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Mixed Motives and the Optimal Size of Voting Bodies

John Morgan University of California, Berkeley

Felix Va´rdy University of California, Berkeley, and International Monetary Fund

We study a Condorcet jury model where voters are driven by instrumental and expressive motives. We show that arbitrarily small amounts of expressive motives significantly affect equilibrium behavior and the optimal size of voting bodies. Enlarging voting bodies always reduces accuracy over some region. Unless conflict between expressive and instrumental preferences is very low, information does not aggregate in the limit, and large voting bodies perform no better than a coin flip in selecting the correct outcome. Thus, even when adding informed voters is costless, smaller voting bodies often produce better decisions.

I. Introduction Why do people vote? Obviously, people vote to influence election outcomes. Yet these “instrumental” motives ðas they are known in the literatureÞ are unlikely to be the sole reason. Voters influence an election when they are pivotal, and this occurs only rarely, even with a modest number of voters. The fact that many people nonetheless bother to cast a ballot suggests that they also derive some consumption utility—known as “expressive” We thank Mu Cai for the proof of lemma 16, as well as Andreas Billmeier, Avinash Dixit, Mitch Hoffman, Meg Meyer, and seminar participants at the University of California at Berkeley and Davis, Penn State, University College London, Cambridge, Oxford, Nottingham, Bielefeld, and the 2010 conference of the European School on New Institutional Economics for helpful comments. Morgan gratefully acknowledges the support of the National Science Foundation. [ Journal of Political Economy, 2012, vol. 120, no. 5] © 2012 by The University of Chicago. All rights reserved. 0022-3808/2012/12005-0002$10.00

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motives—from the act of voting itself, independent of the outcome of the election. The distinction between expressive and instrumental motives is immaterial from a modeling perspective if ðiÞ expressive utility depends only on whether one votes and not on how one votes—this is the situation described by Riker and Ordeshook’s ð1968Þ famous D ðdutyÞ term—or ðiiÞ expressive motives always coincide with instrumental motives. In either case, the standard assumption that voters have only instrumental preferences is harmless because, conditional on showing up, voters behave as if they had only instrumental preferences. As argued by Fiorina ð1976Þ and others, usually people’s expressive utility depends not only on whether they vote but also on how they vote. This is most apparent under an open ballot, where your vote may affect how other people perceive you. The secret ballot shelters a voter from the perceptions of others but does not insulate him from self-perceptions. Therefore, expressive utility may also depend on whether a vote is consistent with personal norms, values, and identity. While intrinsic expressive payoffs might be small, for most voters they are arguably not zero.1 In addition, instrumental and expressive payoffs may not always be perfectly aligned. This is obvious in legislative settings, where representatives may be torn between casting a vote that is “right” from a public policy perspective and casting a vote that plays better with constituents back home. But conflict may arise even at the level of intrinsic expressive motives. To illustrate how this conflict plays out, consider the situation of Republican legislators in the fall of 2008. The financial markets provided worrying signs that a bank bailout would be needed to prevent a collapse of the financial system. However, the idea of government intervention on such a scale was antithetical to the values of these legislators and their constituents. Clearly, casting a vote in favor of the bailout would be personally distasteful and politically costly. Faced with this conflict between instrumental motives ðenacting the bailout was instrumentally beneficialÞ and norms and values ðvoting for government intervention would be morally wrongÞ, the ideal outcome for most Republican legislators was for the bailout to pass while voting against it.2 We show that the resolution of tension between instrumental and expressive motives crucially depends on the size of the voting body. In small voting bodies, there is a reasonable chance that a single vote affects the 1 A growing body of evidence suggests that intrinsic motives to behave in line with aspirational views of oneself are, in fact, not that small. See, e.g., Gneezy and Rustichini ð2000Þ and others. Akerlof and Kranton ð2010Þ offer a rich account of the connection between selfperception and behavior. 2 Indeed, this is largely how H.R. 1424, the Emergency Economic Stabilization Act, played out. The overwhelming majority of Republicans ð145 out of 198Þ voted against it, while only three Democrats were opposed. Nonetheless, the bill passed 268–148.

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election outcome and, hence, instrumental motives dominate. In large voting bodies, the chance of a single vote being decisive is negligible, and expressive motives come to the fore. Finally, for bodies of intermediate size, instrumental and expressive motives jointly determine voting behavior. The idea that the size of a voting body can affect voting behavior has a long history in political economy. In Federalist Paper no. 58 ðHamilton, Jay, and Madison ½1787–88 2000, 376Þ, James Madison pointedly observed that “the more numerous an assembly may be, of whatever characters composed, the greater is known to be the ascendancy of passion over reason.” While Madison viewed this outcome as something to be guarded against, as long as passion and reason are correlated, we show that “passionate” voting need not produce poor policies. Returning to the example of the bailout, what are the chances that a voting body will make the instrumentally correct decision ði.e., bailing out the banks if necessary to save the financial systemÞ? How does it depend on the size of the voting body, and how important is the prevalence of conflict between instrumental and expressive preferences? To study these questions, we amend the canonical Condorcet jury model and incorporate expressive preferences into the voting calculus. We are agnostic about the source of expressive motives ðwhether they are extrinsic or intrinsicÞ and, more importantly, about the degree to which they affect overall payoffs. Even when expressive motives constitute only a small share of payoffs, we show that they dominate all other considerations in large elections. Put differently, when voters have any expressive preferences at all, the reduced-form model of purely instrumental payoffs produces misleading results. Indeed, our main contribution is to show that many results from standard voting models are overturned, or in need of amendment, when expressive motives are a consideration, no matter how small. A key variable, which we call malleability, is the degree to which expressive preferences are influenced by “facts,” that is, information as to what is instrumentally the better option. For example, in the face of a potential meltdown of the financial system, some Republican legislators ðor their constituentsÞ may come around to viewing a bank bailout as the right thing to do. Others may be more rigid in their ideology, regardless of the potential consequences. We show that purely expressive voting can still produce good outcomes if expressive preferences are sufficiently malleable: the Condorcet jury theorem holds and large voting bodies make the correct decision with probability one, despite the fact that no one is voting instrumentally. However, when expressive preferences ðor “norms”Þ are relatively impervious to facts, large voting bodies do no better than a coin flip. This might seem to imply that voting bodies should be as large as possible when expressive motives are malleable and small when they are not. In fact, this would be a mistake. Even when expressive motives are

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989

malleable such that, in the limit, large voting bodies make the correct decision, there always is a ðpotentially largeÞ region where larger voting bodies produce worse decisions. For intermediate-sized voting bodies, the informational gains from adding more voters may be swamped by informational losses from more expressive voting. The paper proceeds as follows. In the remainder of this section, we place our findings in the context of the extant literature. In Section II we present the model. Section III characterizes pure strategy equilibria, while Section IV provides a complete equilibrium characterization. Section V studies the quality of decision making as the size of the voting body grows. Finally, Section VI presents conclusions. All proofs are relegated to the Appendix. Related literature.—The idea that voters must be motivated by considerations other than the purely instrumental dates back to, at least, Downs ð1957Þ. Subsequently, many authors have proposed adding expressive motives to the voting calculus, primarily as a way of explaining turnout. While Riker and Ordeshook ð1968Þ develop an early version of this idea, Fiorina ð1976Þ, Brennan and Buchanan ð1984Þ, Brennan and Lomasky ð1993Þ, Feddersen and Sandroni ð2006Þ, and others have also offered models along these lines. The rationales for expressive voting have varied across models. They include duty, identity, norms, and various other considerations. ðSee Hamlin and Jennings ½2011 for a survey.Þ Our model is in the same spirit. However, we can be agnostic about the exact rationale for expressive voting. Even though we shall couch expressive motives in terms of norms, the particular source driving expressive payoffs is of little consequence for our analysis. All that matters is that voters derive some consumption utility from voting in a particular way and that this utility is less than perfectly correlated with their instrumental utility. Coate and Conlin ð2004Þ and Coate, Conlin, and Moro ð2008Þ present empirical evidence for the importance of noninstrumental considerations in voting. There is also a growing experimental literature on expressive voting. While expressive motives received only weak support in early studies ðsee, e.g., Tyran 2004Þ, more recent studies offer significant evidence. For instance, Feddersen, Gailmard, and Sandroni ð2009Þ find that ethical expressive voters drive turnout and policy outcomes in large elections. Interestingly, Shayo and Harel ð2011Þ find that instrumental and expressive voting depend critically on the probability that a voter is pivotal: more expressive voting is observed when voters are less likely to be decisive. Our work differs from earlier studies in at least two dimensions. First, building on the observation that voting becomes more expressive when the chance of being pivotal falls, we study the implications for the optimal size of voting bodies. Second, the central driver of our results is the prevalence of conflict between instrumental and expressive motives, a notion that is absent from the extant literature.

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990

journal of political economy

Our paper also contributes to the vast literature on information aggregation in voting. In the absence of expressive motives, our model is a special case of Feddersen and Pesendorfer ð1998Þ. That paper, as well as Feddersen and Pesendorfer ð1997Þ, shows that a version of the Condorcet jury theorem holds very generally: large elections succeed in aggregating information.3 However, these papers assume that preferences are purely instrumental. Our main finding is that many of the results in this literature are overturned, or require amendment, when one adds even small amounts of expressiveness to voter preferences. Finally, our concern with the optimal size of voting bodies connects to the literature on the optimal design of committees ðsee, e.g., Mukhopadhaya ½2003, Persico ½2004, and references thereinÞ. This literature highlights free-rider problems in information acquisition—the larger the committee, the less the incentives for an individual voter to become informed—which places a check on the optimal size of voting bodies. We offer a different rationale for limiting committee size, which is relevant even when informational free riding is not a problem. II. Model We study a simple model of voting in which voters are driven by instrumental as well as expressive motives. There are two equally likely states, labeled v ∈ fa; bg, and a simple-majority election with two possible outcomes, o ∈ fA; Bg. Each of n 1 1 voters, where n is even, receives a conditionally independent signal s ∈ fa; bg. With probability r ∈ ð12 ; 1Þ a voter receives a “true” signal: an a signal when the state is a and a b signal when the state is b. Otherwise, the voter receives a “false” signal, defined in analogous fashion. A voter’s payoff is determined by the outcome of the vote, o, the underlying state, v, and his individual vote, v ∈ fA; Bg. Outcome A is objectively better in state a, while outcome B is better in state b. Specifically, all voters receive a payoff of one if the better outcome is selected and a payoff of zero if the worse outcome is selected. We shall refer to this aspect of a voter’s payoffs as his instrumental payoff. In addition, voters also derive direct consumption utility from voting in a particular way. We shall refer to this aspect of a voter’s payoffs as his expressive payoff.

3 For similar results, see, e.g., McLennan ð1998Þ, Myerson ð1998Þ, and Fey ð2003Þ. Goeree and Yariv ð2009Þ offer experimental findings consistent with Condorcet jury theory. On the other hand, Bhattacharya ð2008Þ offers a negative result. He analyzes a class of instrumental models in which information does not aggregate. Callander ð2008Þ also obtains a negative result in a model where, in addition to the usual instrumental preferences, voters care about voting for the winner of the election.

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optimal size of voting bodies

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Expressive payoffs may be intrinsic ði.e., derive from how voting a certain way affects one’s self-imageÞ or extrinsic ðe.g., a representative may have to explain his vote to constituents back homeÞ. We can be agnostic about the precise rationale for these expressive payoffs. What matters is that voters derive some consumption utility from voting in a particular way and that this utility is less than perfectly correlated with their instrumental payoffs. For concreteness and conciseness, we shall couch expressive motives in terms of norms: voting in a fashion consistent with one’s norms—or, as the case may be, the norms of one’s constituents—yields an expressive payoff of one, while casting a vote against one’s norms yields an expressive payoff of zero. Finally, let ε ∈ ½0; 1 denote the relative weight a voter places on expressive payoffs, while complementary weight is placed on instrumental payoffs. Next, we turn to how expressive preferences, or norms, are determined. Suppose that, ex ante, norms are such that, with probability r ≥ 12 and independently across voters, a given voter views voting for A as normative.4 After the state has been realized and the voter has received his signal, his expressive preferences might change. Specifically, we suppose that with probability q ∈ ½0; 1Þ and independently across voters, a voter is influenced by the signal and adopts a norm consistent with his posterior beliefs about which outcome is more likely to be superior. Thus, with probability q, a voter receiving an a signal adopts voting for A as the norm while, with the same probability, a voter receiving a b signal adopts voting for B as the norm. With the complementary probability, 1 2 q, the voter sticks to his ex ante norm. One can think of q as representing how malleable expressive preferences ðor normsÞ are to facts. In environments in which party affiliation plays an important role, one can think of voters with malleable norms as independents and voters with rigid norms as partisans. In this interpretation, partisans, who make up a fraction 1 2 q of the population, receive an expressive payoff from voting for “their” candidate. The remaining fraction, q, are independents. They receive an expressive payoff from voting for whoever they believe is, instrumentally, the better candidate. Formally, a voter’s norm is summarized by his type t ∈ fA; Bg. An A type receives an expressive payoff from voting for A, while a B type receives an expressive payoff from voting for B. With probability q and independently across voters, a voter’s type is determined by his signal; that is, an a signal induces type A, while a b signal induces type B. With probability 1 2 q, a voter’s type is not influenced by his signal such that

4 Assuming r ≥ outcomes.

1 2

is without loss of generality. For the opposite case, simply relabel the

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992

journal of political economy

his type and signal are uncorrelated. In that case, the voter’s type is A with probability r. Voters cast their ballots simultaneously, and the outcome of the vote is decided by majority rule. To summarize, a voter with type t who casts a ballot v in a vote that produces outcome o in state v receives a payoff 5 8 > if o is correct and v 5 t <1 if o is correct and v ≠ t U 5 12ε if o is incorrect and v 5 t > :ε 0 if o is incorrect and v ≠ t: When determining equilibrium voting behavior, we restrict attention to symmetric equilibria. Hence, for the remainder of the paper, when we refer to “equilibrium,” we mean symmetric equilibrium, and when we refer to “unique equilibrium,” we mean unique within the class of symmetric equilibria. Under symmetry, equilibrium is characterized by the voting behavior of each kind of voter, namely, voters with signals and types ðs; tÞ ∈ fa; bg fA; Bg. Without expressive preferences ði.e., ε 5 0Þ, the model is quite standard and easy to analyze: in equilibrium all voters vote according to their signals, and for large n, the probability that the correct outcome is selected converges to one.6 We may divide voters into two classes depending on the realizations of s and t. When s and t coincide—that is, s 5 a and t 5 A or s 5 b and t 5 B—we say that a voter is unconflicted. When s and t differ, we say that a voter is conflicted. After some simplification, it may be readily shown that the probability that a voter is conflicted is 12 ð1 2 qÞ. Notice that when q ↑ 1, type and signal are perfectly correlated, and as a consequence, there are no conflicted voters. As q falls, the probability that a voter is conflicted increases and reaches a maximum of 50 percent at q 5 0. Thus, in expectation, conflicted voters are always a minority in the voting population. We now turn to voting strategies. We first show that voting is straightforward for unconflicted voters: they simply cast a vote consistent with both their signal and their type. In the proof of the following lemma and in the remainder of the paper, ga denotes the equilibrium probability that a random voter casts a vote for A in state a. Likewise, gb denotes the probability that a random voter casts a vote for A in state b. Lemma 1. In all equilibria, unconflicted voters vote according to their type and signal. The lemma says that it never pays for unconflicted voters to forgo their expressive payoffs and vote “strategically” ði.e., against their signalÞ. The 5 In fact, our payoff specification accommodates any preferences of the form U 5 do Ifo is correctg 1 dt Ifv5tg for arbitrary do ; dt > 0. Here, I denotes the indicator function. 6 Because both states are equally likely ex ante, the usual worries about strategic voting highlighted by Austen-Smith and Banks ð1996Þ are absent in this case.

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993

informational symmetry of the model nullifies the motives for strategic voting in the absence of expressive preferences, and the additional cost in terms of expressive considerations merely reinforces this effect. The voting behavior of conflicted voters is considerably more complex and interesting. Before we proceed with an equilibrium characterization, it is useful to define strategies more formally. Let js denote the probability that a conflicted voter with signal s votes for A. From lemma 1 it follows that ga 5 qr 1 r ð1 2 qÞr 1 r ð1 2 qÞð1 2 rÞja 1 ð1 2 r Þð1 2 qÞrjb ; gb 5 qð1 2 r Þ 1 ð1 2 r Þð1 2 qÞr 1 ð1 2 r Þð1 2 qÞð1 2 rÞja 1 r ð1 2 qÞrjb :

ð1Þ

ð2Þ

Note that gb < ga for all fja ; jb g ∈ ½0; 12 . That is, A receives a greater ðexpectedÞ share of the vote when it is the superior option than when it is the inferior option. The same is true for B. While fja ; jb g describe a generic mixed strategy, two polar cases are of interest. When ja 5 1 and jb 5 0, we say that a voter votes instrumentally—that is, purely according to his signal. Similarly, when ja 5 0 and jb 5 1, we say that a voter votes expressively—that is, purely according to his type. A conflicted voter, i, with an a signal who votes instrumentally enjoys a payoff ð1 2 εÞfPr½A winsja; ja;i 5 0Pr½aja 1 Pr½B winsjb; ja;i 5 0gPr½bja while voting expressively yields a payoff ð1 2 εÞfPr½A winsja; ja;i 5 1Pr½aja 1 Pr½B winsjb; ja;i 5 1Pr½bjag 1 ε: Instrumental payoffs differ only when the voter is pivotal. Thus, differencing the two expressions yields ð1 2 εÞPr½PivjaPr½aja 2 Pr½PivjbPr½bja 2 ε: Here, Pr½aja 5 r and Pr½bja 5 1 2 r , while the chance that a voter is pivotal is simply the chance of a tied vote in a given state, which is equal to ! n n ðz v Þn=2 ; 2 z v ; gv ð1 2 gv Þ. Therefore, the difference in payoffs between voting instrumentally and expressively is

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994

journal of political economy

!

n ð1 2 εÞ n ½r ðz a Þn=2 2 ð1 2 r Þðz b Þn=2 2 ε: 2 A similar derivation applies for a conflicted voter with a b signal. Thus, for a conflicted voter with signal s, the difference in expected payoffs between voting instrumentally and voting expressively takes on the same sign as Vs , where ! n ε Va ; n ½r ðz a Þn=2 2 ð1 2 r Þðz b Þn=2 2 ; ð3Þ 12 ε 2

Vb ;

! n ε n ½r ðz b Þn=2 2 ð1 2 r Þðz a Þn=2 2 : 1 2 ε 2

ð4Þ

III. Equilibrium Voting in Pure Strategies Having characterized the equilibrium voting behavior of unconflicted voters, we now turn to the behavior of conflicted voters. As we show below, the behavior of conflicted voters typically varies with the size of the voting body. Intuitively, as the size of the voting body grows, instrumental considerations—which hinge on the probability of being pivotal— become less important, and voting becomes more expressive. While n 1 1 denotes the discrete size of the voting body, it is sometimes convenient to use a continuous analogue of n, which we denote by m. We also adapt the usual floor/ceiling notation for the integer part of m to reflect the restriction that n be an even number. Specifically, let ⌊m⌋ denote the largest even integer less than or equal to m, and let ⌈m⌉ denote the smallest even integer greater than or equal to m. Finally, we use the gamma function to extend factorials to noninteger values. Recall that, for integer values, n! 5 Gðn 1 1Þ, and hence, n n 2

! 5

Gðn 1 1Þ : G ððn=2Þ 1 1Þ 2

The expression Gðm 1 1Þ=G2 ððm=2Þ 1 1Þ represents the continuous analogue. This continuous analogue makes the function Vs —and similar expressions below—well defined for all nonnegative, real-valued m. We now offer a useful technical lemma, which implies that, for fixed values of z a and z b , Vs is monotone in m.

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optimal size of voting bodies

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Lemma 2. Fix z a and z b such that 0 < z a ≤ z b ≤ 14. Then Fðm Þ ;

Gðm 1 1Þ ½r ðz b Þm=2 2 ð1 2 r Þðz a Þm=2 G2 ððm=2Þ 1 1Þ

is strictly decreasing in m. Moreover, limm→` FðmÞ ↓ 0. The restriction that z v be smaller than 14 is automatically satisfied since z v ; gv ð1 2 gv Þ and gv ∈ ½0; 1. Hence, by definition, z v takes on its maximum, 14, for gv 5 12. Instrumental equilibrium.—From an information aggregation perspective, it would be ideal if voters simply voted in line with their signals. As we have shown above, this is not a problem for unconflicted voters. For conflicted voters, whether to vote instrumentally turns on whether the gains from improving the probability of breaking a tie in the right direction outweigh the losses from voting against one’s expressive preferences. Let zaI denote z a under instrumental voting and note that zaI 5 z a jj 51;j 50 5 r ð1 2 r Þ. Lemma 2 implies that the benefits from instrumental voting are strictly decreasing in m. Thus, finding the largest-size voting body for which instrumental voting is an equilibrium simply amounts to determining the value of m such that Va jj 51;j 50 5 Vb jj 51;j 50 5 0 or, equivalently, a

a

b

a

b

b

Gðm 1 1Þ ε ð2r 2 1Þ½r ð1 2 r Þm=2 5 : G2 ððm=2Þ 1 1Þ 12ε

ð5Þ

Notice that Fð0Þ 5 2r 2 1. Decreasingness of FðmÞ in m then implies that for all m > 0, Gðm 1 1Þ ð2r 2 1Þ½r ð1 2 r Þm=2 < 2r 2 1: G2 ððm=2Þ 1 1Þ Hence, a necessary condition for instrumental voting to be an equilibrium for some size of the voting body is that ε=ð1 2 εÞ < 2r 2 1 or, equivalently, ε < 1=r ðr 2 12Þ. If ε ≥ 1=r ðr 2 12Þ, voting expressively is the unique equilibrium, regardless of the size of the voting body. The remainder of the analysis excludes this rather uninteresting by making the following assumption. Assumption 1. ε < 1=r ðr 2 12Þ. Assumption 1 together with lemma 2 guarantee that equation ð5Þ has a unique solution in m, which we denote bym I. Hence, we have shown the following proposition. Proposition 1. Instrumental voting is an equilibrium iff n ≤m I . Proposition 1 implies that, for large voting bodies, instrumental voting is not an equilibrium. Since the probability of being pivotal declines as the number of voters increases, the effective weight of instrumental payoffs, which depends on the chance of a tied election, declines relative to the effective weight of expressive payoffs. Once voters are sufficiently un-

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996

journal of political economy

likely to swing the vote, they are better off voting according to their type and locking in the ε expressive utility rather than voting according to their signal and forgoing this sure gain in exchange for a lottery with only a small chance of success. Inspection of equation ð5Þ reveals that m I does not depend on q and r. That is, the size of the voting body for which instrumental voting is an equilibrium is independent of the prevalence of conflict between instrumental and expressive motives and the level of ex ante bias in expressive motives. The reason is that, under instrumental voting, the payoff from deviating and voting expressively is independent of q and r, and hence, so is the upper bound for instrumental voting, m I . Also, note that m I varies nonmonotonically with the quality of voters’ information. When voters are poorly informed, that is, r < 1=½2ð1 2 εÞ, instrumental voting is never an equilibrium. However, as voters become perfectly informed, that is, r → 1, m I also goes to zero. There are two different forces at work here. When r is low, a voter is relatively likely to be pivotal but unlikely to push the outcome in the right direction. Hence, expected instrumental payoffs are low. When r is high, a voter is very likely to push the outcome in the right direction conditional on being pivotal but very unlikely to be pivotal. Again, this leads to low expected instrumental payoffs. Thus, the size of the voting body for which instrumental voting is an equilibrium is largest when voters are moderately well informed. The fact that instrumental voting is not an equilibrium for voting bodies with more than ⌊m I ⌋ members might seem inconsequential provided that the weight on expressive payoffs is small. Indeed, inspection of equation ð5Þ reveals that m I becomes infinitely large as ε goes to zero. However, a key question is how fast the value of m I grows as ε shrinks. While m I does not have a closed-form solution, Stirling’s approximation offers a way to examine the relationship between m I and ε. Remark 1. For small ε, ! 1 2ln½4r ð1 2 r Þ W ; ð6Þ mI ≈ 2ln½4r ð1 2 r Þ p=2f½1=ð2r 2 1Þ½ε=ð1 2 εÞg2 where W ðÞ is the Lambert W function.7 Consider the sequence εk 5 1=k. Substituting this expression into equation ð6Þ yields the sequence m I;k ≈ y W ððk 2 1Þ2 Þ, where y is a scaling factor independent of k. Now recall that limk→` lnk=W ðkÞ 5 1. Hence, we can conclude that as εk falls,m I;k grows only at rate 2 ln k. In other words, while m I;k increases, it does so only extremely slowly. Example 1 below il-

7

Recall that the Lambert W function is the inverse of f ðW Þ 5 W expðW Þ.

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optimal size of voting bodies

997

lustrates that, even for small ε, instrumental voting is an equilibrium only for modest-sized voting bodies. Example 1. Suppose that r 5 3=5 and ε 5 1=50. Instrumental voting is an equilibrium for voting bodies of up to 23 voters. If ε 5 1=1; 000, then ⌊m I ⌋ 1 1 increases to 129. Expressive equilibrium.—Let us now turn to the other extreme—purely expressive voting. Expressive voting is an equilibrium if and only if ja 5 0 and jb 5 1 is optimal for conflicted voters. This corresponds to Vs jj 50;j 51 ≤ 0, s ∈ fa; bg. Let z Ea ; z a jj 50;j 51 , and let z Eb be likewise defined. It may be readily verified that z Ea < z Eb . Therefore, a

a

Gðm 1 1Þ ε ½r ðz bE Þm=2 2 ð1 2 r Þðz aE Þm=2 2 G ððm=2Þ 1 1Þ 12ε Gðm 1 1Þ ε > 2 ½r ðz aE Þm=2 2 ð1 2 r Þðz bE Þm=2 2 G ððm=2Þ 1 1Þ 12ε

Vb jj 50;j 51 5 a

b

b

b

2

5 Va jj 50;j 51 : a

b

Thus, we need only check the incentive condition for expressive voting for conflicted voters with b signals. Because z aE < z bE , lemma 2 implies that the relative benefits from expressive voting are increasing in m. Hence, finding the smallest-size voting body such that expressive voting is an equilibrium amounts to determining the value of m where Vb jj 50;j 51 5 0 or, equivalently, a

b

Gðm 1 1Þ ε ½r ðz bE Þm=2 2 ð1 2 r Þðz bE Þm=2 5 : G2 ððm=2Þ 1 1Þ 12ε

ð7Þ

Together, assumption 1 and lemma 2 guarantee that equation ð7Þ has a unique solution, which we denote by m E.8 Hence, we have the following proposition. Proposition 2. Expressive voting is an equilibrium iff n ≥ m E . One might have thought that m E 5m I ; that is, once instrumental voting ceases to be an equilibrium, voters immediately revert to expressive voting. Notice, however, that this is generically not the case. This is most easily seen for q 5 0. In that case, equation ð7Þ reduces to Gðm 1 1Þ ε ð2r 2 1Þ½rð1 2 rÞm=2 5 : G2 ððm=2Þ 1 1Þ 12ε 8

While m E does not admit a closed-form solution, a good approximation is ! 2lnð4z bE Þ 1 mE ≈ : W 2lnð4z bE Þ p=2fð1=r Þ½ε=ð1 2 εÞg2

This content downloaded from 128.32.73.103 on Thu, 30 May 2013 21:22:31 PM All use subject to JSTOR Terms and Conditions

998

journal of political economy

Lemma 2 then implies that m E < m I if and only if r > r . Hence, instrumental and expressive equilibria may overlap, or there may be a gap between the two. The gap between m I and m E can be quite large indeed. To see this, let us return to example 1, filling in the remaining parameters of the model. Example 2. Let r 5 3=5, r 5 7=10, q 5 7=10, and ε 5 1=50. Instrumental voting is an equilibrium for n 1 1 ≤ 23, while expressive voting is an equilibrium for n 1 1 ≥ 459. This leaves open the question of what happens in between instrumental and expressive voting. The next section fills in this gap by considering mixed strategies. IV.

Full Equilibrium Characterization

In this section, we allow for mixed strategies and characterize all symmetric equilibria. The following lemma narrows down the kind of voting behavior that can arise in equilibrium. Proposition 3. The following and only the following kinds of equilibria can arise: ð1Þ instrumental, ð2Þ completely mixed, ð3Þ partially mixed, and ð4Þ expressive. In a completely mixed equilibrium, conflicted voters strictly mix between instrumental and expressive voting. In a partially mixed equilibrium, conflicted voters with a signals vote expressively, while conflicted voters with b signals mix. The asymmetry stems from the fact that, since r ≥ 12, voters are ðweaklyÞ biased toward voting for A. To provide a full equilibrium characterization, it is convenient to distinguish between high and low malleability, where the threshold between the two, q * , is defined in equation ðA6Þ in the Appendix. High q corresponds to a low prevalence of conflict between types and signals; that is, instrumental and expressive motives tend to coincide. Low q corresponds to a high prevalence of conflict between types and signals; that is, instrumental and expressive motives are more likely to be at odds with each other. Low conflict ðhigh qÞ.—We now show that when conflict is low ði.e., q > q * Þ, the intervals for which the various classes of equilibria exist partition the set of even integers. In the next proposition, the upper bound for completely mixed voting, m CM , is formally defined in equation ðA4Þ in the Appendix. Proposition 4. Under low conflict, there exists a unique equilibrium for each n. This equilibrium is ð1Þ instrumental for n ≤m I, ð2Þ completely mixed for m I < n < m CM, ð3Þ partially mixed for m CM ≤ n < m E, and ð4Þ expressive for n ≥ m E. Proposition 4 establishes that, as n increases, equilibrium moves smoothly from instrumental to expressive voting. When a voting body is small, in-

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optimal size of voting bodies

999

strumental voting is the unique equilibrium. As the voting body grows larger, equilibrium voting becomes completely mixed. As it grows larger yet, we move to partially mixed voting. That is, voters with a signals vote expressively while voters with b signals continue to mix. Finally, in sufficiently large voting bodies, expressive voting is the unique equilibrium. We say that voting becomes more expressive if ja decreases and jb increases. In the following proposition we rank the expressiveness of voting as a function of the size of the voting body. Proposition 5. Under low conflict, equilibrium voting becomes more expressive as n increases. Let us return to example 2. Because q 5 7=10 > q * , we are in the lowconflict situation, and the analysis above applies. Recall that, for the parameter values in the example, instrumental voting is an equilibrium for 23 voters or less, while expressive voting is an equilibrium for 459 voters or more. Completely mixed voting is an equilibrium for voting body sizes of 25 and 27, while partially mixed voting is an equilibrium for sizes between 29 and 457. High conflict ðlow qÞ.—We now turn to the case in which conflict between types and signals is high ði.e., q < q * Þ. As we shall see, this makes equilibrium behavior more complex. While the classes of equilibria are the same as under low conflict, under high conflict, the ranges for which these classes exist may overlap. Indeed, instrumental and expressive equilibria may coexist for the same value of n. Moreover, equilibrium may no longer be unique within a class: for generic parameter values, two different partially mixed equilibria, which we call “high” partially mixed and “low” partially mixed, may coexist. While a conflicted voter with an a signal votes expressively in both partially mixed equilibria, the equilibria differ in the probability that a conflicted voter with a b signal votes expressively. In a high partially mixed equilibrium, this probability is relatively large, while it is relatively small in a low partially mixed equilibrium. In the next proposition, the upper bound for partially mixed voting, m PM , is formally defined in equation ðA9Þ in the Appendix. Proposition 6. Under high conflict ði.e., q < q * Þ, equilibria are ð1Þ instrumental iff n ≤ m I , ð2Þ completely mixed iff m I < n < m CM , ð3Þ low partially mixed iff m CM ≤ n < m PM , ð4Þ high partially mixed iff m E ≤ n < m PM , and ð5Þ expressive iff n ≥ m E . Moreover, within each ðsub-Þclass, equilibrium is unique. While, typically, expressiveness increases with the size of the voting body, the sequence of high partially mixed equilibria has the somewhat counterintuitive property that expressiveness decreases with n. Moreover, for some parameter values and voting body sizes, instrumental and expressive equilibria coexist. To see this, consider the following amendment of example 2, where we have reduced q from 7=10 to 1=10.

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1000

journal of political economy

Example 3. Let r 5 3=5, q 5 1=10, r 5 7=10, and ε 5 1=50. Instrumental voting is an equilibrium for n 1 1 ≤ 23, while expressive voting is an equilibrium for n 1 1 ≥ 19. There is a completely mixed equilibrium for 25 ≤ n 1 1 ≤ 43, a low partially mixed equilibrium for 45 ≤ n 1 1 ≤ 549, and a high partially mixed equilibrium for 21 ≤ n 1 1 ≤ 549. V.

The Optimal Size of Voting Bodies

Having characterized equilibrium behavior for voting bodies of all sizes, we are now in a position to formally address the question of their optimal size. As our metric, we use selection accuracy, S, the probability that a voting body chooses the ðinstrumentallyÞ correct outcome given the state. Of course, this metric takes no account of the expressive component of preferences. However, as we show later, including this component in welfare calculations does not alter our conclusions about the optimal size of voting bodies. With the exception of the high partially mixed equilibrium, equilibrium voting becomes more expressive when the size of the voting body increases. Thus, the key trade-off for accuracy is between the informational gains from adding an additional voter and the informational losses from more expressive voting. As we show below, whether the former prevails over the latter crucially depends on the prevalence of conflict between expressive and instrumental motives. Fix an equilibrium ðga, gbÞ for a voting body of size n 1 1. In state a, the equilibrium probability that an individual voter casts a vote for the correct outcome, A, is ga. Therefore, the voting body selects the correct outcome with probability ! n11 gka ð1 2 ga Þn112k : Sðn 1 1jaÞ 5 o k k5ðn=2Þ11 In state b, the equilibrium probability that an individual voter casts a vote for the correct outcome, B, is 1 2 gb . Thus, the voting body selects the correct outcome with probability Sðn 1 1jbÞ 5

o

k5ðn=2Þ11

! n11 ð1 2 gb Þk gbn112k : k

Since the two states are equally likely ex ante, Sðn 1 1Þ 5

1 ½Sðn 1 1jaÞ 1 Sðn 1 1jbÞ: 2

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optimal size of voting bodies

1001

It is sometimes convenient to extend S to noninteger values, m. Since the cumulative density function of a binomial distribution may be expressed in terms of beta functions, we have 2 3 m m m m 1 1; 1 1 1 1; 1 1 B g ; B 1 2 g ; a b 16 7 2 1 m 2 m 2 Sðm 1 1Þ 5 4 m 2 5: m 2 B B 1 1; 11 1 1; 11 2 2 2 2 Here, Bðx; yÞ denotes the beta function with parameters x and y, while Bðg; x; yÞ denotes the incomplete beta function. Low conflict ðhigh qÞ.—Suppose that the prevalence of conflict between instrumental and expressive motives is low ði.e., q > q * Þ. In that case, there is a unique equilibrium for each size voting body, and hence, Sðn 1 1Þ is uniquely determined. Once the voting body becomes sufficiently large, voting is purely expressive. Beyond this point, there is no more trade-off between information and expressiveness. As only the informational force persists, it might seem that information should always aggregate in the limit. To see that this is not the case, consider the informational value of a marginal voter when voting is purely expressive. In state a, the chance of casting a vote for the correct outcome is gEa 5 qr 1 ð1 2 qÞr. Since r > 12 and r ≥ 12, we have gEa > 12. This means that the marginal voter always improves accuracy in state a. In state b, the chance of a correct vote is 1 2 gEb 5 1 2 r 1 qðr 1 r 2 1Þ. The marginal voter improves accuracy if and only if 1 2 gEb > 12. Hence, the threshold value of q such that the informational contribution is positive in state b is q>

1 2r 2 1 ; q1 : 2 r 2 ð1 2 r Þ

Therefore, in the limit, the probability of selecting the correct outcome in state a always goes to one. The probability in state b goes to one if and only if q > q1 , where, as proved in lemma 7 in the Appendix, q 1 > q * . We have established the following proposition. Proposition 7. In large voting bodies, information fully aggregates if and only if conflict is very low, that is, q > q1 , where q1 > q * . What happens when conflict is low but not very low ði.e., q * < q < q1 Þ? Because gEa > 12 and 1 2 gEb < 12, each incremental voter increases the chance of selecting the correct outcome in state a but decreases it in state b. Since gEa > gEb > 12, gEa is farther from 12 than 1 2 gEb . This means that the incremental voter is more likely to break a tie correctly in state a than he is to break a tie incorrectly in state b. On the other hand, it also means that the probability of a tie is greater in state b than in state a. When n is small, tie probabilities are relatively similar across states, and hence, adding a voter is beneficial. When n is large, ties are vastly more likely in

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1002

journal of political economy

state b, and thus, the marginal voter has a negative effect on accuracy. In the limit, the correct outcome is chosen with probability one in state a but is never chosen in state b. As a result, accuracy falls to 50 percent. This is stated formally as follows. Proposition 8. Suppose that conflict is low but not very low ði.e., q * < q < q1 Þ. Then, for n sufficiently large, the incremental voter has negative informational value. That is, Sðn 1 1Þ is eventually decreasing in n. Furthermore, in the limit, large voting bodies are no better than a coin flip at selecting the correct outcome. The debate about the optimal size of voting bodies boasts a long history. A standard intuition is that larger voting bodies attract increasingly less qualified individuals, more easily swayed by “honey words” or even outright demagoguery and, hence, are to be guarded against. Indeed, this led Madison to argue that “passion” leads voting bodies to “counteract their own views by every addition to their representatives” ðFederalist Paper no. 58; Hamilton et al. [1787–88] 2000, 376Þ. Proposition 8, however, points out that, even when fully qualified representatives can be found in abundance, adding representatives may worsen policy outcomes. Indeed, unless conflict is very low, eventually, each additional voter reduces accuracy, despite the fact that voters’ preferences are instrumentally aligned. It might seem that when conflict is very low, the best strategy is to always make the voting body as large as possible. Indeed, when n is sufficiently large, incremental voters have positive informational value, and hence, locally, their addition is unambiguously helpful. For smaller values of n, however, the trade-off between information and expressiveness is still present, and the contribution of incremental voters may very well be negative. This holds even when there is no ex ante asymmetry in norms ði.e., r 5 12Þ. The following proposition formalizes this idea. Proposition 9. For all q, accuracy is strictly decreasing in the region of the completely mixed equilibrium. Formally, for m I < n < m CM, Sðn 1 1Þ is strictly decreasing in n. The key trade-off underlying proposition 9 is as follows. On the one hand, when the size of the voting body increases, voting must become more expressive to maintain the completely mixed strategy equilibrium. This causes the informativeness of voting to deteriorate. On the other hand, each additional voter provides information. The proposition shows that, in the region of the completely mixed equilibrium, the former effect always dominates the latter in terms of accuracy. To see why voting must become more expressive, notice that the probability of casting a decisive vote in the completely mixed equilibrium must remain constant as the number of voters increases. Since expressive payoffs are independent of the size of the voting body, n, conflicted voters will mix only if instrumental payoffs remain likewise unchanged. In turn, this necessitates a constant pivot probability. To achieve this, the elec-

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optimal size of voting bodies

1003

F IG . 1.—Iso-pivot curves are steeper than iso-accuracy curves

tion must become closer in expectation, which requires more expressive voting. The heart of the proof is to show that the increase in expressiveness necessitated by the completely mixed equilibrium is so large that it leads to a drop in accuracy. Figure 1 illustrates this effect graphically. On the x-axis, we have ðthe continuous analogue ofÞ the number of voters, m, while the y-axis plots ga, the vote propensity for A in state a. Iso-pivot lines describe the loci of electorate sizes and vote propensities that hold the pivot probability constant. As the figure shows, these loci are downward sloping ði.e., the larger the voting body, the less likely a randomly drawn voter will choose AÞ. Iso-accuracy curves, shown as dashed lines, describe the loci of points where accuracy remains constant. These loci are also downward sloping. Toward the southwest, accuracy falls while the pivot probability rises. A key complication in the proof arises from the fact that both curves are downward sloping. Nevertheless, as the figure illustrates, iso-pivot

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1004

journal of political economy

F IG . 2.—Accuracy under very low conflict ði.e., q > q1 Þ

curves are steeper than iso-accuracy curves at every point. Since, in any sequence of completely mixed equilibria, one moves along a given isopivot line as the electorate increases in size, this implies that accuracy must fall. Interestingly, figure 1 is parameter free: the parameters r, q, and r are embedded in ga. Thus, the figure represents a general description of the behavior of iso-pivot and iso-accuracy curves, which helps to explain why the direction of the trade-off described in proposition 9 holds regardless of parameter values. A similar analysis establishes the result for state b. To illustrate the potential importance of the effect uncovered in proposition 9, we offer an example in which the “valley” of larger voting bodies producing lower accuracy is considerable. Suppose that we amend example 3 to remove any asymmetry in ex ante norms ði.e., r 5 12Þ. Since q 5 1=10 > 0 5 q1 , equilibrium is unique for every n and accuracy converges to one in the limit. However, as figure 2 illustrates, increasing the number of voters is not the same as increasing accuracy. While accuracy increases along the instrumental equilibrium sequence ðup to n 1 1 5 23Þ, it falls along the completely mixed equilibrium sequence ðbetween n 1 1 5 25 and 61Þ. Beyond this point, accuracy once again increases, but it reaches its previous high watermark only at n 1 1 5 2;429. In the region of the completely mixed equilibrium, an increase in the size of the voting body leads to informational losses from

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optimal size of voting bodies

1005

more expressive voting that outpace the informational gains from having more voters. From n 1 1 5 61 onward, voting is purely expressive and the informativeness of votes no longer degenerates as n increases.9 Since q > q1 , additional votes improve equilibrium accuracy, albeit slowly. The point is that, even when conflict is very low and expertise is abundant, expanding the voting body is not necessarily conducive to obtaining better policies. High conflict ðlow qÞ.—When conflict is high ði.e., q < q * Þ, equilibrium multiplicity complicates the determination of the optimal size of voting bodies, as accuracy depends on which equilibrium is selected. Amending our notation, let Sh ðn 1 1Þ denote the selection accuracy of an equilibrium of type h ∈ fI, CM, LPM, HMP, Eg. Here, I, CM, LPM, HMP, and E denote instrumental, completely mixed, low partially mixed, high partially mixed, and expressive equilibrium, respectively. The next proposition shows that, if different types of equilibria coexist for a voting body of a given size, then they can be ordered in terms of accuracy. Proposition 10. If multiple equilibria coexist for given n, then their ranking in terms of selection accuracy is S ∈ fSI ; SCM ; SLPM g > SHPM > SE . Proposition 10 is intuitive: the accuracy ranking corresponds to the expressiveness of equilibria. Thus, an expressive equilibrium is least accurate, while an instrumental equilibrium—provided that one exists for the same-size voting body—is most accurate. Other equilibria are similarly ordered. It can be easily verified that proposition 8 carries over to high-conflict environments ði.e., q < q * Þ. Hence, in large voting bodies, the incremental voter has negative informational value, and in the limit, voting bodies are no better than a coin flip at selecting the correct outcome. For small voting bodies, however, increasing size can increase accuracy. When instrumental voting is an equilibrium, adding more voters is obviously helpful. But even when voting is expressive, initially, adding voters may improve accuracy. This happens as long as the likelihood of a tie remains comparable between the two states. Accuracy properties under high conflict are illustrated in figure 3. The figure depicts the selection accuracy of the equilibria in example 3 as a function of n. Accuracy is increasing in n under instrumental voting and decreasing under completely mixed voting. Accuracy is hump shaped under low partially mixed voting, increasing under high partially mixed voting, and, eventually, decreasing under expressive voting. Figure 3 also illustrates that, under high conflict, equilibrium accuracy can drop discontinuously in n. In other words, accuracy does not degrade 9 When r 5 12, the partially mixed equilibrium region disappears as a consequence of the symmetry of the model.

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1006

journal of political economy

F IG . 3.—Accuracy under high conflict ði.e., low qÞ

“gracefully” as the voting body grows but, at some point, falls off a cliff. Let us denote the sequence of most informative equilibria by C ðmÞ, where we treat m as continuous. From proposition 10 we know that this sequence ðfunctionÞ is uniquely defined even in the presence of multiple equilibria. Next, notice that at m 5m PM , C ðmÞ moves from low partially mixed to expressive voting. Moreover, from proposition 10 we know that, for fixed m, SE ðm 1 1Þ < SLPM ðm 1 1Þ. Thus, we have shown the following proposition. Proposition 11. Suppose that voters coordinate on the most accurate equilibrium. Then, under high conflict, accuracy falls discontinuously at m PM . While accuracy falls discontinuously at m PM when equilibrium selection is optimistic, notice that, under high conflict, accuracy must fall discontinuously at some point, regardless of the equilibrium selection rule. Summary.—Unless conflict is very low, large voting bodies are highly undesirable, producing outcomes no better than a coin flip. While smaller voting bodies do better, they can experience a sudden discrete drop in accuracy when the size of the voting body is expanded or when the prevalence of conflict between instrumental and expressive preferences rises. This is true even if we assume that voters are always able to coordinate on the “best” equilibrium.

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optimal size of voting bodies

1007

When conflict is very low, information fully aggregates in the limit. This does not mean, however, that enlarging the voting body is necessarily a good idea. The reason is that accuracy is nonmonotone in size. Therefore, unless the number of additional voters is sufficiently large, enlarging the voting body may reduce accuracy. Other measures of welfare.—By using accuracy, S, to determine the optimal size of voting bodies, we have ignored expressive payoffs and implicitly assumed that, from a societal point of view, only instrumental payoffs matter. We now show that taking into account expressive preferences does not change our conclusions as to the optimal size of voting bodies. To see this, notice that, in instrumental and expressive equilibria, a voter’s expressive payoff is unaffected by the size of the voting body. Hence, accuracy is the sole determinant of welfare. In completely mixed and partially mixed equilibria, voters who mix are indifferent between voting expressively and voting instrumentally. Thus, for purposes of payoff comparison, we may assume that they vote expressively. When they receive full expressive payoffs, accuracy is then again the sole determinant of these voters’ welfare as the size of the voting body changes. The same is true for voters in completely and partially mixed equilibria who do not mix, because their expressive payoffs are again unaffected by the size of the voting body. Sometimes only expressive payoffs should be counted from a societal point of view. Consider, for instance, situations in which instrumental payoffs correspond to rent-seeking benefits, while expressive payoffs reflect ethical concerns about appropriate policy. In that case, it may well be that social welfare is maximized by aggregating expressive preferences and ignoring instrumental concerns. In such situations, our results operate in reverse: when conflict is high, the problem is with voting bodies that are too small rather than too large. Money will override conscience in small voting bodies, while the “better angels of our nature” will dictate voting in large bodies. Asymmetric priors in large elections.—So far, we have assumed that the two states, a and b, are equally likely. We now relax this assumption and return to the welfare properties of large elections. Let l ∈ ½0, 1 denote the prior probability that the state is a. When voters have purely instrumental preferences, it is well known that strategic voting considerations lead to mixed strategies for individuals receiving the a priori more likely signal, even in the limit ðAusten-Smith and Banks 1996Þ. By contrast, when voters place some weight on expressive motives, then asymptotic voting behavior is in pure strategies and is purely expressive, regardless of prior beliefs. To see why, let ga;n denote the probability that a random voter casts a vote for A in state a when the electorate is of size n and the prior is l. Let gb;n and z v;n be likewise defined. Since z v;n ≤ 14, it follows via Stirling’s approximation that the chance of casting a decisive vote goes to zero in

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1008

journal of political economy

the limit.10 Hence, in large elections, instrumental preferences do not enter the voting calculus and purely expressive voting constitutes the unique equilibrium. Since asymptotic voting behavior is independent of prior beliefs, we may conclude that information continues to aggregate if and only if conflict is very low; thus, proposition 7 is preserved. However, proposition 8 requires amendment. When q < q1 , outcome A is always chosen in the limit of a large election. The chance that this is the correct outcome is simply equal to the prior, l. If l > 12, the outcome of a large election coincides with the choice of a social planner possessing no information. If l < 12, a large election performs strictly worse than an uninformed social planner. Preplay communication.—It is well known that preplay communication can have a large effect on equilibrium voting behavior. Indeed, in a Condorcet model without expressive preferences, a simple straw poll can eliminate strategic voting and associated costs in accuracy, regardless of the voting rule and prior beliefs ðsee, e.g., Coughlan 2000Þ. Since voters have common interests, they have an incentive to truthfully reveal their information. After the straw poll results are announced, all voters share the same posterior beliefs and, consequently, vote for the same alternative. Adding expressive motives causes voter preferences to be only partially aligned: the situation is no longer one of purely common interests. As a consequence, determining whether information exchange is incentive compatible is more problematic. In addition to this strategic complication, there are a host of other difficulties. First, preplay communication need no longer represent pure cheap talk. Indeed, voters may derive direct ðdisÞutility from their votes in the straw poll. Second, since expressive preferences are malleable, the information arising from preplay communication may change a voter’s view about the appropriate norm. This creates an additional strategic effect: voters may tailor their responses in the straw poll to influence others’ norms. Finally, in our model, voters cast only a single vote, and therefore, consistency does not arise as an expressive consideration. However, once multiple votes are taken, expressive payoffs might well depend on the combination of votes cast. For example, a voter might experience losses from “flip-flopping” at successive stages. While preplay communication is clearly relevant in some settings, a full analysis is beyond the scope of the present paper. 10

The asymptotic probability of casting a decisive vote is ! ! pﬃﬃﬃ n n 1 n=2 2 ≈ lim pﬃﬃﬃﬃﬃﬃ 5 0; lim n ðz v;n Þn=2 ≤ lim n n→` n→` n→` 4 pn 2 2

where we have used Stirling’s approximation.

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optimal size of voting bodies VI.

1009

Conclusion

Since Condorcet ([1785] 1976), perhaps the main message from the “informational” voting literature is the remarkable ability of elections to aggregate information and produce the correct decision. Our analysis suggests that, perhaps, we have been overly optimistic in these conclusions. When we enrich the classical model to admit voters who are motivated by expressive as well as instrumental motives, the results are more ambiguous and the conclusions less hopeful. Expressive motives fundamentally change equilibrium behavior and subsequent conclusions about the quality of decisions made through voting. In a sense, purely instrumental models of voting represent a “singularity”: even if the weight put on expressive preferences is arbitrarily small, expressive preferences completely determine voting behavior in large voting bodies. Whether this is for good or for ill depends on how malleable—influenced by facts—expressive preferences are. As long as they are sufficiently malleable, expressive voting is of no real concern as it still leads to the correct outcome in the limit. In contrast, when expressive preferences are rigid and relatively impervious to facts, expressive voting produces dismal results in large voting bodies. In the limit, information is driven out entirely and decisions are no better than chance. However, even when expressive preferences are malleable, there is always a region where informational losses from increased expressive voting dominate the informational gains from adding more voters. The reason is that, while the marginal voter does provide additional information, increased expressiveness drives out instrumentality over the entirety of the voting body. When expressive preferences are more rigid, this effect need not even be gradual: as the voting body expands in size, at some point, there will be a sudden downward jump in performance. A practical implication of our results is that capping the size of voting bodies may be desirable even when logistical or information acquisition constraints are not binding. Thus, our model offers a rationale for the more or less constant size of many legislatures, despite significant population growth and advances in communication and information gathering technologies. Our results also highlight an important downside to increasing the transparency of the voting records of elected representatives. To the extent that transparency increases the need to pander to constituents, it increases expressiveness and, thereby, can have a deleterious effect on the performance of legislatures. Of course, our model is merely suggestive in this regard. While the Condorcet setup offers a parsimonious framework for examining the effect of expressive motives on voting behavior and performance, it is by no means a complete description of real-world legislative settings. One obvi-

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1010

journal of political economy

ous omission is the role of ideology on decision making. In our model, voters have common values: ex post, all agree on the best choice. While there are some situations, such as national defense, in which this may be a reasonable approximation, it is clearly inadequate in many other circumstances, such as when deciding on the appropriate role of government in regulating behavior. Similarly, in our model, all voters bear the same cost of errors. Yet, in many situations, voters may differ along this dimension as well. If one thinks about choices as reflecting conservative versus liberal policies in the service of some commonly agreed-on objective, then it may well be that some voters will find it more costly to be wrong when selecting the conservative policy, while others suffer more when an incorrect liberal policy is implemented. Finally, voters might also differ in the weight they place on expressive preferences. For instance, legislators with safe seats might place less weight on expressive considerations than those whose seats are more hotly contested. Clearly, a richer model would allow greater scope for ideology and for other differences across voters. This, however, remains for the future.

Appendix Proofs Proof of Lemma 1 Consider an unconflicted voter with an a signal. Suppose, contrary to the statement of the lemma, that he prefers to vote for B rather than for A. That is, ! n n ð1 2 εÞ½r ðz a Þn=2 2 ð1 2 r Þðz b Þn=2 1 ε ≤ 0; ðA1Þ 2 where z v 5 gv ð1 2 gv Þ. First, note that a necessary condition for this inequality to hold is that z b > z a . Second, note that the inequality implies that a conflicted voter with an a signal would also strictly prefer to vote for B, that is, Va < 0. Furthermore, an unconflicted voter with a b signal would strictly prefer to vote for B. To see this, note that the difference in that voter’s payoff from voting for B rather than for A is ! n n ð1 2 εÞ½r ðz b Þn=2 2 ð1 2 r Þðz a Þn=2 1 ε; 2 and this expression is strictly positive since z b > z a and r > 12. Finally, a conflicted voter with a b signal would strictly prefer to vote for B since Vb > 2 Va > 0. Hence, we have shown that, if an unconflicted voter with an a signal weakly prefers to vote for B, then all voters strictly prefer to vote for B. In turn, this implies that z a ≥ z b ; this, however, contradicts z b > z a .

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optimal size of voting bodies

1011

The proof that an unconflicted voter with a b signal strictly prefers to vote for B is analogous. Proof of Lemma 2 Differentiating FðmÞ with respect to m, we obtain

0

hm i 1 Gðm 1 1Þ n 2 2H ½m 2 log½z a ð1 2 r Þðz a Þm=2 2H 2 2 G ððm=2Þ 1 1Þ 2 o hm i m=2 2 2H ½m 2 log½z b ; 2 r ðz b Þ 2H 2

F ðmÞ 5

0

where H ½x is the xth harmonic number. Note that F ðmÞ takes the sign of the expression in braces. We claim that 2ðH ½m=2 2 H ½mÞ 2 log½z b > 0 for all m ≥ 2. When m 5 2, we have h1i h1i 3 > 0: 2ðH ½1 2 H ½2Þ 2 log½z b > 2ðH ½1 2 H ½2Þ 2 log 52 12 2 log 4 2 4 Because H ½m is concave in m, the inequality then also holds for all m > 2. 0 This implies that F ðmÞ < 0 iff 1 2 r 2ðH ½m 2 H ½m=2Þ 1 log½z a < r 2ðH ½m 2 H ½m=2Þ 1 log½z b

zb za

!m=2 :

And this inequality indeed holds because r < 12 and z a ≤ z b . To establish the second part of the lemma, use Stirling’s approximation to obtain " pﬃﬃﬃﬃ # pﬃﬃﬃ ð2pﬃﬃﬃﬃ z b Þm ð2 z a Þm FðmÞ ≈ 2 r pﬃﬃﬃﬃﬃﬃﬃ 2 ð1 2 r Þ pﬃﬃﬃﬃﬃﬃﬃ pm pm for large m. Now note that both terms converge to zero as m → ` because z a ≤ z b ≤ 14. Hence, limm→` FðmÞ 5 0. Proof of Proposition 1 A necessary and sufficient condition for instrumental voting to be an equilibrium is that ! n n ð2r 2 1Þ½r ð1 2 r Þn=2 ≥ ε : ðA2Þ 12ε 2 Note that lemma 2 with z a 5 z b 5 r ð1 2 r Þ implies that the left-hand side is strictly decreasing in n. As a consequence, the inequality ðA2Þ holds iff n ≤ m I , where m I is the value of m that solves the continuous analogue of ðA2Þ with equality.

This content downloaded from 128.32.73.103 on Thu, 30 May 2013 21:22:31 PM All use subject to JSTOR Terms and Conditions

1012

journal of political economy

Proof of Remark 1 The term m I solves Gðm 1 1Þ ε : ð2r 2 1Þ½r ð1 2 r Þm=2 5 G2 ððm=2Þ 1 1Þ 12ε

ðA3Þ

By lemma 2, for ε small, m has to be large. For large m, using Stirling’s approximation, we have pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m m pﬃﬃﬃ 2pm ðm =e Þ 2 Gðm 1 1Þ p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃpﬃﬃﬃﬃ : 5 ≈ ð1=2Þm p m G2 ððm=2Þ 1 1Þ f 2pðm=2Þ½ðm=2Þm=2 =e m=2 g2 Hence, approximately, equation ðA3Þ becomes pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃ ½2 r ð1 2 r Þm p 1 ε pﬃﬃﬃﬃ : 5 2 2r 2 1 1 2 ε m Solving for m gives 1 W mI ≈ 2 ln½4r ð1 2 r Þ

2 ln½4r ð1 2 r Þ p=2f½1=ð2r 2 1Þ½ε=ð1 2 εÞg2

! :

Proof of Proposition 2 Under expressive voting, ja 5 0 and jb 5 1. It may be readily verified that this implies that z a < z b and, therefore, Va < Vb . Thus, we need only check the incentive condition to vote expressively for conflicted voters with b signals, that is, Vb ≤ 0. By construction, Vb 5 0 at m E while, by lemma 2, Vb is strictly decreasing in n. Hence, the incentive constraint also holds for all n ≥ m E . Proof of Proposition 3 The fact that each of these kinds of equilibria can indeed arise is proved by example ðsee, e.g., example 3Þ. The proof that no other kinds of equilibria can arise proceeds as follows. First, from lemma 1, we know that all unconflicted voters vote according to their signals. This implies that all equilibria are fully characterized by the mixing probabilities ðja ; jb Þ ∈ ½0; 12 of conflicted voters. To prove the proposition, we have to show that there do not exist equilibria with fja 5 1; jb ∈ ð0; 1Þg, with fja ∈ ð0; 1Þ; jb 5 1g, or with fja ∈ ð0; 1Þ; jb 5 0g. This is proved in lemmas 3, 4, and 5 below. Lemma 3. There is no partially mixed equilibrium with ja 5 1 and jb ∈ ð0; 1Þ. Proof. Suppose, by contradiction, that such an equilibrium does exist. We first show that ja 5 1 implies jga 2 12 j > jgb 2 12 j. One may readily verify that for ja 5 1, ga > 12. Furthermore, gb > 12 iff jb > ðr 2 12Þ=½r ð1 2 qÞr. When jb > ðr 2 1Þ=½r ð1 2 qÞr, jga 2 1 j > jgb 2 1 j follows immediately from 1 < gb < ga . 2 2 2 2 When jb ≤ ðr 2 12Þ=½r ð1 2 qÞr, jga 2 12 j > jgb 2 12 j is equivalent to showing that ga 2 ð1 2 gb Þ > 0. And after some algebra, ga 2 ð1 2 gb Þ 5 ð1 2 qÞrjb > 0. Since jga 2 12 j > jgb 2 12 j, we have z a < z b , and therefore, Vb > Va . Because ja 5 1, it must be that Va ≥ 0, which implies Vb > Va ≥ 0. Thus, conflicted voters with b signals strictly prefer to vote instrumentally, such that jb 5 0. But this is a contradiction because, by assumption, jb ∈ ð0; 1Þ. QED

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optimal size of voting bodies

1013

Lemma 4. There is no partially mixed equilibrium with ja ∈ ð0; 1Þ and jb 5 1. Proof. Suppose, by contradiction, that such an equilibrium does exist. We first show that jb 5 1 implies z a < z b . The algebra establishing this is straightforward and analogous to that given in the proof of lemma 3. Since z a < z b , we have Vb > Va . Because jb 5 1, it must be that Vb ≤ 0, which implies Va < Vb ≤ 0. Thus, conflicted voters with a signals strictly prefer to vote expressively, such that ja 5 0. This is a contradiction because, by assumption, ja ∈ ð0; 1Þ. QED Lemma 5. There is no partially mixed equilibrium with ja ∈ ð0; 1Þ and jb 5 0. Proof. Suppose, by contradiction, that such an equilibrium does exist. We first show that jb 5 0 implies z a > z b . The algebra establishing this is straightforward and analogous to that given in the proof of lemma 3. Since z a > z b , we have Vb < Va . Because jb 5 0, it must be that Vb ≥ 0, which implies Va > Vb ≥ 0. Thus, conflicted voters with a signals strictly prefer to vote instrumentally such that ja 5 1. This is a contradiction because, by assumption, ja ∈ ð0; 1Þ. QED This completes the proof of proposition 3. Proof of Proposition 4 The proof follows from a sequence of lemmas and propositions. We first consider completely mixed equilibria. The following lemma identifies properties that all such equilibria share. Lemma 6. In any completely mixed equilibrium, 1 ε ð1Þ Pr½Pivja 5 Pr½Pivjb 5 2r 2 1 1 2 ε ; ð2Þ ga 5 1 2 gb > 12, and ð3Þ ja 5 1 2 ½r=ð1 2 rÞjb . Proof. The probability of being pivotal in state v is Pr½Pivjv 5

! n n ðz v Þn=2 : 2

For both kinds of conflicted voters to mix, ðja ; jb Þ must solve Va 5 Vb 5 0. This implies that Pr½Pivja 5 Pr½Pivjb 5

1 ε : 2r 2 1 1 2 ε

The equality of pivot probabilities in the two states implies that either ga 5 gb or ga 5 1 2 gb. It may be readily verified that ga 2 gb > 0 for all ja ; jb ∈ ð0; 1Þ. Hence, ga 5 1 2 gb . Finally, ga 5 1 2 gb implies that ja 5 1 2 ½ r=ð1 2 rÞjb , which completes the proof. QED Next, we determine the bounds for which completely mixed voting is an equilibrium. Definem CM to be the ðuniqueÞ value m CM ; fm > 0 : Vb jj 50;j 5ð12rÞ=r 5 0g: a

b

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ðA4Þ

1014

journal of political economy

Proposition 12. A completely mixed equilibrium exists iff n is such that m I < n < m CM . For each such n, there exists exactly one completely mixed equilibrium. Moreover,m CM > m I . Proof. In a completely mixed equilibrium, Va 5 Vb 5 0. From lemma 6 we know that ga 5 1 2 gb , and hence, these equalities reduce to ! n ε n fð2r 2 1Þ½ga ð1 2 ga Þn=2 g 2 5 0: ðA5Þ 12ε 2 Fact 1.—By lemma 2, the left-hand side is strictly decreasing in n for fixed ga. Fact 2.—For fixed n and ga > 12, the left-hand side is strictly decreasing in ga. From lemma 6 we know that, over the range jb ∈ ð0; ½ð1 2 rÞ=rÞ, ja 5 1 2 ½r=ð1 2 rÞjb . Hence, ga ∈ ðga jj 50;j 5ð12rÞ=r ; r Þ, where it is easily verified that ga jj 50;j 5ð12rÞ=r ≥ 12. Facts 1 and 2 imply that the upper bound on voting body sizes for which a completely mixed equilibrium exists, m CM , is the value of m solving equation ðA5Þ at ga 5 ga jj 50;j 5ð12rÞ=r . Similarly, the lower bound is the value of m solving equation ðA5Þ at ga 5 r . Notice that this corresponds to m I . Facts 1 and 2 also imply that m I < m CM . Finally, fact 2 implies that, for all m ∈ ðm I ; m CM Þ, the completely mixed equilibrium is unique. QED Equilibrium uniqueness turns on the monotonicity of Vb jj 50 in jb . Essentially, if q is high such that Vb jj 50 is increasing in jb at jb 5 1 and m 5 m E , then equilibrium is unique for every n. Formally, q * , the threshold between high and low q, is defined as the ðuniqueÞ value ( ) yV b q * ; max q ∈ ½0; 1j 50 ; ðA6Þ yjb j 50;j 51;m5m ðqÞ a

a

b

b

a

b

a

a

a

b

E

where m E ðqÞ reflects the dependence of m E on the prevalence of conflict. The following lemma establishes existence and uniqueness of q * . Lemma 7. The threshold q * exists and is unique. Furthermore, q0 < q * < q1 , where q0 ;

1 2r 2 1 2 r 2 r ð1 2 r Þ

q1 ;

1 2r 2 1 : 2 r 2 ð1 2 r Þ

and

Proof. We prove q0 < q * < q1 by showing that ð1Þ for q ≤ q0 and m > 0, yVb =yjb jj 50;j 51 < 0; and ð2Þ for q ≥ q1 and m > 0, yVb =yjb jj 50;j 51 > 0. Existence of q * then follows from continuity of yVb =yjb jj 50;j 51;m5m ðqÞ in q and the intermediate value theorem, while the max operator in equation ðA6Þ guarantees uniqueness. Notice that a

b

a

a

b

b

E

yVb Gðm 1 1Þ m ð1 2 qÞr½r 2 ðz b Þðm=2Þ21 ð1 2 2gb Þ 5 2 G ððm=2Þ 1 1Þ 2 yjb ðm=2Þ21

2 ð1 2 r Þ ðz a Þ 2

ð1 2 2ga Þ:

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ðA7Þ

optimal size of voting bodies

1015

Hence, yVb =yjb jj 50;j 51 takes on the sign of a

b

r

2

z Eb z Ea

!ðm=2Þ21 ð1 2 2gEb Þ 2 ð1 2 r Þ2 ð1 2 2gEa Þ:

ðA8Þ

By lemma 8 ðbelowÞ, z aE < z bE . Thus, ðA8Þ is strictly smaller than r 2 ð1 2 2gEb Þ 2 ð1 2 r Þ2 ð1 2 2gEa Þ 5 ð2r 2 1Þ½2qðr 2 2 r 1 rÞ 1 1 2 2r; which is negative iff q ≤ q0 . Thus, yVb =yjb jj 50;j 51 for all m. This establishes part 1. For q ≥ q1, ga > 12 and gb ≤ 12. Thus, equation ðA8Þ is greater than zero, which establishes part 2. QED Lemma 8. If ja 5 0 and jb > ð1 2 rÞ=r, then z a < z b . Proof. It is sufficient to show that jga 2 12 j > jgb 2 12 j. If ja 5 0 and jb > ð1 2 rÞ=r, then a

b

ga > qr 1 ð1 2 qÞ½r r 1 ð1 2 r Þð1 2 rÞ ≥ qr 1 ð1 2 qÞ

1 1 > ; 2 2

where the first inequality follows from jb > ð1 2 rÞ=r, the second from r ≥ 12, and the third from r > 12. If gb > 12, then jga 2 12 j > jgb 2 12 j follows immediately from the fact that gb < ga . If gb ≤ 12, then jga 2 12 j > jgb 2 12 j is equivalent to showing that ga 2 ð1 2 gb Þ > 0 . For ja 5 0 and jb > ð1 2 rÞ=r, ga 2 ð1 2 gb Þ > qr 1 r ð1 2 qÞr 1 ð1 2 qÞð1 2 rÞ 1 qð1 2 r Þ 1 ð1 2 r Þð1 2 qÞr 2 1 5 0: This completes the proof. QED Proposition 3 implies that, to prove proposition 4, only partially mixed equilibria remain to be analyzed. Proposition 13. Under low conflict ði.e., q > q * Þ, a partially mixed equilibrium exists iff n is such that m CM ≤ n < m E . For each such n, there exists exactly one partially mixed equilibrium. Moreover, m CM < m E . Proof. By lemma 9 ðbelowÞ, in any partially mixed equilibrium, jb ∈ ½ð1 2 rÞ =r; 1Þ. Next, we claim that Vb jj 50;j 5ð12rÞ=r < 0 iff n > m CM . At m CM , Vb jj 50;j 5ð12rÞ=r 5 0 by construction. Moreover, lemmas 2 and 8 imply that Vb jj 50;j 5ð12rÞ=r is strictly decreasing in n. This proves the claim. We also claim that Vb jj 50;j 51 > 0 iff n < m E . At m E , Vb jj 50;j 51 5 0 by construction. Moreover, lemmas 2 and 8 imply that Vb jj 50;j 51 is strictly decreasing in n. This proves the claim. From Lemma 11 ðbelowÞ—which shows that, under low conflict, Vb is strictly increasing in jb ∈ ½ð1 2 rÞ=r; 1Þ—it then follows that for all m CM ≤ n < m E , there exists a unique value jb ∈ ðð1 2 rÞ=r; 1Þ such that Vb jj 50 ðjb Þ 5 0. It is straightfora

b

a

a

a

b

a

a

b

b

b

a

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b

1016

journal of political economy

ward to verify that, at this value of jb , Va jj 50 < 0. Hence, this constitutes a partially mixed equilibrium. Finally, we establish that m CM < m E . At m CM , Vb jj 50;j 5ð12rÞ=r 5 0. Lemma 11 implies that, at m CM , Vb jj 50;j 51 > 0. Moreover, from lemmas 2 and 8 we know that Vb jj 50;j 51 is strictly decreasing in m. Because, at m E , Vb jj 50;j 51 5 0, this implies that m E > m CM . QED Lemma 9. In any partially mixed equilibrium, jb ≥ ð1 2 rÞ=r. Proof. We prove the lemma by showing that jb < ð1 2 rÞ=r implies z a > z b , which contradicts lemma 10 ðbelowÞ. Recall that ga > gb . First, we find the value of jb that makes ga 5 12. This is readily shown to be a

a

a

a

b

b

b

a

jb0 5

b

1 2 2qr 2 2r ð1 2 qÞr 2ð1 2 r Þð1 2 qÞr

while jb0 2

12r ð2r 2 1Þ½2qð1 2 rÞ 1 2r 2 1 < 0: 5 2 r 2ð1 2 r Þð1 2 qÞr2

Because ga is increasing in jb , for jb ≤ jb0, gb < ga ≤ 12 and, hence, z a > z b . Next, we find the value of jb that makes gb 5 12. This is readily shown to be jb00 5

1 2 2qr 2 2r ð1 2 qÞr 2ð1 2 r Þð1 2 qÞr

while jb00 2

12r ð2r 2 1Þ½2qð1 2 rÞ 1 2r 2 1 > 0: 5 r 2r ð1 2 qÞr2

In the region jb0 < jb < j00b , gb < 12 < ga . As ga and gb are both strictly increasing in jb , z a 2 z b is strictly decreasing in jb . Now notice that, at jb 5 ð1 2 rÞ=r, ga 5 1 2 gb , and therefore, z a 5 z b . Hence, we may conclude that for all jb < ð1 2 rÞ=r, z a > z b . This completes the proof. QED Lemma 10. In any partially mixed equilibrium, z a ≤ z b . Proof. In any partially mixed equilibrium, Va ≤ 0 and Vb 5 0. This implies that Va 1 Vb ≤ 0, which may be rewritten as ! n n ½ðz a Þn=2 2 ðz b Þn=2 ≤ 0: 2 And this inequality holds iff z a ≤ z b . QED Lemma 11. For q ≥ q1, Vb jj 50 is strictly increasing in jb ∈ ½ð1 2 rÞ= r; 1. a

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optimal size of voting bodies

1017

Proof. Differentiating Vb jj 50 with respect to jb yields a

yVb 5 yjb j 50 a

! n n n ð1 2 qÞr½r 2 ðz b Þðn=2Þ21 ð1 2 2gb Þ 2 2 2 ð1 2 r Þ2 ðz a Þðn=2Þ21 ð1 2 2ga Þ;

which takes the sign of the expression in brackets. Notice that gb is increasing in jb and for ja 5 0, gb is decreasing in q. At q 5 q1 , gb jj 50;j 51 5 12. Hence, for q ≥ q1 and jb ∈ ½ð1 2 rÞ=r; 1, we have gb jj 50 ≤ 12. Moreover, for jb ∈ ½ð1 2 rÞ=r; 1, it can be easily verified that ga jj 50 > 12. Finally, together, gb ≤ 12 and ga > 12 imply that the expression in brackets is strictly positive. QED This completes the proof of proposition 4. a

b

a

a

Proof of Proposition 5 Under instrumental and expressive voting, ja and jb remain constant as n increases. Hence, voting becomes ðweaklyÞ more expressive. It remains to show that voting becomes ðstrictlyÞ more expressive under completely mixed and partially mixed voting, as well as when we move from one equilibrium class to the next. Since there exists a unique completely mixed equilibrium for every n in the interval m I < n < m CM , we can define a sequence of completely mixed equilibria, with n running from ⌈m I ⌉ to ⌊m CM ⌋. Note that this sequence is fully characterized by the sequence of mixing probabilities ffja ; jb gn g⌈m ⌉

CM

CM

E

a

a

a

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1018

journal of political economy

expressive when we move from one equilibrium class to the next. This completes the proof of proposition 5. Proof of Proposition 6 Before getting to the heart of the proof, we formally define m PM and derive some of its properties. Define m PM to be the largest value of m such that the indifference condition for conflicted voters with a b signal still has a solution in jb . That is, ( m PM ; max m such that Vb jj 50 5 0 a

"

12r ; 1 has a solution in jb ∈ r

#)

ðA9Þ

:

We denote this solution by jb;m .11 The next lemma establishes existence and uniqueness of m PM and provides sufficient conditions for m PM > m E ðsuch that expressive voting overlaps with partially mixed votingÞ. Lemma 14. m PM exists and is unique. Moreover, for q ≤ q0, m E < m PM . Proof. By lemma 15 ðbelowÞ, for q ≤ q0, the unique jb that maximizes Vb jj 50 ðjb Þ 0 over the interval ½ð1 2 rÞ=r; 1 is strictly interior. Denote this jb by jb. By the envelope theorem, yðVb jj 50 Þ d Vb j 5 : dm j 50;j 5j ðmÞ ym j 5j ðmÞ PM

a

a

b

a

0 b

0

b

b

Lemmas 2 and 8 imply that yðVb jj 50 Þ=ymjj 5j ðmÞ and, therefore, ðd=dmÞVb jj 50;j 5j ðmÞ are strictly negative. From here, the proof of existence and uniqueness of m PM is analogous to that for m E in the main text. To prove thatm PM > m E for q ≤ q0, note that, at m 5 m E , Vb jj 50;j 51 5 0. By lemma 00 15 we know that yVb =yjb jj 50;j 51 < 0. Hence, for some jb strictly smaller than but close to one, Vb jj 50;j 5j > 0. Lemma 2 then implies that there exists an m > m E such that the equation Vb jj 50 5 0 has a solution in jb ∈ ½ð1 2 rÞ=r; 1. Therefore, m PM , which is defined as the largest m for which such a solution exists, must also be strictly greater than m E . QED a

b

0 b

a

a

a

a

b

b

0 b

b

b

00 b

a

11 While neither m PM nor jb;m admits closed-form solutions, approximations are available. For small ε, PM

m PM ≈

2 12ε r p ε

!2

and 1 2 qð1 2 r Þ 12r : 2 jb;m ≈ 2 r r ð1 2 qÞr PM

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optimal size of voting bodies

1019

Lemma 15. For q ≤ q0, Vb jj 50 ðjb Þ is single peaked in jb on the interval ½ð1 2 rÞ=r; 1. Moreover, the peak is strictly interior. Proof. From lemma 7, we know that yVb =yjb jj 50;j 51 < 0. From equation ðA7Þ, we know that yVb =yjb takes the sign of a

a

b

r 2 ðz b Þðn=2Þ21 ð1 2 2gb Þ 2 ð1 2 r Þ2 ðz a Þðn=2Þ21 ð1 2 2ga Þ: This expression is strictly positive at ja 5 0 and jb 5 ð1 2 rÞ=r since ga > gb < 12, where gb < 12 follows from

1 2

and

gb jj 50;j 5ð12rÞ=r 5 qð1 2 r Þ 1 ð1 2 qÞ½ð1 2 r Þr 1 r ð1 2 rÞ 1 1 < qð1 2 r Þ 1 ð1 2 qÞ ≤ : 2 2 a

b

Thus, yVb =yjb jj 50;j 5ð12rÞ=r > 0. The intermediate value theorem now implies that there exists at least one jb ∈ ðð1 2 rÞ=r; 1Þ where yVb =yjb jj 50 5 0. We will show that, at any such point, d 2 Vb =ðdjb Þ2 jj 50 < 0. Therefore, Vb jj 50 is single peaked on jb ∈ ½ð1 2 rÞ=r; 1. First, the first-order condition can be satisfied only when 12 < gb < ga . Next, the first-order condition implies that a

b

a

a

a

1 2 2gb 1 2 2ga

!2

12r r

5

!2

za zb

!n22 :

ðA10Þ

Now notice that d 2 Vb =ðdjb Þ2 is proportional to ! n 2 1 ½r 3 ðz b Þðn=2Þ22 ð1 2 2gb Þ2 2 ð1 2 r Þ3 ðz a Þðn=2Þ22 ð1 2 2ga Þ2 2 1 2½ð1 2 r Þ3 ðz a Þðn=2Þ21 2 r 3 ðz b Þðn=2Þ21 : Since 12 < gb < ga , the term in brackets in the second line is negative. For the term in brackets in the first line to be negative, we need to show that r3 ðz b Þðn=2Þ22 ð1 2 2gb Þ < 1: 3 ð1 2 r Þ ðz a Þðn=2Þ22 ð1 2 2ga Þ2 2

Substituting in equation ðA10Þ, we have za zb

!n=2

12r < 1; r

where the required inequality holds because 12 < gb < ga and r > 12. QED We are now in a position to prove proposition 6. The proofs of parts 1, 2, and 4 are identical to the proofs of proposition 1, lemma 12, and proposition 2, respec-

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1020

journal of political economy

tively. It remains to show that ð1Þ low partially mixed voting is an equilibrium iff m CM ≤ n < m PM ; ð2Þ high partially mixed voting is an equilibrium iff m E ≤ n < m PM ; and ð3Þ if a low, respectively, high partially mixed equilibrium exists, it is unique. First, the proof of lemma 14 implies that m PM constitutes the upper bound on partially mixed voting. Note that lemma 9 holds independently of q. Thus, we may apply the same reasoning as in the proof of lemma 13 to conclude that m CM is the lower bound for low partially mixed voting. The argument as to why Vb jj 50;j 51 < 0 iff n > m E is unchanged from the low-conflict case. Thus, we may conclude that m E is the lower bound for high partially mixed voting. Finally, uniqueness follows from lemma 15. This completes the proof of proposition 6. a

b

Proof of Proposition 8 The second part of the proposition follows immediately from the law of large numbers and the fact that, for q < q1, 1 2 gEb < 12 < gEa . To prove the first part of the proposition, note that adding two voters to a voting body of n 2 1 voters affects the outcome only if, after n 2 1 votes, either ð1Þ the correct choice is lagging by one vote and the next two votes are “successes” or ð2Þ the correct choice is leading by one vote and the next two votes are “failures.” This implies that

Sðn 1 1jaÞ 2 Sðn 2 1jaÞ 5

! n21 n=2 n 2 1 ðz a Þ ð2ga 2 1Þ 2

and Sðn 1 1jbÞ 2 Sðn 2 1jbÞ 5 2

! n21 n=2 n 2 1 ðz b Þ ð2gb 2 1Þ: 2

Hence, 1 Sðn 1 1Þ 2 Sðn 2 1Þ 5 2

! n21 n=2 n=2 n 2 1 ½ðz a Þ ð2ga 2 1Þ 2 ðz b Þ ð2gb 2 1Þ: 2

For n sufficiently large, the sign of this expression is negative iff zaE zbE

!n=2

1 2: < 1 E ga 2 2 gEb 2

Lemma 8 implies that the left-hand side is decreasing in n and goes to zero in the limit. The right-hand side is a positive constant. Thus, for sufficiently large n, Sðn 1 1Þ is decreasing. This completes the proof of proposition 8.

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optimal size of voting bodies

1021

Proof of Proposition 9 In a completely mixed equilibrium,

n n B g; 1 1; 1 1 2 2 ; Sðn 1 1Þ 5 n n 1 1; 1 1 B 2 2 where g ; ga 5 1 2 gb . The proposition follows immediately from the following lemma, which shows that, if the probability of being pivotal remains constant as n increases, then accuracy must fall. Lemma 16. Let 12 < g 2 d < g < 1. If Gðn 2 1Þ Gðn 1 1Þ ½gð1 2 gÞðn=2Þ21 5 2 fðg 2 dÞ½1 2 ðg 2 dÞgn=2 ; G2 ðn=2Þ G ððn=2Þ 1 1Þ

ðA11Þ

then n n n n B g; ; B g 2 d; 1 1; 1 1 2 2 2 2 > 0: 2 n n n n ; 1 1; 1 1 B B 2 2 2 2 Proof. Define the gap between g and 12 to be g ; g 2 12. Then equation ðA11Þ can be rewritten as Gðn 2 1Þ 1 ðn=2Þ21 ð1 2 4g 2 Þðn=2Þ21 G2 ðn=2Þ 4 Gðn 2 1Þnðn 2 1Þ 1 n=2 5 ½1 2 4ðg 2 dÞ2 n=2 : 4 G2 ðn=2Þðn=2Þ2 Simplifying yields the equality n ð1 2 4g 2 Þðn=2Þ21 5 ½1 2 4ðg 2 dÞ2 n=2 : n21

ðA12Þ

Next, note that 1 2ðn 1 1Þ 1 5 : n n n n n B B 1 1; 1 1 ; 2 2 2 2 2

Thus, using the integral representation of the incomplete beta function, we need only show that

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1022

journal of political economy

E

g

½tð1 2 tÞðn=2Þ21 dt 2

0

2ðn 1 1Þ n=2

E

g2d

½tð1 2 tÞn=2 dt > 0:

0

Defining u 5 t 2 12, we may rewrite the left-hand side as 1 ðn=2Þ21 4

"

E

g 2 ðn=2Þ21

ð1 2 4u Þ

21=2

E

n11 du 2 n

#

g 2d

ð1 2 4u Þ du : 2 n=2

21=2

Thus, it suffices to show that the term in brackets is strictly positive. This term may be rewritten as

E

"

g 2d

5

E

2 ðn=2Þ21

ð1 2 4u Þ

21=2

g 2d 2 ðn=2Þ21

4u 2 ð1 2 4u Þ

21=2

# n11 2 ð1 2 4u Þ du 1 12 n

1 du 2 n

E

E

g

g 2d

g 2d

ð1 2 4u Þ du 1 2 n=2

21=2

ð1 2 4u 2 Þðn=2Þ21 du

E

g

ð1 2 4u 2 Þðn=2Þ21 du:

g 2d

Now, integrating the first term of this expression by parts, we obtain 2

1 2 n 52

E

1 1 ðg 2 dÞ½1 2 4ðg 2 dÞ2 n=2 1 n n

E

g 2d

E

g 2d

ð1 2 4u 2 Þn=2 du

21=2

g

ð1 2 4u Þ du 1 2 n=2

21=2

g 2d

1 ðg 2 dÞ½1 2 4ðg 2 dÞ2 n=2 1 n

ð1 2 4u 2 Þðn=2Þ21 du

E

g

ð1 2 4u 2 Þðn=2Þ21 du:

g 2d

Recall that, for all u in the support of the second term, u=g < 1. Hence,

2 > 2

1 ðg 2 dÞ½1 2 4ðg 2 dÞ2 n=2 1 n

E

g

ð1 2 4u 2 Þðn=2Þ21 du

g 2d

1 1 ðg 2 dÞ½1 2 4ðg 2 dÞ2 n=2 1 n g

E

g

uð1 2 4u 2 Þðn=2Þ21

g 2d

ðA13Þ

1 1 1 ð1 2 4g 2 Þn=2 5 2 ðg 2 dÞ½1 2 4ðg 2 dÞ2 n=2 2 n g 4n 1

1 1 ½1 2 4ðg 2 dÞ2 n=2 : g 4n

Using equation ðA12Þ to substitute for ½1 2 4ðg 2 dÞ2 n=2, equation ðA13Þ reduces to

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optimal size of voting bodies 2

1 n 1 1 ðg 2 dÞ ð1 2 4g 2 Þðn=2Þ21 2 ð1 2 4g 2 Þn=2 n n21 g 4n

1

1 1 n ð1 2 4g 2 Þðn=2Þ21 g 4n n 2 1

1023

# " 1 1 1 1 1 2 ðn=2Þ21 2 2 ð1 2 4g Þ 1 : 5 ð1 2 4g Þ 2 ðg 2 dÞg g n21 4n 4n 2 1 It suffices to show that the term in brackets is positive. Rewriting this expression, we have " 1 4

# ! 1 1 1 2 ð1 2 4g 2 Þ 1 4g d ; n21 n n21

which is strictly positive since 1 2 4g 2 > 0 and d > 0. QED Proof of Proposition 10 To prove the proposition, the following lemma is useful. Let Sðga ; gb Þ denote the accuracy of a fixed-size voting body when the probability of a vote for A in state a is equal to ga while the probability of a vote for A in state b is equal to gb. Lemma 17. Fix ga ≥ gb and let 0 ≤ d < 1 2 ga . If ðga 1 dÞ½1 2 ðga 1 dÞ < ½1 2 ðgb 1 dÞðgb 1 dÞ; then d Sðga 1 d; gb 1 dÞ < 0: dd Proof. Using the beta function representation of accuracy, we have d Sðga 1 d; gb 1 dÞ dd n=2 n=2 1 fðga 1 dÞ½1 2 ðga 1 dÞg 2 f½1 2 ðgb 1 dÞðgb 1 dÞg ; 5 1 2 n=2 ½tð1 2 tÞ dt

E

0

which is strictly negative since ðga 1 dÞ½1 2 ðga 1 dÞ < ½1 2 ðgb 1 dÞðgb 1 dÞ: QED

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1024

journal of political economy

We divide the remainder of the proof of proposition 10 into four parts. < gEa , while gHPM < gEb . Next, note that 1. S HPM > S E : Note that gHPM a b 5 ð1 2 r Þð1 2 qÞrð1 2 jb Þ gEa 2 gHPM a < r ð1 2 qÞrð1 2 jb Þ

: 5 gEb 2 gHPM b Lemma 8 implies that, for all 0 < d < 1 2 gHPM , a 1 dÞ½1 2 ðgHPM 1 dÞ < ½1 2 ðgHPM 1 dÞðgHPM 1 dÞ: ðgHPM a a b b > 0. Lemma 17 implies that Now define D 5 gEa 2 gHPM a

; gHPM Þ > SðgHPM 1 D; gHPM 1 DÞ > S E : S HPM 5 SðgHPM a b a b , while gIb < gHPM . Next, note that 2. S I > S HPM : Note that gIa < gHPM a b 2 gIa 5 ð1 2 qÞ½ð1 2 r Þrjb 2 r ð1 2 rÞ gHPM a < ð1 2 qÞ½r rjb 2 ð1 2 r Þð1 2 rÞ

2 gIb : 5 gHPM b Because gIa ð1 2 gIa Þ 5 gIb ð1 2 gIb Þ, we have ðgIa 1 dÞ½1 2 ðgIa 1 dÞ < ½1 2 ðgIb 1 dÞðgIb 1 dÞ . The remainder of the proof is analogous to part 1. for all d ≤ 1 2 gHPM a < ð1 2 rÞ=r < jHPM 3. S CM > S HPM : Since jaCM 5 1 2 ½r=ð1 2 rÞjCM and jCM , we b b b CM < HPM CM > HPM have gb gb . If ga ga , then accuracy deteriorates in both states and, hence, S CM > S HPM . Else, note that HPM CM 2 gCM 2 gCM 2 rjHPM 2 ð1 2 rÞ gHPM a a 2 ðgb b Þ 5 ð1 2 qÞð2r 2 1Þ½2rj b b

< ð1 2 qÞð2r 2 1Þ½2ð1 2 rÞ 2 2ð1 2 rÞ 5 0: CM CM CM Finally, since gCM a ð1 2 ga Þ 5 gb ð1 2 gb Þ, using arguments analogous to those in part 2, S CM > S HPM . < gHPM < gHPM 4. S LPM > S HPM : Note that gLPM , while gLPM . Next, note that a a b b

2 gLPM 5 ð1 2 r Þð1 2 qÞrðjb;HPM 2 jb;LPM Þ gHPM a a < r ð1 2 qÞrðjb;HPM 2 jb;LPM Þ

2 gLPM : 5 gHPM b b The remainder of the proof is analogous to part 1. This completes the proof of proposition 10.

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optimal size of voting bodies

1025

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