Manipulated Electorates and Information Aggregation Mehmet Ekmekci, Boston College Stephan Lauermann, University of Bonn December 1, 2016
Abstract We study information aggregation with a biased election organizer who recruits voters at some cost. Voters are symmetric ex-ante and prefer policy a in state and policy b in state , but the organizer prefers policy a regardless of the state. Each recruited voter observes a private signal about the unknown state but does not learn the size of the electorate. In contrast to existing results for large elections, there are equilibria in which information aggregation fails: As the voter recruitment cost vanishes, the organizer can ensure that policy a is implemented with high probability independently of the state.
We are grateful for helpful comments from Dirk Bergemann, Nina Bobkova, Laurent Bouton, Hulya Eraslan, Christian Hellwig, Nenad Kos, Thomas Mariotti, Wolfgang Pesendorfer, Larry Samuelson, Ronny Razin, and Andrew Newman as well as comments from seminar audiences at Yale, USC, UCLA, ASU, Princeton, Bonn, Bielefeld, Oslo, Helsinki, Cerge-ei, EUI, HEC Paris, Toulouse, Georgetown, Maastricht, UCL, LSE, Boston University, Rochester, Moscow, Rice, Oxford, NYU, UT-Austin, and Harvard/MIT, and audiences at the workshop on Games, Contracts and Organizations in Santiago, Chile, Stony Brook Game Theory Festival, CETC 2014, ESEM Toulouse, Warwick Economic Theory Conference, NBER conference on GE at Wisconsin-Madison, AEA Meetings in Boston, and the ESWC at Montreal. Krisztina Horvath, Zafer Kanik, Deniz Kattwinkel, and Lin Zhang provided valuable proofreading. Lauermann thanks the Cowles Foundation at Yale University and Ekmekci thanks the Toulouse School of Economics for their hospitality. This work was supported by a grant from the European Research Council (ERC 638115).
1
Introduction
Voting is considered an e¤ective mechanism for aggregating information that is dispersed among voters about which available policy is best for society. For instance, consider an election in which voters have to decide between two policies: policy a and policy b. Voters have common interests and prefer policy a in state
and
policy b in state ; i.e., voters prefer that the implemented policy matches the state of the world. However, no individual voter knows the state, and thus the voters are uncertain about the correct policy. Although uncertain about the state, each voter has a small piece of information in the form of a noisy signal. Feddersen and Pesendorfer (1997) showed that in large elections, the majority decision will be as if there were no uncertainty, in all1 equilibria. Thus, simple majority rules allow society to aggregate noisy information in order to correctly choose among available options. In this paper, we uncover an important feature of this and related models of elections that is necessary for majority rules to reliably aggregate information and implement the voters’preferred outcome in all equilibria: The number of voters has to be independent of the state. We show that if the number of participating voters is exogenously state dependent and participation across the states is not identical, then large elections admit a new type of symmetric equilibrium in which information aggregation fails. In both states of the world the same policy receives the majority of the votes with probability close to 1. The policy that wins in this equilibrium is the policy that is preferred by the voters in the state in which the turnout is smaller (for example, whenever voters expect lower turnout in state
, policy a wins with
probability close to 1 in both states). This …nding raises two concerns. First, a simple majority rule does not aggregate information as reliably as may have been thought previously.2 Second, and this is the focus of the paper, the failure of information aggregation opens up the possibility of “manipulation.”Consider an interested agent who prefers policy a independently of the state. He may manipulate the outcome in his favor by creating an expectation that the turnout is lower in state . As discussed, this expectation induces a voting equilibrium that implements his privately preferred outcome a with high probability in both states. Examples of activities that may allow an interested party to a¤ect voter turnout 1
Throughout the introduction and the discussions in the paper, when we refer to equilibrium without further quali…cation we mean symmetric and “responsive” (non-trivial) equilibria. This is the natural class of equilibria to consider in this setting. We will not always repeat this quali…cation. 2 Section 8 discusses other mechanisms by which information aggregation may fail, for example, Feddersen and Pesendorfer (1997, Section 6), Mandler (2012), and Bhattacharya (2013).
are numerous. They include the bussing of voters to polls in elections or in referenda, the activities of a CEO directed at increasing participation in shareholder voting, the general timing and location of elections, and the prodding of colleagues by a department chair. In this paper, we study the e¤ectiveness of such tools to a¤ect election outcomes. To this end, we introduce an election organizer as an additional player in the simple voting model with common interests and noisy information that was outlined above, in which the organizer chooses the number of voters. The organizer privately learns the state of the world,
or
, and then recruits an odd number of voters.
Recruitment is a costly activity, and the total recruitment cost is linear in the number of recruited voters. Each voter then has an equal chance of being selected to participate in the election, after which the recruited voters observe noisy signals and cast votes simultaneously. The election organizer is biased, in the sense that he prefers policy a independently of the state, and the voters are aware of this con‡ict of interest. We assume that the number of recruited voters is not observed by the voters. However, the voters make Bayesian inferences about the state from being recruited because they understand how the organizer’s choice of the number of recruited voters depends on the state. Therefore, a voter receives some information about the state from the fact that she is selected. As an example, consider a referendum to build a bridge in a town. The cost of the bridge is unknown, and the voters prefer that the bridge be built only if the cost of building the bridge is low. The governor knows the building cost and chooses how much e¤ort and money to spend in order to mobilize the voters, which in turn a¤ects the voter turnout. The governor prefers the bridge be built no matter what the cost, maybe because it will increase his popularity and his re-election chances in the next election, or maybe because he will bene…t from doing business with the construction company in charge of building the bridge. Our paper explores how the ability to in‡uence voter turnout can translate into the ability to in‡uence the election outcome. Another example is shareholder voting on a …rm’s managerial decision. Suppose shareholders vote on a compensation package for the management. Shareholders have a common interest in choosing the package that maximizes the company’s value. The management may reasonably be expected to be better informed about the appropriate compensation package and to be biased in the direction of larger compensation. Finally, the management has a number of tools available to manipulate the shareholder voting turnout; see Yermack (2010). We show that the ability to manipulate turnout signi…cantly a¤ects the per-
2
formance of elections: There are equilibria in which the majority chooses policy a almost always, independently of the state, when the recruitment cost is almost zero and the number of potential voters is arbitrarily large. To provide intuition, we state and discuss two observations following the main result. First, as indicated before, if the voters expect the organizer to recruit fewer voters in state , then there is a voting equilibrium given this expectation in which each individual voter is more likely to support policy a than policy b in both states. Second, we show that if each individual voter is more likely to support policy a than policy b in both states, then it is indeed optimal for the organizer to recruit fewer voters in state
. Finally, we use a …xed-point argument to show that this loop of
best replies can be closed. In the second set of results (Theorems 2 and 3), we characterize the equilibrium behavior across all equilibria when the recruitment cost is small and when the population is large. There are no equilibria in which information is fully aggregated in the limit. Then, in Section 5, we tackle an election design question and show via two examples how the extent of the manipulation can be diminished when certain policy tools are used. The tools we consider are, …rst, a participation requirement (a requirement that the number of voters participating be higher than a certain threshold) and, second, a unanimity requirement for policy a to be implemented. In Section 6, we return to the characterization of voting equilibria for large elections in which we abstract from the speci…c recruitment mechanism and instead allow the number of voters to exogenously depend on the state in an arbitrary way. Our main technical contribution is to uncover how the set of voting equilibria depends on the ratio of the number of voters in the two states. Speci…cally, we verify the initial claim that whenever there is any imbalance in the number of voters across states— any ratio di¤erent from 1— information aggregation fails: There are equilibria in which the majority vote is almost independent of the state as the election becomes large. Only if the number of voters in the two states is the same is information aggregated in all (symmetric and interior) equilibria. Equilibria in which information aggregation fails are sustained because of a participation curse that appears when the number of voters is not identical across states. When voters use responsive strategies, a vote is more likely to be pivotal in the state with fewer voters. In Section 7, we discuss extensions and conduct robustness checks (abstention, costly voting, population uncertainty, and competition between multiple organizers).3 In particular, we argue that when voters receive a noisy public signal about 3
We conduct a more extensive robustness check in an older version of this paper, Ekmekci and
3
the size of the election, information aggregation continues to fail when there is an imbalance in the number of voters. Finally, in Section 8, we discuss the paper’s contribution to the existing literature and compare our results especially to those from previous work on elections with an uncertain number of voters. Myerson (1998a) shows that there always exists some equilibria that aggregate information. Compared to this work, we show that additional (symmetric and responsive) equilibria exist in which information fails to aggregate, unless the population size is essentially state-independent.4 In addition, we endogenize the relation between the number of voters and the state through the activity of an election organizer. Given this endogenous relationship, no equilibrium with full information aggregation exists.
2
Model
A …nite number of potential voters, N , has to choose between two available policies, fa; bg. The voters have common interests but are uncertain about which policy
serves their interest better. In particular, there are two possible states of the world, denoted by ! 2
:= f ; g. Voters share the following utility function: u(a; ) = u(b; ) = 1; u(a; ) = u(b; ) =
1;
where u(x; !) denotes the utility if policy x is chosen in state !. In other words, voters prefer that the implemented policy matches the true state of the world, but they do not know what the state is. Note that here we also make the simplifying assumption that u(a; ) u(b; ) = u(b; ) u(a; ), but none of our results depends on this speci…cation. Information Structure: There is a common prior belief
2 (0; 1) that the state is . Each voter receives
a private signal, s 2 S := [0; 1]. The signals are distributed according to a c.d.f.
F (sj!). Conditional on the state, the signals are independent across voters. The distribution F admits a continuous density function, denoted by f (sj!). We assume
the strict Monotone Likelihood Ratio Property (MLRP).5 Lauermann (2014). 4 In a separate paper, Ekmekci and Lauermann (2016), we show the existence of equilibria that fail to aggregate information with a state-dependent participation rate in a model where the number of voters is Poisson distributed, both with compulsory voting and with voluntary voting (allowing abstention). We discuss that paper in Sections 6.2 and 7.3 in more detail. 5 Continuity of the densities f ( j!) and the strict version of the MLRP are for expositional
4
Assumption 1. f (sj ) f (sj )
is strictly decreasing in s:
Assumption 1 implies that voters who receive higher signals attach a strictly larger probability to the state of the world being state
. Another implication of
Assumption 1 is that the signals carry some information about the state of the world; i.e., f (sj ) is not identical to f (sj ) for all s 2 S. Our second assumption
puts a bound on the informativeness of the signals. Assumption 2. There exists a number < f (sj!) <
1
> 0 such that
for all ! 2
and all s 2 S.
An implication of Assumption 2 is that there is no single voter type who has arbitrarily precise information about the state of the world. Organizer’s Actions and Preferences: There is a single election organizer who observes the realization of the state of the world ! and recruits the voters who participate in the election. The organizer prefers that policy a be implemented, irrespective of the state of the world. Recruitment is costly, and in particular, recruiting each additional pair of voters costs the organizer c > 0. Thus, if the organizer recruits n pairs of voters, then the number of participants in the electorate is equal to 2n + 1 2 f1; 3; 5; :::; N g . If the organizer recruits no one, n = 0, then one randomly chosen voter becomes the unique voter. Only the recruited voters participate in the election and so the organizer chooses the turnout.6 Note that the number of voters is always odd, and a tie in the vote count cannot occur. The organizer’s payo¤ is uO (a; n) = 1 uO (b; n) =
cn; cn;
where the …rst argument is the policy that the majority of the electorate chooses to simplicity. All of our results continue to hold without continuity of the density functions and with the weak version of MLRP, together with a condition that states “f ( j ) is not everywhere identical to f ( j ).” 6 In Section 5.1, a general number m of voters participates even if the organizer chooses n = 0.
5
implement, and the second argument is the number of pairs of voters the organizer recruits. We make the following assumption about the relation of c and the number of potential voters, N . Assumption 3. 2 . c This assumption ensures that the size of the population is never a binding conN
straint for the organizer.7 Finally, the choice n is not observed by the voters.8 Timing of the Voting Game: 1. The organizer learns the state. 2. The organizer chooses n. 3. Nature chooses (recruits) 2n+1 voters, each equally likely, from the population. 4. Each recruited voter observes her private signal but does not observe the number of recruited voters, n. 5. Only the recruited voters participate in the election. Each recruited voter casts a vote for policy a or policy b. 6. The policy that receives the most votes is implemented. Strategies and Equilibrium: A strategy for the organizer is a pair of distributions over integers, n ~ = (~ n ;n ~ )2
(f0; 1; :::; (N
1)=2g)2 ;
which denotes the organizer’s recruitment choice in states
and , respectively. We
denote as n = (n ; n ) a pure strategy. A pure strategy9 for voter i is a mapping d : S ! fa; bg; 7 The assumption is a lower bound on the size of the population. Our analysis remains unchanged when the number of voters is in…nite. The advantage of a …nite population is that being recruited is a positive probability event, which facilitates the application of Bayes’formula. 8 This assumption captures the idea that the voters cannot exactly infer the organizer’s recruitment e¤ort (the number of busses, the phone calls made to others etc). Note that voters nevertheless make an inference about n from being recruited, as discussed below. In Section 7, we discuss an extension in which voters receive an imperfect public signal about n. 9 As will become clear, voters’ best replies will have a cuto¤ structure, and therefore, focusing on pure strategies for the voters is without loss of generality.
6
that prescribes which policy the voter supports as a function of her signal if she is recruited. Recall that when a voter is not recruited, she does not have a ballot to cast. A symmetric Nash equilibrium is a tuple (~ n; d) in which the organizer’s strategy n ~ is a best response to a voter strategy pro…le in which each voter uses the same strategy d, and the strategy d is a best response to the strategy pro…le in which the organizer’s strategy is n ~ and all other voters use the strategy d. From here on, we refer to a symmetric Nash equilibrium simply as an equilibrium. For any given symmetric voter strategy d, let the expected vote share for policy a in state ! be q! (d) := Pr (d (s) = aj!) =
Z
1d(s)=a f (sj!) ds.
s2[0;1]
Finally, we sometimes consider the voting game for a …xed recruitment strategy. In this case, we say that symmetric strategy pro…le d is a voting equilibrium given n ~ if d is a best response to the strategy pro…le in which the organizer’s strategy is n ~ and all other voters use strategy d. Inference of Voters and Cuto¤ Strategies: In our model, voters are consequential; i.e., they care only about the implemented policy and not directly about how they vote. A single vote changes the implemented policy only when the number of the other votes that are cast for either alternative is equal. In this event, a single vote is pivotal. A voter acts as if her vote were pivotal, as is typical in voting models with incomplete information. The probability of being pivotal in state ! if the expected vote share is q! and the number of recruited voter pairs is n! is given by 2n! (q! )n! (1 n!
q! )n! .
In our model, there is an additional source of information that the voters use to make their inference about the state of the world, because there is some information contained in the event that a voter is recruited. This is because the number of recruited voters depends on the state of the world, and thus a voter learns some information about the state from being recruited. The probability of being recruited in state ! if the number of recruited voter pairs is n! is given by 2n! + 1 . N The posterior likelihood ratio that the state is , conditional on being recruited
7
and conditional on being pivotal for a voter who received signal s, when all other voters are using strategy d, and the organizer is using a pure strategy n = (n ; n ) is calculated as10 2n n f (sj ) 2nN+1 n (q ) (1 (s; piv; rec; n; d) := ) 2n +1 2n (q )n (1 n |1 {z } |f (sj N {z } | {z } | {z prior signal recruited
pivotal
q )n q )n
,
(1)
}
where we omit the dependence of q! on the voter strategy d for ease of reading, and we assume for the following discussion that either 0 < q! < 1 or n = 0, so that the probability of being pivotal is strictly positive and hence
is well-de…ned.11
This likelihood ratio, which we refer to as the critical likelihood ratio, guides a voter’s decision. In particular, a voter having a signal s supports policy a if her critical likelihood ratio is above 1, and supports policy b otherwise. From the MLRP condition from Assumption 1,
is strictly decreasing. Therefore, voters use cuto¤
strategies in all equilibria. This is a standard result in voting models that we state without proof. Lemma 1. Any equilibrium voting strategy has a cuto¤ structure. There is a signal s^ such that a recruited voter casts a vote for policy b if s > s^ and for policy a if s < s^. From here on, we use s^ 2 S to denote a generic cuto¤ strategy and q! (^ s) to
denote the expected vote share for policy a in state ! when voters use a cuto¤ strategy s^. By Lemma 1, the expected vote share for policy a is q! (^ s) = F (^ sj!) . In an equilibrium with an interior cuto¤, i.e., 0 < s < 1, the cuto¤ type is indi¤erent between voting for a or b (s ; piv; rec; n ~ ; s ) = 1: Conversely, if a cuto¤ s satis…es this condition, then s induces a voting equilibrium given n ~. 10
The extension of the expression to the case in which the organizer uses a mixed strategy is straightforward. For completeness, we write the critical likelihood ratio when n ~ is a mixed strategy in Equation (12), which is in the Appendix. 11 We follow the convention that if n = 0, then 2n = 1. n
8
Remark 1. The recruitment strategy enters
in two places, the likelihood ratio
of being recruited— indicated by “recruited” in (1)— and the likelihood ratio of being pivotal–indicated by “pivotal”. Holding everything else constant, if the number of recruited voters in state
increases, then the likelihood ratio of being recruited
increases. Conversely, the likelihood ratio of being pivotal decreases if the number of recruited voters in state
increases. In general, a change of n! moves the two
likelihood ratios in di¤ erent directions. Which e¤ ect is larger? Note that the recruitment probability is increasing at a linear rate in n . However, unless q = 21 , the pivotality probability is decreasing at an exponential rate. Thus, for large elections, an increase in n will typically decrease
because the pivotality e¤ ect dominates (conversely, an increase in n will
typically increase
).
Organizer’s Best Reply: The organizer chooses n in order to maximize the probability with which policy a is implemented, less the recruitment cost. Recall that the voters do not observe the choice n of the organizer, and thus the organizer’s choice does not a¤ect voter behavior directly. Therefore, in state ! the organizer’s (pure) best-reply correspondence to a given cuto¤ strategy s^ of the voters is: arg max 0;1;:::; 21 (N
n2f
2n+1 X
1)g i=n+1
2n + 1 (q! (^ s))i (1 i
q! (^ s))2n+1
i
nc.
(2)
The …rst term in the organizer’s objective function is the probability that policy a is implemented when the probability that a randomly selected voter supports policy a is q! (^ s), and when the turnout is 2n + 1. The second term is the cost of choosing a turnout of 2n + 1. Remark 2. The number of potential voters, N , appears in the recruitment e¤ ect in Equation (1) and in the organizer’s best reply in Equation (2). However, the term N cancels out in the recruitment e¤ ect. Hence, N has an impact on equilibrium behavior only if N is so small that it becomes a binding constraint in the organizer’s best reply in Equation (2), which we rule out with Assumption 3. Therefore, N plays no further role in the analysis. To get more insight into the organizer’s best reply, we calculate the increase in the probability that policy a gets selected when the organizer recruits an additional
9
pair of voters, that is, the marginal bene…t of increasing n: (n
1; !; s^) :=
2n+1 X
2n + 1 (q! (^ s))i (1 i
i=n+1 2n X1
2n
1 i
i=n
(q! (^ s))i (1
q! (^ s))2n+1
i
q! (^ s))2n
1 i
.
This expression can be rewritten as:12 (n
1; !; s^) =
1 2n (q! )n (1 2 n
q! )n (2q!
1) .
(3)
The increase in the probability that policy a is implemented when the number of recruited voters increases from 2n
1 to 2n + 1 is equal to the probability of a tie in
the vote counts for policies a and b, multiplied by the term 12 (2q
1). It is intuitive
that the marginal bene…t is proportional to the probability that the election is tied, since additional voters matter only if the election is close. The term (2q
1) enters
because the additional pair may either both vote for a (good) or both vote for b (bad). If q! (^ s)
1=2, then
(n
1; !; s^)
0 for every n (the additional pair is more
likely to vote for b than a). Therefore, the organizer recruits no additional voter, since recruitment is costly: When the odds are against him, the organizer recruits as few people as possible in order to maximize the variance in the outcome of the election and to save on recruitment cost. If, however, 1 > q! (^ s) > 1=2, then
(n
1; !; s^) > 0 and
(n
1; !; s^) >
(n; !; s^). Therefore, the objective function is strictly concave. There is a unique n such that
(n
1; !; s^)
c and
(n; !; s^) < c. Notice that when q > 1=2, the
odds are with the organizer, so he wants to minimize the variance of the election outcome by recruiting many people. For instance, if the organizer recruits an in…nite number of voters, then by the weak law of large numbers, policy a is implemented. However, there is a cost of recruiting voters, so the organizer recruits voters until 12
Adding an additional voter pair when there are 2n 1 voters changes the outcome only if either n 1 voters already support a and both of the additional voters support a, or if n voters already support a, and neither of the additional voters support a. Hence, dropping subscript !, ! ! 2n 1 2n 1 n 1 n 2 (n 1; !; s^) = (q) (1 q) (q) (q)n (1 q)n 1 (1 q)2 n 1 n ! 1 2n = (q)n (1 q)n (q (1 q)) . 2 n
10
the marginal bene…t of recruiting an additional pair of voters is below the marginal cost. By the strict concavity of the objective function, in both cases, the organizer’s best reply is either unique (meaning one integer for each state), or a mixed strategy with support on two adjacent integers for one or for both states.
3
Manipulated Electorates
We study election outcomes when c is small. When c is small, the organizer may recruit many voters, and thus we can compare our result to those for exogenously large elections. To this end, we …x the common prior
and some information
structure F that satis…es Assumptions 1 and 2. Let fG(c)gc>0 be a collection of voting games in which for each game G(c), the prior belief is
, the information
structure is F , the recruitment cost to the organizer is c, and the number of potential voters, N (c), is some integer that satis…es Assumption 3. Theorem 1. Let fck gk=1;2;::: be a sequence of positive numbers that converge to
zero. Then, there is a sequence of symmetric Nash equilibria of G(ck ) such that in both states: 1. The probability that policy a is implemented converges to 1. 2. The number of recruited voters increases without bound. 3. The organizer’s payo¤ converges to 1. Theorem 1 states that as the recruitment cost vanishes and the number of potential voters becomes large, there are equilibria in which policy a is elected with a probability that is arbitrarily close to 1 in both states. Moreover, in both states the number of recruited voters becomes large, and the organizer’s expected payo¤ becomes 1. Thus, an endogenously large electorate may lead to the failure of information aggregation, and in the limit the organizer incurs no cost from the recruitment e¤orts although he recruits an unbounded number of voters. As we will see, in all such manipulated equilibria a randomly selected voter supports policy a with a probability strictly larger than 1=2 in both states of the world, and the organizer recruits more voters in state
than in state
. The
following two observations provide intuition. First, if the organizer is expected to recruit more voters in state
than in state
, then it is a voting equilibrium that
voters support policy a with a probability strictly larger than 1=2 in both states. 11
Second, if this is the voters’behavior, then it is optimal for the organizer to recruit more voters in state
than in state .
Observation 1. If the organizer recruits fnk ; nk g1 k=1 voters in the two states
with nk ! 1, nk ! 1, and 0 < limk!1 nk =nk < 1, then there exists a voting equilibrium for every (nk ; nk ) in which the voters use a cuto¤ strategy s^k for which 1 > limk!1 q
s^k > limk!1 q
s^k > 1=2.13
Thus, by creating an expectation of an imbalance in the number of voters across states— no matter how small— the organizer can manipulate the election in his favor and induce an equilibrium in which his favorite outcome wins with a probability approaching 1. To gain some intuition, consider …rst the situation in which the expected vote share is the same in both states (this would be the case if the signals were pure noise), and only the number of voters di¤ers, with more voters in state . The asymmetric number of voters a¤ects voter behavior in two ways. First, because there are more voters in state
than in state
, a voter is more likely to be recruited in state ,
and her posterior belief that the state is
increases when she is recruited. This
is the recruitment e¤ect. The other e¤ect that works in the opposite direction is the pivotality e¤ect. Because there are more voters in state pivotality probability in state
than in state , the
is larger than the pivotality probability in state .
Among the two e¤ects, the pivotality e¤ect is dominant, and the net e¤ect supports voting in favor of policy a. To see why the pivotality e¤ect dominates, recall from Remark 1 that the probability of being recruited is increasing linearly in the number of voters, but the probability of being pivotal is decreasing exponentially. Thus, if the expected vote shares were the same but the turnout was lower in state , then a voter’s posterior probability of state
conditional on being recruited and pivotal
would be close to 1. However, since signals are informative and voters use cuto¤ strategies, the expected vote share of policy a is necessarily smaller in state a cuto¤ for which 1 > q
s^k
> 1=2, it must be that q
s^k
than in state . Given >q
s^k > 1=2. Since
the election is closer in state , as the number of voters becomes large the election is much more likely to be tied in state
than in state
. Thus, if the expected
vote share is larger in state
but the number of voters is the same, then a voter’s
posterior probability of state
conditional on being pivotal would be close to 0. In
an equilibrium, this e¤ect favoring state
can be shown to exactly balance with
the previous e¤ect via the di¤erence in the number of voters that favored state 13
,
This observation is a special case of our characterization of all voting equilibria in large elections in Theorem 5.
12
establishing the existence of interior cuto¤s. Observation 2. If the voters use cuto¤ strategies f^ sk g1 k=1 for which 1 > limk!1 q
limk!1 q
s^k > 1=2, and
nk ; nk
is an optimal recruitment strategy given s^k
and ck for all k as ck ! 0, then it must be that nk ! 1, nk ! 1, and limk!1 nk =nk < 1.14
To see why it is optimal to recruit more voters in state
under the hypothesis
that voters support policy a with a probability larger than 1=2 in both states, consider Figure 1. The …gure depicts the probability that the majority selects policy a as a function of n for an example with q = 0:7 and q = 0:6. When n is large, P
(b2n+1c) 2(bnc)+1 6 2bnc+1 6 j q ; that then Pr the curve given than the curve given is, forj any given n (Majority votesqforis0)steeper = j=(bn+1c) 1 10 10 j that is su¢ ciently large, the marginal bene…t of an additional voter is larger in state
P(b2n+1c)
2(bnc)+1
7
j
7
2bnc+1 j
Pr (Majority 0) =for j=(bn+1c) 1 is . This propertyvotes holdsfortrue all 1 > q >jq > 1=2 and 10 10 a simple consequence
of the fact that both functions must approach 1 eventually, with the function for q Plot:
starting from a lower point.
Prob of a winning
1.0
0.9
0.8
0.7
0.6
0.5 0
5
10
15
20
25
30
35
40
45
50
n
Figure 1: The probability that policy a receives the majority of votes given the number of recruited voter pairs n for q = 0:6 (straight) and for q = 0:7 (dashed). The curve is steeper for q = 0:6 when n is large, implying a larger marginal bene…t. Taken together, the two previous observations imply the following: If the organizer is expected to create an imbalance by recruiting more voters in state
and
the voters behave optimally given this expectation, then it is in fact optimal for the organizer to recruit more actively in state 14
. Our proof of the theorem uses
The observation is a special case of our characterization in Lemma 3 in the Appendix.
13
s^k >
a …xed-point argument to show that this loop of best responses can be closed and establishes the existence of a manipulated equilibrium. Remark 3. Both of the observations continue to hold if there is population uncertainty, and if the number of voters is Poisson distributed; see Subsection 6.2. The limit analysis for the two observations made above is greatly simpli…ed by the use of Stirling’s approximation, which allows us to approximate the probability of being pivotal as follows:15 2n! (q! )n! (1 n!
q! )n!
(4q! (1 q! ))n! p p . n!
Contrast with Voting Models with State-Independent Number of Voters. The existence of equilibria with a limit cuto¤ s = lim s^k for which q (s ) > q (s ) > 1=2 stands in sharp contrast to the results of Feddersen and Pesendorfer (1997), who showed that in a voting model in which the number of voters is state independent, all symmetric equilibrium cuto¤ have a limit cuto¤ s for which q (s )
1=2 = 1=2
close to being
tied.16
q (s ): In both states, the election is expected to be equally This implies that all equilibria aggregate information since
q (s ) > 1=2 > q (s ). What drives Feddersen and Pesendorfer’s result is that, otherwise, the ratio of the pivot probabilities is degenerate in the limit. Note that the ratio of the pivot probabilities is 2n n 2n n
(q (s ))n (1
q (s ))n
(q (s ))n (1
q (s ))n
This expression goes to 0 as n ! 1 if q (s )
if q (s )
1=2 < 1=2
q (s
).17
:
1=2 > 1=2
(4) q (s ) and to 1
Therefore, the critical likelihood ratio cannot be
1 and so there is no sequence of interior equilibria with a limit cuto¤ s for which q (s )
1=2 6= 1=2
q (s ).
In our model, the information contained in the pivotal event is shaped by both the expected vote shares in both states and the relative ratio of the number of participants in each state. This is because an equal split is less likely in the state with a larger number of participants. The organizer’s recruitment decision is linked (n) For two functions, we write f g if limn!1 fg(n) = 1. They consider a setting with private and common values. Large elections with pure common values are analyzed in Feddersen and Pesendorfer (1998), Wit (1998), and Duggan and Martinelli (2001). 17 Note that the expression x (1 x) for x 2 [0; 1] is maximized at 1=2 and symmetric around 1=2. Thus, q > 1 q and q > q implies q (1 q ) < q (1 q ). Similarly, q < 1 q and q > q implies q (1 q ) > q (1 q ). 15
16
14
to the expected vote shares in a way that keeps the inference made by being pivotal moderate, compared to the case in which the number of voters is exogenous. Thus, the existence of the organizer opens up the possibility that the majority votes for policy a. The organizer’s optimal recruitment strategy has implications for the pivot probabilities in di¤erent states. Take any cuto¤ s 2 (0; 1)— not necessarily an equilibrium— and suppose the vote share is 1 > q (s) > 1=2. Then, the organizer
chooses the number of recruited voters, 2n + 1, such that (dropping the argument s)18 2n n q (1 n
q)n (2q
1) = 2c.
The approximation in the above statement represents the error that comes from ignoring the integer constraints. Therefore, if the voters’cuto¤s are any s for which 1 > q! (s) > 1=2, then the ratio of pivot probabilities in each state of the world stays bounded away from 0 and 1 and is approximated as19 2n n 2n n
(q )n (1 (q
)n
q )n
(1
q
2q 2q
)n
1 . 1
(5)
Note that the right-hand side is independent of c. This is because the organizer’s choice of the size of the electorate keeps the pivot probabilities in each state relatively at the same order, and when c vanishes, the relative pivot probabilities stay bounded away from 0 and in…nity. This is unlike the case in which the number of voters is state independent, as depicted by Equation (4). We use a version of (5) to derive a bound on the equilibrium ratio n =n , where Stirling’s approximation simpli…es the left-hand side. In particular, we show in Lemma 7, located in the Appendix, and in the subsequent remark, that the ratio of the number of recruited voters in states
and
stays bounded away from zero
and in…nity in the sequence of the manipulated equilibria of Theorem 1. With nk and nk denoting the expected number of recruited pairs of voters in those equilibria given ck , if nk =nk converges (along some subsequence), then nk k!1 nk
0 < lim 18
1.
Recall the derivation of the marginal bene…t of an additional voter from (3). The approximation error is typically quite large. Inequalities (8) following Lemma 2 give the exact characterization via two bounds. The point here (and in the Appendix when we use those bounds) is that the organizer’s optimal choice implies that the ratio of the pivotality probability is bounded and bounded away from zero. 19
15
4
All Limit Equilibria
In this section, we systematically characterize the limiting equilibrium outcomes that can be generated by any equilibrium sequence, as the recruitment cost vanishes. The recruitment activity limits information aggregation in all symmetric equilibria. Let s! be the median signal in state !, that is, q! = F (s! j!) = 1=2: Trivial Equilibrium: An equilibrium is a trivial equilibrium if the organizer recruits no voter in either state, so that there is only one voter casting a ballot in each state. In a trivial equilibrium, the organizer is passive and information is not aggregated because only one voter makes the decision about the implemented policy. Every equilibrium in which the cuto¤ is not interior is a trivial equilibrium, but there are trivial equilibria with an interior cuto¤. It is easy to see that the voting game G admits a trivial equilibrium for all recruitment costs c if and only if the distribution of signals, F , satis…es the following inequality:
1
f (s j ) f (s j )
1:
(6)
If inequality (6) holds, then in a trivial equilibrium, the single voter who is selected will support policy a with a probability not more than 1=2 in both states of the world.20 This in turn justi…es the organizer’s strategy to recruit no additional voters. Of course, if c is large, a trivial equilibrium exists even if inequality (6) fails. However, if the inequality fails, then a trivial equilibrium does not exist when the recruitment cost c is su¢ ciently small. This is because in a putative trivial equilibrium the probability that the single voter supports policy a is strictly larger than 1=2 in state
if the inequality fails. If the recruitment cost is su¢ ciently small,
the organizer’s best reply is to recruit some voters. All Non-Trivial Equilibria: In Theorem 2 below we show that any limit point of a sequence of non-trivial equilibrium cuto¤s for vanishing recruitment cost has to be either equal to s or strictly larger than s . 20
Thus, if this condition holds, then in a large election sincere voting (voting for the alternative which is more likely to be correct based on one’s individual signal) does not imply information aggregation.
16
Theorem 2. Let fck gk=1;2;::: be a sequence of positive numbers converging to zero and fG(ck )gk=1;2;::: be the induced voting games.
1. There exists a sequence of non-trivial equilibria with limit cuto¤ s > s . 2. There exists a sequence of non-trivial equilibria with limit cuto¤ s = s . 3. For every limit cuto¤ s of non-trivial equilibria, either s = s or s > s . The theorem states that there are only two types of limit points of non-trivial equilibrium cuto¤s as the recruitment cost vanishes and both types of equilibria coexist. None of these equilibria aggregates information fully; thus, information aggregation failure is inevitable in any equilibrium. One type of limit equilibrium outcome cuto¤ is s > s . These equilibria are essentially identical to the equilibrium outcomes of equilibria presented in Theorem 1. In such equilibria, the majority selects policy a; i.e., there is full manipulation.21 The second type of equilibrium sequences feature a close race between the two policies in state . This is because when s = s , the probability that a randomly selected voter supports policy a converges to 1=2. In state , the vote share for a is smaller than 1=2 and consequently, in state
the organizer recruits no one, and
policy b is implemented with probability 1 F (s j ) in state . In the next theorem,
we identify the properties of the limit outcomes of such equilibrium sequences in state : The organizer recruits a large number of voters, and even though there is a close race between the policies, policy a prevails as the winner in state .22 Theorem 3. Let fck gk=1;2;::: be a sequence of positive numbers converging to zero and fG(ck )gk=1;2;::: be the induced voting games. If inequality (6) is not satis…ed, then along all non-trivial equilibrium sequences with limit cuto¤ s = s : The number of voters recruited increases without bounds in state
and is zero
in state . Policy a is implemented in state
with probability converging to 1 and with
limit probability F (s j ) in state . 21
Di¤erent from Theorem 1, the number of voters may remain bounded in some of the manipulated equilibria. 22 For completeness, the proof of the theorem in the appendix also covers the case when inequality (6) is satis…ed and s = s . In this case, either (i) the number of voters recruited in state grows without bound and policy a is implemented in state with probability converging to one; or, (ii) the number of voters recruited in state stays bounded and policy a is implemented with a probability between 0 and 1 in state . In fact, if inequality (6) is satis…ed, then there exist two equilibrium sequences, one with property (i) and one with property (ii).
17
To gain some intuition for these results, we …rst discuss why there can be no equilibria with cuto¤s converging to some s 2 (s ; s ). If s 2 (s ; s ), then in state
only one voter is present while there is an ever-increasing number of
voters in state
, i.e., n = 0 and n ! 1. Thus, the recruitment e¤ect, which is
(2n +1) = (2n +1) = 2n +1, is diverging at a linear rate. However, the probability of being pivotal is 1 in state
whereas in state
the probability of being pivotal is
decreasing to 0 at an exponential rate in n ; a rough estimate implies23 2n n and 4q (s ) (1
q n (1
q )n = (4q (1
q ))n ,
q (s )) < 1 since s 2 (s ; s ). Thus, the (exponential) pivotal
e¤ect is stronger than the (linear) recruitment e¤ect: Conditional on being pivotal and being recruited, the probability of state
vanishes to zero, ruling out interior
equilibrium limit points s 2 (s ; s ).
Conversely, s = s can be sustained as an equilibrium limit point. For this, we
show that if the expected vote share is exactly equal to 1=2 (the cuto¤ is exactly equal to s
along the sequence) then the pivot probability decreases to zero at
the rate at which
p1 n
decreases to 0. Together with the previous discussion for
s > s , this means that there exists a sequence of cuto¤s that are larger than but converging to s for which the pivotality probability is decreasing at a linear rate, exactly balancing the recruitment e¤ect. Finally, for Theorem 3, we utilize the fact that if the pivotality probability is decreasing linearly in n , then the outcome must become deterministic in state .
Finally, we illustrate the ordering of the equilibrium cuto¤s with Figure 2. It shows the median types in the two states, s and s . The cuto¤ corresponding to the limit equilibrium of a large election with exogenous n— FP (Information Aggregation)— must solve q (s)
1=2 = 1=2
q (s). In both states the election is
equally close to being tied; see Feddersen and Pesendorfer (1997). Otherwise, (4) would fail to be interior and would be either in…nity or zero.
5
Robust Election Design
We explore whether election design can be a remedy for an organizer’s ability to manipulate election outcomes. To this end, we analyze two election design tools 23
The proof of Lemma 3 in the appendix provides an exact estimate, using Stirling’s approximation.
18
FP (Info Aggregation) EL (Info Aggregation)
b b
0
sA
EL (Manipulated Electorate)
b b
b
sB
b
1
Figure 2: The Figure: …gure illustrates potential limit|B) cuto¤s in our model (EL) and in a F (sA |A) = F (s = 1/2. B standard Feddersen-Pesendorfer election with common values (FP). Recall that s and s are the median signals, F (s j ) = F (s j ) = 1=2. that provide protection against manipulation, namely, a quorum requirement and the unanimity rule.
5.1
Participation Requirement
We start by relaxing the assumption that the minimum number of voters participating in the election when the organizer is passive is 1. Instead, we consider the minimum number of voters who are participating as an election design tool, denoted by the parameter 2m + 1. Our main result in this setup is that if the number of participants who are present already without any recruitment activity grows large, then there exists an equilibrium sequence in which the majority votes for the correct policy; thus, information aggregates. Speci…cally, suppose the number of voters is 2 (m + n) + 1 for some integer m, where m is the number of required pairs of voters, and n is the number of additional pairs of voters recruited by the organizer. We consider games parameterized by m and a positive number c > 0, denoted G (c; m). We assume that the m required pairs are free to the organizer, and the total recruitment cost is nc.24 The number of potential voters is N (c) = 2 m +
1 c
+ 1 . The following result shows that
information can be aggregated in the limit, and if m grows su¢ ciently quickly, information is aggregated. Theorem 4. Let fck ; mk gk=1;2:: be a sequence of positive numbers ck converging
to zero and integers mk diverging to in…nity. We are interested in the symmetric equilibrium sequences of the voting games fG (ck ; mk )gk=1;2;::: . 24
This assumption is for concreteness. Nothing changes if the costs are (m + n)c.
19
19 / 20
1. There exists a sequence of equilibria such that the probability with which the correct policy is implemented converges to 1 (i.e., policy a in state b in state
and policy
).
2. If mk diverges su¢ ciently quickly relative to the rate at which ck converges to 0, in all sequences of equilibria the probability with which the correct policy is implemented converges to 1. 3. If mk diverges su¢ ciently slowly relative to the rate at which ck converges to 0, then there is a sequence of equilibria in which policy a is implemented with probability converging to 1 in both states (i.e., information aggregation fails, and the outcome is manipulated). The theorem is essentially immediate from the previous discussions. In particular, the …rst part of the result is related to Theorem 2 and its proof. Essentially, we show that there exists a sequence of equilibrium cuto¤s with limit s 2 [s ; s ). For
any such sequence, s < s implies that policy b is chosen in state converging to 1. For state
with probability
, we use an argument as in Theorem 3 to argue that
policy a is chosen with probability converging to 1. The second part of the result is essentially an immediate consequence of the existing results for exogenously large elections (Feddersen and Pesendorfer (1997), Feddersen and Pesendorfer (1998), and especially Duggan and Martinelli (2001)). When mk increases very quickly relative to the speed at which ck vanishes, the organizer does not recruit anyone in either state, and thus we get an election in which the number of participants is independent of the state of the world.25 The third result is an implication of Theorem 1. Recall that there exists a sequence of equilibria along which the number of voters diverges in both states. Thus, if mk diverges su¢ ciently slowly so as not to exceed the number of recruited voters that the organizers choose to recruit optimally anyway, then the constraint on the number of voters does not bind, and the original outcome remains an equilibrium. One interpretation of the parameter m is that there is a “quorum”requirement; i.e., the minimal number of voters that the organizer must recruit is m. In the equilibria that aggregate information stated in the theorem, this constraint on the lower bound on the number of participating voters binds in state
, and if m is
su¢ ciently large, it also binds in state . Thus, the theorem suggests that when the number of voters can be manipulated, quorums are one instrument for improving the performance of elections. 25
It is not a direct corollary of their results, since Duggan and Martinelli (2001) require that f (1j )=f (1j ) is su¢ ciently small; see (7) below.
20
An alternative interpretation of m is that this is the number of voters who choose to participate anyway— maybe because they have particularly low cost of voting— even without the special recruitment of the organizer.
5.2
Unanimity Rule
We analyze what happens with the unanimity rule where all of the participants’ support is required for policy a to be implemented. It is clear that the organizer recruits no additional voters, because recruitment is costly and the probability that policy a is selected weakly decreases with the size of the electorate. Because the organizer recruits no new person regardless of his cost, we drop c as a parameter and consider a participation requirement of 2m + 1 voters as in Section 5.1. Because the organizer recruits no new voter, the turnout does not depend on the state of the world. Therefore, our model corresponds to those analyzed by Duggan and Martinelli (2001) and Feddersen and Pesendorfer (1998). Here, we use their results. The unanimity rule is known to be the only supermajority rule that fails to aggregate information in large electorates. The extent to which information aggregation fails depends on the informativeness of the extreme signals, namely, the likelihood ratio of the highest signal f (1j )=f (1j ). For this section, we assume that
1
f (1j ) < 1: f (1j )
(7)
If this condition fails, then there is a unique equilibrium in which voters support a for all signal realizations. If this condition holds, then there exist interior equilibria for m su¢ ciently large. Moreover, Duggan and Martinelli (2001) show that if (7) holds and m ! 1, then for every sequence of interior equilibria the probability that policy a is selected in state
converges to
1
f (1j ) f (1j )
f (1j ) f (1j ) f (1j )
,
and the probability that policy a is selected in state
1 As
f (1j ) f (1j )
f (1j ) f (1j )
converges to
f (1j ) f (1j ) f (1j )
:
! 0, these probabilities converge to 1 and 0, respectively. Hence, infor-
mation is arbitrarily close to being aggregated when signal 1 is very informative.
Thus, with a participation requirement m, if the voting rule is unanimity then
21
despite the presence of an organizer the information can be almost perfectly aggregated provided the extreme signal is su¢ ciently informative. This contrasts with a simple majority rule. As shown in Theorem 4, in this case even a participation requirement m ! 1 is not su¢ cient to rule out fully manipulated equilibria.
6
Voting Equilibria of Large Elections with Asymmetric Participation
In this section, we consider large elections in which the number of voters depends on the state in an arbitrary way. This allows us to relate our work to Myerson’s work. It also allows us to think about an organizer who can commit to a recruitment policy that creates a state-dependent participation rate. Thus, take a sequence of elections in which the number of voters is exogenously given by nk and nk , with nk = nk for some
> 0 and with nk ! 1 and nk ! 1.
We study the outcomes of the corresponding voting equilibria for the given number of voters (nk ; nk ) in the limit.
6.1
Characterization of All Voting Equilibria
We show that as the number of voters becomes large, the election reliably aggregates information if and only if
= 1. Otherwise, there exists an additional equilibrium in
which the policy that is best for the voters in the state with a smaller participation rate wins independently of the state, in a large election. Observation 1 is a special case of this result; see the remark following the proof in the appendix. The intuition is as described before: If there are more voters in one state than in the other, a voter may be arbitrarily less likely to be pivotal in the state in which the election is larger and hence exclude the possibility of that state when voting. Theorem 5. Fix is
fnk ; nk g1 k=1 ,
> 0. Take a sequence of elections in which the number of voters
with nk = nk , nk ! 1, and nk ! 1.
1. For all , there exists a sequence of voting equilibria with limit cuto¤ s 2 (s ; s ) that aggregates information. For
= 1, there are no other limit
outcomes of interior voting equilibria: Information is always aggregated. 2. For all
< 1, there are additional sequences of voting equilibria with limit
cuto¤ s 2 (s ; 1) in which policy a wins in the limit in both states. There
are no other limit outcomes of interior voting equilibria: Either information is aggregated or a wins in both states. 22
3. For all
> 1, there are additional sequences of voting equilibria with limit
cuto¤ s 2 (0; s ) in which policy b wins in the limit in both states. There
are no other limit outcomes of interior voting equilibria: Either information is aggregated or b wins in both states.
6.2
All Voting Equilibria in Poisson Models and Relation to Myerson
Myerson (1998a) studied a common value environment analogous to ours in which the number of voters is Poisson distributed. The mean of the Poisson distribution depends on a binary state and is k and k, respectively. Myerson (1998a) shows that for all
> 0, there exists a sequence of equilibria that aggregates information
as k becomes large. In a companion paper, Ekmekci and Lauermann (2016), we study elections with population uncertainty where the number of voters is Poisson distributed, and the expected participation rate is exogenously state dependent, as in Myerson’s model. In that paper we look at two scenarios, one where we allow abstention, and another scenario where we do not allow for abstention. In the scenario where abstention is not allowed, we derive a result that is analogous to Theorem 5 for Myerson’s model: For a large election with a Poisson distributed numbers of voters, if
6= 1, then there are additional, non-trivial interior
equilibria that do not aggregate information. In such equilibria, the policy that is best for the voters in the state with smaller participation rate wins with probability close to one independently of the state, in a large election. Thus, information aggregation is guaranteed only when
= 1, i.e., when the expected number of voters
is the same across states.26 In the other scenario where voting is voluntary, we show the existence of an equilibrium that fails to aggregate information in large elections, as in our current paper. In a special case of that scenario where voters have binary signals, we show that such equilibria exhibit non sincere voting behavior. Therefore, state-dependent participation is a friction in elections, both, on the way of information aggregation and also on the way of sincere voting behavior. This friction is not overcome by strategic abstention or by population uncertainty. 26
Myerson (1998a) notes the existence of non-interior equilibria in which voters support a particular policy independently of their signal, for some parameter constellations. To the best of our knowledge, the existence of interior (“responsive”) equilibria that fail to aggregate information in the model by Myerson (1998a) has not been noted.
23
6.3
Commitment
Asymmetric state dependent participation is a source of information aggregation failure as we argued in Theorem 5. In this paper, we highlight the advantage to an informed organizer to be able to in‡uence voter turnout. In general, such an asymmetry in the participation rate may be generated also by committing to policies such as voter ID laws, that create participation di¢ culties whose magnitudes are heterogeneous across the electorate.
A more direct form of voter suppression is
carried out in some Middle Eastern countries, such as Iraq and Syria. To the extent that the impact of the policies depends on a payo¤ relevant state of the world, the forces we identi…ed that result in information aggregation failure will be e¤ective.
7
Robustness, Extensions, and Discussion
In this section, we analyze various extensions of the model to highlight the robustness of our model to variations of our assumptions.
7.1
Information about Voter Turnout
Thus far, we have assumed that voters do not directly observe the realized turnout. Note, however, that in the main model being recruited already contains information about overall turnout. Moreover, in the manipulated equilibrium of Theorem 1 the ratio of number of recruited voters is bounded and bounded away from zero.27 Now, suppose voters observe a public but noisy signal about the realized number of actual voters (say, by observing the outcome of a likely voter survey or seeing the queues on TV) and consider the voting game in which the number of voters n and n is exogenously …xed. Then, as long as the signal stays boundedly informative as the number of voters grows large, the conclusion of Theorem 5 for large elections continues to hold; that is, whenever lim n =n < 1, there are equilibria of the voting game in which information fails to aggregate. This is because a public signal moves the common prior, but we already know that the failure of information aggregation holds independently of the prior. In particular, it follows that if we take as given the organizer’s original recruitment strategies from Theorem 1, the original voter behavior remains close to a best response su¢ ciently deep into the sequence even if there are public noisy signals.28 27
This is discussed in the text after Theorem 1 and proven in Lemma 7 in the Appendix. We do not know how equilibrium looks like with private signals on turnout. In this case, types are two-dimensional (the original signal s and the additional signal about n). This complicates the analysis signi…cantly because voting strategies are no longer characterized by a one-dimensional 28
24
However, if voters receive noisy signals about the recruitment activity and if we now consider the organizer’s optimal recruitment strategy given the noisy signal, then the organizer recruits di¤erently in order to signal the state. We do not know how this signaling incentive a¤ects the equilibrium outcome. Signaling may be considered an additional and somewhat di¤erent mechanism to a¤ect voting from the one we consider here. Thus, we believe our results are robust to adding noisy information about voter turnout when we take the original recruitment strategies as given. However, adding such information implies signaling incentives for the organizer that likely lead to a di¤erent behavior. We leave this analysis for future research since this additional signaling mechanism is likely to function di¤erently from the mechanism that we focus on here.
7.2
Multiple Organizers
In this paper, a single organizer makes all of the recruitment choices. Suppose that there is a second organizer, whom we refer to as O1 , who prefers that policy b be implemented regardless of the state and incurs the same marginal recruitment cost as the organizer, whom we refer to as O0 , who prefers that policy a be implemented regardless of the state. In this scenario, there is always a sequence of manipulated equilibria in which policy a is implemented with a probability that converges to 1 in both states and in which only O0 recruits voters while O1 is passive. There is another sequence of manipulated equilibria in which policy b is implemented with a probability that converges to 1 in both states and in which only O1 is active while O0 is passive. There is, however, one more sequence of equilibria in which O0 chooses to recruit many voters in state , O1 chooses to recruit many voters in state , and information gets aggregated; i.e., the correct policy is implemented with probability that converges to 1. Therefore, competition among organizers opens up the possibility of information aggregation.
7.3
Abstention, Population Uncertainty and Costly Voting
Abstention and Population Uncertainty: Feddersen and Pesendorfer (1996) observed that in an election in which voters have common interests, some voters who are not well informed may have strict incentives to abstain and their abstention has a signi…cant e¤ect on the election cuto¤.
25
outcome. In the equilibrium of our model, however, there is never a strict incentive to abstain: For all signals above the equilibrium cuto¤, a voter strictly prefers to vote for policy b (instead of abstaining or voting di¤erently), and she strictly prefers to vote for a for signals below that cuto¤. With a signal exactly equal to the cuto¤, a voter is indi¤erent between each vote and abstaining. The fact that there is no incentive to abstain in this equilibrium relies on the fact that the number of participating voters is odd, so that there are never any ties. However, once voters can abstain, there may be additional equilibria in which each voter abstains with strictly positive probability, implying a positive probability of an even number of voters and thus a tie. So, our original equilibrium remains if voters can choose to abstain, but additional equilibria may arise. Population uncertainty is a realistic feature of elections. Moreover, abstention is an important strategic tool for voters that may "undo" certain adverse forces in an election. For instance, in Feddersen and Pesendorfer (1996) strategic abstention allows uninformed voters to cancel out the votes of partisans in the elections. In Krishna and Morgan (2012), abstention allows voters to aggregate information through sincere voting, and also allows them to economize on the voting costs when these are positive. The main reason why we do not allow for population uncertainty or abstention in our paper is for tractability. Indeed, Observation 1 in section 3 holds if the number of voters is Poisson distributed or if we allowed for abstention. Observation 2 in the same section would continue to hold if the number of voters is Poisson distributed, and if the organizer could increase the expected number of voters by incurring a constant marginal cost. However, the relationship between the organizer’s best reply correspondence, and the voters’best replies is less tractable to handle once we allow for either population uncertainty or abstention. Costly Voting and Subsidies: Suppose that, in contrast to our model, all citizens can vote but voting is costly. Here, recruitment may correspond to a subsidy by the organizer. Concretely, suppose that there are N citizens and each citizen can vote at a cost r. This cost may correspond to the cost of walking to the voting booth. The organizer can reduce the cost of voting to zero by paying c, for example, by bussing voters to the voting booth. If the voting costs r are not too small, only the citizens who receive a subsidy actually vote.29 In this context, one can interpret the “participation requirement” m as the number of voters who have zero voting costs, while the cost of voting is r for the N
m remaining voters.
29
Note that, in fact, r may be quite low since voters will compare r to the probability of being pivotal.
26
Further analysis of costly voting with subsidies may be an interesting extension of the current model, and such analysis may yield a better understanding of exactly what such scenarios may be and when to expect voter subsidies to have substantial e¤ects on voting behavior.
8
Literature Review
Information aggregation in elections with strategic voters has been studied by AustenSmith and Banks (1996), Feddersen and Pesendorfer (1996, 1997, 1998, 1999a,b), McLennan (1998), Myerson (1998a,b), and Duggan and Martinelli (2001), among others.30 These papers study equilibrium outcomes with an exogenously large number of voters. In particular, Feddersen and Pesendorfer (1997) show that in a model with multiple states— and private and common values— under all supermajority rules except the unanimity rule, large electorates aggregate information. Similar to this paper, they provide a complete characterization of all equilibria. The main di¤erence between their model and ours is that here the number of participating voters is selected by a con‡icted organizer, and thus, the number of voters participating in the election is endogenously state dependent. Myerson (1998a) introduces a Poisson model with population uncertainty in which the expected number of voters may be state dependent. He shows that large electorates aggregate information along some sequence of equilibria. In his model, the ratio of the expected number of voters across states is …xed along the sequence as the expected number of voters grows. In our model, similar to Myerson’s model, the number of voters participating is state dependent. However, the ratio of the number of voters is endogenously determined via the choice of an organizer who incurs a cost for increasing the number of participating voters. A second di¤erence is that we characterize the limiting outcomes of all symmetric equilibria. We show that there is no equilibrium in which information fully aggregates when the number of voters is endogenous and there also exist equilibria in which the organizer’s favorite outcome is implemented regardless of the state. We study Poisson models with population uncertainty in a companion paper, Ekmekci and Lauermann (2016). Information aggregation fails in our setting because whenever the number of voters depends on the state in a non-trivial way, equilibria exist in which in both states the same policy becomes certain to win. This is driven by the e¤ect of the number 30
For example, Bouton and Castanheira (2012) consider information aggregation with more than two candidates.
27
of voters on the inference voters make about the state from being pivotal. To the best of our knowledge, this has not been observed before. The literature has identi…ed other circumstances in which information may fail to aggregate. Feddersen and Pesendorfer (1997) show such a failure in an extension (Section 6) when the aggregate distribution of preferences remains uncertain conditional on the realized state. Mandler (2012) demonstrates a similar failure if the aggregate distribution of signals remains uncertain. In these settings, the e¤ective state is multi-dimensional. Intuitively, this implies an invertibility problem from the relevant order statistic of the vote shares to payo¤-relevant states. A similar problem is identi…ed by Bhattacharya (2013), who observes the necessity of preference monotonicity for information aggregation. A recent generalization was made by Barelli and Bhattacharya (2013). Gul and Pesendorfer (2009) show that information aggregation fails when there is policy uncertainty. In our setting, conditional on the state, there is no aggregate uncertainty, preferences over policies are monotone in the state, and there is no policy uncertainty. Methodologically, information aggregation in elections is related to work on large auctions. Among others, this has been studied by Wilson (1977), Milgrom (1979), Pesendorfer and Swinkels (1997, 2000), and Atakan and Ekmekci (2014). These papers study auctions in which the number of bidders becomes large exogenously. Lauermann and Wolinsky (2012) introduce an auction model in which the number of bidders is random and endogenously state dependent. Related studies of voter (non-)participation in elections include especially Feddersen and Pesendorfer (1996), who identify the swing voters’ curse when voters can abstain, and the vast literature on costly voting, especially Palfrey and Rosenthal (1985) and Krishna and Morgan (2011, 2012). In these models, the number of votes cast depends on the state as well because participation decisions depend on the private signals of the voters. In Feddersen and Pesendorfer (1996), abstention facilitates information aggregation whereas in Krishna and Morgan (2011), the cost of voting helps increasing (utilitarian) welfare by screening according to preference intensities in a model with common and private values. In Krishna and Morgan (2012), voluntary voting also results in signal dependent participation which results in information aggregation across all equilibria. These models emphasize choice on the voters’side, showing how this can improve election outcomes, whereas our model emphasizes the organizer’s ability to a¤ect turnout and how it decreases e¢ ciency. Critically, in these models the underlying population of eligible voters is assumed to be independent of the state. Our companion paper Ekmekci and Lauermann (2016) allows for abstention in such models where the expected number of voters is state
28
dependent. A related paper that endogenizes the issues that are voted on by a strategic proposer is Bond and Eraslan (2010). Similar to us, they show that the unanimity rule may be superior to other supermajority voting rules. In their model, voting behavior under di¤erent rules has di¤erent implications for the proposals put on the table by a strategic proposer. In particular, the unanimity rule disciplines the proposer to make o¤ers preferred by the voters. In contrast, here the alternatives are …xed but the turnout is endogenously determined by a strategic organizer. Moreover, the unanimity rule restricts the organizer’s ability to create the asymmetry of voter turnout across the states. Finally, a large literature analyzes a con‡icted agent’s ability to manipulate one or more decision makers to act in favor of the agent’s interests, either through using informational tools or by taking actions that directly a¤ect the decision makers’ incentives. This includes models of cheap-talk, emanating from Crawford and Sobel (1982), which analyzes a biased sender’s ability to transmit information and induce behavior which is bene…cial to the sender. Our model shares with these models the feature that the organizer has superior information and has biased preferences. Our model di¤ers from the cheap talk literature since information transmission is not through cheap talk messages. The recent literature on Bayesian persuasion, initiated by Kamenica and Gentzkow (2011), and applied to a voting context by Wang (2012) and Alonso and Câmara (2016), assumes that a sender can commit to an information disclosure rule that generates public signals. Similar to that literature, we are interested in an agent’s ability to induce others to undertake his preferred action. However, the agent’s tools are di¤erent, and the agent cannot commit.
9
Conclusion
Understanding the performance of voting mechanisms to pick the best alternatives for society has always received attention, dating all the way back to the Athenian leader Cleisthenes and later to Condorcet. In this paper, we studied the ability of voting mechanisms to aggregate dispersed information among voters when the election takes place in the presence of an organizer who has the tools to change the turnout and whose interests are not aligned with those of the voters. Our main result is that such an organizer can in‡uence the election outcomes in his favor, and thus prevent information aggregation. This result indicates that although voting mechanisms may be very e¤ective in aggregating information, they may be quite
29
susceptible to manipulation activities by outsiders, and thus not robust. An interesting feature of our model is that small electorates in which the organizer is not allowed to intervene may perform better than large electorates with an organizer (in fact, a single voter would choose better than the electorate). More generally, we discuss two tools to design elections that are robust, a participation requirement— quorum— and a unanimity requirement. The organizer’s ability to get his desired outcome relies on his being able to recruit many voters. It does not rely on cherry-picking voters who have information supporting his favorite policy or voters who are a priori more inclined to vote for his favorite policy. In practice, however, many of the manipulation schemes involve the use of additional tools, such as the timing of elections or subsidies that target particular voters. Because in our model the organizer can a¤ect only the overall turnout and cannot distinguish between voters with di¤erent characteristics, our results suggest that the manipulation of elections may be even easier if we a¤orded the organizer additional targeting possibilities that can be found in practice.
30
A
Appendix
A.1
Miscellaneous Results
In this part, we explore several properties of the organizer’s best-reply correspondence and the critical likelihood ratio that are used in proving the theorems. Organizer’s Best Reply: The set of all mixed strategies for the organizer in the voting game with recruit~ (c). Given a generic mixed strategy n ment cost c is denoted by N ~ = (~ n ;n ~ ), the term n ~ ! (i) denotes the probability that the strategy n ~ assigns to integer i in state !. The organizer’s best-reply correspondence to the voter cuto¤ s when the recruit~ (c). Thus, n ~ 2 (s; c) ment cost is c is denoted by (s; c) := ( (s; c); (s; c)) N if and only if each positive integer that is in the support of n ~ ! solves
n2f
max
0;1;:::; 21 (N
2n+1 X
1)g i=n+1
2n + 1 (q! (s))i (1 i
q! (s))2n+1
i
nc.
We abuse notation and write n 2 (s; c) if the pure strategy n is optimal. Properties of (s; c):
Recall the marginal bene…t of an additional voter pair, (n
1; !; s^) =
If s^ is such that q! (^ s)
1 2n (q! )n (1 2 n
1=2, then
(n
1; !; s^)
q! )n (2q!
1) .
0 for every n. Thus,
s; c) ! (^
is
single-valued with n ~ ! (0) = 1. If q! (^ s) > 1=2, then limn!1
(n
(n
1; !; s^) > 0 and
(n
1; !; s^) >
1; !; s^) = 0. So, there is a unique n such that
(n; !; s^). Hence, the support of any n ~! 2
!
(n
(n; !; s^), and 1; !; s^)
c>
contains at most two integers, and if
it includes two integers, they have to be adjacent. We prove the following implication of the optimality condition c
(n!
1; !; x)
(n! ; !; x). Here, n! (x; c) is some pure best reply of the organizer if the voters
use the cuto¤ strategy x. We drop the arguments occasionally to save notation. De…ne s! to be the median signal, F (s! j!) = 1=2. Lemma 2. Given any x 2 (s! ; 1), c > 0, and n! 2 2c 2q!
1
2n! (q! )n! (1 n!
q! )n!
31
q! (1
! (x; c):
c q! ) (2q!
(n! + 1) : 1) (2n! + 1)
Proof. Rewriting the hypothesis, (n! ; !; x) 2n! + 1 (q! )n! +1 (1 q! )n! +1 (2q! 1) n! 2n! (q! )n! (1 q! )n! (2n! + 1) n!
c) c) c(n! + 1) q! (1 q! ) (2q!
1)
,
and 1 2n! (q! )n! (1 q! )n! (2q! 1) 2 n! 2n! (q! )n! (1 q! )n! n!
c) 2c 2q!
1
.
Taken together, the claim follows.
Using the lemma to bound the numerator and the denominator of the ratio of the pivotality probabilities yields, 3q (1
2q q ) 2q
1 1
2n n 2n n
(q )n (1 (q )
n
(1
q )n q
)n
2q 1 3q (1 q ) 2q
1 : 1
(8)
In this Appendix, we use Stirling’s approximation, lim p
n!1
n! 2 n
n n e
= 1.
(9)
Lemma 3. If x 2 (s! ; 1), then for any selection of pure strategy best replies by the organizer, fn (x; c); n (x; c)gc>0 for c ! 0,
n (x; c) ln (4q (1 = c!0 n (x; c) ln (4q (1 lim
q )) > 1: q ))
Proof. If q (x) 2 (0:5; 1), then q (x) 2 (0:5; 1). So, limc!0 n! (x; c) = 1 for ! 2 f ; g.
32
Rewriting the approximation for state , 2n (q )n (1 q )n n (2n )! (q )n (1 q )n (n !)2 p 2n 2 2n 2ne (q )n (1 p n n 2 2 n e (4q (1 q ))n p p . n
= = =
The approximation for state
q )n (10)
is similar; thus, (if the limit exists)
lim
2n n 2n n
(q )n (1
q )n
(11)
(q )n (1 q )n 0 1n r n @ (4q (1 q )) A = lim , n c!0 n (4q (1 q )) n c!0
where the equality is from Stirling’s approximation and the previous algebra. The bounds from Lemma 2 imply a uniform bound on the ratio (11), see (8). Using q! (x) 2 (0:5; 1), both sides of (8) are bounded and bounded away from zero. Hence, the ratio (11) must be bounded and bounded away from zero as well. This requires that lim
c!0
since otherwise, if limc!0
n n
n (x; c) = K 2 (0; 1) , n (x; c)
= 0, then the ratio (11) vanishes, and if limc!0
n n
= 1,
then the ratio explodes. That the ratio (11) is bounded requires therefore that (4q (1 q )) = 1. (4q (1 q ))K Solving this equation for K proves the lemma.
Critical likelihood ratio when the organizer uses a mixed strategy: We extend
from pure to mixed recruitment strategies (with a slight abuse of
notation) and de…ne
33
(s; piv; rec; n ~ ;^ s) :=
1
P f (sj ) i P f (sj ) i
~ 0n
(i)( 2i+1 N )
~ 0n
(i)( 2i+1 N )
2i i 2i i
q (^ s)i (1
q (^ s))i
q (^ s)i (1
q (^ s))i
.
(12)
Lemma 4. Fix s^ 2 (0; 1). For every s 2 [0; 1], max
n ~ 2 (^ s;c)
(s; piv; rec; n ~ ;^ s)
exists, and is attained by some pure strategy n 2 (^ s; c). The set of maximizers is independent of s. Similarly,
min
n ~ 2 (s;c)
(s; piv; rec; n ~ ;^ s)
exists, and is attained by some pure strategy n 2 (^ s; c). The set of minimizers is
independent of s.
Proof. The function
is continuous in n ~ , and the maximum of a continuous function
over a compact domain exists. Independence of the maximizers from s is seen by inspection of the function The extreme values of
.
are attained by a pure strategy n because, the numerator
and denominator of the equation 12 are linear functions of the weights on two adjacent integers, due to the property of the organizer’s best reply correspondence .
Operator ~ : De…nition 1. Let ~ : [0; 1] with x 2 ~ (^ s; c) if and only if
R+
R+ ,
(^ s; piv; rec; n ~ ;^ s) = x for some n ~ 2 (^ s; c) .
The mapping ~ takes a cuto¤ strategy s^ of the voters, calculates the best-reply correspondence of the organizer to s^, and then returns every number that is equal to the critical likelihood ratio of type s^ when all other voters follow the cuto¤ strategy s^ and the organizer is following a strategy that belongs to the set of best replies to s^. Note that ~ is well-de…ned at the boundaries s^ 2 f0; 1g since then (^ s; c) = f0g. Lemma 5. The correspondence ~ (^ s; c) is convex valued and upper hemicontinuous in its …rst argument s^ for s^ 2 (0; 1). 34
Proof. The best-reply correspondence, (^ s; c) is upper hemicontinuous in s^— which follows from Berge’s maximum theorem— and convex valued. The function (^ s; piv; rec; n ~ ;^ s) is continuous in n ~ . Moreover, because the densities f ( j!) are continuous for each
! 2 f0; 1g, the upper hemicontinuity of the organizer’s best-reply correspondence implies that ~ is upper hemicontinuous. Convex-valuedness of ~ follows from the convex-valuedness of (^ s; c), continuity of
in n ~ , and the fact that
is single-
dimensional.
The next lemma is immediate and we skip its proof. Lemma 6. An interior signal s 2 (0; 1) is an equilibrium cuto¤ signal of G(c) if and only if 1 2 ~ (s; c).
A.2
Proof of Theorem 1
The median signal in state ! is s! and satis…es q! (s! ) = F (s! j!) = 1=2.
Our proof strategy is that to …rst show that for all small c there is some s(c) > s + such that 1 2 ~ (s(c); c). This means there are equilibria in which
the voters support policy a with probability more than 1/2 in both states. The second part of the proof shows that in such equilibria, as c vanishes, a gets selected
with probability approaching 1, that the number of voters grows without bound, and that the organizer’s payo¤ approaches 1. We start by showing the existence of equilibria with a large cuto¤, utilizing two claims. We denote by max ~ (s; c) the highest element of the image of the correspondence ~ at (s; c). The maximum exists by Lemma 5. Claim 1: 9 > 0 such that lim max ~ (s + ; c) < 1. c!0
Claim 2: 9
c
> 0; with lim
c!0
c
! 0; such that lim max ~ (1 c!0
c ; c)
= 1:
These two …ndings together with the upper-hemicontinuity and convex-valuedness of ~ (Lemma 5) imply, via a version of the intermediate value theorem for correspondences,31 that for all c smaller than a cuto¤ c > 0, there is a s(c) 2 (s + ; 1 such that 1 2 ~ (s(c); c), which delivers the desired result. 31
c)
Claim 1 in the appendix of Shimer and Smith (2000) states an appropriate extension of the standard intermediate value theorem to convex valued, upper-hemicontinuous correspondences.
35
Claim 1: 9 > 0 such that lim max ~ (s + ; c) < 1: c!0
Proof of Claim 1: From Lemma 2, it follows that max ~ (s; c) =
max
(n ;n )2
(n
=
(n
f (sj ) (2n + 1) )f (sj ) (2n + 1) (s;c) (1
2n n 2n n
(F (sj )(1
F (sj )))n
(F (sj )(1
F (sj )))n
c f (sj ) (2n + 1) q (1 q )(2q max )f (sj ) ;n )2 (s;c) (1 (2n + 1) 2q2c
(n +1) 1) (2n +1) 1
2q 1 f (sj ) 2n + 1 max )f (sj ) 2n + 1 2q (1 q ) (2q ;n )2 (s;c) (1
n +1 . 1) 2n + 1
Note that max ~ (s; c) denotes the biggest element of the correspondence ~ . The term (n ; n ) denotes a pure strategy that puts probability 1 to integers n and n in states
and
respectively. Applying Lemma 3, we obtain that, for any …xed s
such that q (s) 2 (0:5; 1), 2q 1 f (sj ) (1 )f (sj ) 4q (1 q ) (2q
lim max ~ (s; c)
c!0
ln (4 (q ) (1 1) ln (4 (q ) (1
q )) . q ))
Note that the right side vanishes for s ! s from above, because q (s) ! 1=2
(while q (s) stays strictly larger than 1=2). So, there exists some " and c such that for all c
c , max ~ (s + "; c) < 1.
This proves Claim 1. Note also that the number 1 on the right-hand side of the inequality is arbitrary, and the same proof works to show that this inequality holds for any positive number. Proof of Claim 2: Let f (q) := 2q(1 that satis…es f (q (1 small, and limc!0
c
q)(2q
c
> 0 be a number that is close to zero
c > 0 is guaranteed when c is ~ = 0. We show that limc!0 max (1 c ; c) = 1.
The de…nition of 1
c ))
1), and let
c
= 2c. Existence of such a
is that in state , the organizer with a marginal cost c is
indi¤erent between recruiting no additional voters and recruiting one pair of voters.
36
Note that, f 0 (q) = 12q (1
q)
2.
and f 0 (q) < 0 for q su¢ ciently close to 1. Hence, for x close to 1, q (x) < q (x) implies whenever f (q (x)) = 2q (x)(1 q (x))(2q (x) 1) = 2c, f (q (x)) = 2q (x)(1 q (x))(2q (x) in state
1) > 2c. Hence, the organizer’s best reply to the cuto¤ 1
he recruits at least 1 pair and, actually as 1
This is because,
(1; ; x) =
4 2
q
(x)2 (1
q
(x))2 (2q
c
(x)
c
is that
! 1, exactly one pair.
1) < f (q (x)) = 2c for
all x su¢ ciently close to 1. Writing down the pivot probability in state , we get 2q (x)(1
q (x)), and if in state
probability in state and that
f (1j ) f (1j )
the organizer recruits no one, then the pivot
is 1. Because limx!1
q (x) q (x)
= 1, and limx!1
1 q (x) 1 q (x)
=
f (1j ) f (1j ) ,
> 0, we have that
lim max ~ (1
c!0
c ; c)
= lim
c!0
= 1:
f (1 1 cj ) (1 )f (1 c j ) 3 2q (1
1 q (1 c )(1
c ))
Combining Claims 1 and 2 and Lemma 5: Because ~ (s; c) is upper-hemicontinous and convex valued (Lemma 5), and combining this with Claims 1 and 2, it follows via the intermediate value theorem for correspondences (Footnote 31) that there is a c > 0 and c < c, there is an s(c) > s + such that 1 2 ~ (s(c); c).
> 0 such that for every
Hence, there is an equilibrium in which the voters support policy a with a probability more than 1/2. However, the theorem makes the stronger claim that the number of recruited voter pairs grows to in…nity. We now show this: Modifying the Proof of Claim 2 to ensure large turnout: In this part, we modify the second part of the above proof (i.e., the proof of Claim 2) to show that
c
can be chosen in such a way that the organizer, when
faced with voters using a cuto¤ 1 pairs of voters in state
c,
is indi¤erent between m(c) and m(c)
and recruits m(c) pairs of voters in state
limc!0 m(c) = 1.
1
, and that
The alternative mapping that we consider is xm (c), de…ned analogously as the
solution to
2m (q (x))m (1 m
q (x))m (2q (x)
37
1) = 2c.
As before, for any given c that is su¢ ciently small, for x = xm (c) 2m (q (x))m (1 q (x))m (2q (x) 1) m > 2c > 2m + 2 (q (x))m+1 (1 q (x))m+1 (2q (x) m+1
1).
Thus, we can pick some x ^m (c) just above xm (c) such that in state , the organizer recruits m
1 pairs of voters and in state
recruits m pairs of voters. As c ! 0, it
must be that x ^m (c) ! 1 and similar to before,
lim max ~ (^ xm (c); c) = 1.
c!0
Now consider a sequence of equilibria whose existence has been shown with cuto¤s bounded away (above) s ; i.e., limc!0 s(c) = s > s . If s < s < 1 then limc!0 n! (s(c); c) ! 1. If s = 1 then note the following:
Consider the function g(m; x) =
2m m
xm (1
x)m (2x
1) where m is a positive
integer and x 2 [0; 1]. There is some " > 0 such that g(m; x) is decreasing in x and m in the region where x > 1
".32 This property of the function g together with
the property for the equilibrium cuto¤ s(c) that, s(c) that limc!0 s(c) = 1 together imply that n! (s(c); c)
xm (c), and the hypothesis m(c). Since this exercise can
be repeated for any arbitrary m, we can pick the sequence m(c) in such a way that it grows unboundedly. Therefore, the resulting equilibrium turnout grows without bound. Showing that policy a gets selected: Let s (c) denote the equilibrium cuto¤ sequence from the previous parts of this proof. By construction, 1 > s (c) > s + " for all c smaller than c > 0. We show that the probability of the majority voting for policy a approaches 1 as c ! 0. Without loss of generality, suppose s(c) converges. The claim is obvious if lim s(c) = 1.
c!0
If not, then 1 > lim s(c) c!0
32
s +"
To see that g(m; x) is decreasing in m for all x > 1
(2m+2)(2m+1) x(1 (m+1)m
" for some " > 0, notice that
g(m+1;x) g(m;x)
=
x) < 1 for all m when x is close to 1. To see that g(m; x) is decreasing in x, …rst notice that if g(1; x) is decreasing in x then so is g(m; x) for any m 1. Showing that g(1; x) is decreasing in x for x close to one is straightforward.
38
implies lim q (s (c)) > lim q (s (c)) > 0:5,
c!0
c!0
and limc!0 n! (c) ! 1. This implies the claim for the second case because the
weak law of large numbers applies. Thus, policy a gets implemented as c vanishes for this sequence of equilibria. Showing that organizer’s payo¤ is 1 in both states: Let UOc (s(c); n ~ (c)) denote the organizer’s equilibrium payo¤ in the election in which the marginal recruitment cost is c and the equilibrium strategy pro…le s(c); n ~ (c) is the one identi…ed in the previous parts. Consider the following alternative strategy n(c) := (b p1c c; b p1c c); i.e., n(c) is the strategy in which the organizer invites b p1c c
pairs of voters in both states. As c ! 0 the recruitment cost incurred by the organizer given strategy n vanishes. Moreover, because c ! 0, the number of recruited
voters goes to 1, and because s(c) ! s > s + , by the weak law of large numbers
the probability that the majority votes for policy a approaches 1 when the organizer employs strategy n(c). Hence, limc!0 UOc (s(c); n(c)) ! 1. Because n ~ (c) is a best reply to voter cuto¤ strategy s(c), it has to be that UOc (s(c); n ~ (c))
UOc (s(c); n(c));
for every c > 0. Therefore, limc!0 UOc (s(c); n ~ (c)) = 1, as well. Since in each state the organizer’s payo¤ is bounded above by 1, and since each state occurs with positive probability, the organizer’s payo¤ conditional on each state converges to 1, as well. Lemma 7. Suppose (x (c) ; n ~ (c)) is a collection of interior equilibria given c and suppose n (x(c); c) and n (x(c); c) are in the support of n ~ (c) with limc!0 x (c) > s and limc!0 n (x(c); c) = limc!0 n (x(c); c) = 1, then 0 < lim inf c!0
n (x(c); c) n (x(c); c)
lim sup c!0
n (x(c); c) n (x(c); c)
1.
Proof. If limc!0 x(c) = x such that q! (x) 2 (0:5; 1) for ! 2 f ; g, then the ratio of the number of voters in the two states stays bounded as c goes to zero by a
straightforward adjustment of the proof of Lemma 3 using only the organizer’s optimality condition. By hypothesis, this leaves limc!0 x(c) = 1. We show that n n
! 1 in this case. Using the previous observations from the proof of Theorem 1, 2n + 1 c!0 2n + 1 lim
2n n 2n n
(q )n (1
q )n
n
q )n
(q )
(1
= lim
c!0
r
n (4q (1 n (4q (1
q ))n : q ))n
(13)
This approximation stays correct for limc!0 q! (x(c)) = 1, ! 2 f ; g, since we
approximate only the binomial terms. Abbreviate f! := f ( j!). With 39
(c) :=
1
x (c), we have 1
q! (x (c)) =
f! (1) by the continuity and boundedness of f! .
Taking logs of the expression on the right-hand side of (13), with f! = f! (1), ln =
r
n n
1 ln n 2
= n ln
)n )n
(4 f (4 f 1 ln n 2 1 2 ln n n ln
+ n ln 4f + n ln n ln 4f n ln ! ! 1 ln n ln 4f ln 4f 2 + +1 n ln + +1 : ln n ln ln
From n! ! 1 for ! 2 f ; g and ln brackets vanish, and we have lim ln
c!0
r
! ( 1), the …rst two terms in each of the
(4 f )n (4 f )n
n n
= lim (n c!0
Case 1. Suppose lim nn > 1. Then ln imply that lim ln
c!0
r
n n
n ) ln .
! ( 1) and lim (n
(4 f )n (4 f )n
=
n ) = (+1)
1.
Hence, limc!0 (ln(13)) = 1 implies that the limit of (13) is 0. Note that (13) is proportional to ~ . Therefore, lim ~ (x (c) ; c) = 0, in contradiction to x (c) being an interior equilibrium, requiring ~ (x (c) ; c) = 1. Case 2. Suppose limc!0
n n
( 1) imply that lim ln
c!0
< 1. Then, ln r
n n
! ( 1) and limc!0 (n
(4 f )n (4 f )n
n )=
= +1.
Hence, the limit of (13) is +1. Therefore, limc!0 ~ (x (c) ; c) = 1. This is again in contradiction to x (c) being an interior equilibrium. n n
Thus, if limc!0 x(c) = 1 then limc!0
A.3
= 1, as claimed.
Proof of Theorems 2 and 3
This proof characterizes all limit points of non-trivial equilibrium cuto¤s. The proof of the theorems proceeds through a sequence of steps that are combined at the end of the section. To simplify some of the expressions, we sometimes omit the argument c. Moreover, we sometimes omit s (c) as well in expressions like q! = q! (s(c)).
40
We also use the following lemma. Lemma 8. Let fs(c); n (c); n (c)gc>0 be a selection of cuto¤ s s(c) for the voters, and a pair of integers (n (c); n (c)) that are in the support of the organizer’s best reply to voter strategy s(c) with recruitment cost c. If 1 > limc!0 s(c) > s! , then (dropping the dependence of n! on c) lim (2n! + 1)
c!0
2n! q! (s(c))n! (1 n!
q! (s(c)))n! = 0
for ! 2 f ; g .
Proof. First, if 1 > lim s(c) > s! , then 1 > lim q! (s(c)) = q
> 1=2 implies
lim n! (c) = 1. By Stirling’s approximation (see (10)), lim
2n! n!
q! (s(c))n! (1
c!0
(4q (1
q! (s(c)))n!
q ))n!
Because 1 > q > 1=2, we have 4q (1
=
(4q! (1 q! ))n! p p n! lim c!0 (4q (1 q ))n!
= 0.
(14)
q ) < 1. This and n! (c) ! 1 imply33
lim (2n! + 1)(4q (1
c!0
q ))n! = 0:
Combining this with (14) delivers the result.
Step 1: To show s is the only possible limit point of non-trivial equilibria that is not above s . There are 3 cases to consider and rule out: s < s , s 2 (s ; s ), and s = s .
The …rst two cases are easier to rule out while the last case is more subtle. We deal with the …rst two cases …rst. Case 1. Suppose s < s . If this is true, then the probability that a randomly selected voter supports policy a is strictly less than 1=2 in both states, and the organizer recruits no one. However, this is a trivial equilibrium. Case 2. Suppose s < s < s . Then, the organizer recruits no one in state many voters in state
and
as k ! 1. In fact, because s > s , q (s(c)) ! q > 1=2, and
therefore, in any sequence of equilibria, for any selection of integers n (c) that are in the support of the equilibrium recruitment strategy of the organizer, n (c) ! 1. Therefore, by Lemma 8 above, (2n (c) + 1) 33
2n (c) q (s(c))n n (c)
(c)
(1
q (s(c)))n
Recall nxn ! 0 for any …xed number x 2 (0; 1) and n ! 1.
41
(c)
! 0:
Therefore, max ~ (s(c); c) ! 0, which is a contradiction to s(c) being an equilib-
rium cuto¤.
Case 3. Suppose s = s . We argue that this cannot be the case either, by showing that lim max ~ (s(c); c) = 0:
c!0
First, note that lim q (s(c)) = q (s ) > 1=2;
c!0
and this implies 2n n
lim (2n + 1)
c!0
q n (1
q )n = 0:
for every sequence of integers n (c) in the support of the organizer’s best reply, via Lemma 8. Thus, there cannot be any subsequence of cuto¤s in which q (s(c)) 1=2. This is because, otherwise, along such a subsequence, n (c) = 0, and hence lim max ~ (s(c); c) = 0. Therefore, consider a subsequence along which q > 1=2 (so, q ! 1=2 from above). Recall that max ~ (s(c); c) is attained by some pure strategy that is in (s(c); c). Denote such a pure strategy with a pair of integers (n ; n ) that correspond to the integers in the support of the strategy in states
and , respectively.
These integers depend on c, but for the ease of reading we drop the dependence of these integers on c. We now bound limc!0 max ~ (s(c); c) from above, by either putting a lower bound on the multiplication of two terms on the denominator, which is (2n + 1)
2n n
q n (1
q )n ;
or by directly arguing that (2n + 1) (2n + 1)
2n n 2n n
q n (1 q
n
(1
q )n q )n
! 0:
For any given q > 1=2, the function f (q; n) := (2n + 1)
2n n
q n (1
q)n can have
at most one peak, when viewed as a function of n. This is because, 2n + 3 (2n + 2)(2n + 1) f (q; n + 1) = q(1 f (q; n) 2n + 1 (n + 1)2 A simple calculation shows that the expression for
42
q) =
4n + 6 q(1 n+1
f (q;n+1) f (q;n)
q):
is a strictly decreasing
function of n. When q is su¢ ciently close to 1/2, f (q; n) is strictly increasing in n at n = 0. Therefore, for every nonnegative integer N , the minimum of f (q; n) in the domain n 2 f0; 1; :::; N g is attained at one of the extreme points, i.e., either at n = 0 or n = N .
We consider two subsequences. First, consider any in…nite subsequence along which n (c)
n (c). From the above argument when c is small, f (q (s(c)); n (c)
minff (q (s(c)); n (c); f (q (s(c)); 0)g:
Along a subsequence at which the above minimum is attained at n = 0, f (q (s(c)); 0) = 1, and hence our claim follows. This is because,
(2n + 1)
2n n
q n (1
q )n
min
n2f0;1;:::;n g
f (q ; n)
min
n2f0;1;:::;n g
f (q ; n) = 1,
where the …rst inequality follows from the de…nition of f (q; n), and the second one follows from the property of the subsequence that n
n , and hence the min is
taken over a larger set. This together with lim (2n + 1)
c!0
2n n
q )n = 0,
q n (1
delivers that limc!0 max ~ (s(c); c) = 0 along such a sequence. Along the remaining sequence along which the minimum of the above expression is attained at f (q (s(c)); n (c)),
(2n + 1) (2n + 1)
2n n 2n n
q n (1 q
n
(1
q )n
(2n + 1)
q )n
(2n + 1) q n (1 q n (1
= the last line follows from the facts that q (1 q (1
q ) < q (1
q ) because q > q
2n n 2n n )n
q n (1
q )n
q n (1
q )n
q ! 0; q )n
q ) < q (1
q ) and n ! 1. Here,
1=2.
Now the only remaining subsequence is the one along which n (c) < n (c). For such a subsequence, notice that the optimality of the organizer’s best reply delivers: 2n n
q
n
(1
q )n
and 43
2c 2q
1
;
2n n
q n (1
q )n
Notice that, 2q
2q (1
c q )(2q
n+1 1) 2n + 1
2q (1
c q )(2q
1)
.
1 ! 0, and combining this with n (s(c)) < n (s(c)) delivers lim
c!0
(2n + 1) (2n + 1)
2n n 2n n
q n (1 q
n
(1
q )n q )n
= 0,
which then implies that limc!0 max ~ (s(c); c) = 0 along such a sequence as well. Step 2: To show s is an attainable limit point. The proof strategy here is similar to the proof for the existence of manipulated equilibria in Theorem 1. Using Lemma 7 (above) as before, it is straightforward to show that there exists some small " > 0 such that for every 0 < " < ", limc!0 max ~ (s +"; c) = 0. We show that there is an "(c) > 0 with limc!0 "(c) = 0, such that limc!0 max ~ (s +"(c); c) = 1. Then, the intermediate value theorem for correspondences (Footnote 31) implies
that for small c an equilibrium exists that has a cuto¤ s(c) 2 (s +"(c); s +"). By the
previous step, the limit point of s(c) has to be s . So all we have to show is that there is a mapping "(c) > 0 with limc!0 "(c) = 0 such that limc!0 max ~ (s + "(c)) = 1. Note that in state , the organizer recruits no one, for "(c) su¢ ciently small. So
our task is to show that (2n + 1)
2n n
q n (1
q )n can be made arbitrarily large,
for small c, with cuto¤ s + "(c). Given any integer a and su¢ ciently small c, let s(a; c) > s be such that the organizer is indi¤erent between recruiting a and a + 1 pairs of voters when their cuto¤ strategy is s(a; c). In particular, g (a; x) = 2c at x = q (s(a; c)), with g de…ned as in the proof of Theorem 1. Since g (a; 1=2) = 0, g (a; q) > 0 for q > 1=2, and g is continuous in q, we can select s(a; c) such that limc!0 s (a; c) ! s for every …xed integer a.
Let s(a) > s be equal to minf~ s(a); s + "g, where s~(a) is the largest signal that has the property that, for every q 2 [1=2; q (~ s(a))] 2
1)2 q)
a(2q q(1
0.
For every a, such a s~(a) > s exists, by inspection of the inequality. Moreover, for su¢ ciently large a, s(a) = s~(a) < s + ". Note that, limc!0 max ~ (s(a); c) = 0. Moreover, limc!0 max ~ (s(a; c); c) =
44
p O( a), which follows from Stirling’s approximation, 4a q (s(a; c))) = a p p
2a a q (s(a; c))a (1 a
a
a
1 2
a
1 2
a
;
so that lim lim max ~ (s(a; c); c) = 1.
a!1 c!0
Therefore, by the intermediate value theorem for correspondences, for each suf…ciently large a, there is a c such that for all c < c, there is s (a; c) 2 (s(a; c); s(a)) such that 1 2 ~ (s (a; c); c).
Step 3: To show that an equilibrium sequence exists whose limit point is s and
for which in state , the majority selects policy a with probability 1 in the limit. Note that if s = s , then in state
no one is recruited, and hence there is only
one voter for every c. Therefore, as c ! 0, the term (2n + 1)
2n n
q n (1
q )n
converges to a number k 2 (0; 1). Now consider the cuto¤ s (a; c) de…ned in Step 2, above. Note that the organizer’s best reply to this cuto¤ in state
is to recruit at
least a voters, for c small enough. This is because, s (a; c) < s~(a), where for every q 2 [1=2; q (~ s(a))], 2
1)2 q)
a(2q q(1
0:
The marginal bene…t of the organizer is @
2a a
(q(1
q))a (2q @q
1)1=2
=
1 2a (q(1 2 a
q))a [2
a(2q q(1
1)2 ] > 0, q)
for every q 2 [1=2; q (~ s(a))]. Hence, the support of the organizer’s best reply at state
is bounded from below by a pairs of voters, whenever the voters are using
a cuto¤ between s(a; c) and s~(a). Because the equilibrium cuto¤ s (a; c) that we identi…ed is in that interval, the organizer indeed recruits at least a pairs of voters in state . Since a is arbitrary, we can construct a sequence of equilibria along which s(c) !
s , and the number of voters recruited in state
grows without bound.
Now we show that if s(c) ! s and if the number of voters in state
grows
without bound, then the majority selects policy a with a probability that converges to 1. As we stated in the previous paragraph lim (2n + 1)
c!0
2n n
q n (1
q )n = k 2 (0; 1),
where n and q depend on s(c) and c, but we dropped the dependence. Because 45
n ! 1, s(c) > s . The probability that the majority selects policy a in state 2n +1 X
i=n +1
2n + 1 i q (1 i
q )2n
+1 i
is:
:
To show that this probability converges to 1, we use the following lemma. Lemma 9. Let fq(c)gc>0 be a selection of probabilities with limc!0 q(c) ! q , and fn(c)gc>0 be a selection of integers such that limc!0 n(c) ! 1 . If lim (2n(c) + 1)
c!0
2n(c) q(c)n(c) (1 n(c)
q(c))n(c) = k 2 (0; 1),
then n(c) X 2n(c) + 1 lim q(c)i (1 c!0 i
i
q(c))2n(c)+1
= 0.
i=0
Proof. Pick any pair q; n. Let t(i; n) :=
2n+1 n+1 (1 q)n n+1 q 2n+1 i q (1 q)2n+1 i i
Note that t(i; n) > 1 for i
=(
q 1
q
)n+1
i (2n
+ 1 i)(2n i):::(n + 2) : n(n 1):::(i + 1)
n because q > 1=2. Moreover, t(i; n) is decreasing
in i. Pick an arbitrary
> 0. Let 1 + ( ) be a lower bound strictly larger than 1 for
the term
For i
(1
2n + 1 n(1 2 )n, we have that t(i; n)
n X 2n + 1 q(n)i (1 q(n))2n+1 i
i
(n(1 )) : )+1 (1 + ( )) n . Therefore,
((n(1 2 ))(1+ ( ))
i=0
Taking n ! 1, and then using the fact that (2n(c) + 1)
2n(c) q(c)n(c) (1 n(c)
n
+2 n)
2n + 1 n+1 q (1 q)n : n
was arbitrary, and the fact that q(c))n(c) ! k 2 (0; 1)
delivers the result.
Step 4: To prove that if Inequality (6) holds, then there is an equilibrium with limit cuto¤ s and with a bounded number of voters.
46
Note that ~ (s ; c) is single valued for every c > 0, and that value is equal to 1
f (s j ) f (s j ) .
This is because (s ; c) has a single element for every c > 0, and this
single element is a pure strategy that recruits no one in both states. Hence, if inequality (6) holds, then max ~ (s ; c) 1, for every c > 0. By the argument in Step 2, there are some a and c > 0 such that for every c < c, max ~ (s(a; c); c) > 1. Therefore, by the intermediate value theorem for correspondences, there is some s(c) 2 [s ; s(a; c)] such that 1 2 ~ (s(c); c). Because s(c) < s(a; c), and because for all su¢ ciently small c, s(a; c) < s~(a), and because
s(a; c) is the cuto¤ signal to which the organizer’s best reply is to recruit at most a + 1 pairs of voters in state of voters in state
, the organizer recruits not more than a + 1 pairs
when the voters use the cuto¤ s(c). Because this is true for
every c < c, and because limc!0 s(a; c) = s , we can construct a sequence of equilibrium cuto¤s that converge to s , and along such equilibria, the organizer recruits a bounded number of voters in state
(and no one in state ).
Step 5: To prove that if inequality (6) is not satis…ed, then in all sequences of equilibria with limit cuto¤ s the number of voters diverges. On the way to a contradiction, suppose that there is an equilibrium sequence with limit cuto¤ s , which has a bounded number of voters in state , say less than k. Notice that, lim inf c!0
X
n ~ (c)(i)
(2i + 1)
i 0
2i (q (s(c)))(1 i
q (s(c)))i
1;
where n ~ (c) is the equilibrium strategy of the organizer. This is because, …rst, q (s(c)) ! 1=2, second, (2i + 1)
2i i
(1=4)i is strictly increasing in i, and third,
n ~ (c)(i) = 0 for every i > k. Moreover,
f (sj ) f (sj )
is strictly decreasing in s, and hence,
for every s > s , 1
f (sj ) > 1: f (sj )
However, this contradicts the equilibrium requirement that 1 =
(s(c); piv; rec; n ~ (c); s(c)).
Combining the steps to prove Theorems 2 and 3 and Footnote 22: Theorem 2.1 is implied by Theorem 1. Theorem 2.2 is implied by Step 2. Theorem 2.3 is implied by Step 1. Theorem 3 is implied by Step 3 and Step 5. Footnote 22 is implied by Step 3 and Step 4.
47
A.4
Proof of Theorem 4
Proof. As we argued in the main text after the statement of the theorem, Item 3 follows from Theorem 1. Item 2 follows from Theorem 5, noting that if mk diverges su¢ ciently quickly, then the organizer does not add additional voters. Therefore, this corresponds to the case
= 1. In the remainder of the proof, we show that
there is a sequence of equilibria that aggregates information. We now construct a sequence of equilibrium cuto¤s, s~k 2 (s ; s ), such that information aggregates along such equilibria. Consider the mapping ~ k (s; c) that modi…es the mapping ~ (s; c) by incorporating that the minimum number of voters is 2mk + 1, indicated by the superscript k. For large enough k and all sk 2 (s ; s ),
we have n ~ (sk ; ck ) = mk , i.e., the organizer recruits no one in state . If n ~ (sk ; ck ) = mk ; i.e., if in state
the organizer also recruits no one, then it is an equilibrium
that the organizer recruits no one. If n ~ (sk ; ck ) > mk , then because (2m + 1)
2m m
q (sk )m (1 q (sk ))m is decreasing
in m when m is large and sk is su¢ ciently close to some s 2 (s ; s ), we have ~ k (sk ; ck ) < 1. Moreover, ~ k (s ; ck ) > 1 when k is su¢ ciently large. Hence, there is a s~k
s
such that s~k is an equilibrium cuto¤ of the voting game with the
organizer. The proof that when s~k > s , as k ! 1, in state
the probability
that policy a is implemented converges to 1 follows identical reasoning as Lemma 9, so we skip it. Clearly, in state , the organizer recruits no new voters, and since mk ! 1, in state , policy b is implemented with a probability that approaches 1. Thus, information is aggregated in both states.
A.5
Proof of Theorem 5 nk nk
Proof. By hypothesis,
=
for some
> 0. Let s be a limit point of some
sequence of cuto¤s sk 2 (0; 1). Using Stirling’s approximation, if 0 < s < 1, then the limiting probability of being pivotal is (see Equations (9) and (11))
lim
k!1
2nk nk 2nk nk
(q
k
sk )n (1 k
(q (sk ))n (1
q
k
sk )n
k
q (sk ))n
1 = lim p k!1
(4q (s ) (1 4q (s ) (1
q (s ))) q (s ))
nk
. (15)
Moreover, for s = 0, the limit is = 0 if limit is = 0 if
1 and = 1 if
> 1 and = 1 if
< 1. To see why, suppose
1. For s = 1, the sk
! 0. Abbreviate
F! := F ( j!) and f! := f ( j!). Then, using the continuity of F! and f! (0) > 0, 48
the limit of the left-hand side is 4F
1 lim p
k!1
sk
1
nk
sk
F
nk
(4F (sk ) (1
1 = lim p k!1
F (sk )))
The limit on the right-hand side is = 0 if argument for
sk
4
! 1 is analogous.
s^k (f (0)) 4^ sk f (0)
> 1 and = 1 if
!nk
.
1, as claimed. The
The observations for sk ! 0 and sk ! 1 imply in particular that there can be
no voting equilibria with limit point s = 0 or s = 1.
Recall that s^k is an interior equilibrium if and only if the cuto¤ satis…es (sk ; piv; rec; nk ; sk ) = 1. From the previous approximation, the critical likelihood is (dropping s as an argument from q! here and in the following), lim
k!1
k
k
k
(s ; piv; rec; n ; s ) =
Take any
1
f (s j ) p lim k!1 f (s j )
(4q (1 4q (1
q )) q )
nk
:
> 0. Take any sequence of cuto¤s sk . If the limit s = s , then
(sk ; piv; rec; nk ; sk ) ! 0. This follows since then 4q (1 q ) = 1 and 4q (1 q ) <
1, given nk ! 1. If the limit s = s , then (sk ; piv; rec; nk ; sk ) ! 1. This follows
since then 4q (1 q ) = 1, 4q (1 q ) < 1, and nk ! 1. Hence, given the continuity of
for any …xed k, an application of the intermediate value theorem for functions
delivers that given nk , for all k su¢ ciently large, there must exists some s^k such that s
< s^k < s and
(^ sk ; piv; rec; nk ; s^k ) = 1. Thus, there always exists an
equilibrium that aggregates information. (An alternative argument follows from McLennan (1998). Note that information aggregates given any sequence of strategy pro…les with cuto¤s sk with s < limk!1 sk < s . Therefore, the voters’common interest implies that there must be an equilibrium cuto¤ s^k in which voters do at least as well as with sk , establishing the existence of an equilibrium sequence with information aggregation.) Suppose
= 1. Suppose s^k is an interior voting equilibrium. This requires that 4q s^k (1 k!1 4q (^ sk ) (1 lim
q s^k ) = 1. q (^ sk ))
From the observation following (15), s 2 = f0; 1g. Since s 2 (0; 1), then clearly 4q (s ) (1
q (s )) = 4q (s ) (1
q (s )). Therefore, q (s )
1=2 = 1=2
q (s ),
which implies that s 2 (s ; s ). This proves part 1 of the theorem. Suppose that < 1. From before, if the limit s = s , then (sk ; piv; rec; nk ; sk ) !
0. If the limit s = 1, then,
(sk ; piv; rec; nk ; sk ) ! 1. Hence, an application of 49
the intermediate value theorem delivers that given nk , for all k su¢ ciently large, there must be some s^k such that 1 > s^k > s and limk!1 (^ sk ; piv; rec; nk ; s^k ) = 1. Conversely, suppose
s^k
is a sequence of equilibria with limit s for
cannot be that s = s or s = 1, since for any such sequence
< 1. It
would vanish to
zero or diverge to 1. It also cannot be that s < s since for 0 < s < s we have q (s^k )(1 q (s^k ) > 1, and therefore ! 1. Similarly for s = 0. Thus, if < 1, it q (s^k )(1 q (s^k )) must be that either s 2 (s ; s ) or s 2 (s ; 1), where information is aggregated in the …rst case and policy b implemented in both states in the second case. This proves the second part of the theorem. The case
> 1 is symmetric to
< 1. This …nishes the proof.
Remark 4. The proof implies Observation 2 because Stirling’s approximation also works if limk!1 nk =nk =
< 1 but nk 6= nk along the sequence. In addition, a dif-
ferent method of proof can be used to consider sequences for which limk!1 nk =nk = 1 but nk < nk and to show that information aggregation can fail in such cases as well depending on the speed at which nk =nk ! 1; see Case 1 from the proof of Lemma 7.
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