Pivots Versus Signals in Elections

Transcription

1 Pivots Versus Signals in Elections Ada Meirowitz y and Kenneth W. Shotts z Septeber 9, 007 Abstract Incentives in voting odels typically hinge on the event that a voter is pivotal. But voting can also in uence the behavior of elites in subsequent periods. In this paper, we consider a odel in which voters have private inforation about their policy preferences and an election is held in each of two periods. In this setting, a vote in the rst period can have two types of consequences; it ay be pivotal in deciding who wins the rst election and it provides a signal that infors the beliefs that candidates running in the second election use when selecting equilibriu platfors. Pivot events are exceedingly unliely, but when they occur the e ect of a single vote is enorous, since it deterines the electoral outcoe. In contrast, vote totals always have soe signaling e ect on future policies, but the e ect of a single vote is always very sall. We investigate whether the forer, pivot, e ect or the latter, signaling, e ect drives equilibriu voting behavior in large electorates. Prepared for the Annual Conference on Political Econoy at the University of Rochester s Wallis Institutue, October 007. For helpful coents, we than Patric Huel and seinar participants at Bereley, the Hoover Institution, the 006 MPSA annual eeting, and the 007 APSA annual eeting. Marc Meredith provided excellent research assistance on nuerical siulations early in the project. We are indebted to Keith Krehbiel and Nolan McCarty for profound observations on how this wor ts into the larger literature in political science. y Associate Professor of Politics, Princeton University. Princeton, NJ Eail: aeirowi@princeton.edu. z Associate Professor, Stanford Graduate School of Business. 58 Meorial Way, Stanford CA Eail: shotts@stanford.edu.

2 Introduction In nearly all odels of voting the payo fro casting a particular ballot hinges exclusively on pivot events. These are events in which the election is tied or nearly tied, so that a single vote can deterine the outcoe. In decision-theoretic odels a voter decides whether and how to vote based on exogenous probabilities of ties between candidates. Gae theoretic odels endogenize the equilibriu probability that a vote is pivotal. And several recent in uential papers focus on inforation that a voter can infer fro the fact that he is pivotal, and analyze electoral equilibria when voters condition on being pivotal. The pivot-based literature on elections is vast, but the odels have two features in coon: (i) when a voter is pivotal, the action she taes has a large ipact on her payo, but (ii) pivot events are very unliely. The large ipact is due to the fact that in a pivot event, a single vote can deterine the outcoe of the election. The low probability arises fro the fact that in a large election it is exceedingly unliely that two candidates will receive the sae nuber of votes or di er by exactly one vote. Although pivot based odels doinate the gae-theoretic literature on elections, the infrequency of pivot events in all but the sallest elections raises a natural question: is electoral behavior driven by ore than just concerns about being pivotal? If pivot events do not actually drive the calculus of voters then a large and growing literature on voting theory ay be focused on second-order concerns. Why would voters care about anything other than a pivot event? Consider, for exaple, the buildup to the 006 idter election in the United State. Pundits speculated that voters dissatisfaction with President Bush s handing of the war in Iraq would cost the Republican party its ajority in Congress. While Republicans electoral losses ay in fact have been a direct result of voters desire to change the coposition of the legislature, another explanation is that voters cast ballots for Deocrats in order to send Bush a essage, and encourage hi to change policy. An eerging literature, based on the intuition that vote totals atter in elections that don t end in a tie, o ers an alternative perspective to the doinant pivot-based theories of elections. Theorists Decision theoretic odels include the wor of Downs (957), Tulloc (967), Rier and Ordeshoo (968), and Myerson and Weber (993). Exaples of gae theoretic odels include Palfrey and Rosenthal (983), Myerson (998, 000), Capbell (999), and Borgers (004). Models involving inforation aggregation include Feddersen and Pesendorfer (996, 997, 999), Austen-Sith and Bans (996), Deel and Piccione (000), and Battaglini (005).

3 of elections have explored several di erent ways that vote totals could a ect downstrea electoral or policy outcoes. One odelling approach is a coon values setup, in which voters use their votes to convey inforation about the state of the world. Pietty (000) develops a two period odel of referendu voting in which voters counicate policy inforation to each other as they vote. Razin (003) analyzes a odel of andates in which vote totals convey inforation to the winner of an election and thus a ect the policies she enacts. In this odel, the functional for of the election winner s response to signals is a priitive of the gae. Razin investigates liiting behavior for large electorates and characterizes two types of equilibria. In one type of equilibriu voters behavior is conventional in the sense that a voter whose private signal indicates that liberal policies are good tends to vote for a liberal candidate: In any liit of these conventional equilibria, the behavior of voters converges to coin ipping. The other equilibria are unconventional since voters respond perversely to their private signals: upon observing inforation that favors liberal policies, a voter becoes ore liely to vote for the conservative candidate. The equilibria of the Pietty and Razin odels suggest that a desire to in uence the decisiveness of victory, and not just the identity of the winner, can reain in large elections. However, in both odels all voters have identical preferences and the e ects that their votes can have on future policy outcoes are deterined directly by odelling assuptions, rather than being deterined by strategic choices ade by copeting political elites who observe election outcoes. Another odelling approach is a private values setup, in which voters can use their votes to a ect future candidates positions, and thus policy outcoes. Each election thus serves two purposes: to select a winner and to act as a poll about voters preferences. In this vein, Meirowitz and Tucer (007) analyze a odel of alternating priary and presidential elections, in which voters use their votes to signal dissatisfaction with an incubent and thereby induce hi to exert costly e ort to ae hiself ore appealing in a subsequent election. Castanheira (003) adapts Pietty s odel to a private values setting, and uses it to analyze voting for losers in an election with four candidates. In his odel, there are four possible distributions of This result is Razin s Proposition 4, part (i).

4 voters in the electorate, and voters ay choose to vote for a candidate who is alost certain to lose in the rst election since their vote ay deterine which of four positions candidates will adopt in a second period election. In Castanheira s odel, as in Pietty s odel, the signaling e ect of rst period electoral behavior is based on the very low probability event that a vote is inforationally pivotal, i.e., although there is signaling in these odels, the signaling is fundaentally based on pivot events, which are extreely unliely to occur. In contrast, Shotts (006) develops a private values odel of repeated elections in which each voter s actions, by conveying inforation about his preferences, always have soe sall e ect on future policy outcoes. Sall agnitude, high probability signaling e ects wor quite di erently fro large agnitude low probability pivot e ects. However, in large elections, a single vote has a vanishingly sall e ect on politicians beliefs about voter preferences, and Shotts does not address the question of whether signaling otivations are actually relevant in large elections. Moreover, a voter ay nd herself cross-pressured when a vote for the candidate she prefers sends the wrong essage to politicians, i.e., she ay wish to vote for one candidate for signaling reasons and a di erent one for pivot reasons. Because both the lielihood that she is pivotal and the e ect of a single vote on candidates vote shares are sall it is not clear how tradeo s between pivot and signaling e ects balance out in equilibria for large elections. Thus, although recent papers have oved beyond pivot-based theories of voting, they do not shed light on the question of whether equilibriu behavior in large repeated elections with private values is driven by pivot or signaling considerations. In this paper, we analyze a private values odel of repeated elections with both pivot and signaling otivations and show that the latter doinates with a large electorate. While our paper is related to that of Meirowitz (005), which focuses on signaling otivations in public opinion, it is, however, closest to that of Shotts (006). Shotts analyzes elections with a xed population and focuses on equilibria in which oderates abstain, to signal that they are oderate. In contrast, we analyze a odel without the possibility of abstention and focus on liiting behavior for large electorates. It turns out that the assuption that voters cannot abstain is not crucial for our results. Huel (007) builds on the present paper and Shotts (006) to study liiting behavior when voters can choose to abstain. Huel shows that although the type of equilibriu with abstention characterized 3

5 by Shotts exists for any nite population, in the liit abstention vanishes and voter behavior converges to the behavior we characterize here. Huel also derives a useful lower bound for the agnitude of signaling e ects in large elections. Having discussed the relevant literature, we now suarize our contribution. The ey eleents of our odel are as follows. Each voter in our odel has private inforation about his own policy preferences, and in each of two elections he casts a ballot for one of two available alternatives. Following the rst election, two o ce-otivated candidates copete for a second o ce, by staing policy positions, and the second election is held. Candidates in the second period base their policy positions on beliefs about the distribution of preferences in the electorate. In equilibriu these beliefs are infored by the vote totals in the rst election, and each voter s vote in the rst period thus has a sall signaling e ect on second period policy. Our ain result is that as the electorate gets large the equilibriu converges to the equilibriu of a slightly di erent gae in which the rst period outcoe is payo irrelevant, as in the case of a rst period poll. In other words, in large elections behavior is driven by the signaling otivation and not the pivot otivation. This result has potentially iportant iplications for the literature on pivot-based odels of elections, since ost of the interesting equilibria in such odels rely heavily on the fact that a voter only cares about events in which his vote is pivotal. In our odel, in contrast, the e ect of pivot events on equilibriu voter behavior is relatively uniportant copared to the e ect of signaling concerns. At the very least, future research needs to tae seriously the possibility that pivot events are not of rst-order iportance when rational voters tae into account the future e ects of their votes. The paper proceeds as follows. Section introduces the odel and in Section 3 we present two concrete exaples of how signaling and pivot e ects wor. Section 4 proves equilibriu existence. Section 5 describes the intuition behind our ain result, which is proved in Section 6. Section 7 discusses the result. 4

6 The Model Consider an electorate with an odd nuber of voters n 3: It will be convenient to use the fact that n = + for soe integer : Let the set of voters be N, and let each voter i N have an ideal point, v i [0; ]: We assue that the ideal points are iid draws fro a distribution function, F (); which is strictly increasing and continuously di erentiable, and has a continuous density f() on the support [0; ]. Each voter s utility over policy, x; in a given period is u i (x) = (jx v i j) where : [0; ]! R + is strictly increasing, convex and di erentiable. Since expected utility is de ned only up to positive a ne transforations we ae the innocuous but convenient assuption that 0 () = : The voter s total utility is siply the su of his policy utility in the two periods. In the rst period election, two xed alternatives are available. We denote the locations of the alternatives by L; R [0; ] (with L R): If voters care only about the rst period, or are yopic, eliination of wealy doinated strategies yields a unique equilibriu, in which all voters to the left of x p = L+R vote L and all voters to the right of x p vote R: We call this the pivot cutpoint. We, however, are interested in the dependencies across elections, and thus consider a odel with two periods, building on Shotts (006). In the second period, two o ce otivated candidates select policy platfors and then the electorate votes. The candidates are assued to now only the distribution F () fro which the n ideal points are drawn, the size of the electorate, n, and the voters rst period actions. Fro Calvert (985), we now that for a gae in which two o ce otivated candidates believe that F () is the distribution of the voter s ideal point, in any Nash equilibriu with wealy undoinated voting the candidates will both locate at F ( ):3 In the two-period signaling gae that we study, in any Perfect Bayesian equilibriu, the distribution of the depends on the rst-period votes via Bayes Rule. At any history in which F () is a distribution consistent with Bayes Rule following the observed rst period voting, the second period candidates both locate at the point F ( ): While Shotts (006) focuses on equilibria with abstention in the rst period, we restrict the set of actions available to voters so that they ust vote either L or R; this enables us to focus on a particularly siple class of equilibria, 3 One such equilibriu has each voter ipping a fair coin when indi erent. Given strategies for the rst period, the second period behavior is standard, and well understood (Calvert 985, Shotts 006). 5

7 involving only a single cutpoint. In particular, we focus on a class of equilibria in which all voters use the sae type speci c onotone voting strategy. In such an equilibriu, rst-period voting strategies are characterized by a cutpoint x c with voters to the left ( v i < x c ) voting L and voters to the right (v i > x c ) voting R. If voting satis es this cutpoint, i.e. it is onotone, then the nuber of votes for R, denoted #R; captures all of the publicly available inforation about voter ideal points, and is a su cient statistic for the second-period candidates proble of inferring the distribution of the voter s ideal point fro rst-period behavior. We denote such a posterior distribution as F ( j #R; x c ): Before proceeding we provide a few coents about this function. Given that #R of n voters have ideal points to the right of (greater than) x c ; the is less than x c if and only if #R = n. Siilarly the is greater than x c if and only if #R +. In the forer case, the is the ( + ) th lowest ideal point of the n #R voters with ideal points less than x c, i.e., the ideal point is the ( + ) th order statistic fro n #R draws fro the distribution H (x; x c ) = F (x) F (x c) with support [0; x c ]. Siilarly in the latter case, the is the ( + (n #R)) th order statistic fro #R draws fro the distribution H + (x; x c ) = F (x) F (xc) F (x c) with support [x c ; ]: As previously entioned, Calvert s result shows that given x c, #R, and a belief apping F (j#r; x c ) sequential rationality of the candidates and wealy undoinated voting by the voters iplies that the second period policy is F ( j #R; x c): In characterizing a cutpoint perfect Bayesian equilibriu with wealy undoinated second period voting strategies it is su cient to characterize a rst period cutpoint x c [0; ] such that if every voter other than i is using the strategy with cutpoint x c it is optial for voter i to do so as well. Checing this condition hinges on the fact that in an equilibriu of this for second period candidates both locate at the point F ( j #R; x c): The equilibriu cutpoint balances two e ects that in uence rst period voting. The pivot e ect captures the incentive to vote for L if jl v i j < jr v i j and R if the opposite is true. The signaling otivation captures the incentive to vote for R if, given i s expectations about the actions of the other voters, increasing #R is liely to ove the second period policy F ( j #R; x c) towards v i, and to vote for L if increasing #R is liely to ove the second period policy away fro v i. The pivot e ect is the product of the probability that i is pivotal and the payo di erence between the policy L and R: 6

8 In contrast to the pivot e ect, which captures a low probability event with a non trivial payo in that event, the signaling otivation taes into account the fact that i s vote will always have an e ect on the second period policy. However, the signaling e ect is sall for each of the possible realizations of the votes cast by Nnfig. For soe realizations of the votes by Nnfig increasing #R will be attractive to i, while for other realizations of the these votes increasing #R will be unattractive to i. The goal of this paper is to copare high ipact, low probability pivot events versus low ipact, high probability signaling e ects and deterine which type of e ect doinates in large elections. In particular we investigate the liiting behavior of the cutpoint x c as n tends to in nity. We nd that the liiting cutpoint corresponds to the equilibriu cutpoint in a di erent gae in which the rst period is irrelevant (or, equivalently, L = R) so that, in the liit, the cutpoint for voter behavior is identical to what it would be if voters were otivated purely by signaling concerns. Thus, we nd that while equilibriu voting involves a balancing of these two otivations, in a very strong sense, equilibriu voting in large elections is driven by voters desire to in uence the inferences of observers and not by their desire to in uence the election at hand. 3 Two Exaples Before analyzing the odel, we illustrate it with two siple exaples. Exaple. We start with what is essentially a decision-theoretic version of the odel, in which there is just one voter, with ideal point v i and a linear loss function (jx v i j) = jx v i j. Suppose the exogenously- xed rst period candidate locations are L = and R = : The second period candidates believe that the single voter s ideal point is drawn fro a unifor distribution on [0; ]. Thus, if the voter s strategy is onotone, with cutpoint x c ; the second period policy will be xc if i votes for L and +x c if i votes for R. For i to be indi erent between voting L and R when her ideal point is v i = x c, the following equality ust hold: jx c Lj x c x c = jr x c j x c + x c : For L = and R = this equality is solved at x c = 3 : Exaple. To illustrate how pivot and signaling e ects wor in the odel, we now consider the 7

9 siplest variant where a vote has a probabilistic e ect on both rst and second period outcoes. While this exaple cannot resolve the horse race between the signaling and pivot e ects as the nuber of voters gets large, all of the relevant incentives and quantities of interest are present. Consider n = 3 and assue that voters i f; ; 3g have ideal points that are i:i:d: draws fro a unifor distribution on [0; ]. Assue that each voter has a linear loss function (jx v i j) = jx v i j : The rst-period election is between two candidates, with exogenously- xed policy positions L = and R = : We rst consider two benchar cases: a pure pivot odel and a pure signaling odel. In a pure pivot odel there is a unique voting equilibriu in wealy undoinated strategies: a voter votes for the closer candidate, i.e., she votes for L if her ideal point is to the left of L+R = 0:75 and votes R if her ideal point is to the right of 0:75: So x p = 0:75 is the pivot cutpoint. For a pure signaling odel, all that atters is how a vote a ects F ( j #R; x c) through #R: If voters only care about the outcoe of the second period election then in the three voter exaple there is a unique equilibriu, speci ed by the signaling cutpoint, x s = =: To chec that this cutpoint is an equilibriu in the gae in which only the second period outcoe a ects voter payo s, we con r that a voter with v i = = is indi erent between voting L and R; given the other actors strategies: Focusing on voter i = ; assue that the other voters are using this cutpoint strategy. The signaling e ect of i 0 s rst-period vote thus depends on the other voters rst-period actions: With probability F (x s )F (x s ) = = 0:5 the other two voters vote L. In this case the secondperiod policy outcoe will be 0:5, if i votes L. This is true because the second-period candidates posterior belief given #R = 0 and x s = = is that all three voters ideal points are unifor draws fro [0; 0:5]: If i votes R the second-period policy outcoe will be F j; = = 0:35, since there is a 50% chance that both of the L voters, and hence the, will be to the left of 0:35: Thus, the signaling e ect of voting R if both other voters vote L is to ove the second-period policy outcoe fro 0:5 to 0:35: With probability F (x s ) ( F (x s )) + ( F (x s )) F (x s ) = = 0:5 the other two voters split their votes. In this case if i votes L the second-period policy outcoe will be 0:35 and if she votes R the policy outcoe will be 0:65: 8

10 With probability ( F (x s )) ( F (x s )) = = 0:5 the other two voters vote R. In this case if i votes L the second-period policy outcoe will be 0:65 and if i votes R the policy outcoe will be 0:75: Thus, if a voter with ideal point v i votes L her expected second-period utility is 0:5 jv i 0:5j 0:5 jv i 0:35j 0:5 jv i 0:65j : If she votes R her expected utility is 0:5 jv i 0:35j 0:5 jv i 0:65j 0:5 jv i 0:75j : A voter at v i = x s = = is indi erent between voting L and voting R: It is straightforward to con r that any voter left of = strictly prefers to vote L and a voter to the right prefers to vote R: It is worth noting three features of signaling e ects that will show up in our later analysis of large elections. First, which action, L or R; better prootes the voter s policy interests in the second period depends on the other voters actions, as well as the cutpoint x s : For a voter with v i = =; if the other two voters vote L then voting R is optial, whereas if the others vote R then voting L is optial, and if the others split their votes then the voter is indi erent. Second, the di erent signaling e ects are not equally liely to occur, but rather occur with di erent probabilities. Third, since the other voters actions are siply draws fro a binoial, in a large election, the ost liely realized vote totals are those where L receives a share close to F (x s ) of the votes and R receives a share close to F (x s ) of the votes. All three of these properties of signaling e ects hold regardless of the cutpoint for voter behavior in the rst period. In a odel with both pivot and signaling e ects, equilibriu behavior hinges on the cobined cutpoint x c : As shown in Figure, in the three-voter exaple, x c 0:65: 4 For this x c the pivot probability is 0:65 ( 0:65) = 0:455: For a voter with v i = x c the utility di erence between the two possible rst-period policy outcoes, L and R; is jv i Lj + jv i Rj = 0:65 + j0:65 j = 0:: So i receives, in expectation, 0: 0:455 0:09 ore rst-period utility by voting L than by voting R: [Insert Figure about here] The second-period signaling e ect is a bit ore coplicated to copute: 4 It is just coincidence that this value for x c is the sae, to two decial places, as the second period policy outcoe under one of the action pro les in the pure signaling odel. For the general odel, they need not be the sae. 9

11 With probability F (x c ) F (x c ) = 0:65 0:65 = 0:45 the other two voters vote L. In this case the second-period policy outcoe will be 0:35 if i votes L: If i votes R the second-period policy outcoe will be 0:463. With probability F (x c ) ( F (x c )) + ( F (x c )) F (x c ) = 0:65 ( 0:65) = 0:455 the other two voters split their votes. In this case if i votes L the second-period policy outcoe will be 0:463 and if i votes R the policy outcoe will be 0:756: With probability ( F (x c )) ( F (x c )) = ( 0:65) ( 0:65) = 0:5 the other two voters vote R. In this case if i votes L the second-period policy outcoe will be 0:756 and if i votes R the policy outcoe will be 0:85: Thus, if a voter with ideal point v i votes L his expected second-period utility is 0:45jv i 0:35j 0:455jv i 0:463j 0:5jv i 0:756j ; which equals 0:4 for v i = 0:65: And if i votes R his expected utility is 0:45jv i 0:463j 0:455jv i 0:756j 0:5jv i 0:85j ; which equals 0:5 for v i = 0:65: The di erence is equal to 0:09; and for v i = x c it exactly counteracts the rst-period utility gain that the voter receives by voting L rather than R: Thus at x c the pivot and signaling e ects cancel each other out and the voter is indi erent. This exaple illustrates the basic tension between pivot and signaling e ects in our odel. In this three voter exaple, the equilibriu cutpoint is x c 0:65; which lies between the signaling cutpoint, x p = 0:5; and the pivot cutpoint, x s = 0:75: The question is how a sequence of equilibriu cutpoints fx g will behave in the liit as the population size n = + gets large. The di culty in answering this question is that in large elections both the pivot e ect and the signaling e ect becoe sall; the probability of a pivot event goes to zero and the distance that second-period candidates ove in response to a single vote also goes to zero. The question is which converges faster. 4 Preliinary Results In this section we establish two leas that are useful in establishing existence of a particular type of equilibriu for any n as well as in proving the ain result about the liiting behavior of this type 0

12 of equilibriu. We then present the existence result. Our analysis focuses on a particular class of equilibria. De nition (Syetric Cutpoint Strategy) Voters use a syetric cutpoint strategy if there exists a point x c [0; ] such that for all i N () if x c = 0 then i votes L if v i = 0; and i votes R if v i > 0 () if x c (0; ]; then i votes L if v i < x C ; and votes R if v i x c : Given that all other voters use a syetric cutpoint strategy with cutpoint x c ; optial behavior for a voter with ideal point v i depends on the di erence in her expected utility between voting R and voting L in the rst period. Using a i fl; Rg to denote voter i0 s rst period action, we can express this di erence as u dif (v i ) u(a i = Rjv i ) u(a i = Ljv i ) = u dif (v i ) + u dif (v i ) () where u dif (v i ) F (x c ) ( F (x c )) ((jl v i j) (jr v i j)) () and u dif (v i ) X =0 (F (x c )) ( F (x c )) 0 ( F ( j; x c) v i ) ( F ( j + ; x c) v i) C A : (3) Thus u dif (v i ) captures the rst period e ect of voting: the pivot probability is (F (xc )) ( F (x c )) and the utility di erence between the two candidates is (jl v i j) (jr v i j) for a voter with ideal point v i. Liewise, u dif (v i ) captures the second period e ect: the probability that other voters vote R is (F (xc )) ( F (x c )) and the utility di erence between voting R versus L in this event is ( F ( j ; x c) v i ) ( F ( j + ; x c) v i ): Rear : Deriving F (yj#r; x c ) for x c (0; ) To understand u dif (v i ) it is iportant to see how F (yj#r; x c ) depends on #R and x c. This function can be characterized in ters of order statistics. We note that for xed x c (0; ); the distribution of the is constructed as follows. Given that there are n #R draws with values strictly less than x c and #R draws with values greater than or

13 equal to x c we now that the is less than x c if #R is strictly less than + and it is greater than x c otherwise. If #R < +; the is the + largest of n #R draws fro the conditional (on y < x c ) distribution H if #R < + ; F (yj#r; x c ) = H +;n (y; x c ) = axf0; inf; F (y) F (x c) gg: Accordingly #R (y; x c ), which is the distribution of the ( + ) th order statistic fro n #R draws fro the distribution function H (; x c ): Siilarly if #R +, the is the (+ (n #R)) th order statistic fro #R draws fro the conditional (on y > x c ) distribution H + (y; x c ) = axf0; inf; F (y) F (xc) F (x c) gg. So if #R + then F (yj#r; x c ) = H + (n #R);#R (y; x c), which is the distribution of the ( + (n #R)) th order statistic fro #R draws fro the distribution function H + (; x c ). Rear : Deriving F (yj#r; x c ) for x c f0; g Now consider extreal cutpoints, x c f0; g: If x c = 0 then according to De nition, all voters with ideal points in (0; ] vote R and voters with ideal point v i = 0 vote L: Accordingly, if #R < + then the voter s ideal point is 0 with probability and F (yj#r; 0) is constant at for all y [0; ]: In this case, we de ne F ( j #R; 0) = 0 and it is clear that equilibriu second period candidate locations are at 0. If #R + then the is the ( + (n #R)) th order statistic fro #R draws fro F (). This distribution corresponds to H + (y; x c ) = F (y) F (0) F (0) with support [0; ]: If x c = then according to De nition, all voters with ideal point vote R and all voters with ideal points v i [0; ) vote L: Accordingly, if #R + then the voter s ideal point is at with probability and F (yj#r; ) is constant at 0 for all y [0; ) and equal to at y = : In this case we de ne F ( j #R; ) =. If #R < + then the is the ( + ) th order statistic fro n #R draws fro F (): This distribution corresponds to H (y; x c ) = F (y) F () on [0; x c]: One way to see how the distribution function F (j; ) behaves as the arguents #R and x c change is to consider the case of the unifor, F (y) = y on [0; ]: Figure plots the function H (y; x c ) for x c f0; 0:65; g. Figure 3 plots H +;n #R (y; x c ) for n = ; = 5 and #R f; 3g : This gure shows how increasing #R shifts the second period candidates beliefs about the location of the

14 to the right, thereby causing F ( j #R; x c) to increase. [Insert Figures and 3 about here] Cobining Rears and, we can express the second period policy location as a function of x c and #R when voters use a syetric cutpoint strategy. 8 0 if x c = 0 and #R < + >< (x c ; #R) = >: if x c = and #R + fy : H +;n #R (y; x c ) = g if x c (0; ] and #R < + fy : H + + (n #R);#R (y; x c) = g if x c [0; ) and #R + : The rst lea builds on this derivation to establish properties of the distribution of the and the above apping Lea (Properties of Second Period Policy Outcoes) If voters use a syetric cutpoint strategy with cutpoint x c then () For each #R < + and y [0; ]; F (yj#r; x c ) is wealy decreasing in x c for x c [0; y) and strictly decreasing for x c [y; ] and for each #R + and y (0; ); F (yj#r; x c ) is wealy decreasing in x c for x c (y; ] and strictly decreasing in x c [0; y] : () If #R < #R (both in 0; ; ; :::; n) then for each x c (0; ), for soe set A xc [0; ] with positive lebesgue easure F (yj#r ; x c ) > F (yj#r ; x c ) if y A xc and F (yj#r ; x c ) F (yj#r ; x c ) for all y [0; ]: For x c f0; g ; F (yj#r ; x c ) F (yj#r ; x c ) for all y [0; ]: (3) For any #R f0; ; ; :::; n g and x c [0; ]; F ( j #R; x c) F ( j #R + ; x c): (4) F (yj#r; x c ) is continuous in x c on (0; ) for each #R f0; :::; ng and y [0; ] as well as continuous in y on [0; ] for each #R f0; :::; ng and x c (0; ): x c. (5) The apping (x c ; #R) is a function fro [0; ] f; ::::; ng into [0; ] and it is continuous in Proof: () Assue #R < + : Fro our derivation of F (yj#r; x c ) in Rear this distribution taes on the value if y x c and H +;n #R (y; x c ) otherwise. Thus the conclusion that it is wealy 3

15 decreasing for x c [y; ] is iediate. Consider x c < x 0 c: Since F (y) F (x 0 c) < F (y) F (x c ) H (y; x 0 c) < H (y; x c ) for all x < x c and thus the forer rst order stochastically doinates the latter on [0; x c ]. This ordering of H (y; x 0 c) and H (y; x c ) iplies that the distributions of order statistics, H +;n #R (y; x 0 c) and H +;n #R (y; x c ) are also ordered by rst order stochastic doinance (see for exaple Theore 4.4. of David and Nagaraja, p. 75). An analogous arguent holds in the case of #R +, establishing that H + + (n #R);#R (y; x0 c) and H + + (n #R);#R (y; x c) are ordered by rst order doinance. () To establish strict onotonicity in #R, consider two integers, #R and #R, with 0 #R < #R n. If #R < + #R then the support of F (j#r ; x c ) is [0; x c ] and the support of F (j#r ; x c ) is [x c ; ]: Since the distribution F () is strictly increasing on [0; ]; F (yj#r ; x c ) > 0 for all y (0; x c ) while F (yj#r ; x c ) = for all y x c. Siilarly, F (yj#r ; x c ) = 0 for all y x c and F (yj#r ; x c ) < for y (x c ; ) : Thus, F (yj#r ; x c ) F (yj#r ; x c ); with a strict inequality for any y = f0; g : Suppose instead that #R < #R < +: The relevant coparison is now between H +;n #R (y; x c ) and H +;n #R (y; x c ): To see that these two distributions are ordered by rst order stochastic doinance, we can partition n #R draws fro F () into two sets: rst n #R draws are taen and then another #R #R are taen. Because F () is strictly increasing on [0; ]; the probability that one of the #R #R draws is less than the + highest draw of the rst #R #R draws is strictly positive, and thus H +;n #R (y; x c ) < H +;n #R (y; x c ) for y on [0; x c ]. This iplies that F (yj#r ; x c ) F (yj#r ; x c ) with a strict inequality if y A xc = [0; x c ) if #R < #R < + : A siilar arguent holds for A xc = (x c ; ] and + #R < #R : The result for x c f0; g follows fro Rear. (3) Follows iediately fro (). (4) Continuity of F (yj#r; x c ) in x c on (0; ) for each #R f0; :::; ng and y [0; ] as well as continuity in y on [0; ] for each #R f0; :::; ng and x c (0; ) follows fro the assuption that F () is strictly increasing and continuously di erentiable and the fact that the distribution of an order statistic fro a di erentiable distribution function has a density. In particular, for #R < + the distribu- 4

16 h tion F (yj#r; x c ) has density h +;n #R (y; x c i F (y) F (y) a F (x c) F (x c) ( F (y) F (x c) )b for integers ; a; b: For #R + ; the distribution F (yj#r; x c ) has density h + + (n #R);#R (y; x c) = h i a 0 F (y) F (y) ( F (y) ) b0 for soe 0 ; a 0 ; b 0 : Since we have assued F (xc) F (x c) F (xc) F (x c) F (xc) F (x c) F () has a continuous density, for xed #R; as long as x c (0; ) the above densities are well de ned and thus the distribution functions are continuous. (5) To show that (x c ; #R) is de ned on its doain we ust show that fy : H +;n #R (y; x c ) = g is non-epty if x c (0; ] and #R < + and that fy : H + + (n #R);#R (y; x c) = g is non-epty if x c [0; ) and #R +: In the rst case, consider x c (0; ] and #R < +: Fro the proof of part 4 of this lea we see that H +;n #R (y; x c ) has a continuous density function that is strictly positive as long as y < x c : So the function H +;n #R (y; x c ) is continuous and strictly increasing in y on [0; x c ] with 0 = H +;n #R (0; x c ) < < H +;n #R (x c; x c ) = : This eans that the set S (x c ; #R) = fy [0; ] : H +;n #R (y; x c ) (0; )g is non-epty for x c (0; ] and #R < + : Moreover, by the interediate value theore this eans that the set fy : H +;n #R (y; x c ) = g is non-epty if x c (0; ] and #R < +: An analogous arguent establishes that fy : H + + (n #R);#R (y; x c) = g is non-epty if x c [0; ) and #R + : To show that (x c ; #R) is a function it is su cient to note that F (j#r; x c ) has the property that for each value of x c and #R there are nubers, a ; a [0; ] such that F (j#r; x c ) is constant at 0 on [0; a ]; F (j#r; x c ) is strictly increasing on [a ; a ] and F (j#r; x c ) is constant at on [a ; ]: This eans that the equation F (j#r; x c ) = has at ost one solution (and it is in [a ; a ]). To establish continuity we consider two cases. First assue that #R < + : By part 4 of this lea, for a xed y, H +;n #R (y; x c ) is continuous in x c on (0; ) and thus this and the fact that it is strictly increasing (and has a density) in y on a neighborhood of the point fy : H +;n #R (y; x c ) = g iplies by way of the iplicit function theore that the solution (x c ; #R) is continuous in x c if x c (0; ) and #R < + : Continuity at x c = 0 follows fro the fact (x c ; #R) x c if #R < + and thus li xc!0 (x c ; #R) = 0 and (0; #R) = 0. Continuity at x c = follows fro the fact that H +;n #R (y; ) is de ned and for each y; H +;n #R (y; x c ) is continuous in x c at : An analogous arguent about H + + (n #R);#R (y; x c) establishes continuity in the case of #R + : 5

17 The next result establishes soe properties of the utility di erence function in Equation. Since this result siply uses conclusions fro Lea in standard ways the proof is in the appendix. Lea (Properties of Utility Di erence Function) If voters use a syetric cutpoint strategy with cutpoint x c then () u dif (v i ) is continuous and wealy increasing in v i : () (Lipschitz property) 8~v i ; ^v i [0; ], ju dif (~v i ) u dif (^v i )j 4 j~v i ^v i j : (3) u dif (0) 0 and u dif () 0: We can now state our rst ain result. The proof, which applies a standard xed point arguent to the function u dif (), is in the appendix. Proposition There exists an equilibriu in which voters use a syetric cutpoint strategy in the rst period. 5 Intuition for the Convergence Result Having established existence, we now turn to the question of equilibriu behavior in large electorates, i.e., as!. We suppress the c subscript and let x denote the cutpoint in a syetric cutpoint strategy equilibriu with n = + voters. Our interest is then in li! x (if it exists). We show that this liit is equal to the point F ( ): Since this liit does not depend on the rst period candidate locations L and R, it is also the liit of cutpoints for equilibria in the pure signaling gae. The proof proceeds by contradiction. We show that if a sequence of cutpoints does not converge to F ( ) then these cutpoints cannot be equilibriu cutpoints for in nitely any values of because voters at the cutpoint x ; who ust be indi erent in equilibriu, will strictly prefer to vote for one candidate over the other. To be ore precise, we show that if any subsequence of equilibriu cutpoints converges to a point other than F ( ) then for su ciently large a voter with ideal point x will strictly prefer to vote for one candidate over the other. Once it is established that no subsequence converges to a point other than F ( ) it follows that every subsequence, and thus the actual sequence, converges to F ( ): This brief section serves as a roadap for the proof, presenting an inforal version of the arguent. The next section contains a proof of the ain result. 6

18 Suppose that in a large electorate voters behave according to a cutpoint x which is converging to a nuber Z > F ( ): We show that for large values of a voter at x will strictly prefer to vote R. There are three types of e ects that the voter ust consider. The rst consideration is a pivot e ect, which we label P V: Since the election is not expected to be a tie, i.e., x 6= F ( ), and the population size is large, the probability of this pivot event is exceedingly sall in a large electorate. The second consideration involves bad signaling e ects fro voting R; whenever ore than half of the other voters vote R, the second period policy will be to the right of x, so if a voter with v i = x c votes R this will ove second period policy to the right, i.e., away fro his ideal point, as established in part 3 of Lea. However, because x > F ( ); ore than half of the votes are expected to go to L; and thus in a large electorate bad signaling e ects are extreely unliely to occur. We nd an upper bound on the probability-weighted su of these bad signaling e ects, and label it U BBS (upper bound for bad signaling ). The third consideration involves good signaling e ects fro voting R; whenever ore than half of the other voters vote L, the second period policy will be to the left of x, so if a voter with v i = x votes R this will ove second period policy to the right, i.e., towards his ideal point. Since x > F ( ); ore than half of the votes are expected to go to L; and thus in a large electorate it is extreely liely that the signaling e ect of voting R will be good. We nd a lower bound on the probability-weighted su of these good signaling e ects, and label it LBGS (lower bound for good signaling ). We consider the ratio of bad signaling plus pivot e ects to good signaling e ects, and show that this ratio P V + UBBS LBGS can be expressed as a liit of the for li! P tie + ( + ) P tie : (4) cp In this expression, P tie = F (x ) ( F (x )) is the probability of an exact tie aong the other voters given the cutpoint x. In the denoinator, P is the probability of a certain type of good signaling e ect, and P goes to zero uch ore slowly than P tie : The c in the denoinator is a constant that does not depend on. At the end of the proof we show that Equation 4 is bounded by an 7

19 expression of the for ( + ) q ( c ) with constants q (0; ), and c (0; =) : Thus the liit of Equation 4 is 0, which eans that for a voter with ideal point x (and voters with ideal points close enough to x ) it will be optial to deviate and vote R: 6 The Convergence Result Our ain result is Proposition li! x = F ( ): Proof: Assue by way of a contradiction that the cutpoints do not converge to the point M F ( ): Because x [0; ], 8; the Bolzano-Weierstrass Theore iplies that there exists soe nuber Z [0; ] with Z 6= M such that a subsequence fx 0g! Z: We focus on such a subsequence, ignoring the residual portion of the original sequence. Thus the assuption that Proposition is false equates to the clai that fx g! Z. Either Z < M or Z > M and in the reainder of the proof we focus on the latter case; the arguent for the forer case is virtually identical and is thus oitted. Our goal is to show that there exists a such that if > then a voter with ideal point x has a strict preference to vote for R: Once this clai is established, the continuity of the utility functions established in Lea iplies that for > there exists a > 0 such that if v i (x ; x + ) a voter with ideal point v i prefers to vote for R when everyone else uses the cutpoint x : Thus for soe voters to the left of x voting L is not a best response, contradicting the hypothesis that x is an equilibriu cutpoint when the population size is + : This contradiction eans that we cannot have a subsequence of cutpoints converging to any Z 6= M and thus the sequence of cutpoints converges to M: For each, consider a voter, i, with ideal point x. Given that voters to the left of x vote L and voters to the right of x vote R, the probability of any individual voting R is p F (x ): Since x > F (=) we now that p < : We start by analyzing the utility function of a voter with ideal point x : If exactly voters other than i vote R then the election is tied, and i s vote is pivotal in deterining the rst period policy. 8

20 However, in ters of rst period otivations, which depend on the candidate locations L and R; it is not clear whether i prefers to vote L or vote R in the event that he is pivotal. In ters of second period otivations, which depend on candidate locations given #R; it is also unclear whether i prefers to vote L or vote R in the event that he is pivotal. In contrast, the voter s preferences are clear for events in which he is not pivotal. If or fewer of the other voters vote for R the second period policy will be to the left of x regardless of i s vote and if at least + of the other voters vote for R then the second period policy will be to the right of x regardless of i s vote (as established in Rear ). These facts and the onotonicity of the second period policy in #R (part 3 of Lea ) iply that if or fewer of the other voters vote for R then a vote for R oves the second period policy closer to i s ideal point. On the other hand, if at least + of the other voters vote for R; then a vote for R oves the second period away fro i s ideal point. Note that in either of these cases; i s vote cannot ove policy far enough to leapfrog her ideal point, x (see Rear ). Following Equations,, and 3, given the conjectured equilibriu for population size n = + ; the utility di erence between voting R versus voting L for a voter with ideal point v i = x in the equilibriu with population size + is u dif (x ) u dif (x ) + u dif (x ) which we re-write as u dif (x ) = ( p ) p ((jl x j) (jr x j)) (5) X + =0 + + X =+ ( p ) p ( p ) p ( p ) p ( F j; x x ) ( F j + ; x ( F j; x x ) ( F j + ; x ( F j; x x ) ( F x ) j + ; x The rst line of Equation 5 is the pivot e ect. The second line represents good signaling e ects of x ) x ) : voting R when or fewer other voters vote R. The third line represents the indeterinate signaling e ect when the other voters split their votes equally between L and R. The fourth line represents bad signaling e ects, when + or ore other voters vote R: 9

21 Our ultiate goal is to show that there exists an such that for > ; u dif (x ) > 0: To siplify the expression in Equation 5, we rst siplify each coponent of the utility function, by nding bounds that liit how severe the bad e ects of voting R can be, along with a lower bound on good e ects of voting R: Pivot e ect. The pivot e ect can be either positive or negative, depending on the positions of the two rst-period candidates. A bound based on the fact that L; R, and x are all in the interval [0; ]; will be su cient: ( p ) p ((jl x j) (jr x j)) > ( p ) p 0 () = ( p ) p : (6) Recall that we have noralized utility so that 0 () = and we have assued that this function is convex, allowing us to treat the axial possible utility di erence between L and R as. Bad and indeterinate signaling e ects. For any f; ::; n g, F j; x (0; ) and F j + ; x (0; ), so X ( p ) p ( F =+ + ( p ) p ( F X > = j; x j; x x ) ( F j + ; x x ) ( F j + ; x X ( p ) p ( p ) p 0 () = = x ) We will use repeatedly the fact that the binoial expansion is onotonic. In particular, since p < =; x ) for any f + ; :::; g ; ( p ) p < ( p ) p : As a consequence of this fact, the total of the bad and indeterinate signaling e ects ust be strictly greater than ( + ) ( p ) p : (7) Good signaling e ects. We now develop a lower bound on good signaling e ects. Fix any points A and B in the unit interval such that M < A < B < Z: For any ; let A represent the largest nuber less than A such that for soe integer a < + it is the case that A = F ja ; x : Liewise, let B represent the largest nuber less than B such that for soe integer b < + it is the case that B = F jb ; x : For the b identi ed in the de nition of B let C = F jb + ; x : 0

22 For su ciently large it turns out that M < A < B < B C < Z, and since we are interested in the liit as! we henceforth focus on values of that are large enough for this inequality to hold. Since A B < B < Z are true by construction, establishing that these inequalities eventually hold requires only that we show that (i) M < A, (ii) B C < Z and (iii) A < B for large enough. Each of these results follows fro the fact that for any q (0; ) the sequence F j( + )q + ; x F j( + )q; x converges to 0. To see this note that in the rst instance this fact allows us to conclude that if A < M in nitely often then it is not the case that for every ; A < F ja + ; x ; which contradicts the de nition of A (and a ). In the second instance, if Z < C in nitely often then this fact eans that we ust have B < B for in nitely any values of (contradicting the de nition of B ) and if C < B in nitely often then this fact eans that for in nitely any values of ; B < C < B; contradicting the de nition of B : In the third instance the fact that A < B iplies that if A = B then A > A or F jb + ; x < B for in nitely any (contradicting the de nition of A or B ). For xed the set of pro les for other voters that, given i 0 s vote, can result in a policy between A and C consists of pro les for which the nuber of other voters that vote R is in the set fa ; a + ; ::::; b ; b g: Although we cannot analytically solve for the policy distance between F j; x and F j + ; x for particular values of ; we do now that b X =a F j + ; x F j; x = C A > B A: Since we have A < B < B C the last inequality above is due to the fact that A < A < B: We re-write the good signaling e ects ter fro Equation 5 as X =0 ( p ) p ( F j; x x ) ( F j + ; x x ) = ax =0 + + b X =a ( p ) p X =b + ( F ( p ) p ( F ( p ) p ( F j; x j; x j; x x ) ( F x ) ( F x ) ( F j + ; x j + ; x j + ; x x ) x ) x ) :

23 Because B < Z and () is convex the slope of the utility di erence to i with v i > Z+B for points in [0; B] is at least := 0 ( Z B ): Since () is strictly increasing 0 ( Z B ) > 0: Because x is assued to converge to Z; for su ciently large v i > Z+B : Thus, the good signaling e ects ter is eventually greater than b X =a ( p ) p F j; x x F j + ; x x : Since M < A < C and p < =; we now fro onotonicity of the binoial expansion that the event in which b others vote for R is the least liely of the set of pro les of the other voters in which i s vote can result in a policy in the interval [A ; C ]. Thus the above expression is greater than ( p ) b p b b b X =a F j; x Note that because 8 fa ; a + ; :::; b g ; F b X =a F j; x x F j + ; x j; x < F j + ; x < x ; x F j + ; x x x : (8) = F jb + ; x F ja ; x = C A > B A: Thus, we can rewrite Equation 8 to get the following lower bound for good signaling e ects: (B A) ( p ) b p b : (9) b Having derived bounds on pivot e ects, bad signaling e ects, and good signaling e ects (Equations 6, 7, and 9, respectively) we now substitute these bounds into the utility di erence expression in Equation 5 to get u dif (x ) > ( p ) p ( + ) ( p ) p + (B A) b ( p ) b p b : To show that there exists an, such that for > ; u dif (x ) > 0, it is su cient to show that li! ( p ) p + ( + ) ( p ) p (B A) b ( p ) b p b = 0: