 3 (θ1 + θ2 − 1), 2
0,
if approved if rejected if approved
,
if rejected
which is coincide with three times of DM 2’s utility function. For the surplus maximizing expert, we redefine the 2-element partition as follows. Definition 10. A partition M = {T + , T − } is the 2-element partition when
T + = {θ ∈ Θ|θ1 + θ2 > 1}
and T − = {θ ∈ Θ|θ1 + θ2 ≤ 1}
With this particular partition, we show the existence of PBE..
29
4.1
Unanimity
Proposition 10 (The Existence of a rejection equilibrium). Consider the unanimity rule, r = n. For any value of p < 56 , a PBE is a pair of strategy (σ, v) and a set of belief µ, such that 1. for each θ,
σ(θ) =
T + , θ ∈ T + T − , θ ∈ T −
2. for any i and s, vi (T + , s) = (1, 1) and vi (T − , s) = (0, 0) 3. for all Tˆ ∈ M and sˆ ∈ Si p , θ ∈ (Tˆ ∩ Disˆ) |Tˆ∩Disˆ| 1−p µi (θ|m = Tˆ, si = sˆ) = , θ ∈ (Tˆ/Disˆ) |Tˆ/Disˆ| 0, θ ∈ Θ/Tˆ
As with the highly biased expert, for all the other equilibria in this section, the expert’s sequentially rational messaging strategy and the consistent belief of a decision maker are identical to them in Remark above. From now on, hence, we identify any equilibrium only with decision makers’ sequentially rational voting strategy. Proposition 11 (The Existence of an approval equilibrium). Consider the unanimity rule. 1. If p ≥ 65 , there is a 2-element partition equilibrium with positive probability of approval. 2. Specifically, if p > 56 , there is a 2-element partition equilibrium with {T + , T − } + (a) for T + , v˜1+ = v˜n+ = (1, 0) and v˜1 14 17 − 1 and s = (s+ 1 , s2 , s3 ). − + − + − Without communication, V (s+ 1 , s1 , s2 , s2 , s3 , s3 )?
1. V = (1, 0, 1, 0, 1, 1) when p > 0.72 2. V = (0, 0, 1, 0, 1, 1) when p < 0.72 3. V = (0, 0, 1, 0, 1, 0) when p < 0.69 4. V = (0, 0, 1, 1, 1, 1) when p < 0.83 5. V = (0, 0, 0, 0, 1, 0) (never pivotal) 6. V = (0, 0, 0, 0, 0, 0) (never pivotal) s+ 1
s− 1
s+ 2
s− 2
s+ 3
s− 3
Existence
1
1
1
1
1
1
N/A
v2 (s− 2)=0
1
0
1
1
1
1
N/A
v2 (s− 2)=0
1
0
1
0
1
1
p > p¯
1
0
1
0
1
0
N/A
0
0
1
1
1
1
p < p˜(> p¯)
0
0
1
0
1
1
p < p¯
0
0
1
0
1
0
p < pˆ < p¯
0
0
0
0
1
0
p ∈ [ 12 , 1]
never pivotal
0
0
0
0
0
0
p ∈ [ 12 , 1]
never pivotal
Deviation incentive
v2 (s− 2)=1
Those strategy profiles are sequentially rational for DM, but for the third one, an expert’s cheap talk strategy is not sequentially rational because M + and M − is not indifferent over some θ. 32
A.1.2
Majority
+ − With communication? The proposal approved if s = (s+ 1 , s2 , s3 ) is equal to
1. s = (+, +, +) 2. s = (+, +, −) 3. s = (+, −, +) 4. s = (−, +, +) − + − + − Without communication, V (s+ 1 , s1 , s2 , s2 , s3 , s3 )?
1. V = (1, 1, 1, 1, 1, 1) (never pivotal) 2. V = (0, 0, 1, 0, 1, 1) 3. V = (0, 0, 0, 0, 0, 0) (never pivotal) s+ 1
s− 1
s+ 2
s− 2
s+ 3
s− 3
Existence
Deviation incentive
1
1
1
1
1
1
p ∈ [ 12 , 1]
never pivotal
1
0
1
1
1
1
N/A
v2 (s− 2)=0
1
0
1
0
1
1
N/A
v1 (s+ 1)=0
1
0
1
0
1
0
N/A
v1 (s+ 1)=0
0
0
1
1
1
1
N/A
v2 (s− 2)=0
0
0
1
0
1
1
p ∈ [ 12 , 1]
0
0
1
0
1
0
N/A
v3 (s− 3)=1
0
0
0
0
1
0
N/A
v2 (s+ 2)=1
0
0
0
0
0
0
p ∈ [ 12 , 1]
never pivotal
Strategy profile: V = (1, 0, 1, 0, 1, 0) Given this strategy profile, DM i is pivotal when (sj = h, sk = l) or (sj = l, sk = h) for i 6= j, i 6= k
33
and j 6= k.
E[u(v1 = 1)|M+ , s+ 1] − + + + − =P r(s+ 2 , s3 |M+ , s1 )E[u(v1 = 1)|M+ , s1 , s2 , s3 ] + + + − + + P r(s− 2 , s3 |M+ , s1 )E[u(v1 = 1)|M+ , s1 , s2 , s3 ]
=
2(p − 1)p 1 p−1 1 (2p2 + p − 2) + (2p − 3) ≥ 0 2p − 3 6 2p − 3 12
+ − + P r(s+ 1 , s2 , s3 , M ) P r(M + , s+ 1) P + + − P r(s , s , s |D+ )P r(Di+ ) = i P 1 2+ 3+ i + i P r(s1 |Di )P r(Di ) p−1 = 2p − 3
− + + P r(s+ 2 , s3 |M , s1 ) =
− + + P r(s+ 1 , s2 , s3 , M ) + P r(M + , s1 ) P P r(s+ , s− , s+ |D+ )P r(Di+ ) = i P 1 2+ 3+ i + i P r(s1 |Di )P r(Di ) 2(p − 1)p = 2p − 3
+ + + P r(s− 2 , s3 |M , s1 ) =
B
Appendix: Technical Appendix
N = {1, 2, ...., n} + For all i ∈ N , Bi+ := (Di+ ∩ M + ) \ (Di−1 ∩ M + ). + Bn+1 = M + \ (Dn+ ∩ M + )
s = (s1 , s2 , ...., sn )
34
P (s|θ ∈ Bi+ )P (θ ∈ Bi+ ) P (θ ∈ Bi+ |θ ∈ M + , s) = Pn+1 + + j=1 P (s|θ ∈ Bj )P (θ ∈ Bj ) −
+
=P
pni (1 − p)ni P (θ ∈ Bi+ ) n+1 n+ j j=1 p (1
−
− p)nj P (θ ∈ Bj+ )
where
+ − n+ i = ]{j ∈ N |(j ≥ i and sj = sj )or(j < i and sj = si )} − + n− i = ]{j ∈ N |(j ≥ i and sj = sj )or(j < i and sj = si )}.
Lemma 8. Consider unanimity rule. When p >
5 6,
with 2-element symmetric partition M =
{M + , M − }, there cannot be an equilibrium with more than two decision makers with vi = (1, 0). Proof of Lemma 8. We prove by contradiction. Suppose there is a equilibrium voting strategy v + with more than two decision makers with (1, 0). Denote three decision makers with (1, 0) i, j > i
35
and l > j, i.e. vi = vj = vl = (1, 0). Then for the decision maker j, + + − E[˜ ul |M + , Sˆl (v−l ), s− ul |M + , s+ i , sj , sl ] = l ] ≥ E[˜ 1 6γi γj (γj ((p−1)γi ((4γl −2)p+1)+γl p(1−2p))+γl (p−1)(2p−1)γi ) h 2 p(2p − 1)γ 2 − (p − 1)γ 2 8pγ 2 + 2p − 1 × γ j i j l i 2γ2 +2γ γ γ ((p − 1)γ (2p(γ + 1) − 1) + p(1 − 2p)γ ) + ((3 − 2p)p − 1)γ i j i j j l i j , if γl > 12 1 6γi γj ((p−1)γi (γl (4(p−1)γj −2p+1)−2pγj +2p+3γj −1)+(γl −1)p(2p−1)γj ) h × γj2 −(p − 1)γi2 (4γl (2γl − 3)(p − 1) + 6p − 5) + (γl − 1)p(2p − 1)(2γi − γl ) −2(γl − 1)(p − 1)(2p − 1)γi2 γj + (γl − 1)γl (p − 1)(2p − 1)γi2 , if γl < 21 and γj > 12 1 6γi (γj −1)(2p2 (γl (2γi γj −γi −γj +1)−3γi γj +2γi +γj −1)−p(γl (4γi γj −γi −γj +1)−7γi γj +4γi +γj −1)+γi (γl −γj )) × γl2 p 2p γi2 (−4(γj − 2)γj − 3) + (γj − 1)2 + γi2 (8(γj − 2)γj + 5) − (γj − 1)2 + γi2 , +γl 2p2 2γi2 (γj (3γj − 7) + 3) − 2γi (γj − 1)2 − (γj − 1)2 +γl p −6γi2 (γj − 2)(2γj − 1) + 2γi (γj − 1)2 + (γj − 1)2 − 2γi2 γj i 2 4(γ − 1)2 − 2γ ((γ − 4)γ + 2) + p γ ((γ − 8)γ + 4) − 2(γ − 1)2 + γ γ 2 +γ p i j i j j i j j j i j 1 1 , if γj < 2 and γi > 2 1 − 21 − 6(γl (p(p(4γi γj −6γi −2γj +4)+3γi −γj −2)−γi +1)+γ j (p(4−(2p+3)γi )+γi −1)+p(p(4γi −2)−1)) h 1−γ γ −1 1 1 2 2 l l × (γl − 1)(γi − 1)(γj − 1) 2p + + − 8 + + 4γ − l γl −1 1−γ γj −1 (γi −1)2 (γj −1)2 i i (γl −γj )(γl +2γj −3) 3(γl −1) γl −1 6 1 3 +(γ − 1)(γ − 1)(γ − 1) p − + + − + i j l 2 2 2 γl −1 γi −1 γj −1 (γi −1) (γj −1) (γl −1)(γj −1) 1 , if γi < 2
> 0.
So, vl should be (1, 1). Contradiction.
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western University, Center for Mathematical Studies in Economics and Management Science Discussion Papers 1359. Battaglini, Marco. 2002. “Multiple Referrals and Multidimensional Cheap Talk.” Econometrica, 70(4): 1379–1401. Board, Oliver, and Tiberiu Dragu. 2008. “Expert Advice with Multiple Decision Makers.” , (242). Caillaud, Bernard, and Jean Tirole. 2007. “Consensus Building: How to Persuade a Group.” American Economic Review, 97(5): 1877–1900. Chakraborty, Archishman, and Rick Harbaugh. 2010. “Persuasion by Cheap Talk.” American Economic Review, 100(5): 2361–82. Crawford, Vincent P., and Joel Sobel. 1982. “Strategic Information Transmission.” Econometrica, 50(6): pp. 1431–1451. de Barreda, In´ es Moreno. 2013. “Cheap Talk With Two-Sided Private Information.” Mimeo. Feddersen, Timothy, and Wolfgang Pesendorfer. 1997. “Voting Behavior and Information Aggregation in Elections with Private Information.” Econometrica, 65(5): 1029–1058. Guo, Yingni. 2013. “Information Transmission and Voting.” Mimeo. Ishida, Junichiro, and Takashi Shimizu. 2013. “Cheap Talk with an Informed Receiver.” Mimeo. Lai, Ernest K. 2013. “Expert Advice for Amateurs.” Mimeo. ´ Rivas, Javier, and Carmelo Rodr´ıguez-Alvarez. 2013. “Deliberation, Leadership and Information Aggregation.” Mimeo. Schnakenberg, Keith E. 2015. “Informational Lobbying and Legislative Voting.” mimeo. Yeung, Timothy Y.C. 2013. “A Cheap Talk with Multiple Free-riding Audiences. An Application to Environmental Lobbying.” Mimeo.
37
Abstract We investigate a cheap talk model with a collective decision making. In our model multiple decision makers vote on a proposal which determines their payoffs, and an expert tries to persuade them to choose the outcome she prefers. We allow decision makers to possess not all but some of information regarding the state of nature, which determines the gross utility of them based on the voting outcome. Two different types of experts has been considered: a heavily biased expert who always wants a rejection and a surplus maximizing expert who tries to maximize the total surplus of the group of decision makers. We show that experts can transmit credible and influential information to voters by using their respective optimal cheap talk strategies and try to prevent voters from taking informative actions. This limited information aggregation induced by each type of experts results in either polarization or unification of the voters: the highly biased expert polarizes the voters to achieve her aim, while the surplus maximizing expert unifies them. Furthermore, with the highly biased expert, polarization among voters has been observed under any voting rules, but voters tend to be more conservative under a voting rule that requires less votes in favor.
JEL Classification Numbers: Keywords: ∗
The Ohio State University, 410 Arps Hall, 1945 North High Street, Columbus, Ohio, [email protected].
1
Contents 1 Introduction 1.1
3
Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Model
8
3 Highly Biased Expert 3.1
3.2
4
12
Unanimity with n-Decision Makers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.1
Babbling Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2
2-element Partition Equilibrium
. . . . . . . . . . . . . . . . . . . . . . . . . 15
Majority with n-Decision Makers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.1
2-element Partition Equilibrium
. . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3
Existence of Rejection Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4
Continuum of voters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Surplus maximizing expert 4.1
29
Unanimity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Conclusion
31
A Appendix
32
A.1 HBE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.1.1 Unanimity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.1.2 Majority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 B Appendix: Technical Appendix
34
2
1
Introduction
In many economic situations, a group makes a collective decision, which determines the payoffs for all members. For example, the Supreme Court justices decide whether a bill is constitutional, and a board of directors or a group of policy makers have to decide whether to approve a proposal. Each decision maker may have different criteria for choices, bills or proposals, and this variety of criteria may cause some conflicts over their collective decision. 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 policy-lobbying group or an affiliated office of a private company does not usually have the right to decide even though they have more precise information on the proposals than the decision makers. This wellinformed 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. The primary question of the “cheap talk” literature is whether an expert with informational superiority could send credible messages, cheap talk, to a single decision maker, even when messages are unverifiable and there is no cost for lying. In my basic model, on the other hand, multiple decision makers vote on a proposal which determines their payoffs, and an expert tries to persuade them to choose the outcome she prefers. This feature of collective decision making in a cheap talk model allows me to explore several interesting research questions. First, this model investigates the expert’s optimal information transmission behavior, which may vary with the voting rule as well as the preference of the expert. I find that, under unanimity rule, the expert who always prefers rejecting the proposal can polarize the decision makers and increase the chances of rejection by sending a simple but credible message. I also show that the same messaging strategy may not help the expert with extreme preferences much under the simple majority rule. Different types of experts, such as an expert who tries to maximize the total welfare of the group of decision makers could also be considered. For example, in a jury trial, a prosecutor could be the welfare-maximizing expert, while a defense attorney is an expert with extreme preferences for acquittal. By comparing their different communication strategies that may
3
polarize or unify the opinions of the decision makers, I can provide economic implications for the persuasion of the prosecutor and defense attorney or of the lobbying behavior of different interest groups. Second, this study also explores whether information transmission helps voters make better decisions. Relevant policy implications can be derived from understanding the general setting I present with collective decision making that cannot be derived from the single decision maker case. Finally, I study the impact of the degree of the information dominance of the expert over the group of decision makers. The amount of information transmitted and the agents’ profits may depend on the gap between the amounts of information the expert and the group of decision makers initially have. Therefore, by investigating the impact of the informational gap between the agents, I am able to qualify the importance of the initial information distribution over economic agents when decisions are collective.
1.1
Related Literature
Even though most of cheap talk literature consider one-dimensional state variable, there are some studies with multiple dimensions. Battaglini (2002) studies multidimensional cheap talk with multiple senders and shows that there exists a full revelation equilibrium. Chakraborty and Harbaugh (2010) examines 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 equilibrium which influential cheap talk of very biased expert on decision making. Though they mainly focus on cheap talk models with a single sender, they also discuss, with a simple example, the possibility of the welfare loss of voters in a cheap talk equilibrium with a highly biased expert. They conclude that 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. Though the basic setting of two dimensional state of nature is the same as theirs, we consider a voting body of which each individual has a unique preference as well as two different types of experts, while they consider only a highly biased expert and two groups of voters who have identical preference within groups. Moreover, we allow voters to possess not all but some private information about the state of the world. With this feature, we can investigate the effect of cheap talk on information aggregation 4
among voters. 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) shows that under the unidimensional state of nature, voting with a large number of voters effectively aggregates information among voters 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 to voters under two dimensional state of nature. 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. This result provides a bridge between the study of information aggregation and polarization. Literature on polarization studies polarized opinion of people who observe the same information. For example, Andreoni and Mylovanov (2012) develop a theoretical model and conduct an experiment to investigate polarization of opinions and its persistence when people would be provided more information over time. They conclude that polarization of opinions could exist and persist over time when the dimensionality of uncertainty or information, as in our model, exceeds that of opinions. While each individual makes her own decision in their model, people make collective decision in our setting. In spite of this difference, our result with a highly biased expert is consistent with theirs. We show that polarization of voters’ opinions is the result of limited information aggregation among the voters, which induced by a highly biased expert’s strategic information transmission. In addition to that, we conclude that this polarization highly depends on the style or the types of additional information, by comparing the result of polarization with a highly biased expert to that of unification with a surplus maximizing expert. Several papers explore the possibility of persuasion of a biased informed agent on a voting body. Caillaud and Tirole (2007) and Schnakenberg (2015) explore an expert’s effort to persuade a voting body in the context of a mechanism design and a cheap talk, respectively. However, interestingly, they seem to merely show that an expert as a supporter of the proposal can increase the probability 5
of approval by transmitting information to existing allies among a voting body. In 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. There are several working, not yet published, papers which study a cheap talk model with an informed decision maker. Lai (2013) assumes that a decision maker knows whether the state is high or low relative to a private threshold. He shows that the expert’s message is more biased compare to the situation with uninformed receiver. Ishida and Shimizu (2013) investigate a cheap talk with incompletely informed agents. Unlike general cheap talk models, they 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 information transmitted. de Barreda (2013) assumes that a decision maker receives a signal which is affiliated with the state of nature. She shows the existence of CS type (partition) equilibrium and compare with basic uninformed receiver situation. Previous studies has also examined a cheap talk model with multiple decision makers. Board and Dragu (2008) 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 (2013) 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. Other studies examine a cheap talk model with voting. Austen-Smith and Feddersen (2002) explore the effect of deliberation prior to a collective decision making via voting. The deliberation is a type of cheap talk among decision makers who have private information of the state of nature. They compare majority and unanimity rules, and conclude that the former induces more information transmission and fewer errors in decision making than the latter. Guo (2013) analyzes an informed voter’s incentive of information transmission. In her model, there are three voters: a left biased, a right biased, and an unbiased voter. She conclude that a biased voter has more incentive to transmit her private information to the others than an unbiased voter. Rivas and Rodr´ıguez´ Alvarez (2013) examine the impact of a leadership in a deliberation among decision makers before 6
a voting. They conclude that a very influential leader affect more 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. 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.
7
2
Model
There are an expert (female) and an odd number of decision makers (male) i ∈ N = {1, 2, ..., n} where n ≥ 3. Decision makers (DMs) vote on a given proposal. We rename the voting rules from unanimity to simple majority rule as follows. Definition 1 (r-Majority Rule). r-Majority Rule requires at least r/n of support to approve a proposal. When r = n, l-Majority Rule is a unanimity rule, and when r =
n+1 2 ,
it is a simple
majority rule. The space of the state of nature is two-dimensional, θ = (θ1 , θ2 ) ∈ Θ ≡ [0, 1] × [0, 1], the aspect 1 and 2, respectively. Each dimension represents a different aspect of the state of nature. A DM i’s preference conditional on approval is characterized by the weighted average of the state of nature.
udi =
γi θ1 + (1 − γi )θ2 − 1 , 2
0,
if approved if rejected
γi captures the intensity of DM i’s personal interest (or concern) on the aspect 1. γi = 1 means DM i only cares about the aspect 1 (the first dimension), and γi = 0 means DM i only cares about the aspect 2 (the second dimension). A decision maker i has payoff γi θ1 + (1 − γi )θ2 −
1 2
from approval and payoff 0 from rejection.1 To economize on notation, without loss of interesting generality, we assume γi > γj if and only if i > j. We assume that a decision maker’s preference parameter γi is publicly informed. 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, si ∈ Si = {h, l}, which may contains partial information about θ. For all θ 1 P si = h|γi θ1 + (1 − γi )θ2 − > 0 =p ∈ 12 , 1 2 1 P si = l|γi θ1 + (1 − γi )θ2 − > 0 =1 − p 2 p, which is commonly known, represents an accuracy of DM’s private information. To economize 1
Chakraborty and Harbaugh (2010)
8
on notation, we define
Dih ≡ {θ ∈ Θ|γi θ1 + (1 − γi )θ2 −
1 2
> 0}
and
Dil ≡ {θ ∈ Θ|γi θ1 + (1 − γi )θ2 −
1 2
≤ 0} = Θ/Dih .
By Bayes’ rule, we get
P θ∈
Dih |si
P θ ∈ Dih P si = h|θ ∈ Dih =h = P (si = h) P θ ∈ Dih P si = h|θ ∈ Dih =P t t t∈Si P (θ ∈ Di ) P (si = h|θ ∈ Di )
=1
2p
+
1 2p 1 2 (1
− p)
=p
P θ ∈ Dil |si = h =1 − p. Similarly, P θ ∈ Dil |si = l = p and P θ ∈ Dih |si = l = 1 − p. Therefore, if p = 12 , the signal does not convey any information, but if p > 12 , it conveys some information. The game proceeds as follows: 1. Nature draws the sate θ ∼ U [0, 1] × [0, 1]. 2. 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 si ∈ Si = {h, l}. 5. Each decision maker votes on the proposal, vi ∈ {0, 1}. 6. Under r-Majority rule, agents receive payoffs as follows. P • udi (vi |v−i , γi , θ) = 1{ ni vi ≥ r}(γi θ1 + (1 − γi )θ2 − 12 ) • ue (v|θ) where v = (vi )ni=1 = (v1 , v2 , ....vn ) 9
Different types of experts would have different functional forms of payoff function ue (v|θ). In the basic model, we assume a heavily biased expert who always prefer the rejection. Definition 2 (Highly biased expert). A highly biased expert is an expert whose objective is to get rejection regardless of the state of nature. The utility function of the highly biased expert is ue (v|θ) = 1 − 1{
n X
vi ≥ r}.
i
Thus, the expert’s preference does not depend on the state of nature. The solution concept employed in this paper is perfect Bayesian equilibrium. Definition 3 (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)], Z
∗
ue (v(m, s), θ)f (s|θ)ds
m ∈ arg max m
where s = (si )ni=1 and S =
Qn i
S
Si .
2. for each m and si Z
∗
vi (m , si ) ∈ arg max vi
udi (vi |θ, γi )µi (θ|m∗ , si )dθ
Θ
3. for all m ∈ M and all si ∈ Si
µi (θ|m, si ) =
R
Θ
σ(m|θ)φi (θ|si ) , σ(m|θ0 )φi (θ0 |si )dθ0
ψi (θ|si ),
if Θσ (m) 6= ∅ if Θσ (m) = ∅
where ψi (θ|si ) is any distribution. Assumption 1 (Tie breaking rule). If DM is indifferent between approval and rejection of the proposal, he is more likely to vote for it. 10
To simplify the analysis, we occasionally assume “symmetry” in a group of decision makers. Assumption 2 (Symmetric decision makers). In the group of decision makers, the decision makers are symmetric around
n+1 2
≡ r-th decision maker, and there are always extreme types γn = 1 and
γ1 = 0. More precisely, γk = 21 , γn = 1, γ1 = 0, and γn−j+1 = 1 − γj for any j such that k > j > 1. 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 4 (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, µi (θ|m, s) ≥ φi (θ|s) for all i with at least one strict inequality. Definition 5 (Influential equilibrium). An equilibrium is influential if there exists different messages m and m0 where m 6= m0 induce different actions of decision makers, vi (m, s) 6= vi (m0 , s) for some i and s. Definition 6 (Influential in outcomes). An equilibrium is influential in outcomes if there exists different messages m and m0 where m 6= m0 induce different outcomes of decision makers, i.e., P P 6 1{ ni vi (m0 , s) ≥ r} for some i and s. 1{ ni vi (m, s) ≥ r} = 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 order to sharpen predictions and results, we only focus on non-trivial equilibrium in which there exists an individual decision maker with positive probability of being pivotal at some information set.2 Definition 7 (Non-trivial equilibrium). An equilibrium is non-trivial if the probability of being pivotal for some decision maker i is positive at some information set. In order to economize the notion, we may abuse the term “equilibrium” for “non-trivial equilibrium”. So, in our analysis, if there is no other comment, an “equilibrium” means a “non-trivial equilibrium.” 2
Note that we do not require the existence of pivotal decision maker in all information set.
11
3
Highly Biased Expert
To economize the notion,
1 u ˜i ≡ γi θ1 + (1 − γi )θ2 − . 2 With any message M k , strategy profile v k and signal si , k k E(ui (vi (si ) = 1)|M k , v−i , si ) − E(ui (vi (si ) = 0)|M k , v−i , si ) k k =P r(Sˆi (v−i )|M k , si )E(˜ ui |M k , Sˆi (v−i ), si ),
k )⊆× k where Sˆi (v−i j6=i Sj is the set of events where DM i is pivotal under vi . So, if
k E(˜ ui |M k , Sˆi (v−i ), si ) > 0,
then v k (si ) = 1 is a strictly (weakly, with equality) dominant strategy.
3.1
Unanimity with n-Decision Makers
Proposition 1. [A property of a non-trivial partition equilibrium] Consider the unanimity rule. If there exists a non-trivial equilibrium with a partition {M k }K k=1 and a corresponding strategy profile {v k }K k=1 , then for any decision maker i who is pivotal with positive probability given a message M k and the corresponding strategy profile v k , vik (shi ) ≥ vik (sli ). Proof of Lemma 1. Since DM i is pivotal with positive probability under a message M k and the corresponding strategy k ) 6= ∅. So, it is clear that profile v k , Sˆi (v−i
k k E[˜ ui |M k , Sˆi (v−i ), shi ] ≥ E[˜ ui |M k , Sˆi (v−i ), sli ],
which proves the lemma. Corollary 1.1. Consider unanimity, r = n, if there exists a non-trivial equilibrium {(M k , v k )}K k=1 12
with a positive probability of approval, then for any i and k, vik (shi ) = 1 ≥ vik (sli ). 3.1.1
Babbling Equilibrium
Proposition 2. [The Existence of Babbling Equilibrium] Consider unanimity, r = n. For any value of p >
1 2,
there always exists a babbling equilibrium.
Specifically, there is a pivotal babbling equilibrium equilibrium with
− vi (s+ i , si ) =
(1, 0),
if i = 1, n
.
(1, 1), if 1 < i < n Moreover, this equilibrium grants expert the highest ex-ante (and interim) 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. Since these two DMs’ 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 pair of DMs. Therefore, from the expert’s point of view, this equilibrium would be optimal. Lemma 1. Consider unanimity r = n. In a babbling equilibrium with v, for any i, vih = 1. Proof of Lemma 1. For any v−i which makes DM i pivotal, E[˜ ui |Sˆi (v−i ), shi ] ≥ E[˜ ui |shi ] > 0
Lemma 2. Consider unanimity, r = n. There always exist a babbling equilibrium with v = (1, 0, 1, 1, ...1, 1, 1, 0). Proof of Lemma 2.
13
From Lemma 1, we know vih = 1 for any i. For any s1 and sn E[˜ u1 |Sˆ1 (v−1 ), s1 ] = E[˜ u1 |s1 , sn ] = E[˜ u1 |s1 ],
and E[˜ un |Sˆn (v−n ), sn ] = E[˜ un |s1 , sn ] = E[˜ un |sn ].
So,
E[˜ u1 |sn , sl1 ] = E[˜ u1 |sl1 ] < 0,
and
E[˜ un |s1 , sln ] = E[˜ un |sln ] < 0. For any i such that 1 < i ≤ 21 , E[˜ ui |Sˆi (v−i ), sli ] = E[˜ ui |sh1 , shn , sli ] =
Similarly, for any i such that
1 2
(1 − γi )(1 − 7γi )(2p − 1) >0 6(1 − 2(1 − γi )p − 5γi )
< i < n, we can show that
E[˜ ui |Sˆi (v−i ), sli ] = E[˜ ui |sh1 , shn , sli ] > 0.
Lemma 3. Consider unanimity, r = n. There is no symmetric babbling equilibrium with more than 2 decision makers with vi = (1, 0). Proof. Suppose there exists a symmetric equilibrium with vˆi = vˆn−i+1 = (1, 0) and vˆj = vˆn−j+1 = (1, 0), which is different from the voting strategy from Lemma 2, v = (1, 0, 1, 1, ...1, 1, 1, 0). By
14
construction, E[˜ un−i+1 |Sˆn−i+1 (ˆ v−(n−i+1) ), sln−i+1 ] > E[˜ un−i+1 |Sˆn−i+1 (v−(n−i+1) ), sln−i+1 ],
and from Lemma 2, we know E[˜ un−i+1 |Sˆn−i+1 (ˆ v−(n−i+1) ), sln−i+1 ] > E[˜ un−i+1 |Sˆn−i+1 (v−(n−i+1) ), sln−i+1 ] > 0.
Therefore, vˆn−i+1 should be (1, 1). So, contradiction. Lemma 4. Consider unanimity, r = n. If there exists a symmetric pivotal babbling equilibrium with vi>1 = vn−i+1 = (1, 0) and vj6=i = (1, 1), then the expert’s payoff is less than 34 . Proof.
3.1.2
2-element Partition Equilibrium
As in pure persuasion game, the highly biased expert’s preference is sate-independent. Then the rationale behind the partition equilibrium in Crawford and Sobel (1982), which makes the cheap talk credible, does not work in our situation. Instead, as in Chakraborty and Harbaugh (2010) where also have mutidimensional information structure, we argue that, the 2-element symmetric partition cheap talk could be credible even if the expert is highly biased, since it has a intrinsic trade-off for the expert. Definition 8. A partition M = {M + , M − }, is a 2-element symmetric partition when
M + = {θ ∈ Θ|θ1 < θ2 }
and M − = {θ ∈ Θ|θ1 ≥ θ2 }.
A message in the 2-element symmetric partition cheap tale says the intensity of a state of nature in one dimension is (relatively) higher than in the other dimension. So, by sending that, the expert
15
sacrifice the informational advantage in one dimension for the greater advantage in the other. This trade-off establishes the credibility of the cheap talk strategy. This argument also provides the intuition behind the impossibility of information transmission in one dimension. If there is only one dimension, unlike in the multidimension case, any generic partition strategy would not be credible. Since one element in a partition generally carries (weakly) higher expected values of the state of nature than the other, it would never be credible. We now argue that the optimality of the 2-element partition equilibrium for the expert. Lemma 5. For uninformed decision makers, p =
1 2,
the 2-element symmetric partition is the
optimal messaging strategy for the expert. The interim expected payoff of the expert depends on the interim probability of being approved. Under the unanimity,3
(3.1)
P r(approved|θ, Mk , vk ) i Yh = P r(shi |θ)vik (shi ) + P r(sli |θ)vik (sli ) i
i Yh k k = P r(shi |θ)1(E[˜ ui |M k , Sˆi (v−i ), shi ] > 0) + P r(sli |θ)1(E[˜ ui |M k , Sˆi (v−i ), sli ] > 0) i
≡
Y
qi (θ, Mk , vk ).
i
Notice that the expressions in the indicator functions are independent on the state of nature θ. On the other hand, all the terms of conditional probabilities, P r(shi |θ) and P r(sli |θ), do depend on θ. In addition to that, the former depends on Mk and vk,−i , but the later does not. In any equilibrium with {(Mk , vk )}K k=1 , Equation (3.1) should be constant over Mk for any given θ ∈ Θ. Lemma 6. Consider the unanimity rule. If there exists a rejection equilibrium with a (weakly) finer partition + − k K than 2-element partition, {M k }K k=1 ≤ {M , M }, and the corresponding strategy profile {v }k=1 , + − then there exists a rejection equilibrium with {M k }K v k }K k=1 ≤ {M , M } and {˜ k=1 such that 3 Without loss of interesting generality, we ignore the case in which vik (si ) = 1 when E[udi (θ)|Mk , vk , si ] > 0, since we focus on the expert optimal equilibrium.
16
1. for any k with M k ⊂ M + , v˜1k = (0, 0) and v˜i6k=1 = (1, 1) and 2. for any k with M k ⊂ M − , v˜nk = (0, 0) and v˜i6k=n = (1, 1). Proof of Lemma 6. Since it is a rejection equilibrium, vjk should be (0, 0) for some j. So, k E[˜ uj |M k , shj ] ≤E[˜ uj |M k , Sˆj (v−j ), shj ] ≤ 0.
Consider any pair of signal (shi , shj ). For DM i and j such that i < j, we know E[˜ ui |M k , shi ] × p(M k , shi ) =p(Di+
∩M
k
)p(shi |Di+
k
Z
∩M ) + k ˆ θ∈D i ∩M
u ˜i dθˆ
Z + + + + k h k + p( Dj \ Di ∩ M )p(si | Dj \ Di ∩ M )
+ + k ˆ θ∈(D j \Di )∩M
+ p(M k \ Dj+ )p(shi |M k \ Dj+ )
Z k \D + ˆ θ∈M j
u ˜i dθˆ
u ˜i dθˆ
and
Eθˆ[˜ uj |M k , shj ] × p(M k , shj ) =p(Di+ ∩ M k )p(shj |Di+ ∩ M k )
Z + k ˆ θ∈D i ∩M
u ˜j dθˆ
Z + p( Dj+ \ Di+ ∩ M k )p(shj | Dj+ \ Di+ ∩ M k )
+ + k ˆ θ∈(D j \Di )∩M
k
+ p(M \
Dj+ )p(shj |M k
\
Dj+ )
Z k \D + ˆ θ∈M j
u ˜j dθˆ
ˆ u ˜j dθ.
The right hand sides of both equations have three terms. For any M k ⊂ M + , each term, from the first to the third, is no greater in the former equation than in the later equation. Therefore,
E[˜ uj |M k , shj ] ≤ 0
17
implies k E[˜ ui |M k , Sˆi (˜ v−i ), shi ] = E[˜ ui |M k , shi ] ≤ 0.
So, for any M k ⊂ M + , there could be an equilibrium with v˜k such that v˜1k = (0, 0) and v˜ik = (1, 1) for any i > 1 is an equilibrium strategy profile. Similarly, we can show that, for any M k ⊂ M − , there could be an equilibrium with v˜k such that v˜nk = (0, 0) and v˜ik = (1, 1) for any i < n is an equilibrium strategy profile. Proposition 3 (The Existence of rejection equilibrium). Consider the unanimity rule. 1. If p ≤
5 6,
there exists a rejection equilibrium with 2-element partition {M + , M − } which is
optimal for the highly biased expert. 2. Specifically, if p ≤ 65 , there exist a equilibrium with a partition {M + , M − } and a corresponding strategy {v + , v − } such that • v˜1+ = (0, 0) and v˜i6+=1 = (1, 1) and • v˜n− = (0, 0) and v˜i6−=n = (1, 1). 3. If p > 56 , there is no rejection equilibrium with 2-element partition {M + , M − }. Proof of Proposition 3. To prove this proposition, we use Lemma 6. Check, for some p, if k E[˜ ui |M k , Sˆi (˜ v−1 ), sh1 ] =
6p − 5 ≤ 0. 6(3 − 2p)
So, there exists a rejection pivotal equilibrium with 2-element partition if and only if p ≤ 56 . Proposition 4 (The Existence of an approval equilibrium). Consider the unanimity rule. 1. If p > 56 , there is a 2-element partition equilibrium with positive probability of approval. 18
2. Specifically, if p > 56 , there is a 2-element partition equilibrium with {M + , M − } + (a) for M + , v˜1+ = v˜2+ = (1, 0) and v˜i>2 = (1, 1) and − − (b) for M − , v˜n− = v˜n−1 = (1, 0) and v˜i γ3 ≥
1 2
+ − k E[˜ u3 |M + , Sˆi (˜ v−3 ), s− u3 |M + , s+ 3 ] = E[˜ 1 , s2 , s3 ]
=
γ32 ((3 − 2p)p − 1) − γ22 (p(γ3 (γ3 (6p − 7) + 2) + 2p − 3) + 1) + 2γ3 (p − 1)(2p − 1)γ2 ≥ 0. 6γ2 (γ3 (p(2p − 3)(γ2 + 1) + 1) + ((3 − 2p)p − 1)γ2 )
So, there cannot be more than 3 zeros in an equilibrium strategy v. Moreover, any other 2-element symmetric partition equilibrium with two zeros in v is worse for the expert than the
19
equilibrium characterized in Proposition 4. Hence, there is no 2-element symmetric partition equilibrium with other voting strategy that grants the expert a higher payoff than with v˜. Dominance over the finer partition equilibrium. First of all, there cannot be any rejection equilibrium with a finer partition. (∵ if there exists a rejection equilibrium with a finer partition, from Lemma 6, there should be a rejection equilibrium with 2-element partition.) Second, for any partition equilibrium with positive probability of approval, M k ∩ D1 6= ∅, and there should be same number of zeros. (If not for any θ ∈ D1 , one is better than the other, so the message is not credible.) Now, suppose there is any other pivotal equilibrium with a finer partition and a voting strategy v 0 . If v 0 = v˜, then the payoff is the same with the characterized equilibrium. Suppose v 0 6= v˜. If there is one zero, then the payoff cannot be higher than the 2-element partition equilibrium. (The best equilibrium with one zero is v = (1, 0) which gives the payoff less than the 2-element partition equilibrium.) Similar for the case with two zeros. For the last step, we show that there cannot be more than two voters with vik = (1, 0) in v k + − for any element M k in a finer partition {M k }K k=1 ≤ {M , M }. To have more than two zeros, we
should have k E[˜ ul |M k , Sˆl (v−l ), s− l ] 56 , the expert optimal 2-element partition equilibrium is better for the expert than the optimal symmetric babbling equilibrium.
20
Proof. In the equilibrium specified in Proposition 4, 1 3 1 2 p + (1 − p) 3(1 − p) − (1 − 5p) ≥ . E[u (v|θ)] = 1 − 4 γ2 4 e
Proposition 5 (Optimality of the 2-element symmetric partition equilibrium). For any p ∈ [ 12 , 1], the 2-element symmetric partition equilibrium gives the expert weakly higher payoff than any other pivotal equilibrium with a finer or coarser partition. This means that the more precise information transmission compare to the 2-element partition equilibrium does not help the expert. First, the finer partition, keeping the order of toughness of DMs, just provides more precise information for them. So, there cannot be a further advantage from the schemed (or contrived) ordering, but could be a loss from excessive information transmission. Second, if the element in any finer partition could be ordered in the expected values of them, it would not be credible since the expert could be better off with one signal than with another. Therefore, only equilibrium with finer partition which is credible should have the same or similar level of expected values. This expected value (or these expected values) would not be so different with that in an element with 2-element symmetric partition. So, the expert cannot be better off with a finer partition. In a 2-element partition equilibrium, the DMs would be ordered in toughness. And in the expert optimal equilibrium, they delegate their decision to the voter (or voters) with toughest criterion (or criteria).
21
3.2 3.2.1
Majority with n-Decision Makers 2-element Partition Equilibrium
Proposition 6 (The existence of 2-element partition equilibrium). Consider the simple majority rule, r =
n+1 2 .
For any n and p,
1. there is a 2-element symmetric partition equilibrium. 2. Specifically, there is a 2-element partition equilibrium with {M + , M − } + + + (a) for M + , vi>r = (1, 1), vir = (0, 0), vi γr = 21 . Moreover, + ˆ Eθˆ[ui (v = 1|θ)|M , sr = l, si = h] < E[u(v = 1)|M + , sr = l, si = h].
Therefore, any i < r dose not have incentive to deviate. For any i > r, i is pivotal only when sl = h. 1 + ˆ Eθˆ[ui (v = 1|θ)|M , sr = h, si = l] = (1 − 2γi ) > 0 6 , since γi < γr = 21 . Moreover, + ˆ Eθˆ[ui (v = 1|θ)|M , sr = h, si = h] > E[u(v = 1)|M + , sr = h, si = l].
Therefore, any i > r dose not have incentive to deviate.
22
Voter i = r is always pivotal, and it is easy to show that he has no incentive to deviate for any p > 21 . 2p − 1 + ˆ ≥ 0. Eθˆ[ul (v = 1|θ)|M , sr = h] = 6
+ ˆ Eθˆ[ul (v = 1|θ)|M , sr = l] =
1 − 2p ≤ 0. 6
Part 3. One zero other than 1/2? Lower than 1/2 implies lower payoff Higher than 1/2. to have higher payoff, there should be at least (n − i) − (r − 1) (0, 0) Impossible, since from r, it should be (1, 0) or (1, 1). maximum (n − i) − r (0, 0) Two zeros? If both γi > γj > 21 , g 2 p 2 − 8r2 − 1 + 2gr(2p(r − 1) + 1) + (2p − 1)r2 21 , g 2 p 2 − 8r2 − 1 + 2gr(2p(r − 1) + 1) + (2p − 1)r2 n−(r−1) = (1, 1) and − − (b) for M − , v˜i≥r = (0, 0) and v˜in−(r−1) = (1, 1). For any i ≤ n − (r − 1), who is pivotal,
we need to show E[˜ ui |M + , Sˆi (˜ v−i ), s+ i ] < 0, which also implies E[˜ ui |Sˆi (˜ v−i ), s− i ] < 0. We know E[˜ ui |M + , Sˆi (˜ v−i ), s+ ui |M + , s+ i ] = E[˜ i ]=
4γi (−2γi (p − 1) + p − 1) − 2p + 1 , 6(4γi (p − 1) − 2p + 1)
and this is decreasing in γi , in other words, increasing in i. So it is enough to show that E[˜ un−(r−1) |M + , Sˆn−(r−1) (˜ v−(n−(r−1)) ), s+ n−(r−1) ] < 0, which is guaranteed by the condition p < p¯(r; n). For M − , it is symmetric, so we prove the existence. Now, we want to show there is no rejection equilibrium if p ≥ p¯(r; n). We prove by contradiction. Suppose p ≥ p¯(r; n). Assume there is a rejection equilibrium with a voting strategy v + and v − . To have any rejection pivotal equilibrium with 2-element partition, there should be n − r voters who always vote no, i.e. vi = (0, 0), and the others should have (1, 0) or (0, 0). Let’s denote j the DM with the lowest value of γ among DM’s who always vote no for M + . Then, the index j should be
24
greater than n − (r − 1). In order to be an equilibrium strategy, we should have E[˜ uj |M + , Sˆj (v−j ), s+ j ] < 0.
However, we know E[˜ un−(r−1) |M + , s+ uj |M + , s+ uj |M + , Sˆj (v−j ), s+ j ] ≤ E[˜ j ], n−(r−1) ] ≤ E[˜ and
E[˜ un−(r−1) |M + , s+ n−(r−1) ] ≥ 0, since p ≥ p¯(r; n). Therefore, vj+ = (0, 0) cannot be an equilibrium strategy for DM j. Contradiction.
Remark 1. 2.
∂ p¯(r;n) ∂n
1. > 0 if
3. p¯(r = n; n) =
∂ p¯(r;n) ∂r n+1 2 5 6
> 0.
< r < n.
and p¯(r =
n+1 2 ; n)
= 12 .
pˆ(r; n) 2 2 2 2 2 2 2 γn−(r−2) γn−(r−1) −4γn−r +1)−2γn−(r−1) γn−r +γn−r −γn−(r−1) γn−r +2γn−(r−1) γn−r γn−(r−2) (8γn−r = 2 2 2 2 2 2 2 2γn−(r−2) 4γn−(r−1) +1 γn−r +γn−(r−1) −2(γn−(r−1) +1)γn−(r−1) γn−r −2γn−(r−1) γn−r +4γn−(r−1) γn−r γn−(r−2)
Proposition 7 (Existence of Approval Equilibrium). Assume there are n decision makers. Consider r-majority rule where
n+1 2
≤ r ≤ n.
1. If p¯(r; n) ≤ p < pˆ(r; n), there exists a 2-element partition approval equilibrium. 2. Specifically, if p¯(r; n) ≤ p < pˆ(r; n), there is a 2-element partition equilibrium with {M + , M − }
+ + + + (a) for M + , v˜in−(r−2) = (1, 1)
and 25
− − − − (b) for M − , v˜i>r = (0, 0), v˜i=r = v˜i=r−1 = (0, 0) and v˜i
1 2,
n+1 2
< r ≤ n. For any value
there always exists a babbling equilibrium. Specifically, there is a babbling equilibrium
equilibrium with
− vi (s+ i , si ) =
(1, 1), if r < i < n − (r − 1) (1, 0),
.
o.w.
Moreover, this equilibrium grants expert the highest ex-ante (and interim) expected payoff among symmetric babbling equilibria.
26
3.4
Continuum of voters
In a babbling equilibrium, all voters take informative action. p portion of voters vote for it. So, if p ≥ r, always approved, and if p < r, not. Proposition 9. Let’s assume there are continuum of voters with γi ∈ [0, 1] where i ∈ [0, 1]. Consider the r-majority rule which requires at least r ∈ ( 21 , 1] portion of support to approve a proposal. Then, for any value of p ∈ ( 12 , 1], there is a 2-element symmetric partition equilibrium with (0, 0), γi ∈ (¯ γ (p), 1] vi+ = (1, 0), γi ∈ [γ(p), γ¯ (p)] , and (1, 1), γi ∈ [0, γ(p))
(0, 0), γi ∈ [0, γ(p)) vi− = (1, 0), γi ∈ [γ(p), γ¯ (p)] (1, 1), γi ∈ (¯ γ (p), 1]
where
γ(p) =
q 3 − 1 (3p−1) , p < 4 4 (1−p)
5 6
p≥
5 6
0,
, and
γ¯ (p) = 1 − γ.
Proportion of approval φ(p) is
φ(p) =
h q q i 1 (3 − 2p) − (3p−1) + 2p (3p−1) , p < 4 (1−p) (1−p)
5 6
p≥
5 6
p,
ue (p, r) =
1, φ(p) < r 0, φ(p) ≥ r
Remark 2. 1.
dE(ue ) dp
≤0
2.
dE(ue ) dr
≥0
3. E(ue ) > E(ue |Babbling) if φ(p) < r ≤ p
27
.
4. E(ue ) = E(ue |Babbling) if r ≤ φ(p) or r > p
28
4
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
ues =
=
n X
udi
i=1 θ1 Pn γi + θ2 Pn (1 − γi ) − n , i=1 i=1 2
if approved
0,
With three decision makers with γ1 = 1, γ2 =
ues =
=
1 2
if rejected and γ3 = 0, the utility function turns out to be
θ1 3 + θ2 3 − 3 , 2 2 2 0, 3 (θ1 + θ2 − 1), 2
0,
if approved if rejected if approved
,
if rejected
which is coincide with three times of DM 2’s utility function. For the surplus maximizing expert, we redefine the 2-element partition as follows. Definition 10. A partition M = {T + , T − } is the 2-element partition when
T + = {θ ∈ Θ|θ1 + θ2 > 1}
and T − = {θ ∈ Θ|θ1 + θ2 ≤ 1}
With this particular partition, we show the existence of PBE..
29
4.1
Unanimity
Proposition 10 (The Existence of a rejection equilibrium). Consider the unanimity rule, r = n. For any value of p < 56 , a PBE is a pair of strategy (σ, v) and a set of belief µ, such that 1. for each θ,
σ(θ) =
T + , θ ∈ T + T − , θ ∈ T −
2. for any i and s, vi (T + , s) = (1, 1) and vi (T − , s) = (0, 0) 3. for all Tˆ ∈ M and sˆ ∈ Si p , θ ∈ (Tˆ ∩ Disˆ) |Tˆ∩Disˆ| 1−p µi (θ|m = Tˆ, si = sˆ) = , θ ∈ (Tˆ/Disˆ) |Tˆ/Disˆ| 0, θ ∈ Θ/Tˆ
As with the highly biased expert, for all the other equilibria in this section, the expert’s sequentially rational messaging strategy and the consistent belief of a decision maker are identical to them in Remark above. From now on, hence, we identify any equilibrium only with decision makers’ sequentially rational voting strategy. Proposition 11 (The Existence of an approval equilibrium). Consider the unanimity rule. 1. If p ≥ 65 , there is a 2-element partition equilibrium with positive probability of approval. 2. Specifically, if p > 56 , there is a 2-element partition equilibrium with {T + , T − } + (a) for T + , v˜1+ = v˜n+ = (1, 0) and v˜1 14 17 − 1 and s = (s+ 1 , s2 , s3 ). − + − + − Without communication, V (s+ 1 , s1 , s2 , s2 , s3 , s3 )?
1. V = (1, 0, 1, 0, 1, 1) when p > 0.72 2. V = (0, 0, 1, 0, 1, 1) when p < 0.72 3. V = (0, 0, 1, 0, 1, 0) when p < 0.69 4. V = (0, 0, 1, 1, 1, 1) when p < 0.83 5. V = (0, 0, 0, 0, 1, 0) (never pivotal) 6. V = (0, 0, 0, 0, 0, 0) (never pivotal) s+ 1
s− 1
s+ 2
s− 2
s+ 3
s− 3
Existence
1
1
1
1
1
1
N/A
v2 (s− 2)=0
1
0
1
1
1
1
N/A
v2 (s− 2)=0
1
0
1
0
1
1
p > p¯
1
0
1
0
1
0
N/A
0
0
1
1
1
1
p < p˜(> p¯)
0
0
1
0
1
1
p < p¯
0
0
1
0
1
0
p < pˆ < p¯
0
0
0
0
1
0
p ∈ [ 12 , 1]
never pivotal
0
0
0
0
0
0
p ∈ [ 12 , 1]
never pivotal
Deviation incentive
v2 (s− 2)=1
Those strategy profiles are sequentially rational for DM, but for the third one, an expert’s cheap talk strategy is not sequentially rational because M + and M − is not indifferent over some θ. 32
A.1.2
Majority
+ − With communication? The proposal approved if s = (s+ 1 , s2 , s3 ) is equal to
1. s = (+, +, +) 2. s = (+, +, −) 3. s = (+, −, +) 4. s = (−, +, +) − + − + − Without communication, V (s+ 1 , s1 , s2 , s2 , s3 , s3 )?
1. V = (1, 1, 1, 1, 1, 1) (never pivotal) 2. V = (0, 0, 1, 0, 1, 1) 3. V = (0, 0, 0, 0, 0, 0) (never pivotal) s+ 1
s− 1
s+ 2
s− 2
s+ 3
s− 3
Existence
Deviation incentive
1
1
1
1
1
1
p ∈ [ 12 , 1]
never pivotal
1
0
1
1
1
1
N/A
v2 (s− 2)=0
1
0
1
0
1
1
N/A
v1 (s+ 1)=0
1
0
1
0
1
0
N/A
v1 (s+ 1)=0
0
0
1
1
1
1
N/A
v2 (s− 2)=0
0
0
1
0
1
1
p ∈ [ 12 , 1]
0
0
1
0
1
0
N/A
v3 (s− 3)=1
0
0
0
0
1
0
N/A
v2 (s+ 2)=1
0
0
0
0
0
0
p ∈ [ 12 , 1]
never pivotal
Strategy profile: V = (1, 0, 1, 0, 1, 0) Given this strategy profile, DM i is pivotal when (sj = h, sk = l) or (sj = l, sk = h) for i 6= j, i 6= k
33
and j 6= k.
E[u(v1 = 1)|M+ , s+ 1] − + + + − =P r(s+ 2 , s3 |M+ , s1 )E[u(v1 = 1)|M+ , s1 , s2 , s3 ] + + + − + + P r(s− 2 , s3 |M+ , s1 )E[u(v1 = 1)|M+ , s1 , s2 , s3 ]
=
2(p − 1)p 1 p−1 1 (2p2 + p − 2) + (2p − 3) ≥ 0 2p − 3 6 2p − 3 12
+ − + P r(s+ 1 , s2 , s3 , M ) P r(M + , s+ 1) P + + − P r(s , s , s |D+ )P r(Di+ ) = i P 1 2+ 3+ i + i P r(s1 |Di )P r(Di ) p−1 = 2p − 3
− + + P r(s+ 2 , s3 |M , s1 ) =
− + + P r(s+ 1 , s2 , s3 , M ) + P r(M + , s1 ) P P r(s+ , s− , s+ |D+ )P r(Di+ ) = i P 1 2+ 3+ i + i P r(s1 |Di )P r(Di ) 2(p − 1)p = 2p − 3
+ + + P r(s− 2 , s3 |M , s1 ) =
B
Appendix: Technical Appendix
N = {1, 2, ...., n} + For all i ∈ N , Bi+ := (Di+ ∩ M + ) \ (Di−1 ∩ M + ). + Bn+1 = M + \ (Dn+ ∩ M + )
s = (s1 , s2 , ...., sn )
34
P (s|θ ∈ Bi+ )P (θ ∈ Bi+ ) P (θ ∈ Bi+ |θ ∈ M + , s) = Pn+1 + + j=1 P (s|θ ∈ Bj )P (θ ∈ Bj ) −
+
=P
pni (1 − p)ni P (θ ∈ Bi+ ) n+1 n+ j j=1 p (1
−
− p)nj P (θ ∈ Bj+ )
where
+ − n+ i = ]{j ∈ N |(j ≥ i and sj = sj )or(j < i and sj = si )} − + n− i = ]{j ∈ N |(j ≥ i and sj = sj )or(j < i and sj = si )}.
Lemma 8. Consider unanimity rule. When p >
5 6,
with 2-element symmetric partition M =
{M + , M − }, there cannot be an equilibrium with more than two decision makers with vi = (1, 0). Proof of Lemma 8. We prove by contradiction. Suppose there is a equilibrium voting strategy v + with more than two decision makers with (1, 0). Denote three decision makers with (1, 0) i, j > i
35
and l > j, i.e. vi = vj = vl = (1, 0). Then for the decision maker j, + + − E[˜ ul |M + , Sˆl (v−l ), s− ul |M + , s+ i , sj , sl ] = l ] ≥ E[˜ 1 6γi γj (γj ((p−1)γi ((4γl −2)p+1)+γl p(1−2p))+γl (p−1)(2p−1)γi ) h 2 p(2p − 1)γ 2 − (p − 1)γ 2 8pγ 2 + 2p − 1 × γ j i j l i 2γ2 +2γ γ γ ((p − 1)γ (2p(γ + 1) − 1) + p(1 − 2p)γ ) + ((3 − 2p)p − 1)γ i j i j j l i j , if γl > 12 1 6γi γj ((p−1)γi (γl (4(p−1)γj −2p+1)−2pγj +2p+3γj −1)+(γl −1)p(2p−1)γj ) h × γj2 −(p − 1)γi2 (4γl (2γl − 3)(p − 1) + 6p − 5) + (γl − 1)p(2p − 1)(2γi − γl ) −2(γl − 1)(p − 1)(2p − 1)γi2 γj + (γl − 1)γl (p − 1)(2p − 1)γi2 , if γl < 21 and γj > 12 1 6γi (γj −1)(2p2 (γl (2γi γj −γi −γj +1)−3γi γj +2γi +γj −1)−p(γl (4γi γj −γi −γj +1)−7γi γj +4γi +γj −1)+γi (γl −γj )) × γl2 p 2p γi2 (−4(γj − 2)γj − 3) + (γj − 1)2 + γi2 (8(γj − 2)γj + 5) − (γj − 1)2 + γi2 , +γl 2p2 2γi2 (γj (3γj − 7) + 3) − 2γi (γj − 1)2 − (γj − 1)2 +γl p −6γi2 (γj − 2)(2γj − 1) + 2γi (γj − 1)2 + (γj − 1)2 − 2γi2 γj i 2 4(γ − 1)2 − 2γ ((γ − 4)γ + 2) + p γ ((γ − 8)γ + 4) − 2(γ − 1)2 + γ γ 2 +γ p i j i j j i j j j i j 1 1 , if γj < 2 and γi > 2 1 − 21 − 6(γl (p(p(4γi γj −6γi −2γj +4)+3γi −γj −2)−γi +1)+γ j (p(4−(2p+3)γi )+γi −1)+p(p(4γi −2)−1)) h 1−γ γ −1 1 1 2 2 l l × (γl − 1)(γi − 1)(γj − 1) 2p + + − 8 + + 4γ − l γl −1 1−γ γj −1 (γi −1)2 (γj −1)2 i i (γl −γj )(γl +2γj −3) 3(γl −1) γl −1 6 1 3 +(γ − 1)(γ − 1)(γ − 1) p − + + − + i j l 2 2 2 γl −1 γi −1 γj −1 (γi −1) (γj −1) (γl −1)(γj −1) 1 , if γi < 2
> 0.
So, vl should be (1, 1). Contradiction.
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