Electoral Competition with Privately-Informed ... - Semantic Scholar

Electoral Competition with Privately-Informed Candidates¤ Dan Bernhardt Department of Economics University of Illinois 1206 S. Sixth Street Champaign, IL 61820 John Duggany Department of Political Science and Department of Economics University of Rochester Rochester, NY 14627 Francesco Squintani Department of Economics University of Rochester Rochester, NY 14627 February 29, 2003

¤

We thank Jean-Francois Mertens for helpful discussions during his visit to the Wallis Institute of Political Economy at the University of Rochester. y The second author gratefully acknowledges support from the National Science Foundation, grant number SES-0213738

Abstract We consider a model of elections in which two o±ce-motivated candidates receive private signals about the location of the median voter's ideal point prior to taking policy positions. We show that at most one pure strategy equilibrium exists and provide a sharp characterization, if one exists: After receiving a signal, each candidate locates at the median of the distribution of the median voter's location, conditional on the other candidate receiving the same signal. It follows that a candidate's position, conditional on his/her signal, is a biased estimate of the true median, with candidate positions tending to the extremes of the policy space. We provide su±cient conditions for the existence of a pure strategy equilibrium. Essentially, the pure strategy equilibrium exists if for each signal, the other candidate is su±ciently likely to receive a signal in the same direction that is at least as extreme. We prove generally that, though the electoral game leads to discontinuous candidate payo®s, mixed strategy equilibria do exist. We provide bounds on the support of mixed strategy equilibria and restrictions on possible atoms. We further show that equilibrium expected payo®s are continuous in the parameters of the model and that mixed strategy equilibria are upper hemicontinuous.

1

Introduction

The most familiar and widely-used model of elections in political science and political economy is the classical Downsian model (Hotelling (1929), Downs (1957), Black (1958)). The central result is the median voter theorem: The unique Nash equilibrium in an election between two o±ce-motivated candidates is that both candidates locate at the median voter's ideal point. The logic is simple for symmetric equilibria: If both candidates locate at a position that is not the median voter's, then the result is a tie, but either candidate could move slightly toward the median and win for sure. The same logic captures non-symmetric equilibria once we note that, because the electoral game is constant-sum, equilibria are interchangeable. Implicit in this logic is the assumption that candidates are perfectly informed about the location of the median voter. The political science literature on \probabilistic voting" relaxes the assumption of perfect information, assuming instead that candidates share a common prior distribution about the location of the median voter. It is a \folk theorem" that, in this environment, the unique Nash equilibrium is for both candidates to locate at the median of the prior distribution of the median voter's ideal point. The median voter theorem therefore extends to the probabilistic voting model in an intuitive way. It is still implicitly assumed, however, that the candidates have symmetric information about the location of the median. Symmetric information, while a useful simplifying assumption, is clearly very strong. We would expect private information about voter preferences to arise from di®erences in the candidates' personal experiences or from the di®erent backgrounds of their political advisors. A better-documented source of private information is the use of private pollsters: 46 percent of all spending on U.S. Congressional campaigns in 1990 and 1992 was devoted to the hiring of political consultants, and the use of political pollsters was more common than any other type of consultant. Of 805 U.S. House of Representatives races in 1990, 209 campaigns employed private pollsters; of 856 races in 1992, 396 campaigns did. In races for open seats, the fraction of candidates using pollsters was about one half in both election cycles (Medvic (2001)). This is in addition to the polling services o®ered by the major parties, particularly the Republican party. In this paper, we develop a general model of elections that allows for privately-informed candidates. Before selecting a platform, each candidate receives a signal drawn from an arbitrary ¯nite set of possible signals; each candidate updates about the location of the

1

median voter and the platform of the opponent and then chooses a platform; the median voter's location is then realized, and the candidate closest to the median wins. In the natural setting where candidates have access to identical polling technologies, we prove uniqueness and fully characterize the pure strategy equilibrium of the model, if it exists. We ¯nd that the logic of the median voter theorem does not extend to the general case in the expected way: After receiving a signal, a candidate updates the prior distribution of the median voter, conditioning on both candidates receiving that same signal, and locates at the median of that posterior distribution.1 In the Downsian and probabilistic voting models, the candidates have symmetric information, and conditioning on one candidate receiving a signal is the same as conditioning on both receiving it, so we obtain the known results for those models as special cases. When information is asymmetric, though one might expect a candidate simply to target the median voter conditional on his/her own signal, this is not what happens. In fact, a candidate's equilibrium platform is a biased estimator of the median voter, and strategic competition between the candidates often leads the candidates to take positions more extreme than their estimates of the median voter's ideal point. To understand this result, consider a symmetric pure strategy equilibrium. As we vary a candidate's platform after receiving some signal s, the candidate's probability of winning varies continuously, except for possibility that the other candidate chooses the same platform after s: In that case, continuity holds if and only if that platform is the median of the distribution conditional on both receiving s. But candidates could exploit any discontinuity by moving slightly toward the conditional median, so the only possible symmetric pure strategy equilibrium is the one claimed. Finally, because the electoral game is constant-sum, symmetry and interchangeability imply that there are no other asymmetric pure strategy equilibria. We give su±cient conditions for existence of the pure strategy equilibrium under two sets of conditions. First, we consider the case where information of the candidates' is coarse, in the sense that polling generates either a signal that the median voter is likely to the left or likely to the right. Here, where only two signals are possible, the key condition guaranteeing existence is quite weak: Conditional on a candidate receiving a signal, the probability that the opponent receives the same signal should be at least one half. In other words, signals 1 This result is reminiscent of the ¯ndings of Milgrom (1981), who shows that in a common-value secondprice auction, the equilibrium bid of a type µ corresponds to the expected value of the good conditional on both types being equal to µ. Here, since candicates maximize the probability of winning, the relevant statistic is the median.

2

should not be negatively correlated. We then allow for multiple signals. The key condition extends to the following: Conditional on a candidate receiving a signal, the probability that the opponent receives a signal weakly to the \left" should exceed the probability that the opponent receives a signal strictly to the \right," and vice versa. This limits the incentive for a candidate to move away from the equilibrium platform after any signal, and, with other background conditions, it ensures the existence of the pure strategy equilibrium. While these are su±cient|not necessary|conditions, we construct examples showing that the pure strategy equilibrium can fail to exist when the above su±cient conditions are violated. In fact, we can add an arbitrarily small amount of asymmetric information to the Downsian model in such a way that the pure strategy equilibrium ceases to exist. The possibility of doing so raises the issue of robustness of the median voter theorem, to which we return later. A feature of the Downsian and probabilistic voting models is that \policy convergence" obtains in equilibrium, i.e., the candidates adopt exactly the same policy position, giving voters essentially no choice.2 While di®erences in positions of real-world candidates may be limited, this stark policy convergence result is not strictly supported by real-world campaigns, where we inevitably see some divergence of policy platforms. Policy convergence is a very robust result in the Downsian model, surviving even if candidates place weight on policy preferences (Calvert (1985), Duggan and Fey (2000)). In contrast, adding policy motivation to the probabilistic voting model precludes policy convergence, introducing a wedge between the candidates' platforms, when a pure strategy equilibrium exists (Wittman (1983), Calvert (1985)). Divergence is also generated if one candidate has a \valence advantage," so that all voters would vote for that candidate even if the opponent o®ers a slightly better policy. In that case, there do not exist equilibria in pure strategies, but mixed strategy equilibria obviously can lead candidates to adopt distinct platforms (Aragones and Palfrey (2002), Groseclose (2001)). In our model, policy divergence arises in a very natural way even if candidates are purely o±ce-motivated and neither possesses a valence advantage: Even in a symmetric pure strategy equilibrium, di®erences in platforms are generated simply by di®erences in information, due to randomness inherent in political polling. More importantly, we show 2 Policy convergence obtains in a variety of electoral models. Banks and Duggan (1999) show that it obtains quite generally in a related class of probabilistic voting models, where candidates seek to maximize plurality. See also Duggan (2000), Bernhardt, Hughson, and Dubey (2002), and Banks and Duggan (2002) for convergence results in models of repeated elections.

3

that for a wide class of models, strategic competition between the candidates can further increase the dispersion between the candidate's platforms by inducing them to take positions more extreme than would be the case if they simply targeted the median voter. This is most starkly highlighted when candidates receive only binary signals|\left" or \right"|about the median voter's location. Then, as long as candidates are more likely than not to receive the same signed signal, the pure strategy equilibrium exists, and in it, a candidate receiving a given signal locates at the median conditional on both candidates receiving similarly signedsignals. Since the median given two \left" signals will typically be more extreme than the median given one \left" signal, it follows that, in equilibrium, candidates bias their platforms away from the median voter in the direction of their private signals, increasing platform dispersion. To adapt Barry Goldwater's dictum, \Extremism in the pursuit of victory is no vice." The pure strategy equilibrium does not exist generally, so we move to the analysis of mixed strategy equilibria: We prove that mixed strategy equilibria exist; we prove that the (unique) mixed strategy equilibrium payo®s of our model vary continuously in its parameters; and we use this result to prove upper hemicontinuity of equilibrium mixed strategies. Lastly, we bound the support of mixed strategy equilibria by the interval de¯ned by the smallest and largest conditional medians, and we show that the only possible atoms of equilibrium mixed strategies are at conditional medians. In the context of the probabilistic voting model, Ball (1999) proves the existence of a mixed strategy equilibrium, and our existence result can be viewed as extending his to models with asymmetrically-informed candidates. An implication of our results is that in the Downsian model, the only equilibrium is that in which candidates locate at the median voter's ideal point|there are no mixed strategy equilibria. By our upper hemicontinuity result, it follows that in models \close" to the Downsian model, mixed strategy equilibria must be \close" (in the sense of weak convergence) to the median voter's ideal point. Thus, while we show that the pure strategy equilibrium may cease to exist when small amounts of asymmetric information are added, we regain robustness in mixed strategies: Even when the pure strategy equilibrium does not exist, mixed strategy equilibria do, and they are all close to the median.3 Elsewhere, we explicitly solve for a mixed strategy equilibrium in a tractable version of our model, we conduct comparative statics analysis, and we investigate the e®ect of electoral 3

Banks and Duggan (1999) derive a similar ¯nding in the probabilistic voting model with expected plurality-maximizing candidates. There, the unique pure strategy equilibrium may cease to exist when probability of vote functions are perturbed slightly, though mixed strategy equilibria must vary continuously.

4

competition on voter welfare. A number of other papers have independently considered aspects of elections with privately-informed candidates. Chan (2001) develops a related three-signal model, and assuming a pure strategy equilibrium exists, he provides partial characterizations and welfare analysis. Ottaviani and Sorensen (2002) consider a model of ¯nancial analysts who receive private signals of a ¯rm's earnings and simultaneously announce forecasts, with rewards depending on the accuracy of their predictions. The case of two analysts can be interpreted as a model of electoral competition with privately informed candidates. They assume a normally distributed median, and o®er a numerical analysis of pure strategy equilibrium based on necessary ¯rst order conditions, but they do not address existence analytically. More distantly related are Heidhues and Lagerlof (2001) and Martinelli (2001,2002).

2

The Electoral Framework

2.1

The Model

Two political candidates, A and B, simultaneously choose policy platforms, x and y, on the real line, 0 and P(t) > 0. Conditional probabilities P(¢js) and P(¢jt) are then de¯ned using Bayes rule. The model is completely general with respect to 5

the correlation between candidates' signals, allowing for conditionally-independent signals and perfectly-correlated signals as special cases. We assume that for all a; b; c 2 < with a < b < c, 0 < F^s;t (a) and F^s;t(c) < 1 implies that F^s;t(a) < F^s;t (b) < F^s;t (c). As a result, if F^s;t admits a density, denoted fs;t, then it has convex support. Let ms;t be the uniquely-de¯ned median of F^s;t . The subsequent analysis admits a particular form of discontinuity in the speci¯cation of conditional distributions to incorporate the Downsian model as a special case. That generality introduces a complication in de¯ning the probability, given a platform for each candidate, that a particular candidate wins the election. Let F^s;t(z)¡ denote the lefthand limit of F^s;t , i.e., F^s;t(z)¡ = limw"z F^s;t (w). We then de¯ne the non-decreasing function Fs;t : < ! < by

´ 1³^ Fs;t(z) = F^s;t(z)¡ + Fs;t(z) ¡ F^s;t(z)¡ : 2

Of course, Fs;t is identical to F^s;t if the latter is continuous. Given a subset T 0 µ T, we de¯ne Fs;T 0 by

Fs;T 0 (z) =

X P(tjs) 0js) F s;t (z); P(T 0 t2T

when P(T 0 js) > 0. If F^s;t is continuous for each t 2 T 0, then Fs;T 0 coincides with the distribution of ¹ conditional on s and T 0 , and we let ms;T 0 denote the unique median of Fs;T 0 . We de¯ne FS0 ;t and mS0 ;t analogously. The advantage of working with Fs;t is that it will allow us to easily de¯ne the probability that a particular candidate wins the election. Speci¯cally, under conditions to follow, the probability that candidate A wins when A uses platform x and receives signal s and B uses platform y and receives signal t, will be Fs;t

³x + y ´ 2

if x < y, 1 ¡ Fs;t

³x + y ´ 2

if y < x,

and

1 if x = y. 2

(1)

The probability that B wins will have an analogous form and will, of course, equal one minus the probability that A wins. Thus, we de¯ne a Bayesian game between the candidates, in which pure strategies for the candidates are vectors X = (xs ) and Y = (yt ), and, given pure strategies X and Y ,

6

candidate A's interim expected payo® conditional on signal s is ³x + y ´ ³ ³ x + y ´´ X X s t s t ¦A(X; Y js) = P (tjs)Fs;t + P(tjs) 1 ¡ Fs;t 2 2 t2T :x s 0 and P (^ s; ^t0 ) > 0, we have P (s; t0 )Fs;t0 = P(^s; ^t0 )F^s;t^0 . Thus, »S is actually an equivalence relation. Of course, we can de¯ne an equivalence relation »T on T similarly. We therefore partition S into equivalence classes

S(s) = f^s 2 S j s »S ^sg; and we partition T into equivalence classes T(t) = f^t 2 T j t »T ^tg; with S^ and T^ denoting generic equivalence classes. An implication of symmetric information is that, given s 2 S and t 2 T with P(s; t) > 0,

it is common knowledge at (s; t) that the pair of signals realized lies in S(s) £ T(t), i.e.,

P(T(t)js) = P (S(s)jt) = 1. The electoral game with sets S(s) and T (t) of signals can therefore be analyzed independently. Since t 0 2 T(t) implies that P(s; t) = P(s; t 0) and Fs;t = Fs;t0 , it follows that the distribution of the cut point ¹ conditional on signal s and t0

is invariant over t0 2 T(t), and therefore it is simply Fs . Similar remarks hold for candidate B. Thus, (X; Y ) is a pure strategy Bayesian equilibrium if and only if for each pair S^ and T^ ^ > 0, the restricted strategies (X ^; Y ^ ) = (xs ; yt ) ^ ^ form an equilibrium with P(S^ £ T) S T s2S;t2T of the restricted game with payo®s ¦A (XS^; YT^ js) =

X

P (tjs)Fs

t2T^:x s y, ¹ is strictly closer to A with probability ¡ ¢ 1¡Fs;t x+y and equidistant with probability zero. Thus, the probability of winning indeed 2 takes the form described in (1) above.

We sometimes strengthen (C0) by adding four further conditions. Conditions (C0){(C4) then de¯ne our Canonical Model of polling, in which candidates employ identical polling technologies and signals exhibit a natural ordering structure. We ¯rst impose symmetry conditions (C1) and (C2). (C1) S = T . Thus, the same set I, with elements i; j, etc., can be used to index these sets. We then write P(i; j) for P(si; tj ), Fi;j for Fsi ;tj , and so on. (C2) For all signals i; j 2 I, P(i; j) = P (j; i) and Fi;j = Fj;i . Condition (C2) implies that signal i of candidate A can be identi¯ed with signal i of candidate B in the sense that they are equally informative. While the general model allows for asymmetries between candidates, it is natural to expect candidates to have equal access to polling technologies, in which case conditions (C1) and (C2) are appropriate. In that case, we will be especially interested in equilibria in which candidates use information similarly: A symmetric pure strategy Bayesian equilibrium is an equilibrium (X; Y ) in which xi = yi for all i 2 I. Under (C1)-(C2), candidates' ex ante payo®s are symmetric in the sense that ¦A(X; Y ) = ¦B (Y; X); for all X and Y . Thus, pure strategy Bayesian equilibria of electoral competition are equilibria of a two-player, symmetric, constant sum game. Symmetry implies that if (X; Y ) is an equilibrium, then so is (Y; X). With interchangeability, we conclude that, if (X; Y ) is an equilibrium, then (X; X) and (Y; Y ) are symmetric equilibria. 10

The next condition, which assumes (C1), says that if one candidate receives a signal, then it must also be possible for the other candidate to receive it. (C3) For all signals i 2 I, P(i; i) > 0. The last condition de¯ning the Canonical Model imposes a natural ordering structure on signals. Again assume (C1) holds.4 (C4) There exists an ordering - of I, with asymmetric part Á and symmetric part », such that for all signals i; j 2 I:

(i) i Á j if and only if, for all K µ I with P(Kji)P(Kjj) > 0, we have mi;K < mj;K . (ii) i » j if and only if, for all k 2 I, Fi;k = Fj;k . The ordering - will always be taken to be the natural ordering · whenever we explicitly identify signals with numbers.

Several implications of (C4) for the Canonical Model are noteworthy. If mi;K = mj;K for one subset K with P(Kji) > 0 and P(Kjj) > 0, then i » j. In that case, we have

Fi;i = Fj;i = Fi;j = Fj;j, so that i » j implies mi;i = mj;j . If i Á j, then the conditional medians satisfy

mi;fig < mj;fig = mi;fjg < mj;fjg ; so that i Á j implies mi;i < mj;j . Combining these observations, we see that i Á j if and only if mi;i < mj;j .

Condition (C4) is natural if \higher" signals are correlated with higher values of the cut point. The second part of the condition says that two signals are equivalently ranked by - only if they generate the same conditional distributions. It is trivially satis¯ed if is \linear," i.e., if i 6 = j implies i Á j or j Á i, which may be a reasonable assumption in many applications. By admitting the possibility of conditionally equivalence, however, we allow for candidates to condition their platforms on informationally redundant signals: The same consultant's report received in the morning or in the afternoon will convey the In the following de¯nition, we call - an ordering to indicate that it is complete and transitive. The asymmetric part Á is de¯ned as i Á j if and only if i - j and not j - i, and the symmetric part » is de¯ned as i » j if and only if i - j and j - i. We write i % j if j - i, and similarly for Â. 4

11

same information, but a candidate could conceivably condition his/her platform on such informationally irrelevant detail; collapsing the two reports into one signal would implicitly rule out this kind of conditioning.

2.4

The Downsian Model

We will sometimes consider an alternative to the continuity condition (C0) that captures the idea that polling results, when pooled, are fully informative; in other words, the conditional distribution of the cut point conditional on any pair of signal realizations is a point mass. (D0) For all signals s 2 S and t 2 T, we have Fs;t (ms;t) ¡ Fs;t (ms;t )¡ = 1. It is straightforward to verify that the candidates' probability of winning under (D0) is as described in (1). Note also that, because (C0) does not impose any more than continuity, it admits conditional distributions arbitrarily close to point mass, and it allows us to obtain the class of models satisfying (D0) as limit points of the class of models satisfying (C0). While (D0) captures one aspect of the traditional Downsian model, namely that the information of the candidates uniquely determines the location of the median voter, it is more general in that it allows for incomplete information about the candidates' signals. We may therefore sometimes refer to it as the \Generalized Downsian Model." We de¯ne the Downsian Model by adding to (D0) the restriction of complete information, formalized next. (D1) For all s 2 S and all t 2 T , if P(s; t) > 0, then P(tjs) = P(sjt) = 1. Thus, given a signal realization for one candidate, the other candidate's is known with probability one. Note that (D1) implies jSj = jTj, and by identifying a signal s with the signal t such that P(tjs) = 1, we see that (C1)-(C4) are satis¯ed, and so we may use the notation developed for the Canonical Model in the analysis of the Downsian Model. Moreover, (D1) implies that symmetric information holds, an Proposition 1 gives a full characterization of pure strategy Bayesian equilibria in the Downsian Model: For all i 2 I,

we must have xi = mi = mi;i in equilibrium. This, of course, is the well-known Median Voter Theorem.

12

Finally, we note that the expected payo®s for candidate A take an especially simple form in the Downsian Model: 8 1 > > < 1 ¦A(xi ; yi ji) = 1 > > 2 : 0

i if xi < y i and xi+y < mi;i 2 i if yi < xi and mi;i < xi+y 2 i if xi = y i or xi+y = mi;i 2 else.

That is, candidate A wins with probability one if the candidate's platform is closer to the median than is B's platform; the candidate wins with probability 1=2 if the platforms are equidistant from the median; and the candidate loses with probability one if B's platform is closer.

3

Examples of the Canonical Model

We now present four special cases of the Canonical Model. Symmetric-Information Canonical Model. In the Canonical Model, symmetric information holds if for all i; j 2 I with P(i; j) > 0, we have P(i; k)Fi;k = P(j; k)Fj;k ; for all k 2 I. As before, if signal realizations i and j are consistent, then one candidate's information following signal i is exactly that of the other following signal j. Again write i »I j if there exists k 2 I with P (i; k) > 0 and P(j; k) > 0 such that P(i; k)Fi;k =

P(j; k)Fj;k , but now note that this is equivalent to P(i; j) > 0. Thus, I can be partitioned into equivalence classes I(i) = fj 2 I j P(i; j) > 0g; with I^ denoting a generic equivalence class. Again, given a signal pair (i; j) with P(i; j) > 0, I(i) £ I(j) is common knowledge, but now I(i) = I(j), and these common knowledge components are \diagonal."

Moreover, for all j 2 I(i), (C1) and (C2) imply that P(i; j) = P (i; i), so that P is

constant on I(i) £ I(i). Thus, we have P(i; j) =

1 jI(i)j2

and P(jji) =

1 . jI(i)j

The restricted

electoral game with set I(i) of signal realizations can be analyzed independently, so that ^ (X; Y ) is a pure strategy Bayesian equilibrium if and only if for each equivalence class I, with corresponding distribution FI^, the restricted strategies (XI^; YI^) = (xi; yj )i;j2I^ are an 13

equilibrium of the restricted game with payo®s ¦A(XI^; YI^ji) µ ¶ µ µ ¶¶ jfj 2 I^ j xi < yj gj xi + yj jfj 2 I^ j yj < xigj x + yj = FI^ + 1 ¡ FI^ i ^ ^ 2 2 jIj jIj jfj 2 I^ j xi = yj gj + ; ^ 2jIj ^ and likewise for candidate B. for all i 2 I, Note that (C4) holds if and only if distinct equivalence classes I^ and I~ are associated ^ ~ with distinct medians, i.e., I^ 6 = I~ implies mI 6 = mI . In this case, the ordering ¹ on I is such that lower signal realizations correspond to equivalence classes with lower medians, and then the relation »I will be the symmetric part of ¹. Probabilistic Voting Model. The traditional probabilistic voting model with an unknown median voter is captured as the special case of the symmetric information model in which information between the candidates is complete, meaning that P(iji) = 1 for all i 2 I. From the observations above, it follows that the electoral game following signal pair (i; i) can be

analyzed independently as the symmetric information game with strategy set X for each candidate and payo®s ¦A(xi ; y iji) =

8 > >
> :

³

´

x i +yj ³2 ´ j ¡ Fi;i xi+y 2 1 2

Fi;i

if xi < yi if yi < xi if xi = yi ;

where the conditional distribution Fi;i corresponding to signal pair (i; i) is usually assumed to be continuous. The unique pure strategy Bayesian equilibrium is given by Proposition 1, with the candidates locating at mi after signal i. The model is usually de¯ned without reference to a realized signal pair, essentially taking that as given and suppressing the informational foundations of the model. Stacked-Uniform Model. This model captures situations with a number of \preliminary" locations of the median voter, where candidate polling generates signals about these preliminary locations, and subsequent to polling there is a uniform disturbance to the location of the median voter. Let ¹ = ® + ¯, where ® is uniformly distributed on [¡a; a] and ¯ is an independently-distributed discrete random variable with support on b 1 < b 2 < ¢ ¢ ¢ < bN . Let Q(bk ) denote the probability of bk .5 Candidates share the same set of signals. Signals 5

We assume ¯niteness only for simplicity. Our analysis of the Stacked Uniform Model would hold if preliminary locations were drawn from an arbitrary probability measure space.

14

depend stochastically on the realization of ¯: P(i; j) =

N X

Q(i; jjb k)Q(bk );

k=1

where Q(i; jjbk ) is the probability conditional on b k that candidates receive signals i and j. Given bk , ¹ is uniformly distributed on [bk ¡ a; bk + a], with piece-wise linear distribution ½ ½ ¾ ¾ z ¡ bk + a Fk (z) = max min 0; ;1 : 2a By Bayes rule, the probability of bk conditional on signals i and j is Q(i; jjbk )Q(bk ) P (i; j)

Q(bk ji; j) =

when well-de¯ned, and the distribution of ¹ conditional on signals i and j is Fi;j (z) =

N X k=1

Q(b kji; j)Fk (z):

Condition (C2) is satis¯ed if Q(i; jjbk ) = Q(j; ijb k) for all i, j, and k. Condition (C3) holds if, for all i 2 I, there exists a bk such that Q(i; ijbk )Q(bk ) > 0. An example of conditional densities for the simplest case of N = 2 is portrayed in Figure 1.

[ Figure 1 about here. ] For general numbers of preliminary cut-point locations and signal realizations, our equilibrium characterizations are sharpest when a is large, speci¯cally a ¸ bN ¡ b 1. Then bN ¡a · b1 +a, so that conditional densities are single-plateaued. Indeed, a ¸ bN ¡b1 implies

that the conditional medians lie in the center interval: For all i 2 I, bN ¡ a · mi;i · b 1 + a. To see this, note that Q(b 1ji; i) = 1 yields a lower bound for mi;i . In that case, Fi;i is

uniformly distributed with median b 1 ¸ bN ¡ a. Therefore, b N ¡ a · mi;i . Similarly,

Q(bn ji; i) = 1 provides an upper bound for mi;i . In that case, Fi;i is uniformly distributed with median bN · b1 + a. Thus, mi;i · b1 + a, as claimed. As a consequence, Fi;j (z) =

N X k=1

Q(bk ji; j)

µ

z + a ¡ bk 2a

¶

1 z¡ = + 2

for all z 2 [b N ¡ a; b 1 + a]. Therefore, when a ¸ bN ¡ b1, mi;j =

N X k=1

Q(bk ji; j)bk :

15

PN

k=1 Q(bk ji; j)bk ;

2a

Substituting for mi;j into Fi;j using equation (2) yields a ¡ mi;j + z ; 2a

Fi;j (z) = for all z 2 [b N ¡ a; b 1 + a], and hence,

Fi;j (z) ¡ Fi;j (w) =

z ¡w ; 2a

for all z; w 2 [bN ¡a; b1 +a]. Note that, from (C4), mh;i < mj;k implies that Fh;i(z) > Fj;k (z) for all z 2 [b N ¡ a; b 1 + a], i.e., Fj;k stochastically dominates Fh;i over the interval.

Shape-Invariant Model. In this model, the shape of the distribution of ¹ conditional on pairs of signal realizations is the same regardless of the signal realizations. This is equivalent to identifying the median conditional on the signal realizations as a scale parameter of a family of shape-invariant distributions. Formally, we assume that Fi;j (z + mi;j) = Fk;`(z + mk;`); for all z 2 < and all i; j; k; ` 2 I: The assumption that conditional distributions form a family

of shape-invariant distributions is pervasive in classical statistics|the most celebrated example of such a family is the homoskedastic econometric models with normally-distributed errors.

4

Pure Strategy Equilibrium

We have already characterized, in Proposition 1, the unique pure strategy Bayesian equilibrium of the model under symmetric information. The main result of this section is a full characterization of the pure strategy equilibria of the Canonical Model: If a pure strategy Bayesian equilibrium exists, then it is unique; and after receiving a signal, a candidate locates at the median of the distribution of ¹ conditional on both candidates receiving that signal. This is consistent with our result for symmetric information, because under that restriction the distribution of the cut point conditional on one i signal is the same as that conditional on two i signals, i.e., mi = mi;i , but Proposition 1 extends to the case of non-symmetric information in a perhaps unexpected way. Given a natural restriction on conditional medians, a corollary is that candidates take policy positions that are extreme relative to their expectations of ¹ given their own information. In other words, a candidate's location is a biased estimator of ¹: Candidates who receive high signals overshoot 16

¹, while those who receive low signals undershoot. We also provide su±cient conditions for the existence of a pure strategy equilibrium. The characterization result relies on the following lemma, which is proved in the appendix. Lemma 3 In the Canonical Model, let (X; Y ) be a pure strategy Bayesian equilibrium. If xi = yj for some i; j 2 I with P(i; j) > 0, then xi = yj = mi;j . The intuition behind this lemma is simple. Suppose that candidates A and B locate at the same point following two signal realizations, i and j, and suppose for simplicity that these are the only realizations for which they locate there; the proof uses (C4) to rule out other cases. Then, conditional on those realizations, each candidate expects to win the election with probability one half. If the candidates are not located at the median conditional on signals i and j, then the payo® of either candidate would be increased by a small move toward that conditional median: If A deviates in this way, then the candidate's expected payo® given other signal realizations for B varies continuously with A's location, but A's payo® given realization j would jump discontinuously above one half. Therefore, a su±ciently small deviation would raise A's payo®, something that is impossible in equilibrium. Theorem 1 (Necessity) In the Canonical Model, if (X; Y ) is a pure strategy Bayesian equilibrium, then xi = yi = mi;i for all i 2 I. Proof: First, consider a symmetric equilibrium (X; Y ), where xi = yi for all i 2 I. By (C3)

and Lemma 3, xi = yi = mi;i , as required. Now suppose there is an asymmetric equilibrium (X; Y ), where xi 6 = mi;i for some i 2 I, and de¯ne the strategy Y 0 = X for candidate B.

Then, by symmetry and interchangeability, (X; Y 0 ) is a symmetric Bayesian equilibrium with xi 6 = mi;i, contradicting Lemma 3. In many situations, it is reasonable to suppose that lower signals indicate lower values of ¹ and higher signals indicate higher ones. Then Theorem 1 implies that polling leads candidates to extremize their locations.

17

Corollary 1 In the Canonical Model, suppose there exists a signal c 2 I such that i Á c

implies mi;i < mi and c Á i implies mi < mi;i. If (X; Y ) is a pure strategy Bayesian equilibrium, then xi < mi for i Á c and mi < xi for i  c.

This result is su±ciently important to highlight. If c corresponds to an uninformative signal in the sense that the conditional median mc;c is equal to the unconditional median, then it follows that in a pure strategy equilibrium, private polling magni¯es platform divergence: Candidates bias their locations in the direction of their private signals past the median given their own signal and away from the unconditional median. Theorem 1 gives a necessary, not a su±cient, condition for the existence of a pure strategy equilibrium. While we soon give su±cient conditions for existence, the next example shows that pure strategy equilibria do not always exist. The example begins with the Downsian Model and demonstrates arbitrarily close Canonical Models in which choosing mi;i after signal i is not an equilibrium. An implication of Theorem 1 and Proposition 1 is that, even if the unique pure strategy equilibrium obtains in the Downsian Model, pure strategy equilibria may not exist in models arbitrarily close the original, demonstrating the fragility of pure strategy equilibrium. We return to the issue of robustness of equilibria in our analysis of mixed strategies. An implication of results there is that even if pure strategy equilibria cease to exist in models close to the Downsian Model, mixed strategy equilibria do exist and will necessarily be \close" to the pure strategy equilibrium of the original model. Example 1 Fragility of Pure Strategy Equilibrium in the Downsian Model. Consider the Downsian Model in which I = f¡1; 1g and P (¡1; ¡1) = P (1; 1) =

1 2,

with conditional

distributions F¡1;¡1 and F1;1 with point mass on m¡1;¡1 and m1;1 = m¡1;¡1+1, respectively. Let F¡1;1 and F1;¡1 be degenerate on m¡1;1 = m1;¡1 = (m¡1;¡1 + m1;1)=2. By Proposition 1, the unique equilibrium is (X; Y ) de¯ned by x¡1 = y¡1 = m¡1;¡1 and x1 = y1 = m1;1 . n Now de¯ne the sequences fFi;j j i; j = ¡1; 1g of conditional distributions as follows. For

each n ¸ 2, let P n(¡1; ¡1) = P n (1; 1) =

1 2

¡

1 n

and P n (¡1; 1) = P n (1; ¡1) =

1 n.

Let

n n F¡1;¡1 be the uniform distribution on < with density n centered at m¡1;¡1; let F1;1 be the

n n uniform distribution with density n centered at m1;1; and let F¡1;1 = F1;¡1 be the uniform

distribution with density n2 centered at m¡1;1 = m1;¡1. Note that the conditional medians n n are ¯xed at m of F¡1;¡1 and F1;1 ¡1;¡1 and m1;1, respectively, for all n. Furthermore, the n n upper bound of the support of F¡1;1 = F1;¡1 is m¡1;1 +

1 . 2n2

By Theorem 1, the only

possible equilibrium in the nth perturbed model is (X; Y ) de¯ned above. But we claim that 18

(X; Y ) is not an equilibrium of the perturbed game, because A can deviate pro¯tably to ^ n de¯ned by x strategy X ^n = m¡1;¡1 + 12 and x ^n = xn . To see this, note that ¡1

1

n

1

^ n; Y j ¡ 1) ¡ ¦A(X; Y j ¡ 1) ¦A(X · µ ¶¸ m¡1;¡1 + x ^n¡1 1 n = P n (¡1j ¡ 1) 1 ¡ F¡1;¡1 + P n (1j ¡ 1)(1) ¡ ; 2 2 ¡ ¢ 1 1 1 where³we use the fact that m + ¡1;1 2 = 2 m¡1;¡1 + n2 + m1;1 , which in turn implies 2n ´ xn ^ n ¡1+m1;1 F¡1;1 = 1. After substituting, this equals 2 µ ¶µ ¶ 2 1 1 2 1 1 1 1¡ ¡ + ¡ = + > 0; n 2 2n n 2 2n n2

k !F k establishing the claim. Since Fi;j i;j weakly and P (i; j) ! P(i; j) for i; j = 1; ¡1, the

sequence of perturbed models can be chosen arbitrarily close to the original. Thus, even perturbations from the Downsian Model that are arbitrarily small, in the sense of weak convergence, can lead to the non-existence of pure strategy equilibrium. We next provide conditions that ensure existence of the pure strategy equilibrium characterized in Theorem 1. We consider two cases of the Canonical Model: with two possible signals our condition is weak, but in the multi-signal case we require more structure. In both cases, loosely speaking, a pure strategy equilibrium exists if conditional on receiving signal i, the probability that the other candidate also receives signal i is high enough. Theorem 2 (Su±ciency for Binary Signals) In the Canonical Model, let I = f¡1; 1g.

Then a su±cient condition for the existence of the unique pure strategy Bayesian equilibrium is that µ

¶ µ ¶ z + m1;1 z + m¡1;¡1 P (1j1)f1;1 ¸ P(¡1j1)f1;¡1 2 2 µ ¶ µ ¶ z + m¡1;¡1 z + m1;1 P(¡1j ¡ 1)f ¡1;¡1 ¸ P(1j ¡ 1)f1;¡1 ; 2 2 for all z 2 [m¡1;¡1; m1;1]. In that equilibrium, candidates locate at mi;i following signal i 2 I.

Proof: We will show that (X; Y ) is an equilibrium, where xi = yi = mi;i for i = ¡1; 1. Consider candidate A's best response problem, conditional on signal 1. If A deviates to

19

x 2 [m¡1;¡1; m1;1 ], then the change in A's interim expected payo® is · µ ¶ µ ¶¸ m1;1 + m¡1;¡1 x + m¡1;¡1 P (¡1j1) F1;¡1 ¡ F1;¡1 2 2 · µ ¶ ¸ x + m1;1 1 +P(1j1) F1;1 ¡ 2 2 µ ¶ µ ¶¸ Z m1;1 · z + m¡1;¡1 z + m1;1 = P(¡1j1)f1;¡1 ¡ P(1j1)f1;1 dz 2 2 x · 0: Thus, the deviation does not increase A's expected payo®. It is easily veri¯ed that deviations x < m¡1;¡1 and x > m1;1 are also unpro¯table. A similar argument holds for signal i = ¡1, and a symmetric argument for candidate B establishes that (X; Y ) is an equilibrium.

The su±cient condition detailed in Theorem 2 is weak. If signals are not negatively correlated, so that P(1j1) ¸ P(¡1j1) and P(¡1j ¡ 1) ¸ P(1j ¡ 1), then a pure strategy equilibrium exists if

f1;1

µ

z + m1;1 2

¶

¸ f 1;¡1

µ

z + m¡1;¡1 2

¶

(2)

for all z 2 [m¡1;¡1; m1;1], and µ ¶ µ ¶ z + m¡1;¡1 z + m1;1 f¡1;¡1 ¸ f1;¡1 2 2 for all z 2 [m¡1;¡1; m1;1 ]. Inequality (2) compares f1;1, shifted to the left by f1;¡1 shifted to the left by

m¡1;¡1 . 2

(3) m1;1 2 ,

with

To make the nature of the comparison clearer, note that

inequality (2) holds over [m¡1;¡1; m1;1 ] if and only if µ ¶ m1;1 ¡ m¡1;¡1 f1;1 +z ¸ f1;¡1 (z) (4) 2 h i ¡m1;1 m1;1 +m¡1;¡1 holds over 3m¡1;¡1 ; . Thus, the su±cient condition is that f1;1 shifted to 2 2 the left by

m1;1¡m¡1;¡1 2

> 0 weakly exceed f1;¡1 over this range. This su±cient condition

holds for the Stacked-Uniform Model with a > b N ¡b 1, as conditional densities all equal

1 2a

over the relevant range. The condition also holds in the shape-invariant model when f1;¡1 is the translation of f1;1 with median f1;¡1 has its median near

m1;1 +m¡1;¡1 2

m1;1 +m¡1;¡1 . 2

More generally, the condition holds if

and is somewhat more dispersed than f 1;1 , as one

would expect if identical signals decrease the variance of the distribution of ¹ and opposing signals, which o®set each other, lead to a higher variance. 20

To simplify our su±ciency argument for existence in the multi-signal Canonical Model, we provide separate conditions on the the priors over signal pairs and on the distribution of ¹ conditional on signal realizations. Condition (C5) is a regularity condition on the conditional distributions that reinforces the symmetry already present in the Canonical Model. (C5) For all signals i; j 2 I with P(i; j) > 0, we have mi;j =

mi;i +mj;j . 2

Condition (C5) is trivially satis¯ed under symmetric information. It is satis¯ed in the Stacked-Uniform Model when a ¸ b N ¡ b1 , if Q(bk ji; j) =

Q(bk ji; i) + Q(b kjj; j) ; 2

or equivalently if Q(i; jjbk ) P(i; j)

1 = 2

µ

Q(i; ijbk ) Q(j; jjbk ) + P(i; i) P (j; j)

¶

;

for each bk . We also impose a stochastic dominance-like restriction on the conditional distributions. (C6) For all signals i; j; k 2 I with i - j - k, we have µ ¶ µ ¶ mi;i + z mk;k + z Fi;j ¸ Fj;k 2 2

(5)

for all z 2 [mi;i; mk;k]. Note that, in the Canonical Model, i - j - k implies mi;i · mj;j · mk;k , so the range

over which inequality (5) holds is necessarily nonempty. To better interpret the inequality, note that it holds over [mi;i ; mk;k ] if and only if µ ¶ mk;k ¡ mi;i Fi;j (z) ¸ Fj;k +z 2 h i m m +m holds over 2i;i ; i;i 2 k;k . That is, the distribution conditional on signals j and k when shifted to the left by

mk; k ¡mi;i 2

¸ 0 must dominate the distribution conditional on signals i

and j. Condition (C6) is stronger than stochastic dominance in that Fj;k is shifted to the left, but it is weaker in that the inequality must hold only over a given range. Again, the

condition is trivially satis¯ed under symmetric information, and at the end of this section 21

we will show that it is satis¯ed in the stacked-uniform model when the disturbance term has a su±ciently large support. Finally, we impose a restriction on priors over signals, formalizing the idea that conditional on a candidate's own signal, the probability the other candidate received the same signal is su±ciently high. In fact, the condition is weaker than that, because it only restricts \net" probabilities. (C7) For all signals i 2 I, X

j2I:j-i

P(jji) ¸

X

P(jji)

X

and

j2I:j Âi

j2I:jÁi

An equivalent statement of (C7) is that

P(jji) ·

P

j2I:j-i P(jji)

¸

1 2

X

P(jji):

j2I:j %i

and

P

j 2I:i-j P(jji)

¸ 12 .

In words, for any signal i, it must be that i is a \median" of the distribution P(¢ji) on I. A stronger condition is that P(iji) ¸

1 2

for all i 2 I. Clearly, (C7) is most restrictive for the

\extremal" signals, for which P (iji) ¸

1 2

is implied by the condition, and its restrictiveness

depends on the number of possible signals. In the binary signal model, for example, it is satis¯ed whenever signals are not negatively correlated. Note also that (C7) is trivially satis¯ed under symmetric information, because given any i 2 I, the only signals j with P(jji) > 0 are such that i » j.

Theorem 3 (Su±ciency for Multiple Signals) In the Canonical Model, conditions (C5)(C7) are su±cient for the existence of the unique pure strategy Bayesian equilibrium. In that equilibrium candidates locate at mi;i following signal i 2 I. Proof: We show that (X; Y ) is an equilibrium, where xi = y i = mi;i for all i 2 I. Without

loss of generality, we focus on candidate B's best response problem after receiving signal j. Consider a deviation to strategy Y 0. There are two cases: yj0 < mj;j and mj;j < yj0 . In the ¯rst case, de¯ne G = fi 2 I : mi;i · yj0 g and L = fk 2 I : mj;j · mk;k g: Note that for all i 2 I n (G [ L), we have y0j < mi;i < mj;j . Hence, for i with P(ijj) > 0, Fi;j

µ

yj0 + mi;i 2

¶

·

¡ 1 ¡ Fi;j 22

µ

mi;i + mj;j 2

¶¸

· 0;

where we use (C5) to deduce that Fi;j

³ y0 +m ´ i;i

j

2

·

1 2

and Fi;j

³

mi;i +mj;j 2

´

= 12 . That is, B's

gains from deviating when A receives signal i 2 I n (G [ L) are non-positive. Therefore, the change in B's interim expected payo® satis¯es

¦B (X; Y 0 jj) ¡ ¦B (X; Y jj) · µ ¶ µ µ ¶¶¸ X mi;i + y0j mi;i + mj;j · P(ijj) 1 ¡ Fi;j ¡ 1 ¡ Fi;j 2 2 i2G · µ 0 ¶ µ ¶¸ X yj + mk;k mj;j + mk;k + P(kjj) Fj;k ¡ Fj;k 2 2 k2L · µ ¶¸ X · µ 0 ¶ ¸ X mi;i + y0j yj + mk;k 1 1 = P(ijj) ¡ Fi;j + P (kjj) Fj;k ¡ ; 2 2 2 2 i2G

k2L

which is non-positive as long as · µ ¶¸ · µ 0 ¶¸ X X mi;i + y0j y j + mk;k 1 1 P(ijj) ¡ Fi;j · P(kjj) ¡ Fj;k : 2 2 2 2 i2G

Let i¤ minimize Fi;j

(6)

k2L

³m

i;i

´ +y0

2

j

over G, and let k¤ maximize Fj;k

³m

0 i;i +yj

2

´

over L. Then

inequality (6) holds if · µ ¶¸ X · µ 0 ¶¸ X mi¤;i¤ + y 0j yj + mk¤ ;k¤ 1 1 ¡ Fi¤ ;j P (ijj) · ¡ Fj;k¤ P(kjj): 2 2 2 2 i2G

(7)

k2L

Note that, by (C4), we have G µ fi 2 I j i Á jg and L = fi 2 I j j - ig, so (C7) P P implies i2G P(ijj) · i2L P (ijj). Furthermore, i¤ - j - k¤ and yj0 2 [mi¤;i¤ ; mk¤ ;k¤ ], ³ m ¤ ¤ +y0 ´ ³ y0 +m ¤ ¤ ´ i ;i k ;k j j so (C6) implies Fi¤;j ¸ F ¤ . Thus, inequality (7) holds, and it is j;k 2 2 unpro¯table for B to deviate to yj0 < mj;j . A symmetric argument applies for deviations y0j > mj;j. We have shown that condition (C7) on the signals' conditional correlation together with regularity conditions (C5) and (C6) ensure the existence of a pure strategy equilibrium. We now establish that, under (C5) and a strengthening of (C6), condition (C7) is necessary for equilibrium existence. Condition (C60 ) strengthens (C6) by stating it with equality. (C60 ) For all signals i; j; k 2 I with i - j - k, µ ¶ µ ¶ mk;k + z mi;i + z Fi;j = Fj;k 2 2 for all z 2 [mi;i; mk;k]. 23

It is immediate that, under (C5), condition (C60 ) is satis¯ed in the shape-invariant model. To see this, take any signals i; j, and note that by (C5) we have mi;i + z z ¡ mj;j mk;k + z z ¡ mj;j = mi;j + and = mj;k + : 2 2 2 2 Thus, µ ¶ µ ¶ µ ¶ µ ¶ mi;i + z z ¡ mj;j z ¡ mj;j mk;k + z Fi;j = Fi;j mi;j + = Fj;k mj;k + = Fj;k ; 2 2 2 2 where the second equality uses shape-invariance. Under (C5), condition (C60 ) also holds in the Stacked-Uniform Model when a su±ciently high, speci¯cally, a ¸ bN ¡ b1: To see this, take i; j; k as in (C60 ) and z 2 [mi;i ; mk;k ]; then, because Fi;j is linear with slope

a 2

over

[mi;i; mk;k], we have µ ¶ µ ¶ mi;i + z z ¡ mj;j 1 z ¡ mj;j Fi;j = Fi;j mi;j + = + ; 2 2 2 4a ³ ´ mk;k +z and similarly for Fj;k . 2

Theorem 4 In the Canonical Model, given conditions (C5) and (C60 ), condition (C7) is necessary and su±cient for the existence of the pure strategy Bayesian equilibrium. In that equilibrium, candidates locate at mi;i , following signal i 2 I. The proof follows the proof of Theorem 3 and is omitted.

5

Mixed Strategy Equilibria

Our results for pure strategy equilibria suggest that if there are many signals, then pure strategy equilibria may well fail to exist. We now consider mixed strategy equilibria in the electoral game. We let candidate A randomize over campaign platforms following signal s according to a distribution Gs . A mixed strategy for A is a vector G = (Gs ) of such distributions, and a mixed strategy for B is a vector H = (Ht). We follow the above convention and let Gs (z)¡ and Ht (z)¡ be the left-hand limits of these distributions, e.g., Gs (z)¡ = limw"z Gs (w). Accordingly, Gs has an atom at x if and only if Gs (x)¡Gs (x)¡ > 0. To extend our de¯nition of interim expected payo®s, we denote the probability that A wins using platform x following signal s when B uses platform y following signal t as 8 ¡ x+y ¢ < Fs;t ¡2 ¢ if x < y ¼ A(x; yjs; t) = 1 ¡ Fs;t x+y if y < x 2 : 1 if x = y; 2 24

and we let ¼B (¢js; t) = 1 ¡ ¼ A(¢js; t). Then, given mixed strategies (G; H ), candidate A's interim expected payo® conditional on signal s is Z X ¦A(G; Hjs) = P (tjs) ¼A(x; yjs; t)Gs(dx)Ht(dy); t2T

and B's interim payo® ¦B (G; Hjt) is de¯ned analogously. Abusing notation slightly, let ¦A(X; H js) be A's expected payo® from the degenerate mixed strategy with Gs (xs ) ¡

Gs (xs )¡ = 1 for all s 2 S, and let ¦B (G; Y jt) be the analogous expected payo® for B. A mixed strategy Bayesian equilibrium is a strategy pair (G; H) such that ¦A (G; Hjs) ¸ ¦A(G0 ; Hjs); for all signals s 2 S and all strategies G0 , and ¦B (G; Hjt) ¸ ¦B(G; H 0 jt); for all signals t 2 T and all strategies H 0 . Note that X can be a discontinuity point of ¦A(¢; Hjs) only if Ht puts positive probability on a point y such that ¼ A(¢; yjs; t) is discontinuous at xs for some t 2 T. Under (C0), there is only one such point, namely, y = xs . Under (D0), there are two such points: y = xs

and y = 2ms;t ¡ xs . Therefore, since each Ht can have at most a countable number of

atoms, candidate A's expected payo® function is continuous on all but perhaps a countable set of pure strategies. Furthermore, in equilibrium, if xs is a continuity point of ¦A(¢; Hjs) in the support of Gs , then the expected payo® from xs must be ¦A(G; Hjs). Candidate A must therefore be indi®erent over all such points. As with pure strategies, we can de¯ne ex ante expected payo®s as ¦A(G; H) =

X

P (s)¦A(G; Hjs)

and

s2S

¦B (G; H) =

X

P(t)¦B (G; Hjt):

t2T

Thus, mixed strategy Bayesian equilibria of the electoral game are equilibria of a twoplayer, constant-sum game. In the canonical model, the game is symmetric and we de¯ne a symmetric mixed strategy Bayesian equilibrium as an equilibrium pair (G; H) of strategies such that G = H. The next theorem provides a general existence result for mixed strategy equilibria in which candidates use mixed strategies with supports bounded as follows. Let m = maxfms;t : s 2 S; t 2 Tg and m = minfms;t : s 2 S; t 2 T g. The interval de¯ned by 25

these \extreme" conditional medians is M = [m; m]. We say (G; H) has support in M if the candidates put probability one on M following all signal realizations: For all s 2 S, Gs (m) ¡ Gs (m)¡ = 1; and for all t 2 T , Ht(m) ¡ Ht (m)¡ = 1.

Theorem 5 Under (C0), there exists a mixed strategy Bayesian equilibrium with support in M. Adding (C1) and (C2), there exists a symmetric mixed strategy Bayesian equilibrium with support in M. Proof: We use the existence theorem of Dasgupta and Maskin (1986) for multi-player games with one-dimensional strategy spaces. To apply this result, view the electoral game as a jSj +jT j-player game in which each type (corresponding to di®erent signal realizations) of each candidate is a separate player. Player s (or t) has strategy space M µ m. Note that for all t 2 T and all y 2 M, we have ¼A(m; yjs; t) ¸ ¼ A(x; yjs; t). Let G0 be any deviation such that G0s puts probability one on xs > m, and let G00 put probability one on m instead. Then

¦A (G0; Hjs) · ¦A (G00; Hjs) · ¦A (G; Hjs): A similar argument applies when x < m, yielding the claim. Adding (C1) and (C2), we see that the electoral game is a two-player, symmetric constant-sum game. Therefore, by existence of equilibrium and by interchangeability, there exists a symmetric mixed strategy equilibrium.

27

Existence of a mixed strategy equilibrium follows from Theorem 5 when conditional distributions are continuous. If discontinuities are allowed for, then the weak lower semicontinuity condition of Dasgupta and Maskin (1986) may be violated. A weaker su±cient condition for existence in symmetric games that might be applied is Reny's (1999) diagonal better reply security. The next example shows, however, that Reny's condition can be violated even if only the very restricted discontinuities allowed by (D0) are present. Thus, the prospects for a more general result using known su±cient conditions for existence in discontinuous games seem poor. Example 2 Diagonal Better Reply Security Violated with Multiple Discontinuous Conditional Distributions. Consider the canonical model in which I = f1; 2; 3g and, for all i; j 2 I,

Fi;j is the point mass on mi;j , given in the table below. We assign priors on I £I as indicated in the table below. De¯ne the mixed strategy G as follows: G1 (m1;1) ¡ G1 (m1;1)¡ = ®,

G1(m1;2) ¡ G1 (m1;2)¡ = 1 ¡ ®, G2(2) ¡ G2(2)¡ = 1, and G3(m3;3) ¡ G3(m3;3 )¡ = 1, and set ® = :8 and H = G. That is, after signal 1, the candidates mix between two conditional medians, m1;1 and m1;2 ; after signal 2, the candidates adopt the platform 2 (which does not correspond to a conditional median); and after signal 3, the candidates adopt the conditional median m3;3. Reny's (1999) diagonal better reply security requires that, if (G; H) is not a mixed strategy Bayesian equilibrium, then candidate A has a mixed strategy deviation that is pro¯table, even if B's mixed strategy is allowed to vary within some open ^ for A and an open set H of mixed set. Speci¯cally, there must exist a mixed strategy G ^ H) ^ > 1 .6 strategies for B such that H 2 J and infH2H ¦A (G; ^ 2

Note that this strategy is clearly a best response following signal 3, if we set ² > 0 su±ciently small. Following signal 2, candidate A would tie with candidate B in case B received signal 1 and positioned at m1;1 = 0, would lose to B in case B received signal 1 and positioned at 1, and would tie with B in case B received signals 2 or 3: A's expected payo® would be P(2)¦A(G; Hj2) = ®P(1; 2)(:5) + (1 ¡ ®)P(1; 2)(0) + P(2; 2)(:5) + P(2; 3)(:5) = ²(:225® + :23) = :41²; where we weight the interim payo® by the marginal probability of signal 2. If candidate A moved to the left following signal 2, the candidate could move a small enough amount 6

Here, we give candidate B's strategy space the product topology, where each factor, the set of distributions over the real line, is given the weak* topology.

28

j=3

P(1; 3) = :16² P (2; 3) = :41² m1;3 = 1:8 m2;3 = 2:3

P(3; 3) = 1 ¡ 2:89² m3;3 = 2:6

j=2

P(1; 2) = :45² P (2; 2) = :05² m1;2 = 1 m2;2 = 1:9

P(3; 2) = :41² m3;2 = 2:3

j=1

P(1; 1) = :8² m1;1 = 0

P (2; 1) = :45² m2;1 = 1

P(3; 1) = :16² m3;1 = 1:8

i=1

i=2

i=3

to x02 2 (1:8; 2) to win against candidate B in case B received signal 1 and positioned at

m1;1 = 0 or received signal 2, but would then lose against B in case B received signal 3: A's expected payo® would again be P (2)¦A(X 0 ; H j2) = ®P(1; 2)(1) + (1 ¡ ®)P(1; 2)(0) + P(2; 2)(1) + P(2; 3)(0) = ²(:45® + :05) = :41²: Moving further to the left, A could do no better than locate at m1;2 = 1, which yields an expected payo® of P (2)¦A(X 0 ; H j2) = ®P(1; 2)(1) + (1 ¡ ®)P(1; 2)(:5) + P(2; 2)(0) + P(2; 3)(0) = ²(:225® + :225) = :405²: Moving to the right, the best A could do would be to win against B in case B received signal 3 and lose otherwise, which yields an expected payo® of P (2)¦A(X 0 ; H j2) = ®P(1; 2)(0) + (1 ¡ ®)P (1; 2)(0) + P(2; 2)(0) + P(2; 3)(1) = :41²: Thus, G is a best response to H following signal 2 as well. De¯ne G0 as G, but with G01(m1;1 ) ¡ G01(m1;1)¡ = 1; that is, according to G0 , candidate

A plays as in G but chooses the conditional median m1;1 with probability one following signal 1. Letting X be the pure strategy with x1 = m1;2, we have P(1)¦A(X; H j1) = P (1; 1)(®(0) + (1 ¡ ®)(:5)) + P (1; 2)(1) + P (1; 3)(:5) = ²(:1® + :53) = :61² 29

and P (1)¦A(G0; Hj1) = P(1; 1)(®(:5) + (1 ¡ ®)(1)) + P(1; 2)(:5) + P(1; 3)(0) = ²(:6® + :225) = :705²: Thus, since G1 puts positive probability on x1 = m1;2, we conclude that (G; H ) is not a ^ and H, as described Bayesian equilibrium. Thus, diagonal better reply security requires G above. Note that H 2 H, and that G2 and G3 are best responses to H conditional on signals 2 and 3, respectively.

Furthermore, we claim that the only position that increases A's expected payo® conditional on signal 1 is, in fact, x1 = m1;1. If A took a position x01 2 (m1;1; m1;2 ) following signal 1, then A's expected payo® would be

P(1)¦A (X 0 ; Hj1) = P(1; 1)(®(0) + (1 ¡ ®)(1)) + P(1; 2)(1) + P(1; 3)(0) = ²(1:25 ¡ :8®) = :61²: If A took a position x01 2 (m1;2 ; 2) following signal 1, then A's expected payo® would again be

P(1)¦A (X 0 ; Hj1) = P(1; 1)(®(0) + (1 ¡ ®)(0)) + P(1; 2)(1) + P(1; 3)(1) = :61²: And positioning further to the right would yield an even lower expected payo®. Therefore, ^ must involve the transfer of probability mass from m1;2 to m1;1 following signal 1. G The di±culty for diagonal better reply security is that such a change no longer increases ^ by specifying that A's ex ante expected payo® above one half if we perturb H slightly to H ^ 2 put probability one on a point to the left of, and close to, 2: In that case, A now loses H ^ = G0 . to B when A receives signal 1 and B receives signal 2. To see the claim, consider G

30

^ H) ^ ¡ 1 , is Then the increase in A's ex ante expected payo®, ¦A (G; 2 ^ ^ j2) + P(3)¦A(G0; Hj3) ^ ¡ :5 P(1)¦A(G0 ; Hj1) + P (2)¦A(G0 ; H = [P(1; 1)(®(:5) + (1 ¡ ®)(1)) + P(1; 2)(0) + P(1; 3)(0)] +[P (1; 2)(®(:5) + (1 ¡ ®)(0)) + P(2; 2)(0) + P(2; 3)(:5)] +[P (1; 3)(®(1) + (1 ¡ ®)(:5)) + P(2; 3)(1) + P(3; 3)(:5)] ¡ :5 = ®[(:5)(:8²) ¡ :8² + (:5)(:45²) + :16² ¡ (:5)(:16²)] +:8² + (:5)(:41²) + (:5)(:16²) + (:41²) + (:5)(1 ¡ 2:89²) ¡ :5 = ²[®(¡:095) + :05] = ¡:026²; ^ = G0 . which is negative. Thus, diagonal better reply security is not ful¯lled by G ^ = G0 , as in the Reny's diagonal better reply security condition does not require that G above calculation: Candidate A could, for example, move probability mass from 2 or m3;3 following signals 2 and 3, respectively; as already con¯rmed, this would not increase A's ex ante expected payo® when B uses H, but it could conceivably mitigate the problem illustrated in the preceding, \protecting" A from B's slight move to the left following signal 2. A closer look shows, however, that no such protection is available. Following signal 3, of course, any change in G03 will lead to a discontinuous decrease in A's expected payo®, weighted by 1 ¡ 2:89², which can be made arbitrarily close to one. Following signal 2, A might move probability mass from 2 to the left to defeat candidate B in case B also receives

signal 2, but such a change means that A would lose to B in case B received signal 3, and we have seen that the two e®ects cancel. Finally, after signal 1, A might move probability mass from m1;1 to the right in order to defeat B when B receives signal 2, but we have seen that such a move does not increase A's interim expected payo® conditional on signal 1. This completes the example. An alternative approach to equilibrium existence when the limited discontinuities of (D0) are allows is to ¯rst restrict the strategies of the candidates to the set fms;t j s 2 S; t 2 Tg

of conditional medians, to ¯nd an equilibrium of the restricted game, and to then prove that these strategies form an equilibrium of the unrestricted game. While this approach may seem plausible under (D0), the next example demonstrates that it does not work. Example 3 An Equilibrium of the Restricted Generalized Downsian Model Is Not an Equi31

librium of the Unrestricted Game. Let I = f1; 2; 3g, where each Fi;j is a point mass and priors and conditional medians are as below.

j =2

P (0; 2) = 0

P (1; 2) = 2² 5 m1;2 = 1:7

P(2; 2) = 12 ¡ ² m2;2 = 2

j =1

P (0; 1) = 2² 5 m0;1 = :8

P (1; 1) = 2² 5 m1;1 = 1

P (2; 1) = 2² 5 m2;1 = 1:7

j =0

P(0; 0) = 12 ¡ ² m0;0 = 0

P (1; 0) = 2² 5 m1;0 = :8

P (2; 0) = 0

i=0

i=1

i=2

De¯ne the pure strategy X by x0 = 0, x1 = 1, and x2 = 2, let Y = X, and note that (X; Y ) is a pure strategy Bayesian equilibrium of the electoral game restricted to the conditional medians f0; :8; 1; 1:7; 2g. This strategy is clearly a best response following

signals 0 and 2. Following signal 1, playing x1 = 1 produces a win when B receives signal 0, a tie when B receives signal 1, and a loss when B receives signal 2. Moving to x01 = 1:7, for example, produces a tie, a loss, and a win, respectively. Since these outcomes have equal weight, A's expected payo® is maximized at x1 = 1 over the conditional medians. In the unrestricted game, however, x01 = 1:5 yields a higher expected payo® of ¡1¢ ¡1¢ ¡ 1¢ ¡1 ¢ 1 (1) + 3 3 2 + 3 (1) > 2 . We now study the continuity properties of the mixed strategy equilibrium correspondence as we vary the parameters of the model, speci¯cally the candidates' marginal prior on S £ T and the conditional distributions of ¹. We index speci¯cations of the model by

°, where the marginal probability of (s; t) in game ° is P ° (s; t), and the distribution of ¹ °

conditional on s and t in ° is Fs;t . To consider continuity properties, we assume ° lies in a metric space that we partition into ¡C0 , the speci¯cations satisfying (C0), and ¡D01 , the speci¯cations satisfying (D0) and (D1), i.e., the class of Downsian Models. We assume that indexing is continuous: For each s 2 S and t 2 T , if °n ! °, then P °n (s; t) ! P ° (s; t) and °n ° Fs;t ! Fs;t weakly. Denote the interval de¯ned by the extreme conditional medians in game

32

° by M(°), and note that by the assumption of continuous indexing, the correspondence M : ¡C0 [ ¡D01 ¶ < so-de¯ned is continuous. Theorem 5 and Proposition 1 establish the existence of a mixed strategy equilibrium for all ° 2 ¡C0 [ ¡D01 . Since the electoral game is constant-sum, the ex ante expected payo® of

a candidate in game ° is the same in all mixed strategy equilibria. Denote these payo®s or \values" by vA(°) and vB(°). Furthermore, each candidate has an \optimal" mixed strategy that guarantees the candidate's value, no matter which strategy the opponent uses. If (C1) and (C2) hold for game °, then the game is symmetric, so that vA(°) = vB (°) = 12 . The next theorem establishes that vA(°) and vB(°) vary continuously in the parameters of the game, even when asymmetries are allowed. Theorem 6 The mapping vA : ¡C0 [ ¡D01 ! < is continuous. Proof: First, take ° 2 ¡ D01 and a sequence f° ng converging to °. Since ° satis¯es symmetric information, we know from Proposition 1 that v A(°) =

1. 2

We will show that

lim inf v A(° n) ¸ 12 , and with a symmetric argument for candidate B we will conclude that

n denote the distribution of ¹ conditional on signal pair lim vA (° n) = 12 . For each n, let Fs;t

(s; t) and let P n (s; t) denote the prior probability of signal pair (s; t) in °n, and let mns;t n . De¯ne the mapping ¿ : S ! T so that, for each s 2 S, we have denote the median of Fs;t

P(¿(s)js) = 1 in °. That is, in the Downsian Model °, A's signal s corresponds to B's signal ¿(s). Now let X n be de¯ned by xns = ms;¿ (s) for all s 2 S, and let Y n be an arbitrary

pure strategy for B. Then A's expected payo® from (X n ; Y n) conditional on signal s in game °n is ¦nA(X n; Y n js) µ n ¶ X xs + ynt n n = P (tjs)Fs;t + 2 n n t2T :x s F if mns;¿(s) < y¿n(s) > < s;¿(s) 2 ³ ´ n n ms;¿ (s)+y¿ (s) ©n = n 1 ¡ Fs;¿(s) if yn¿(s) < mns;¿ (s) > 2 > : 1 else. 2 33

It follows that lim inf ¦nA(X n ; Y njs) ¸ lim inf ©n ¸ where the last inequality follows from ©n ¸

1 2

1 ; 2

for all n. Thus, A can guarantee an expected

payo® arbitrarily close to one half as n goes to in¯nity, and we conclude that lim inf vA(°n ) ¸ 1 2,

as required.

Now take ° 2 ¡C0 n ¡D01 and a sequence f°n g converging to °. Since ¡D01 is closed,

it follows that ° n 2 ¡C0 for high enough n. We prove lower semi-continuity of vA at °. A symmetric argument proves lower semi-continuity of vB = 1 ¡ vA, which, in turn, gives us

upper semi-continuity of vA. Suppose vA(°) > lim inf v A(°n). Again let ¦nA denote A's ex ante expected payo® function corresponding to °n , and let ¦A denote the ex ante payo®s ^ be any corresponding to °. Let M n denote the interval M(° n), let M = M(°), and let M ^ for high enough compact set containing M in its interior. By continuity, therefore, M n µ M n. For each n, let (Gn ; H n ) be an equilibrium with support in M n for the electoral game ^ indexed by °n, so ¦n (Gn; H n ) = vA(°n) and ¦n (Gn ; H n) = vB (°n ). By compactness of M, A

B

there exists a weakly convergent subsequence of f(Gn ; H n)g, also indexed by n, with limit (G; H ). Going to a further subsequence if necessary, we may assume fvA(°n )g converges to limit v < v A(°). Let (G¤; H ¤) be an equilibrium of the electoral game indexed by °, so G¤

is an optimal strategy for A, which guarantees a payo® of at least vA(°) in game °. Thus, ¦A(G¤; H) ¸ vA(°). In particular, there exists a pure strategy X ¤ such that ¦A (X ¤ ; H ) ¸ vA(°) > v: We claim that, as a consequence, there exists a pure strategy X 0 such that ¦nA(X 0 ; H n) >

¦A(X ¤ ; H) + v ; 2

for high enough n. But this, with vA(°n ) ! v, contradicts the assumption that Gn is a best response to H n for candidate A. We establish the claim in three steps. Step 1. By Lemma 1 (in the appendix), for every s 2 S, either X t2T

or

X t2T

P (tjs)[Ht (x¤s ) ¡ Ht (x¤s )¡][Fs;t (x¤s )] ¸

P(tjs)[Ht (x¤s ) ¡ Ht (x¤s )¡ ][1 ¡ Fs;t (x¤s )] ¸ 34

1X P(tjs)[Ht(x¤s ) ¡ Ht (x¤s )¡] 2 t2T 1X P (tjs)[Ht (x¤s ) ¡ Ht(x¤s )¡ ]: 2 t2T

(8)

(9)

Let S ¡ be the set of s 2 S such that (8) holds, and let S+ be the set of s 2 S n S¡ such that (9) holds. For s 2 S ¡, let fxks g be a sequence increasing to x¤s , and for s 2 S +, let fxksg

be a sequence decreasing to x¤s . In addition, we choose each xks to be a continuity point of Ht for all t 2 T; this is possible because T is ¯nite and each Ht has a countable number of discontinuity points. Thus, Ht (xks ) ¡ Ht (xks )¡ = 0 for all t 2 T. For each k, de¯ne the strategy X k = (xks ) for candidate A.

Step 2. We now argue that X k satis¯es lim inf ¦A(X k; H) ¸ ¦A(X ¤; H ). For each

t 2 T, let ¸t denote the probability measure generated by the distribution Ht , let ¹t denote the degenerate measure with mass Ht (x¤s ) ¡ Ht(x¤s )¡ on each x¤s , and let º t = ¸t ¡ ¹t. Let ¤ ¼s;t (z) = ¼A(x¤s ; zjs; t)

denote A's probability of winning using x¤s conditional on signal s when B receives signal t and chooses platform z, and let k ¼s;t (z) = ¼A(xks ; zjs; t)

denote A's analogous probability of winning using xks. Note that ¦A (X k ; H) ¡ ¦A(X ¤ ; H) Z X X k ¤ = P(s) P (tjs) [¼s;t (z) ¡ ¼s;t (z)] ¸t (dz) s2S

t2T

XX

P(s; t)[Ht (x¤s ) ¡

¶ ¸ x¤s + xks 1 = Fs;t ¡ 2 2 ¡ t2T s2S · µ ¶ ¸ XX x¤s + xks 1 ¤ ¤ ¡ + P (s; t)[Ht(xs ) ¡ Ht (xs ) ] 1 ¡ Fs;t ¡ 2 2 + s2S t2T Z XX ¤ + P(s; t) [¼ks;t (z) ¡ ¼s;t (z)] ºt (dz): Ht(x¤s )¡ ]

·

µ

s2S t2T

¤ Since ¼ ks;t ¡¼s;t ! 0 almost everywhere (ºt ), the corresponding integral terms above converge

to zero. Thus, by construction of X k ,

lim inf ¦A(X k ; H) ¸ ¦A(X ¤; H ) > v; m!1

as desired. Step 3. Choose k such that ¦A(X k ; H) >

¦A (X ¤ ;H)+v 2

and set X 0 = X k. To prove the

claim that ¦nA(X 0 ; H n ) > v for high enough n, de¯ne the functions Á0s;t (z) = ¼A (x0s ; zjs; t) and Áns;t(z) = ¼°An (x0s ; zjs; t): 35

Note that 0

¦A(X ; H) =

Z

XX

P(s; t)

XX

P n (s; t)

s2S t2T

Á0s;t (z) Ht (dz);

and, letting P n = P °n , ¦nA (X 0 ; H n) =

s2S t2T

Z

Áns;t (z) Htn (dz):

Since ¦A(X 0; H) > v, it su±ces to show that Z Z n n Ás;t (z) dHt ! Á0s;t (z) dHt ; for each s 2 S and t 2 T. To prove this, ¯x ² > 0. Because x0s = xks is not a mass point of

Ht , we may specify an interval Z = [z; z] with x0s 2 (z; z) such that Ht (z) ¡ Ht (z)¡ < 4² . By weak convergence, Htn (z) ¡ Htn(z)¡
0 or

lim sup Htn(b) < 1. Without loss of generality, assume the latter. Using the notation from the proof of Theorem 6, let s = ¿ ¡1 (t), and de¯ne X n so that xns = mns;t for each n. As in the proof of Theorem 6, we can show that lim inf

inf ¦nA(X n; Y js0 ) ¸

n!1 Y 2

37

1 : 2

To see the last strict inequality, take c such that ms;t < c < ms;t2 +b , and note that µ ¶ ms;t + yt 1 n n n lim inf © = lim inf Fs;t ¸ lim inf Fs;t (c)¡ ¸ Fs;t (c)¡ > ; 2 2

(11)

where the second-to-last inequality follows from weak convergence and the strict inequality follows from c < ms;t and our assumption of a unique median. From the above observations, we have 1 lim inf ¦nA (X n ; H njs) ¡ 2 Z · ¸ Z · ¸ 1 1 n n n n ¸ ¼ A(ms;t; yjs; t) ¡ H t(dy) + ¼A(ms;t ; yjs; t) ¡ Ht (dy) 2 2 (¡1;b) [b;1) Z · ¸ 1 ¡ ¸ Fs;t (c) ¡ Ht(dy) 2 [b;1) > 0; where the second weak inequality follows from (10) and the strict inequality follows from (11) and Ht (b) < 1. It follows that lim inf ¦nA (Gn; H n) > 12 , i.e., lim inf vA(°n ) > 12 . But vA(°) = 12 , contradicting the continuity result of Theorem 6. Now assume that ° 2 ¡C0 n ¡D01. If (G; H) 2 = E(°), then one candidate, say A, has a

pure strategy X such that

¦A(X; H) > v A(°):

(12)

But then, as in the proof of Theorem 6, we can ¯nd a strategy X 0 satisfying (12) such that no x0s is a mass point of any H t, and then we can show that ¦A(X; H ) + vA(°) ; 2

¦nA(X 0; H n ) >

for high enough n. But vA(°n) ! vA(°) by Theorem 6, so it follows that ¦nA(X 0; H n ) > vA(°n); for high enough n, contradicting the assumption that Gn is a best response to H n for A in the electoral game indexed by °n . The next example shows that the upper hemicontinuity result of Theorem 7 cannot be generalized to allow for even the limited discontinuities in (D0). Example 5 Upper Hemicontinuity Violated at the Generalized Downsian Model. Let I = f¡1; 0; 1g, with uniform priors, i.e., P(i; j) =

1 9

38

for each i; j 2 I. For each i; j 2 I, let Fi;j be

n the point mass on mi;j , and let fFi;j g be a sequence of uniform distributions with density

n, where all conditional medians are depicted below.

j =1

m¡1;1 = 0 mn¡1;1 = n1

m0;1 = 0 mn0;1 = 0

m1;1 = 0 mn1;1 = 1n

j =0

m¡1;0 = 0 mn¡1;0 = 0

m0;0 = 0 mn0;0 = 0

m1;0 = 0 mn1;0 = 0

j = ¡1

m¡1;¡1 = ¡1 mn¡1;¡1 = ¡1

m0;¡1 = 0 mn0;¡1 = 0

m1;¡1 = 0 mn1;¡1 = n1

i = ¡1

i=0

i=1

Thus, each n de¯nes a version of the Canonical Model, and the conditional distributions n , F n , and F n converge weakly to the degenerate distributions F F1;¡1 1;¡1 , F ¡1;1 , and ¡1;1 1;1 n n F1;1, respectively. Furthermore, the supports of F¡1;0 and F¡1;1 are contiguous, as are the

n n . For each n, de¯ne the pure strategy X n as follows: x n = ¡1, supports of F0;¡1 and F1;¡1 ¡1

xn0 = 0, and xn1 =

1 n,

and let Y n = X n . Then (X n; Y n ) is a Bayesian equilibrium of the

nth game in the sequence. To see this, note that following signal ¡1, xn¡1 = ¡1 produces

a tie if B receives signal ¡1, a loss if B receives signal 0, and a loss if B receives signal 1, yielding a conditional expected payo® for candidate A of 16 . Moving from xn¡1 = ¡1 to the right, A's conditional expected payo® is maximized for x0¡1 2 [0; 1n ], which yields µ ¶ µ ¶ µ 0 ¶ µ ¶µ µ n ¶¶ x¡1 ¡ xn0 x1 ¡ x0¡1 1 1 1 1 (0) + F¡1;0 + 1 ¡ F¡1;1 = ; 3 3 2 3 2 6

and thus xn¡1 = ¡1 is a best response. Following signal 0, xn0 = 0 produces a win if B

receives signals ¡1 or 1 and a tie if B receives signal 0, and this is clearly a best response. Following signal 1, xn1 =

1 n

produces a win if B receives signal ¡1, a loss if B receives

signal 0, and a tie if B receives signal 1, and this is also a best response, establishing the

claim. Clearly, (X n; Y n ) converges to (X; Y ) de¯ned by x¡1 = y¡1 = ¡1, x0 = y0 = 0, and x1 = y1 = 0. But this is not an equilibrium of the original model, because x¡1 = ¡1 is not a best response: while x¡1 = ¡1 produces a tie and two losses, moving to x0¡1 = 0 produces two ties and one loss, increasing A's expected payo®.

39

6

Mixed Strategy Characterization Results

To this point, our results have established existence of mixed strategy equilibria with support in M, but we have not provided a necessary condition to bound the supports of all equilibrium mixed strategies. Our next result does just that for the Canonical Model, even without (C4), showing that all mixed strategy equilibria have support in M. The result holds also for the symmetric version of the generalized Downsian Model, though existence has not been established there. Theorem 8 Under (C1), (C2), and either (C0) or (D0), if (G; H) is a mixed strategy Bayesian equilibrium, then it has support on M. Proof: Let (G; H) be a mixed strategy Bayesian equilibrium, let xi = supfx 2 < : Gi (x) =

0g be the lower bound of the support of Gi for each i 2 I, and let x = mini2I xi be the

minimum of these lower bounds. Suppose that x < m, and take i such that xi = x. By symmetry and interchangeability, (G; G) is also an equilibrium, so we may assume that H = G. Consider a sequence of pure strategies fX ng satisfying the following. If Gi puts positive probability on x, i.e., Gi (x) ¡ Gi (x)¡ > 0, then let xni = x for all n. Otherwise, let

fxni g be a sequence decreasing to x such that each xni is in the support of Gi . Furthermore, choose xni so that ¦A(X n ; Hji) = ¦A (G; Hji) for all n. To see that this can be done, set x0i

arbitrarily and, if possible, let xni be any continuity point of A's expected payo® function in the support of Gi and in the interval [x;

x+xn¡1 i ] 2

to satisfy the desired condition. Since there

is at most a countable number of discontinuity points of A's payo® function, such a point can be found unless the support of Gi in [x;

x+xn¡1 i ] 2

is countable. In that case, however,

any point in the support of Gi in this interval satis¯es the desired condition, and there is at least one such point since xi = x. In any case, we have ¦A (X n ; H ji) = ¦A(G; Hji) for all n and limn!1 Gi (xni)¡ = 0. Now consider a pure strategy X 0 satisfying x0i = m, and note

40

that for n such that xin < m, ¦A(X 0 ; Hji) ¡ ¦A(X n; Hji) "Z · µ n ¶ µ ¶¸ X xi + z m+z = P (jji) Fi;j ¡ Fi;j Hj (dz) 2 2 [x;x n i) j2I · µ n ¶¸ 1 xi + m n n ¡ +(Hj (xi ) ¡ Hj (xi ) ) ¡ Fi;j 2 2 · µ ¶ µ n ¶¸ Z m+z xi + z + 1 ¡ Fi;j ¡ Fi;j Hj (dz) 2 2 (x n i ;m) · µ n ¶¸ 1 xi + m ¡ + (Hj (m) ¡ Hj (m) ) ¡ Fi;j 2 2 # Z · µ ¶ µ n ¶¸ m+z xi + z + Fi;j ¡ Fi;j H j(dz) : 2 2 (m;1)

(13) (14) (15)

For each j 2 I, the ¯rst integral goes to zero, because limn!1 Hj (xni )¡ = limn!1 Gj (xni )¡ = 0. Further, the last integral is clearly non-negative. So, too, the other terms are non¡ ¢ 1 negative, because Fi;j w+z < 2 for all w · m and all z < m. This establishes that for all 2 j 2 I, the expression in brackets is non-negative. It is strictly positive for j = i, because

the bracketed terms in (13)-(15) have strictly positive limits; and the total probability mass on these terms is strictly positive since lim Hj (m) ¡ H j(xni )¡ = Gi (m) ¡ Gi (x)¡ = Gi(m) > 0:

n!1

Because P (iji) > 0 by (C3), we conclude that ¦A(X 0 ; Hji) > ¦A(X n; Hji) = ¦A(G; H ji) for high enough n, contradicting the assumption that (G; H) is an equilibrium. An identical argument establishes that the supports of equilibrium strategies are bounded from above by m. The next example shows that the symmetry assumed in Theorem 8 is essential for the bounds on equilibrium strategies given there. Example 6 Symmetry Needed for Equilibrium Bounds. Let I = f¡1; 1g, and let each Fi;j

be a uniform distribution with density 2, where priors on I £I and conditional medians are

41

depicted below. j=1

P(¡1; 1) = ² m¡1;1 = ¡1

P(1; 1) = ²2 m1;1 = 1

j = ¡1

P(¡1; ¡1) = 1 ¡ ² ¡ ²2 ¡ ²3 m¡1;¡1 = 0

P(1; ¡1) = ²3 m1;¡1 = ¡1

i = ¡1

i=1

Note that the conditional distribution following signal pairs (¡1; 1) or (1; ¡1) has support

[¡1:25; ¡:75]; the conditional distribution following (¡1; ¡1) has support [¡:25; :25]; and the conditional distribution following (1; 1) has support [:75; 1:25]. When ² is small, the conditional probability that candidate A receives signal i = ¡1 is close to one, regardless of

B's signal. In contrast, the conditional probability that B receives signal j is close to one when A receives signal i = j. Let x¡1 = 0, x1 = 1:25, y¡1 = 0, and y1 = ¡1:25. We claim that the strategy pro¯le (X; Y ) so-de¯ned is a Bayesian equilibrium, despite the fact that x1 = 1:25 > m = 1 and y1 = ¡1:25 < m = ¡1, violating the bound given in Theorem 8. To

see this, ¯rst note that candidate A maximizes probability of winning following signal ¡1: moving to the left from x¡1 = 0 only decreases A's probability of winning in case B receives signal ¡1; and moving far enough to increase A's probability of winning in case B receives signal 1 means A must position at x0¡1 < ¡:75, but then A would win with probability zero in case B receives the more likely signal ¡1. Similarly, A maximizes probability of winning

following signal 1: A already wins with probability one in case B receives signal 1; and A cannot increase the probability of winning in case B receives signal ¡1 without moving to

the left of y¡1 = 0, but then A would win with probability zero in case B receives the more likely signal 1. A symmetric argument for B establishes the claim. Theorem 8 applies to the Downsian Model. Moreover, decomposing the Downsian Model into its component games, it applies to each one separately. Since the set of medians for the component game corresponding to any signal pair (s; t) with P(s; t) > 0 is just the singleton consisting of ms = mt , we conclude the unique mixed strategy Bayesian equilibrium of the component game is the point mass on ms = mt . In other words, the unique pure strategy Bayesian equilibrium characterized in Proposition 1 is also unique among all mixed strategy Bayesian equilibria. 42

Corollary 2 In the Downsian Model, if (G; H) is a mixed strategy Bayesian equilibrium, then the candidates locate at mi with probability one following signal i 2 I, i.e., Gi(mi) = Gi (mi )¡ = Hi (mi ) ¡ Hi (mi )¡ = 1 for all i 2 I.

With Theorem 7, Corollary 2 implies that in models close to the Downsian Model, mixed strategy equilibria must be close, in the sense of weak convergence, to the pure strategy equilibrium. This is relevant, in particular, for our earlier example of fragility of the pure strategy equilibrium of the Downsian Model: Though pure strategy equilibria do not exist close to the Downsian Model, for every open set around m¡1;¡1, for every open set around m1;1 , and for high enough n, mixed strategy equilibria exist and put probability arbitrarily close to one on those sets following signals ¡1 and 1, respectively. Thus, while pure strategy

equilibria in symmetric information models may be fragile, Theorem 8 delivers a robustness result in mixed strategies. We have yet to consider whether the distributions used by candidates in equilibrium may contain atoms. Our next result shows that, with reasonable structure on the electoral game, candidates can have atoms only at the conditional medians, mi;i . In addition to (C1)-(C4), we impose the following \monotone likelihood ratio" condition on conditional signal probabilities. (C8) For all signals i; j; i0 ; j0 2 I, if i Á i0 and j Á j 0 , then P (jji0 )P(j 0ji) · P(jji)P (j 0 ji0 ): Provided that P(jji) and P(jji0 ) are positive, condition (C8) can be written more intuitively as P(j 0ji) P (j 0 ji0 ) · ; P(jji) P(jji0 ) which gives the monotone likelihood condition its name. A reasonable interpretation of the signals in our model is that they indicate the ideological leanings of the electorate, i.e., whether ¹ is likely to be located more to the left or more to the right. Under that interpretation, the following stochastic dominance condition, which presumes (C1)-(C4), is natural. (C9) For all signals i; i0 2 I with i Á i0 , for all j 2 I, and for all z 2 M with 0 < Fi0;j (z) < 1, we have Fi0 ;j (z) < Fi;j (z).

43

By (C0), condition (C9) implies that for signals i Á i0 , Fi0 ;j (z) · Fi;j (z), for all z 2 M. Condition (C9) is implied by (C4) in the Stacked-Uniform Model when a ¸ bN ¡b1: In that case, i Á i0 implies mi0;j > mi;j , so that Fi0 ;j (z) =

1 z ¡ mi0;j 1 z ¡ mi;j + < + = Fi;j (z) 2 2a 2 2a

which yields the condition. Lemma 4, which we prove in the appendix, establishes a key consequence of (C8) and (C9). Lemma 4 In the Canonical Model, assume that conditions (C8) and (C9) hold. For each j 2 I, let ®j 2 [0; 1]. Then, for all i; i0 2 I with i Á i0 and for all z 2 M with 0 < ®j P (jji)P(jji0)Fi0 ;j (z) < ®j P(jji)P (jji0 ) for at least one j, we have P j 2I ®j P(jji)Fi;j (z) P > j2I ® j P(jji)

P

0

)Fi0 ;j (z) : 0 j2I ®j P(jji )

j2I ® j P(jji

P

Note that Lemma 4 reinforces (C4): Setting ®j = 1 for all j 2 K and ®j = 0 other-

wise, we see that Fi0 ;K stochastically dominates Fi;K . Lemma 4 allows us to prove (in the

appendix) a ¯nal lemma on the location of mass points of equilibrium mixed strategies. It parallels, under the extra conditions of (C8) and (C9), Lemma 3. Lemma 5 In the Canonical Model, assume that conditions (C8) and (C9) hold. Let (G; H) be a mixed strategy Bayesian equilibrium. For all z 2 M, if both candidates place positive probability mass on z, that is, Gi(z) ¡ Gi (z)¡ > 0 for some i 2 I and Hj (z) ¡ Hj (z)¡ > 0 for some j 2 I with P(i; j) > 0, then z = mi;j .

The intuition behind this lemma is simple. Suppose that candidates A and B both put positive mass on the same point z 2 M following signal realizations, i and j. Lemma 4

allows us to assume that i and j are the only signal realizations after which the candidates put positive mass on z. The argument then proceeds as in Lemma 3. Conditional on signals i and j, each candidate expects to choose z with positive probability, and if z is not equal to mi;j , then a candidate, say A, can transfer probability mass from z and move it toward mi;j by an arbitrarily small amount. This increases A's expected payo® discretely when 44

B chooses z, and it a®ects A's expected payo® continuously otherwise. Therefore, a small enough deviation increases A's expected payo®. We now derive our restriction on atoms of mixed strategy equilibria: In the Canonical Model with (C8) and (C9), the only possible atom of an equilibrium distribution, Gi or Hi, is the conditional median mi;i. Save for the added assumptions of (C8) and (C9), this result generalizes Theorem 1. Theorem 9 In the Canonical Model, assume that conditions (C8) and (C9) hold. Let (G; H ) be a mixed strategy Bayesian equilibrium. If Gi(z) ¡ Gi(z)¡ > 0 for some i 2 I, then z = mi;i. If Hj(z) ¡ Hj (z)¡ > 0 for some j 2 I, then z = mj;j .

Proof: Let (G; H ) be a mixed strategy Bayesian equilibrium, and suppose Gi(z)¡Gi(z)¡ > 0 for some i 2 I, but z 6 = mi;i . By symmetry and interchangeability, (G; G) is an equilibrium, and (C3) imposes P(i; i) > 0. By Theorem 8, we must have z 2 M. But then Lemma 5 implies z = mi;i, a contradiction.

Theorem 9 does not quite allow us to use di®erentiable methods to analyze mixed strategy equilibria. While the result limits the potential discontinuities of equilibrium mixed strategies to a ¯nite set, there may be other points at which an equilibrium distribution Hj is non-di®erentiable, albeit continuous. The Cantor-Lebesgue function (see Wheeden and Zygmund, 1977) is an example of a continuous distribution that puts probability one on its points of non-di®erentiability, so this technical problem is potentially signi¯cant. In applications, it may be helpful to restrict attention to a subset of mixed strategy equilibria: We say a strategy pair (G; H) is regular if for all i 2 I and all z 2 0 and ®j P(jji0) > 0 for some j, so cross multiply and rewrite the desired inequality as X

j;j 02I

®j ®j 0 P(jji)P (j 0 ji0 )Fi;j (z) >

X

j;j0 2I

®j ®j0 P(jji 0)P(j 0ji)Fi0 ;j (z):

We compare the two sides of the inequality one pair fj; j 0g at a time. For j = j0 , we have ®2j P (jji)P(jji0)Fi;j (z) ¸ ®2j P(jji)P(jji0 )Fi0;j (z) from (C9). Moreover, there is at least one j such that ®2j P(jji)P (jji) > 0 and Fi0 ;j (z) 2

(0; 1), which implies Fi;j (z) > Fi0;j (z) and gives us a strict inequality. For distinct j and j 0 , say j < j 0 , we want to show that ®j ®j0 [P(jji)P(j0 ji0 )Fi;j (z) + P (j 0 ji)P(jji0 )Fi;j0 (z)]

¸ ®j ®j 0 [P(jji0 )P (j 0 ji)Fi0;j (z) + P(j 0ji0 )P(jji)Fi0 ;j0 (z)]: 49

Note that by (C9), we have Fi;j (z) ¸ maxfFi;j0 (z); Fi0;j(z)g ¸ minfFi;j0 (z); Fi0;j(z)g ¸ Fi0;j0 (z); and therefore Fi;j (z) ¡ Fi0;j0 (z) ¸ Fi0;j(z) ¡ Fi;j0 (z): Then (C8) implies P (jji)P(j 0ji0 )(Fi;j (z) ¡ Fi0 ;j0 (z)) ¸ P(jji0)P(j 0ji)(Fi0 ;j (z) ¡ Fi;j 0 (z)); which yields the desired inequality.

Lemma 5 In the Canonical Model, assume (C8) and (C9). Let (G; H) be a mixed strategy Bayesian equilibrium. For all z 2 M, if Gi(z) ¡ Gi (z)¡ > 0 for some i 2 I and Hj (z) ¡ Hj (z)¡ > 0 for some j 2 I with P(i; j) > 0, then z = mi;j .

Proof: Let (G; H ) be a mixed strategy Bayesian equilibrium, and take any z 2 M. De¯ne the sets

I 0 = fi 2 I : Gi (z) ¡ Gi(z)¡ > 0g

J 0 = fj 2 I : Hj (z) ¡ Hj (z)¡ > 0g: Take any i 2 I 0 and j 2 J 0 such that P(i; j) > 0. Lemma 2 implies X j0 2I

P (j 0 ji)[Hj0 (z) ¡ Hj0 (z)¡][Fi;j0 (w)] =

1X P(j0 ji)[Hj0 (z) ¡ Hj0 (z)¡ ]: 2 0

(18)

j 2I

If there exists i0 2 I 0 with i0 6 = i and P(i0; j) > 0, then (19) must hold for i0 as well. Setting

®j0 = Hj0 (z) ¡ Hj0 (z)¡, we see that (C8), (C9), and Lemma 4 imply that i and i0 have the

same conditional distributions. An analogous argument for candidate B establishes that j 0 2 J 0 and P (i; j 0) > 0 imply that j and j 0 have the same conditional distributions. Now

take any j 0 2 J 0 such that P (jji0 ) > 0. This implies P (i0 ; j) > 0, so Fi0 ;j = Fi0 ;j0 . Therefore, (19) reduces to Fi0 ;j0 (z) = 1=2, i.e., z = mi0 ;j0 .

50

References [1] E. Aragones and T. Palfrey (2002) \Mixed Equilibrium in a Downsian Model with a Favored Candidate," Journal of Economic Theory, 103: 131-161. [2] R. Ball (1999) \Discontinuity and Non-existence of Equilibrium in the Probabilistic Spatial Voting Model," Social Choice and Welfare, 16: 533-556. [3] J. Banks and J. Duggan (1999) \The Theory of Probabilistic Voting in the Spatial Model of Elections," mimeo. [4] J. Banks and J. Duggan (2002) \A Multidimensional Model of Repeated Elections," mimeo. [5] D. Bernhardt, E. Hughson, and S. Dubey (2002) FILL IN REFERENCE [6] D. Black (1958) Theory of Committees and Elections, Cambridge: Cambridge University Press. [7] R. Calvert (1985) \Robustness of the Multidimensional Voting Model: Candidate Motivations, Uncertainty, and Convergence," American Journal of Political Science, 29: 69-95. [8] J. Chan (2001) \Electoral Competition with Private Information," mimeo. [9] A. Downs (1957) An Economic Theory of Democracy, New York: Harper and Row. [10] J. Duggan (2000) \Repeated Elections with Asymmetric Information," Economics and Politics, 109-136. [11] J. Duggan and M. Fey (2002) \Repeated Downsian Electoral Competition," mimeo. [12] R. Fauli Oller, E.A. Ok, I. Ortuno-Ortin (2001) \Delegation and Polarization of Platforms in Political Competition" mimeo, CEPR [13] T. Groseclose (2001) \A Model of Candidate Location when One Candidate Has a Valence Advantage," American Journal of Political Science, 45: 862-886. [14] P. Heidhues and Johan LagerlÄo f (2001) \Hiding Information in Electoral Competition," mimeo. [15] H. Hotelling (1929) \Stability in Competition," Economic Journal, 39: 41-57. 51

[16] C. Martinelli (2001) \Elections with Privately Informed Parties and Voters," Public Choice, 108: 147-167. [17] C. Martinelli (2002) \Policy Reversals and Electoral Competition with Privately Informed Parties," Journal of Public Economic Theory, 4: 39-62. [18] S. Medvic (2001) Political Consultants in U.S. Congressional Elections, Columbus: Ohio State University Press. [19] P. Milgrom (1981) \Rational Expectations, Information Acquisition, and Competitive Bidding" Econometrica 49: 921-43 [20] M. Ottaviani and P. Sorensen (2002) \Forecasting and Rank Order Contests," mimeo. [21] R. Wheeden and A. Zygmund (1977) Measure and Integral, New York: Marcel Dekker. [22] D. Wittman (1983) \Candidate Motivation: A Synthesis of Alternatives," American Political Science Review, 77: 142-157.

52