Electoral Competition with Privately-Informed Candidates∗

(Preliminary: Rochester-Pasadena draft)

Dan BernhardtDepartment of Economics

University of Illinois1206 S. Sixth Street

Champaign, IL 61820

John DugganDepartment of Political Scienceand Department of Economics

University of RochesterRochester, NY 14627

Francesco SquintaniDepartment of EconomicsUniversity of RochesterRochester, NY 14627

August 24, 2002

∗We thank Jean-Francois Mertens for helpful discussions during his visit to theWallis Institute of Political Economyat the University of Rochester.

Abstract

We consider a model of elections in which two office-motivated candidates receive privatesignals about the location of the median voter’s ideal point prior to taking policy positions. Weshow 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 distributionof 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 thetrue median, with candidate positions tending to the extremes of the policy space. We providesufficient conditions for the existence of a pure strategy equilibrium. Essentially, the purestrategy equilibrium exists if for each signal, the other candidate is sufficiently likely to receive asignal in the same direction that is at least as extreme. If the correlation in candidates’ signals isless strong, then we prove that even though a pure strategy equilibrium does not exist, a mixedstrategy equilibrium does exist. We show that in any mixed strategy equilibrium, candidates donot locate more extremely than they would in a pure strategy equilibrium. We further show thateven though the electoral game leads to discontinuous candidate payoffs, expected equilibriumcandidate payoffs are continuous in the game specification. We then explicitly characterizemixed strategy equilibria in a tractable version of the model. Qualitatively, candidates whoreceive sufficiently moderate signals locate more extremely than their information suggests,while candidates who receive more extreme signals temper their locations; and as candidates’signal correlation rises, candidates locate more extremely. Finally, we consider the consequencesof electoral competition for voter welfare. In particular, we characterize the optimal amount ofnoise in the polling technology, and show that candidates do not extremize their location by asmuch as voters would like.

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 office-motivated

candidates is that both candidates locate at the median voter’s ideal point. The logic is simple if

we consider symmetric equilibria. If both candidates locate at a position that is not the median

voter, then the result is a tie, but either candidate could move slightly toward the median and win

for sure. We capture non-symmetric equilibria with the same logic once we note that equilibria of

this game are interchangeable.1 Implicit in this logic is the additional 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 the

candidates share a common prior distribution about the location of the median voter, who votes for

the closest candidate. 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 differences in the candidates’ per-

sonal experiences or from the different 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. Con-

gressional 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. (See Medvic (2001).) This is in addition to polling services offered

by the major parties, particularly the Republican party.

In this paper, we develop a general model of elections that allows for privately-informed candi-

dates. Before selecting a platform, each candidate receives a signal drawn from an arbitrary finite

set of possible signals; each candidate updates about the location of the 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 equi-1This follows because the electoral game is zero-sum.

1

librium of the model, if it exists. While we obtain the Downsian and probabilistic voting models

as special cases of ours, we show that the logic of the median voter theorem does not extend in the

expected way. Indeed, we show that strategic competition between the candidates quite generally

leads the candidates to take positions more extreme than their estimates of the median voter’s ideal

point. We verify existence and upper hemicontinuity of mixed strategy equilibria, even though the

electoral game gives candidates discontinuous payoffs, and we give a full characterization of mixed

strategy equilibria in a tractable version of our model. Finally, we analyze voter welfare in a special

case: We show that, while candidates tend to take positions more extreme than their estimates of

the median voter, all voters would be better off (in ex ante terms) if candidates took even more

extreme positions; and we give comparative statics with respect to polling parameters, showing

that signal correlation is always bad for voters, whereas signal precision is sometimes good and

sometimes bad.

When the pure strategy equilibrium exists, its characterization is simple: After receiving a sig-

nal, a candidate updates the prior distribution of the median voter, conditioning on both candidates

receiving that same signal, and locates at the median of the posterior distribution.2 In the Down-

sian 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 results for those models as special cases. When information is asymmetric, though one might

expect a candidate to simply target the median voter condition on his/her own signal, that is not

what happens in equilibrium. In fact, the equilibrium platform of a candidate is a biased estimator

of the median voter! 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 we show that a candidate could exploit any discontinuity

by moving slightly toward that conditional median, so the only possible symmetric pure strategy

equilibrium is the one claimed. Finally, because the electoral game is constant-sum, we conclude

by symmetry and interchangeability that there are no other asymmetric pure strategy equilibria.

We give sufficient 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 quite2This result is reminiscent of the findings of Milgrom (1981), who shows that in a common-value second-price

auction, the equilibrium bid of a type θ corresponds to the expected value of the good conditional on both typesbeing equal to θ. Here, since candicates maximize the probability of winning, the relevant statistics is the median,instead of the expected value.

2

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 should not be negatively correlated.

Second, we 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 sufficient—not necessary—conditions, we construct examples where

the pure strategy equilibrium fails to exist (and the above sufficient conditions are violated). In

fact, we show that if redundant signals are introduced into the Downsian model, then we can

add an arbitrarily small amount of asymmetric information 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 results for the Downsian and probabilistic voting models is that, in equilibrium,

“policy convergence” obtains, i.e., the candidates adopt exactly the same policy position, giving

voters essentially no choice.3 While differences in positions of real-world candidates may be lim-

ited, 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 very robust result in

the Downsian model, surviving even if candidates place some (or all weight) on policy preferences

(Calvert (1985)). In contrast, adding policy motivation to the probabilistic voting model precludes

policy convergence, introducing a wedge between the candidates’ platform, if a pure strategy equi-

librium indeed exists (Wittman (1983), Calvert (1985)). Divergence is also generated if one of the

candidates has a “valence advantage,” meaning that all voters are willing to vote for that candidate

even if the opponent offers 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

office-motivated and neither possesses a valence advantage, even in a symmetric pure strategy

equilibrium: Differences in platforms are simply generated by differences in information, due to

randomness inherent in political polling. We show that, in fact, for a wide class of models, strategic

competition between the candidates can actually increase the dispersion between the candidate’s

platforms by inducing them to take positions more extreme than would be the case if they simply3Policy convergence is a result found 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 convergenceresults in models of repeated elections.

3

targeted the median voter. This is most starkly highlighted in the simplest model, where the

candidates receive only binary signals, “left” or “right,” about the location of the median voter.

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 signed-signals. Since the median given two “left”

signals will 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 the existence of mixed strategy equilibria; we prove that the (unique)

mixed strategy equilibrium payoffs of our model vary continuously in its parameters; and we use

that result to prove upper hemicontinuity of equilibrium mixed strategies. Existence of mixed

strategy equilibrium is also proved by Ball (1999) in the context of the probabilistic voting model,

and our existence result can be viewed as extending his to the model with asymmetrically informed

candidates.4 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: If a

small amount of asymmetric information is added to the Downsian model, then the pure strategy

equilibrium may no longer exist, but mixed strategy equilibria will, and they will be close to the

median.5

We then explicitly solve for a mixed strategy equilibrium in a tractable version of our model, the

“stacked uniform model,” in which the median voter’s ideal point is given by α+β, where candidates

receive signals conditional on the realization of β and α is an independent, uniformly distributed

noise term. A possible interpretation of this is that candidates learn about an initial location of the

median voter, given by β, after which electoral preferences are subject to a random shock, given by

α. We solve for an equilibrium in which the supports over which candidates mix following different

signals are adjacent and non-overlapping. Candidates with sufficiently moderate signals adopt

their pure strategy locations, locating more extremely than their information suggests. If signal4Though we generalize Ball (1999) in this respect, his model, unlike ours, allows for policy-motivated candidates.5Banks and Duggan (1999) give a similar finding in the probabilistic voting model with expected plurality maxi-

mizing candidates. There, the unique pure strategy equilibrium may cease to exist when probability of vote functionsare perturbed slightly, though mixed strategy equilibria must vary continuously.

4

correlation is low enough that the pure strategy equilibrium does not exist, then candidates who

receive more and more extreme signals mix over their location, tempering their location by more

and more toward the unconditional median. This then is the general interpretation of our model:

Candidates who receive sufficiently moderate signals “extremize” their location, while candidates

with sufficiently extreme signals moderate their location. We then consider the special case with

only three possible signals, to be understood as “left,” “center,” and “right,” and we prove that

the mixed strategy equilibrium solved for above is (essentially) unique. We explicitly calculate this

equilibrium, and we derive comparative statics: We show, for example, that candidates tend to

locate more extremely as the quality of their signals rises and as the probability of receiving an

uninformative signal falls.

We end by exploring a seemingly puzzling complaint in both the public press and academic

research. To whit, it is generally recognized that improved polling techniques lead to a convergence

in candidate platforms, as candidates try to hone in on the median voter’s preferred platform

given their current information. Such convergence is often criticized as being bad for the median

voter, and for voters in general—candidates are “too similar,” there is “not enough choice” between

candidates, and “they are all the same.” Yet, on first pass, it is hard to understand why and how it

can hurt the median voter for candidates to target his/her location with increased precision; and it

is hard to understand how this could reduce the welfare of risk averse voters with other ideologies,

because such improved targeting is variance-reducing.

We take up this issue in the simplest possible framework, the stacked uniform model with two

possible signals. We first show that, while candidates extremize their location relative to their

private information, they do not do so by as much as voters would like. The logic behind this

is that even if the candidates’ signals correctly identify the initial location of the median voter,

the median voter’s final position may become more extreme in the interim. Because the winning

platform is the one ultimately preferred by the median voter, increased dispersion of the candidates’

platforms can therefore be welfare-enhancing. In other words, candidates do not provide voters as

much choice as they would like. We then show that greater signal correlation always reduces

voter welfare: It decreases the dispersion between the candidates’ platforms, and it reduces the

probability that candidates receive different signals and, hence, choose different locations. Finally,

we show that the effect of signal precision on welfare is, indeed, not monotonic: Increased polling

accuracy increases the probability that the signals of both candidates correctly identify the median

voter’s initial location, and, hence, it increases the probability that the candidates locate at the

same point, again reducing the probability that the candidates give voters a choice.

A number of other papers have independently considered some aspects of elections with privately

informed candidates. Chan (2001) develops a related three-signal model, assumes a pure strategy

5

equilibrium exists, and then provides partial characterizations and welfare analysis. Ottoviani

and Sorenson (2002) consider a model of financial analysts who receive private signals of a firm’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. In contrast to our paper, they assume a normally distributed

median, they assume a pure strategy equilibrium exists, and they offer a numerical analysis based

on necessary first order conditions. Heidhues and Lagerlof (2001)

Martinelli (2001,2002) ...

2 The Model

Let two political candidates, A and B, simultaneously choose policy platforms, x and y, on the real

line, �. For simplicity, we model the electorate as a cut point, µ, that determines which candidate

wins: Candidate A wins the election if |x − µ| < |y − µ| and loses if the inequality is reversed; if

|x − µ| = |y − µ|, then the election is decided by a fair coin toss, so that A wins with probability

one half. Assuming symmetric utilities, this formulation captures representative voter models and,

as long as a median is uniquely defined, models with multiple voters. In particular, the median

is uniquely defined if there is an odd number of voters, or there is a continuum of voters with

ideal points distributed according to a density with convex support. In such models, policy z is

majority-preferred to w if and only if z is preferred by the median voter.

Candidates do not observe the location of µ, but receive private signals, s and t, of its true

location. Let S denote the finite set of possible signals for A, and let T denote the finite set of

possible signals for B. Candidates have a common prior distribution on � × S × T , where the

distribution of µ conditional on signals s and t is Fs,t, and the marginal probability of signal pair

(s, t) is P (s, t). We denote the marginal probabilities of signal s by P (s) and signal t by P (t),

where we assume that P (s) > 0 and P (t) > 0. Conditional probabilities P (·|s) and P (·|t) are

defined using Bayes rule. The model is completely general with respect to the correlation between

candidates’ signals, allowing for conditionally-independent signals and perfectly-correlated signals

as special cases.

We assume the following regularity conditions hold on the conditional distributions: (i) for all

s ∈ S and all t ∈ T , Fs,t is continuous; and (ii) for all a, b, c ∈ � with a < b < c, 0 < Fs,t(a) and

Fs,t(c) < 1 implies that Fs,t(a) < Fs,t(b) < Fs,t(c). As a result, Fs,t admits a density, fs,t, with

convex support. Let ms,t be the uniquely-defined median of Fs,t. The distribution of µ conditional

6

on s ∈ S and T ′ ⊆ T is

Fs,T ′(x) =Xt∈T ′

P (t|s)P (T ′|s)

Fs,t(x),

when P (T ′|s) > 0, and the unique median of Fs,T ′ is ms,T ′ . We write Fs for Fs,T , the distribution of µ

conditional only on s, and let ms represent the associated median. We define Ft and mt analogously.

We say signal realizations s and s′ are conditionally equivalent if, for all t, Fs,t = Fs′,t, and similarly

for signals t and t′. Conditional equivalence does not restrict the posteriors on T conditional on s

and s′, but Fs,T ′ = Fs′,T ′ nevertheless holds for all T ′ ⊆ T if s and s′ are conditionally equivalent.

Given platforms x and y, the probability that A wins conditional on s and t, is

Fs,t

�x + y

2

�if x < y and 1 − Fs,t

�x + y

2

�if y < x,

and is 12 if x = y. The probability that B wins is defined symmetrically, and is equal to one minus

the probability that A wins. This defines a Bayesian game between the candidates, where pure

strategies for the candidates are vectors X = (xs) and Y = (yt), and a candidate’s payoff is the

probability of winning the election. Given pure strategies X and Y , candidate A’s interim expected

payoff conditional on signal s is

ΠA(X,Y |s) =X

t∈T :xs<yt

P (t|s)Fs,t�xs + yt

2

�+

Xt∈T :yt<xs

P (t|s)�

1 − Fs,t

�xs + yt2

��+

12

Xt∈T :xs=yt

P (t|s).

Candidate B’s interim expected payoff is defined symmetrically. A pure strategy Bayesian equilib-

rium is a strategy pair (X,Y ) such that

ΠA(X,Y |s) ≥ ΠA(X ′, Y |s),

for all signals s ∈ S and all strategies X ′; and

ΠB(X,Y |t) ≥ ΠB(X,Y ′|t),

for all signals t ∈ T and all strategies Y ′. This formalizes the idea that the candidates’ campaign

platforms are optimal given all information available to them.

At times, we impose additional conditions. Conditions (C1)–(C4) define our “canonical model”

of polling, in which candidates employ identical polling technologies and signals exhibit a natural

ordering structure. We first 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.

7

(C2) For all signals i, j ∈ I, P (i, j) = P (j, i) and Fi,j = Fj,i.

Condition (C2) implies that signal i of candidate A can be identified with signal i of candidate B

in the sense that they are equally informative. While our 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 interested in equilibria in

which candidates use information similarly: A symmetric pure strategy Bayesian equilibrium is an

equilibrium (X,Y ) such that xi = yi for all i ∈ I.

The next condition says that if one candidate receives signal i, then it must also be possible for

the other candidate to receive it.

(C3) For all signals i ∈ I, P (i, i) > 0.

The last condition defining the canonical model imposes a natural ordering structure on signals.

We say an ordering ≺ of I preserves conditional equivalence if for all i, j ∈ I, i �= j if and only if i

and j are not conditionally equivalent.

(C4) There exists an ordering ≺ of I such that for all signals i, j ∈ I and all K ⊆ I with P (K|i) > 0

and P (K|j) > 0, i ≺ j if and only if mi,K < mj,K.

Condition (C4) is natural if “higher” signals are correlated with higher values of µ. An implication

is that if i ≺ j, then the conditional medians satisfy

mi,{i} < mj,{i} = mi,{j} < mj,{j},

so that i ≺ j implies mi,i < mj,j. Under (C4), given signal i, denote the next signal realization

according to ≺ by i + 1.

Define candidate A’s ex ante expected payoff, his expected payoff before receiving a signal, as

ΠA(X,Y ) =Xs∈S

P (s)ΠA(X,Y |s).

Candidate B’s ex ante expected payoff, ΠB(X,Y ), is defined similarly. Clearly,

ΠA(X,Y ) + ΠB(X,Y ) = 1,

for all strategies X and Y . Note that (X,Y ) is a pure strategy Bayesian equilibrium if and only if

ΠA(X,Y ′) ≥ ΠA(X,Y ) ≥ ΠA(X ′, Y ),

8

for all X ′ and all Y ′. Under (C1)-(C2), candidates’ ex ante payoffs are symmetric in the sense that

ΠA(X,Y ) = ΠB(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. A consequence of the constant sum property is

the “interchangeability” of equilibria: If (X,Y ) and (X ′, Y ′) are equilibria, then so are (X,Y ′) and

(X ′, Y ). With symmetry, if (X,Y ) is an equilibrium, then so is (Y,X).

3 Examples

Symmetric-Information Model. In the framework of the canonical model, symmetric information

holds if for all i, j ∈ I with P (i, j) > 0, we have

P (i, k)Fi,k = P (j, k)Fj,k,

for all k ∈ I. Thus, 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. Under symmetric information, I

can be partitioned into equivalence classes

I(i) = {j ∈ I | P (i, j) > 0},

with the property that for all j, k ∈ I(i), P (i, k) = P (j, k) and Fi,k = Fj,k, so that signals i and

j are conditionally equivalent. Thus, following consistent signal realizations, the distribution of

µ is common knowledge. Moreover, for all j ∈ I(i), P (i, j) = 1|I(i)|2 and P (j|i) = 1

|I(i)| . Clearly,

symmetric information holds if information is complete, in which case P (i|i) = 1, for all i ∈ I.

The electoral game with set I(i) of signal realizations can be analyzed independently. That

is, (X,Y ) is a pure strategy Bayesian equilibrium if and only if for each equivalence class I, the

restricted strategies (XI , YI) = (xi, yj)i,j∈I′ are an equilibrium of the restricted game with payoffs

ΠA(XI , YI |i) =|{j ∈ I | xi < yj}|

|I | F

�xi + yj

2

�+|{j ∈ I | yj < xi}|

|I|

�1 − F

�xi + yj

2

��+|{j ∈ I | xi = yj}|

2|I | ,

for all i ∈ I, and likewise for candidate B.6 We call this the component game of I . Note that

symmetric information essentially implies condition (C4). To see this, let F � be the distribution

corresponding to component game , and let m� be the associated median. As long as the medians of

the component games are distinct, the elements of I can be ordered so that lower signal realizations

correspond to component games with lower medians.6In fact, because the posteriors above are independent of i, every pure strategy Bayesian equilibrium of the

restricted game is a mixed strategy equilibrium of a complete information game between the candidates.

9

Downsian and Probabilistic Voting Models.

[ to be filled in. ]

Stacked-Uniform Model. In the framework of the canonical model, let µ = α + β, where α is

uniformly distributed on [−a, a] and β is an independently-distributed discrete random variable

with support on b1 < b2 < · · · < bN . Let Q(bk) denote the probability of bk. Candidates share the

same set of signals. Signals depend stochastically on the realization of β:

P (i, j) =NXk=1

Q(i, j|bk)Q(bk).

where Q(i, j|bk) is the probability conditional on bk that candidates receive signals i and j. Given

bk, µ is uniformly distributed on [bk − a, bk + a], with piece-wise linear distribution

Fk(z) = max�

min�

0,z − bk + a

2a

�, 1�

.

By Bayes rule, the probability of bk conditional on signals i and j is

Q(bk|i, j) =Q(i, j|bk)Q(bk)

P (i, j)

when well-defined, and the distribution of µ conditional on signals i and j is

Fi,j(z) =NXk=1

Q(bk|i, j)Fk(z).

Since condition (C3) holds, for all i ∈ I, there exists a bk such that Q(i, i|bk)Q(bk) > 0. Condition

(C2) is satisfied if Q(i, j|bk) = Q(j, i|bk) for all i, j, and k.

This model describes a situation with a finite number of preliminary locations of the median

voter (finiteness is unimportant), candidate polling generates a finite number of possible signals

about these preliminary locations, and subsequent to polling there is a uniform disturbance to the

location of the median voter.

A nice special case features three possible preliminary locations of the median voter, b1 = −b,

b2 = 0, b3 = b > 0, and three possible signals, I = {−1, 0, 1}. If a > b, then conditional densities

are “single-plateaued” as in Figure 1, with density over the center range equal to a2 .

[ Figure 1 here. ]

If b < a, then the distribution Fi,j is as in Figure 2.

[ Figure 2 here. ]

10

Thus, the posterior distribution of µ has a “stacked” uniform distribution shape with shifted sup-

ports, where the shape of this distribution depends on the sign of a − b. If a > b, then a useful

expression for the conditional distribution of µ,

Fi,j(z) = Q(−b|i, j)�

z + b + a

2a

�+ Q(0|i, j)

�z + a

2a

�+ Q(b|i, j)

�z + a− b

2a

�

=12

+z + [Q(−b|i, j) −Q(1|i, j)]b

2a,

obtains for z ∈ [b − a, a − b]. In particular, if mi,j ∈ [b − a, a − b], then solving for Fi,j(mi,j) = 12

yields the conditional median:

mi,j = [Q(b|i, j) −Q(−b|i, j)]b. (1)

For general numbers of preliminary cut-point locations and signal realizations, our equilibrium

characterizations are sharpest when a is large, specifically a ≥ bN − b1. Then bN − a ≤ b1 + a, so

that conditional densities are single-plateaued, as in Figure 1. Moreover, a ≥ bn − b1 implies that

the conditional medians lie in the center interval: For all i ∈ I, bN − a ≤ mi,i ≤ b1 + a. To see this,

note that Q(b1|i, i) = 1 yields a lower bound for mi,i. In that case, Fi,i is uniformly distributed with

median b1 ≥ bN − a. Therefore, bN − a ≤ mi,i. Similarly, Q(bn|i, 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 a consequence,

Fi,j(z) =NXk=1

Q(k|i, j)�

z + a− bk2a

�=

12

+z −PN

k=1 Q(bk|i, j)bk2a

,

for all z ∈ [bN − a, b1 + a]. Therefore, when a ≥ bN − b1,

mi,j =NXk=1

Q(bk|i, j)bk . (2)

Equation (2) generalizes the three-signal expression for the conditional median given in equation

(1), and illuminates the joint restriction on Q and the bk’s imposed by (C4). Substituting for mi,j

into Fi,j using equation (2) yields

Fi,j(z) =a −mi,j + z

2a,

for all z ∈ [bN − a, b1 + a]. Hence,

Fi,j(z) − Fi,j(w) =z −w

2a,

for all z,w ∈ [bN − a, b1 + a]. Note that, assuming (C4), mh,i < mj,k implies that Fh,i(z) > Fj,k(z)

for all z ∈ [bN − a, b1 + a]; i.e., Fj,k stochastically dominates Fh,i over the interval.

11

Shape-Invariant Model. In this model, the shape of the distribution of µ 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,l(z + mk,l),

for all z ∈ � and all i, j, k, l ∈ 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

The main result of this section is a full characterization of the pure strategy Bayesian 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 µ conditional on both

candidates receiving that signal. Given a natural restriction on conditional medians, a corollary is

that candidates take policy positions that are extreme relative to their expectations of µ given their

own information. In other words, a candidate’s location is a biased estimator of µ: Candidates who

receive high signals overshoot µ, while those who receive low signals undershoot. We also provide

sufficient conditions for the existence of a pure strategy equilibrium.

The characterization result relies on the following lemma, which is proved in the appendix in a

somewhat more general form.

Lemma 3 In the canonical model, let (X,Y ) be a pure strategy Bayesian equilibrium. If xi = yj

for some i, j ∈ 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 payoff of

either candidate would be increased by a small move toward that conditional median: If A deviates

in this way, then his expected payoff given other signal realizations for B varies continuously with his

location, but his payoff given realization j would jump discontinuously above one half. Therefore,

a sufficiently small deviation would raise A’s payoff, something that is impossible in equilibrium.

12

Theorem 1 (Necessity) In the canonical model, if (X,Y ) is a pure strategy Bayesian equilibrium,

then xi = yi = mi,i for all i ∈ I.

Proof: First, consider a symmetric equilibrium (X,Y ), where xi = yi for all i ∈ I. By (C4) and

Lemma 3, xi = yi = mi,i, as required. Now suppose there is an asymmetric equilibrium (X,Y ),

where xi �= mi,i for some i ∈ I, and define the strategy Y ′ = X for candidate B. Then, by

symmetry and interchangeability, (X,Y ′) is a symmetric Bayesian equilibrium with xi �= mi,i, a

contradiction of lemma 3.

If symmetric information holds in the canonical model, then mi = mi,i. To see this, note that

Fi(mi) =Xj∈I

P (j|i)Fi,j(mi) =12,

and that for all j with positive probability, P (j|i) = 1I(i) and Fi,j(mi) = Fi,i(mi,i). Therefore,

Xj∈I

P (j|i)Fi,j(mi) =Xj∈I(i)

1I(i)

Fi,j(mi) =Xj∈I(i)

1I(i)

Fi,i(mi) = Fi,i(mi),

establishing the claim. Thus, Theorem 1 has the following familiar implication.

Corollary 1 In the canonical model with symmetric information, if (X,Y ) is a pure strategy

Bayesian equilibrium, then xi = yi = mi for all i ∈ I.

Theorem 1 and Corollary 1 give a necessary, not a sufficient, condition for the existence of

a pure strategy equilibrium. It is clear—and will follow from later results—that, with symmetric

information, the strategies specified in Corollary 1 do indeed form an equilibrium. The next example

shows, however, that pure strategy equilibria do not exist generally. The example begins with an

arbitrary symmetric-information model and demonstrates arbitrarily close models with no pure

strategy equilibria, illustrating the fragility of the pure strategy equilibria in Corollary 1. We

return to the issue of robustness of equilibria in our analysis of mixed strategy equilibria. An

implication of results there is that even if pure strategy equilibria cease to exist in models close to

symmetric information, mixed strategy equilibria do exist and are necessarily “close” to the pure

strategy equilibrium of the original model.

Example: Non-existence of Pure Strategy Equilibrium. In the canonical model, let I = {−1, 0, 1},

let P (i, j) = 19 for all i, j ∈ I, and let Fi,j = F for all i, j ∈ I. Thus, symmetric information holds,

and the unique pure strategy Bayesian equilibrium is (X,Y ) such that xi = yj = m for all i, j ∈ I,

where m is the median of F . Now let {F k} and {F k} be sequences of distributions converging

13

weakly to F defined by F k(z) = F (z + 1k ) and F k(z) = F (z − 1

k ) for all z ∈ �. Denoting the

medians of these distributions by mk and mk, we have mk = m − 1k and mk = m + 1

k for all k.

Now perturb the original specification of the game so that F−1,−1 = F k and F1,1 = F k, and the

conditional distribution following other signal combinations remains F .

By Theorem 1, if a pure strategy equilibrium exists, then it is (X,Y ) defined by x−1 = y−1 = mk,

x0 = y0 = m, and x1 = y1 = mk. But, (X,Y ) is not an equilibrium to the perturbed game because

A can deviate profitably to strategy X ′ defined as X, but with x′−1 = m: In the perturbed game,

ΠA(X,Y | − 1) =13

�12

+ F (mk + m

2) + F (

mk + mk

2)�

=13

�12

+ F (m− 12k

) +12

�<

12,

but

ΠA(X ′, Y | − 1) =13

�(1 − F k(

mk + m

2)) +

12

+ F (m + mk

2)�

=13

�1 − F (m +

12k

) +12

+ F (m +12k

)�

=12.

Thus, the deviation to X ′ increases candidate A’s expected payoff. It follows that there does not

exist a pure strategy Bayesian equilibrium for games arbitrarily close to the original game.

In many situations, it is reasonable to suppose that lower signals indicate lower values of µ

and higher signals indicate higher ones. Then Theorem 1 implies that polling leads candidates to

extremize their locations.

Corollary 2 In the canonical model, suppose there exists a signal n ∈ I such that i < n implies

mi,i < mi and n < i implies mi < mi,i. If (X,Y ) is a pure strategy Bayesian equilibrium, then

xi < mi for i < n and mi < xi for i > n.

This result is sufficiently important to highlight. If n corresponds to an uninformative signal

in the sense that the conditional median mn,n is equal to the unconditional median, then it follows

that in a pure strategy equilibrium, private polling magnifies 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.

14

We next provide conditions that ensure the pure strategy equilibrium characterized in Theorem

1 exists. We consider two environments. In 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 (Sufficiency for Binary Signals) In the canonical model, let I = {−1, 1}. Then asufficient condition for the existence of the unique pure strategy Bayesian equilibrium is that

P (1|1)f1,1

�z + m1,1

2

�≥ P (−1|1)f1,−1

�z + m−1,−1

2

�

P (−1| − 1)f−1,−1

�z + m−1,−1

2

�≥ P (1| − 1)f1,−1

�z + m1,1

2

�,

for all z ∈ [m−1,−1,m1,1]. In that equilibrium, candidates locate at mi,i following signal i ∈ 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 x ∈ [m−1,−1,m1,1],

then the change in A’s interim expected payoff is

P (−1|1)�F1,−1

�m1,1 + m−1,−1

2

�− F1,−1

�x + m−1,−1

2

��+ P (1|1)

�F1,1

�x + m1,1

2

�− 1

2

�

=Z m1,1

x

�P (−1|1)f1,−1

�z + m−1,−1

2

�− P (1|1)f1,1

�z + m1,1

2

��dz ≤ 0.

Thus, the deviation does not increase A’s expected payoff. It is easily verified that deviations

x < m−1,−1 and x > m1,1 are also unprofitable. A similar argument holds for signal i = −1, and a

symmetric argument for candidate B establishes that (X,Y ) is an equilibrium.

The sufficient condition detailed in Theorem 2 is weak. If signals are not negatively correlated, so

that P (1|1) ≥ P (−1|1) and P (−1| − 1) ≥ P (1| − 1), then a pure strategy equilibrium exists if

f1,1

�z + m1,1

2

�≥ f1,−1

�z + m−1,−1

2

�, for all z ∈ [m−1,−1,m1,1], (3)

f−1,−1

�z + m−1,−1

2

�≥ f1,−1

�z + m1,1

2

�, for all z ∈ [m−1,−1,m1,1]. (4)

Inequality (3) compares f1,1, shifted to the left by m1,1

2 , with f1,−1 shifted to the left by m−1,−1

2 . To

make the nature of the comparison clearer, rewrite inequality (3) as:

f1,1

�m1,1 −m−1,−1

2+ z

�≥ f1,−1(z), for all z ∈ [

3m−1,−1 −m1,1

2,m1,1 + m−1,−1

2]. (5)

15

Thus, the sufficient condition is that f1,1 shifted to the left by m1,1−m−1,−1

2 > 0 weakly exceed f1,−1

over this range. This sufficient condition holds for the stacked-uniform model with a > bN − b1, as

conditional densities equal 12a over the relevant range. The condition holds in the shape-invariant

model when f1,−1 is the translation of f1,1 with median m1,1+m−1,−1

2 . More generally, the condition

holds if f1,−1 has its median near m1,1+m−1,−1

2 and is somewhat more dispersed than f1,1, as one

would expect if identical signals decrease the variance of the distribution of µ; and opposing signals,

which offset each other, lead to a higher variance.

To simplify our sufficiency argument for existence in the multi-signal case, we provide separate

conditions on (i) the distribution of µ conditional on signal realizations, and (ii) the priors over

signal pairs. Condition (C5) is a regularity condition on the conditional distributions that reinforces

the symmetry present in the canonical model.

(C5) For all signals i, j ∈ I with P (i, j) > 0, mi,j = mi,i+mj,j

2 .

Condition (C5) is trivially satisfied under symmetric information. It is satisfied in the stacked-

uniform model when a ≥ bN − b1, if

Q(bk|i, j) =Q(bk|i, i) + Q(bk|j, j)

2,

or equivalently if

Q(i, j|bk)P (i, j)

=12

�Q(i, i|bk)P (i, i)

+Q(j, j|bk)P (j, j)

�,

for each bk. We also impose a stochastic dominance-like restriction on the conditional distributions.

(C6) For all i, j, k ∈ I with i ≤ j ≤ k, we have mi,i ≤ mj,j ≤ mk,k, and

Fi,j(mi,i + z

2) ≥ Fj,k(

mk,k + z

2), for all z ∈ [mi,i,mk,k]. (6)

The meaning of inequality (6) is more transparent if it is rewritten as:

Fi,j(z) ≥ Fj,k

�mk,k −mi,i

2+ z

�, for all

�mi,i

2,mi,i + mk,k

2

�.

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 satisfied under symmetric information,

and at the end of this section we will show that it is satisfied in the stacked-uniform model when

the disturbance term has a sufficiently large support.

16

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 sufficiently

high. In fact, the condition is weaker than that, because it only restricts “net” probabilities.

(C7) For all i ∈ I,Xj∈I:j≤i

P (j|i) ≥X

j∈I:j>iP (j|i) and

Xj∈I:j<i

P (j|i) ≤X

j∈I:j≥iP (j|i).

An equivalent statement of (C7) is thatPj∈I:j≤i P (j|i) ≥ 1

2 andPj∈I:i≤j P (j|i) ≥ 1

2 . In words, for

any signal i, it must be that i is a “median” of the distribution P (·|i) on I. A stronger condition

is that P (i|i) ≥ 12 for all i ∈ I. Clearly, (C7) is most restrictive for the “extremal” signals, for

which P (i|i) ≥ 12 is necessary, and its restrictiveness depends on the number of possible signals. In

the binary signal model, for example, it is satisfied whenever signals are not negatively correlated.

Note also that (C7) is trivially satisfied under symmetric information, because given any i ∈ I, the

only signals j such that P (j|i) > 0 are conditionally equivalent to i.

Theorem 3 (Sufficiency for Multiple Signals) In the canonical model, conditions (C5)-(C7)

are sufficient for the existence of the unique pure strategy Bayesian equilibrium. In that equilibrium

candidates locate at mi,i following signal i ∈ I.

Proof: We show that (X,Y ) is an equilibrium, where xi = yi = mi,i for all i ∈ I. Without loss

of generality, we focus on candidate B’s best response problem after receiving signal j. Consider a

deviation to strategy Y ′. There are two cases: y′j < mj,j and mj,j < y′j. In the first case, define

G = {i ∈ I : mi,i ≤ y′j} and L = {k ∈ I : mj,j ≤ mk,k}.

Note that for all i ∈ I \ (G ∪ L) such that P (i|j) > 0, we have y′j < mi,i < mj,j. Hence, for such i,

Fi,j

�y′j + mi,i

2

�−�1 − Fi,j

�mi,i + mj,j

2

��≤ 0,

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

�y′j+mi,i

2

�≤ 1

2 and Fi,j

�mi,i+mj,j

2

�= 1

2 . Intuitively, B’s

gains from deviating when A receives signal si with i ∈ I \ (G ∪L) are non-positive. Therefore, the

change in B’s interim expected payoff satisfies

ΠB(X,Y ′|j) − ΠB(X,Y |j) ≤Xi∈G

P (i|j)�1 − Fi,j

�mi,i + y′j

2

�−�

1 − Fi,j

�mi,i + mj,j

2

���

+Xk∈L

P (k|j)�Fj,k

�y′j + mk,k

2

�− Fj,k

�mj,j + mk,k

2

��

=Xi∈G

P (i|j)�

12− Fi,j

�mi,i + y′j

2

��+Xk∈L

P (k|j)�Fj,k

�y′j + mk,k

2

�− 1

2

�,

17

which is non-positive as long asXi∈G

P (i|j)�

12− Fi,j

�mi,i + y′j

2

��≤Xk∈L

P (k|j)�

12− Fj,k

�y′j + mk,k

2

��. (7)

Let i∗ minimize Fi,j

�mi,i+y′j

2

�over G, and let k∗ maximize Fj,k

�mi,i+y′j

2

�over L. Then inequality

(7) holds if�12− Fi∗,j

�mi∗,i∗ + y′j

2

��Xi∈G

P (i|j) ≤�

12− Fj,k∗

�y′j + mk∗,k∗

2

��Xk∈L

P (k|j),

which follows from (C5)-(C7). Thus, it is unprofitable for B to deviate to y′j < mj,j. A symmetric

argument applies for mj,j < y′j.

The following corollary for symmetric-information models is immediate.

Corollary 3 In the canonical model with symmetric information, the strategy pair in which can-

didates locate at mi following signal i ∈ I is the unique pure strategy Bayesian equilibrium.

We have shown that condition (C7) on the signals’ conditional correlation, together with reg-

ularity conditions (C5) and (C6), ensure the existence of a pure strategy equilibrium. We now

show that (C7), under (C5) and a stronger version of (C6), are necessary for equilibrium existence.

Condition (C6′) strengthens (C6) by stating it with equality.

(C6′) For all i, j, k ∈ I with i ≤ j ≤ k,

Fi,j

�mi,i + z

2

�= Fj,k

�mk,k + z

2

�, for all z ∈ [mi,i,mk,k].

It is immediate that under (C5), condition (C6′) is satisfied in the shape-invariant model. To

see this, take any signals i, j, and note that by (C5) we havemi,i + z

2= mi,j +

z −mj,j

2and

mk,k + z

2= mj,k +

z −mj,j

2.

Thus,

Fi,j

�mi,i + z

2

�= Fi,j

�mi,j +

z −mj,j

2

�= Fj,k

�mj,k +

z −mj,j

2

�= Fj,k

�mk,k + z

2

�,

where the second equality uses shape-invariance. Under (C5), condition (C6′) also holds in the

stacked-uniform model when a sufficiently high, i.e., a ≥ bN − b1: To see this take i, j, k as in (C6′)

and z ∈ [mi,i,mk,k]. Then, because Fi,j is linear with slope a2 over [mi,i,mk,k], we have

Fi,j

�mi,i + z

2

�= Fi,j

�mi,j +

z −mj,j

2

�=

12

+z −mj,j

4a,

and similarly for Fj,k

�mk,k+z

2

�.

18

Theorem 4 In the canonical model, given conditions (C5) and (C6′), condition (C7) is necessary

and sufficient for the existence of the pure strategy Bayesian equilibrium. In that equilibrium,

candidates locate at mi,i, following signal i ∈ I.

5 Mixed Strategies

Our results for pure strategy equilibria suggest that if there are many signals then pure strategy

equilibria may not 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 use the shorthand Gs(z)− or Ht(z)− for the left-hand limits of these

distributions: Gs(z)− = limw↑z Gs(w). Accordingly, Gs has a mass point at x if Gs(x)−Gs(x)− > 0.

To extend our definition of interim expected payoffs, define the probability that A wins using

platform x following signal s when B uses platform y following signal t, as

πA(x, y|s, t) =

8<:

Fs,t�x+y

2

�if x < y

1 − Fs,t�x+y

2

�if y < x

12 if x = y,

and let πB(·|s, t) = 1 − πA(·|s, t). Then, given mixed strategies (G,H), candidate A’s interim

expected payoff conditional on signal s is

ΠA(G,H|s) =Xt∈T

P (t|s)Z

πA(x, y|s, t)Gs(dx)Ht(dy),

and B’s interim payoff ΠB(G,H|t) is defined analogously. Abusing notation slightly, let ΠA(X,H|s)

be A’s expected payoff from the degenerate mixed strategy with Gs(xs)−Gs(xs)− = 1 for all s ∈ S.

ΠB(G,Y |t) is the analogous expected payoff for B. A mixed strategy Bayesian equilibrium is a

strategy pair (G,H) such that

ΠA(G,H|s) ≥ ΠA(G′,H|s),

for all signals s ∈ S and all strategies G′, and

ΠB(G,H|t) ≥ ΠB(G,H ′|t),

for all signals t ∈ T and all strategies H ′. Note that, in equilibrium, if xs is a continuity point of

ΠA(X,H|s) in the support of Gs, then the expected payoff from xs must be ΠA(G,H|s). Candidate

A must therefore be indifferent over all such points.

As with pure strategies, we can define ex ante expected payoffs as

ΠA(G,H) =Xs∈S

P (s)ΠA(G,H|s) and ΠB(G,H) =Xt∈T

P (t)ΠB(G,H|t).

19

Thus, mixed strategy Bayesian equilibria of the electoral game are equilibria of a two-player,

constant-sum game. In the canonical model, the game is symmetric and we define a symmetric

mixed strategy Bayesian equilibrium as an equilibrium pair (G,H) of strategies such that G = H.

The next theorem establishes for the general model the existence of a mixed strategy equilibria

in which candidates use mixed strategies with support bounded as follows. Let m = max{ms,t :

s ∈ S, t ∈ T} and m = min{ms,t : s ∈ S, t ∈ T}. The interval defined by the 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 ∈ S, Gs(m)−Gs(m)− = 1; and for all t ∈ T , Ht(m)−Ht(m)− = 1.

Theorem 5 There exists a mixed strategy Bayesian equilibrium with support in M . Under (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 |S|+ |T |-player

game in which each type (corresponding to different signal realizations) of each candidate is a

separate player. Player s (or t) has strategy space M ⊆ �, a compact set, with pure strategies xs

(or yt). Then (X,Y ) = (xs, yt)s∈S,t∈T is a pure strategy profile, one for each type. Let (X−s, Y )

denote the result of deleting xs from (X,Y ). The payoff function of player s ∈ S is

Us(X,Y ) = P (s)ΠA(X,Y |s),

and the payoff function of player t ∈ T is

Ut(X,Y ) = P (t)ΠB(X,Y |t).

The space of mixed strategies for each player type s (or t) is M, the Borel probability measures

on M , with mixed strategies denoted Gs (or Ht). Then (G,H) is a mixed strategy profile, one for

each type. Note thatXs∈S

Us(X,Y ) +Xt∈T

Ut(X,Y ) = 1,

for all X and Y , so that the total payoff is trivially upper semi-continuous. Furthermore, payoffs

are between zero and one, so they are bounded. Note that Us is discontinuous at (X,Y ) only if

xs = yt for some t ∈ T . Therefore, the discontinuity points of Us lie in a set that can be written

as A∗(s), as in Dasgupta and Maskin’s equation (2). The discontinuity points of Ut lie in a similar

set. It remains to show that Us (likewise Ut) is weakly lower semi-continuous in xs; that is, for all

xs ∈ M , there exists a λ ∈ [0, 1] such that for all (X−s, Y ),

Us(X,Y ) ≤ λ lim infz↓xs

Us(z,X−s, Y ) + (1 − λ) lim infz↑xs

Us(z,X−s, Y ).

20

It is straightforward to verify that this condition holds with equality for λ = 12 . Let T+ = {t ∈ T :

xs < yt}, let T− = {t ∈ T : yt < xs}, and let T 0 = {t ∈ T : xs = yt}. Since

Us(X,Y ) =Xt∈T−

P (s, t)�

1 − Fs,t

�xs + yt

2

��+Xt∈T 0

P (s, t)2

+Xt∈T+

P (s, t)Fs,t

�xs + yt

2

�,

it follows that

lim infz↑xs

Us(z,X−s, Y ) =Xt∈T−

P (s, t)�

1 − Fs,t

�xs + yt

2

��+

Xt∈T 0∪T+

P (s, t)Fs,t

�xs + yt

2

�

and

lim infz↓xs

Us(z,X−s, Y ) =X

t∈T−∪T 0

P (s, t)�

1 − Fs,t

�xs + yt

2

��+Xt∈T+

P (s, t)Fs,t

�xs + yt

2

�.

The claim

Us(X,Y ) =12

lim infz↓xs

Us(z,X−s, Y ) +12

lim infz↑xs

Us(z,X−s, Y )

then follows immediately. The same argument can be used to verify that Ut is weakly lower semi-

continuous. Then by Dasgupta and Maskin’s (1986) Theorem 5, there exists a mixed strategy

equilibrium of the multi-player game, and therefore of the electoral game when strategies are

restricted to M . To see that following a signal s, candidate A has no profitable deviations outside

M , take any x > m. Note that for all t ∈ T and all y ∈ M , we have πA(m, y|s, t) ≥ πA(x, y|s, t).Let G′ be any deviation such that G′

s puts probability one on xs > m, and let G′′ put probability

one on m instead. Then

ΠA(G′,H|s) ≤ ΠA(G′′,H|s) ≤ ΠA(G,H|s).

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.

We now study the continuity properties of the mixed strategy equilibrium correspondence as

we vary the parameters of the model, specifically the candidates’ marginal prior on S × T and

the conditional distributions of µ. To state our continuity result, we index these parameters by

a metric space Γ, with typical element γ representing a model specification. Let the marginal

probability of (s, t) in game γ be P γ(s, t), and let the distribution of µ conditional on s and t in

γ be F γs,t. We assume that this indexing is continuous: For each s ∈ S and t ∈ T , if γn → γ,

then P γn(s, t) → P γ(s, t) and F γns,t → F γs,t weakly. Let M(γ) denote the interval defined by the

21

extreme conditional medians in game γ, and note that by the assumption of continuous indexing,

the correspondence M : Γ ⇒ � so-defined is continuous.

Theorem 5 establishes the existence of a mixed strategy equilibrium for all γ ∈ Γ. Therefore,

since the electoral game is constant-sum, the ex ante expected payoff of a candidate in game γ is

the same in all mixed strategy equilibria. Denote these payoffs or “values” by vA(γ) and vB(γ).

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 γ, then the

game is symmetric, so that vA(γ) = vB(γ) = 12 . The next theorem establishes that vA(γ) and vB(γ)

vary continuously in the parameters of the game, even when asymmetries are allowed.

Theorem 6 The mapping vA : Γ → � is continuous.

Proof: We prove lower semi-continuity of vA. A symmetric argument proves lower semi-

continuity of vB = 1 − vA, which, in turn, gives us upper semi-continuity of vA. Let γn → γ, and

suppose vA(γ) > lim inf vA(γn). Let ΠnA denote A’s ex ante expected payoff function corresponding

to γn, and let ΠA denote the ex ante payoffs corresponding to γ. Let Mn denote the interval

M(γn), let M = M(γ), and let M be any compact set containing M in its interior. By continuity,

therefore, Mn ⊆ M for high enough n. For each n, let (Gn,Hn) be an equilibrium with support

in Mn for the electoral game indexed by γn, so ΠnA(Gn,Hn) = vA(γn) and ΠnB(Gn,Hn) = vB(γn).

By compactness of M , there exists a weakly convergent subsequence of {(Gn,Hn)}, also indexed

by n, with limit (G,H). Going to a further subsequence if necessary, we may assume {vA(γn)}converges to limit v < vA(γ). Let (G∗,H∗) be an equilibrium of the electoral game indexed by γ,

so G∗ is an optimal strategy for A, which guarantees a payoff of at least vA(γ) in game γ. Thus,

ΠA(G∗,H) ≥ vA(γ). In particular, there exists a pure strategy X∗ such that

ΠA(X∗,H) ≥ vA(γ) > v.

We claim that, as a consequence, there exists a pure strategy X ′ such that

ΠnA(X ′,Hn) >ΠA(X∗,H) + v

2,

for high enough n. But this, with v(γn) → v, contradicts the assumption that Gn is a best response

to Hn for candidate A. We establish the claim in three steps.

Step 1. By Lemma 1 (in the appendix), for every s ∈ S, eitherXt∈T

P (t|s)[Ht(x∗s) −Ht(x∗

s)−][Fs,t(x∗

s)] ≥ 12

Xt∈T

P (t|s)[Ht(x∗s) −Ht(x∗

s)−] (8)

or Xt∈T

P (t|s)[Ht(x∗s) −Ht(x∗

s)−][1 − Fs,t(x∗

s)] ≥ 12

Xt∈T

P (t|s)[Ht(x∗s) −Ht(x∗

s)−]. (9)

22

Let S− be the set of s ∈ S such that (8) holds, and let S+ be the set of s ∈ S \ S− such that (9)

holds. For s ∈ S−, let {xks} be a sequence increasing to x∗s, and for s ∈ S+, let {xks} be a sequence

decreasing to x∗s. In addition, we choose each xks to be a continuity point of Ht for all t ∈ T ; this

is possible because T is finite and each Ht has a countable number of discontinuity points. Thus,

Ht(xks) −Ht(xks)− = 0 for all t ∈ T . For each k, define the strategy Xk = (xks) for candidate A.

Step 2. We now argue that Xk satisfies lim inf ΠA(Xk,H) ≥ ΠA(X∗,H). For each t ∈ T , let λt

denote the measure generated by the distribution Ht, let µt denote the degenerate measure with

probability Ht(x∗s) −Ht(x∗

s)− on each x∗s, and let νt = λt − µt. Let

π∗s,t(z) = πA(x∗

s, z|s, 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

πks,t(z) = πA(xks , z|s, t)

denote A’s analogous probability of winning using xks . Note that

ΠA(Xk,H) − ΠA(X∗,H) =Xs∈S

P (s)Xt∈T

P (t|s)Z

[πks,t(z) − π∗s,t(z)]λt(dz)

=Xs∈S−

Xt∈T

P (s, t)[Ht(x∗s) −Ht(x∗

s)−]�Fs,t

�x∗s + xks

2

�− 1

2

�

+Xs∈S+

Xt∈T

P (s, t)[Ht(x∗s) −Ht(x∗

s)−]�1 − Fs,t

�x∗s + xks

2

�− 1

2

�+Xs∈S

Xt∈T

P (s, t)Z

[πks,t(z) − π∗s,t(z)] νt(dz).

Since πks,t − π∗s,t → 0 almost everywhere (νt), the corresponding integral terms above converge to

zero. Thus, by construction of Xk,

lim infm→∞ΠA(Xk,H) ≥ ΠA(X∗,H) > v,

as desired.

Step 3. Choose m such that ΠA(Xk,H) > ΠA(X∗,H)+v2 and set X ′ = Xk. To prove the claim

that ΠnA(X ′,Hn) > v for high enough n, define the functions

φ′s,t(z) = πA(x′

s, z|s, t) and φns,t(z) = πγn

A (x′s, z|s, t).

Note that

ΠA(X ′,H) =Xs∈S

Xt∈T

P (s, t)Z

φ′s,t(z)Ht(dz),

23

and, letting Pn = P γn ,

ΠnA(X ′,Hn) =Xs∈S

Xt∈T

Pn(s, t)Z

φns,t(z)Hnt (dz).

Thus, it suffices to show that Zφns,t(z) dHn

t →Z

φ′s,t(z) dHt,

for each s ∈ S and t ∈ T . To prove this, fix ε > 0. Because x′s = xks is not a mass point of Ht, we may

specify an interval Z = [z, z] with x′s ∈ (z, z) such that Ht(z)−Ht(z)− < ε

4 . By weak convergence,

Hnt (z)−Hn

t (z)− < ε2 for sufficiently high n. Furthermore, {φns,t} is a sequence of functions that are

non-decreasing on [m, z] and converge pointwise to φ′s,t on this interval, so they converge uniformly

to φ′s,t on the interval. Similarly, each φns,t is non-increasing on [z,m], so the functions converge

uniformly to φ′s,t on this interval. Choosing n high enough that |φns,t(z) − φ′

s,t(z)| < ε2 for all

z ∈ [m, z] ∪ [z,m], yields ����Z

φns,t(z) dHnt −

Zφ′s,t(z) dHt

���� < ε,

as required.

Letting B denote the Borel probability measures over X, define the mixed strategy equilibrium

correspondence E : Γ ⇒ BS∪T so that E(γ) consists of all mixed strategy equilibrium pairs (G,H).

We have shown that this correspondence has non-empty values. The next result establishes an

important continuity property of the equilibrium correspondence.

Theorem 7 The correspondence E : Γ ⇒ BS∪T has closed graph.

Proof: Let γn → γ, let (Gn,Hn) ∈ E(γn) for each n, and suppose that (Gn,Hn) → (G,H). If

(G,H) /∈ E(γ), then, using the notation from the proof of Theorem 6, one candidate, say A, has a

pure strategy X such that

ΠA(X,H) > vA(γ). (10)

But then, as in the proof of Theorem 6, we can find a strategy X ′ satisfying (10) such that no x′s

is a mass point of any Ht, and then we can show that

ΠnA(X ′,Hn) >ΠA(X,H) + vA(γ)

2,

24

for high enough n. But vA(γn) → vA(γ) by Theorem 6, so it follows that

ΠnA(X ′,Hn) > vA(γn),

for high enough n, contradicting the assumption that Gn is a best response to Hn for A in the

electoral game indexed by γn.

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 equilibrium mixed

strategies. Our next result does just that for the canonical model, showing that all mixed strategy

equilibria have support in M .

Theorem 8 Under (C1)-(C3), 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 = sup{x ∈ � : Gi(x) = 0}be the lower bound of the support of Gi for each i ∈ I, and let x = mini∈I 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. Define the sequence

of pure strategies {Xn} as follows. If Gi puts positive probability on x, i.e., Gi(x) − Gi(x)− > 0,

then let xni = x for all n. Otherwise, let {xni } be a sequence decreasing to x such that each xni

is in the support of Gi. Furthermore, choose xni so that ΠA(Xn,H|i) = ΠA(G,H|i) for all n. To

see that this is possible, set x1i arbitrarily and note that we can let xni be any continuity point

of A’s expected payoff function in the support of Gi and in the interval [x, x+xn−1i

2 ] to satisfy the

desired condition. Since there is at most a countable number of discontinuity points of A’s payoff

function, such a point can be found unless the support of Gi in [x, x+xn−1i

2 ] is countable. In that

case, however, any point in the support of Gi in this interval satisfies the desired condition, and

there is at least one such point since xi = x. In any case, we have ΠA(Xn,H|i) = ΠA(G,H|i) for

all n and limn→∞ Gi(xni )− = 0. Now consider a pure strategy X ′ satisfying x′i = m, and note that

ΠA(X ′,H|i) − πA(Xn,H|i) =Xj∈I

P (j|i)"Z

[x,xni )

�Fi,j

�xni + z

2

�− Fi,j

�m + z

2

��Hj(dz)

+ (Hj(xni ) −Hj(xni )−)�

12− Fi,j

�xni + m

2

��+Z

(xni ,m)

�1 − Fi,j

�m + z

2

�− Fi,j

�xni + z

2

��Hj(dz)

+ (Hj(m) −Hj(m)−)�

12− Fi,j

�xni + m

2

��+Z

(m,∞)

�Fi,j

�m + z

2

�− Fi,j

�xni + z

2

��Hj(dz)

#,

25

for n such that xin < m. For each j ∈ I, the first integral goes to zero, because limn→∞ Hj(xni )− =

limn→∞ Gj(xni )− = 0. Further, the last integral is clearly non-negative. So, too, the other terms

are non-negative, because Fi,j�w+z

2

�< 1

2 for all w ≤ m and all z < m. This establishes that for all

j ∈ I, the expression in brackets is non-negative. It is strictly positive for j = i, because the second

third and fourth terms in square brackets have strictly positive limits; and the total probability

mass on these terms is strictly positive since

limn→∞Hj(m) −Hj(xni )− = Gi(m) −Gi(x)− > 0.

Because P (i|i) > 0 by (C3), we conclude that

ΠA(X ′,H|i) > πA(Xn,H|i) = πA(G,H|i)

for high enough n, contradicting the assumption that (G,H) is an equilibrium. An identical argu-

ment establishes that the supports of equilibrium strategies are bounded from above by m.

We can apply Theorem 8 to symmetric-information models to strengthen substantially our ex-

istence and uniqueness results. The next corollary follows from Theorem 8 by decomposing any

symmetric-information game into its component games. If F is the distribution corresponding to

one component, then M , defined for the component game, is just the singleton consisting of the

median of F . Theorem 8 establishes that the unique mixed strategy equilibrium of the component

game is the strategy pair such that each type chooses the median with probability one.

Corollary 4 In the canonical model with symmetric information, the strategy pair in which candi-

dates locate at mi with probability one following signal i ∈ I is the unique mixed strategy Bayesian

equilibrium.

Together with Theorem 7, Corollary 4 reveals that in games close to symmetric information,

mixed strategy equilibria must be close, in the sense of weak convergence, to the pure strategy

equilibrium. In particular, this applies to our example of non-existence of pure strategy equilibria:

Though pure strategy equilibria do not exist, it is the case that for every open set around m and

for high enough k, mixed strategy equilibria exist and all put probability arbitrarily close to one on

that open set. Thus, while pure strategy equilibria 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.

26

(C8) For all i, j, i′, j′ ∈ I, if i < i′ and j < j′, then

P (j|i′)P (j′|i) ≤ P (j|i)P (j′ |i′).

Provided that P (j|i) and P (j|i′) are positive, condition (C8) can be written more intuitively as

P (j′|i)P (j|i) ≤ P (j′|i′)

P (j|i′) ,

which gives the monotone likelyhood 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 µ 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 i, i′ ∈ I with i < i′, for all j ∈ I, and for all z ∈ M with 0 < Fi′,j(z) < 1, we have

Fi′,j(z) < Fi,j(z).

By continuity, condition (C9) implies that for signals i < i′, Fi′,j(z) ≤ Fi,j(z), for all z ∈ M .

Condition (C9) is implied by (C4) in the stacked-uniform model when a ≥ bN − b1: In that case,

i < i′ implies mi′,j > mi,j, so that

Fi′,j(z) =12

+z −mi′,j

2a<

12

+z −mi,j

2a= Fi,j(z).

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 ∈ I, let

αj ∈ [0, 1]. Then, for all i, i′ ∈ I with i < i′ and for all z ∈ M with

0 < αjP (j|i)P (j|i′)Fi′,j(z) < αjP (j|i)P (j|i′)

for at least one j, we havePj∈I αjP (j|i)Fi,j(z)P

j∈I αjP (j|i) >

Pj∈I αjP (j|i′)Fi′,j(z)P

j∈I αjP (j|i′) .

Note that lemma 4 strengthens (C4): Just set αj = 1 for all j ∈ K, and αj = 0 otherwise. Lemma

4 allows us to prove (in the appendix) a final 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 ∈ M , if Gi(z) − Gi(z)− > 0 for some i ∈ I and

Hj(z) −Hj(z)− > 0 for some j ∈ I with P (i, j) > 0, then z = mi,j.

27

The intuition behind this lemma is again simple. Suppose that candidates A and B both put

positive mass on the same point z ∈ 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 payoff discretely when B chooses z, and it affects A’s expected payoff

continuously otherwise. Therefore, a small enough deviation increases A’s expected payoff.

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 ∈ I, then z = mi,i. If

Hj(z) −Hj(z)− > 0 for some j ∈ 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 ∈ I, but z �= 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 ∈ M . But then Lemma 5 implies z = mi,i, a

contradiction.

Theorem 9 does not quite allow us to use differentiable methods to analyze mixed strategy

equilibria. While the result limits the potential discontinuities of equilibrium mixed strategies to a

finite set, there may be other points at which an equilibrium distribution Gi is non-differentiable,

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-differentiability, so this

technical problem is potentially significant. In our analysis of equilibrium uniqueness in the three-

signal stacked-uniform model, we therefore restrict attention to a subset of mixed strategy equilibria:

We say a strategy pair (G,H) is regular if for all i ∈ I and all z ∈ �, either Gi is differentiable at z

or it is discontinuous at z, and similarly for Hi. This restriction eliminates the problem anticipated

above, at the cost of possibly omitting pathological equilibria.

Suppose (G,H) is a regular mixed strategy Bayesian equilibrium. Under the conditions of

Theorem 9, we can decompose the probability measure generated by Hj into a degenerate measure

28

with mass Hj(mj,j) − Hj(mj,j)− on mj,j and an absolutely continuous measure with density hj .

Candidate A’s expected payoff from pure strategy X against H, conditional on signal i, is

ΠA(X,H|i) =X

j:mj,j<xi

P (j|i)�1 − Fi,j

�xi + mj,j

2

��(Hj(mj,j) −Hj(mj,j)−)

+X

j:mj,j=xi

P (j|i)(Hj(mj,j) −Hj(mj,j)−)2

+X

j:xi<mj,j

P (j|i)Fi,j�

xi + mj,j

2

�(Hj(mj,j) −Hj(mj,j)−)

+Xj∈I

P (j|i)�Z xi

−∞

�1 − Fi,j

�xi + z

2

��hj(z) dz +

Z ∞

xi

Fi,j

�xi + z

2

�hj(z) dz

�.

Note that this expected payoff is differentiable at xi if there is no j such that xi = mj,j. Indeed,

it is enough that if xi = mj,j, then Hj does not put positive mass on mj,j. In this case, the usual

first order condition must be satisfied at xi:

0 =X

j:mj,j<xi

P (j|i)�−fi,j

�xi + mj,j

2

�(Hj(mj,j) −Hj(mj,j)−)

2

�

+X

j:xi<mj,j

P (j|i)�fi,j

�xi + mj,j

2

�(Hj(mj,j) −Hj(mj,j)−)

2

�(11)

+Xj∈I

P (j|i)�Z xi

−∞−fi,j

�xi + z

2

�hj(z)

2dz + (1 − Fi,j(xi))hj(xi) +

Z ∞

xi

fi,j

�xi + z

2

�hj(z)

2dz − Fi,j(xi)hj(xi)

�.

We next use these observations, first to construct a regular mixed strategy equilibrium in the

stacked-uniform model, and then to derive the properties of this equilibrium.

6 The Stacked-Uniform Model

To construct this mixed strategy equilibrium we first conjecture the general form of the equilibrium,

then deduce strong necessary conditions for the conjectured strategies to be an equilibrium, and

finally verify that the strategies are indeed mutual best responses. We consider the canonical

stacked-uniform model assuming sufficiently large support of the disturbance term, i.e., a ≥ bN−b1,

together with added structure on the marginal prior on signal pairs. Our first condition is:

(C10) For all i, k ∈ I, if k < i, then

(a)Xj:j<k

P (j|i) ≤Xj:j<k

P (j|k) ≤Xj:j<i

P (j|i), and (b)Xj:j>i

P (j|k) ≤Xj:j>i

P (j|i) ≤Xj:j>k

P (j|k).

The first inequality in sub-condition (a) is a stochastic dominance condition that formalizes the

obvious intuition: The higher is a candidate’s signal, the more likely the other candidate is to receive

a higher signal. The second inequality in sub-condition (a) limits the extent of this dominance:

29

Moving from k to a higher signal i, signals below k lose probability, but not too much. Sub-

condition (b) has an identical interpretation. Condition (C10) implies that for signals k < i, the

net probability to the left of k is greater given signal k than given higher signal i,

Xj:j<k

P (j|k) −Xj:j>k

P (j|k) ≥Xj:j<k

P (j|i) −Xj:j>k

P (j|i),

with a similar implication for the net probability to the right.

An important consequence of (C10) is that the subset of signals i such that

Xj:j≤i

P (j|i) ≥Xj:j>i

P (j|i) andXj:j<i

P (j|i) ≤Xj:j≥i

P (j|i)

is an “interval.” That is, letting C denote this subset, with c = minC and c = maxC, if c ≤ i ≤ c,

then i ∈ C. To see this, suppose that

Xj∈I:j<i

P (j|i) >X

j∈I:j≥iP (j|i),

and consider signal k > i. Then

12

<X

j∈I:j<iP (j|i) ≤

Xj∈I:j<k

P (j|k),

where the first inequality follows by supposition and the second by (C10).

Our next condition says that a candidate is most likely to receive signal i when the other

candidate also receives it.

(C11) For all i, j ∈ I, P (i|i) ≥ P (i|j).

The next condition is self-explanatory.

(C12) 0 ∈ [mc,mc].

Note that (C12) implies that C �= ∅. The final condition is a technical one on extreme signals (i.e.

signals i, j �∈ C), which is used to derive the existence of an equilibrium in which the candidates’

distributions following signal pairs do not “overlap.”

(C13) For all pairs of signals i, j �∈ C such that either c < j < i, or i < j < c, we have

P (j|j)mi,j ≤ P (j|i)mj,j .

30

Since P (j|j) > 0, condition (C13) implies

P (j|j)P (j|i) ≤ mi,j

mj,j.

Thus, condition (C13) limits (C11), saying that for extreme signals the relative likelihood of P (j|j)

to P (j|i) is bounded by the ratio of the conditional medians. Condition (C13) is satisfied both when

signals are uninformative, in which case P (j|j) = P (j|i), and when signals are perfectly correlated,

for then C = I. It is also vacuously satisfied in the three signal model, with I = {−1, 0, 1} and

C = {0}, as there do not exist distinct signals to the right of c = 0 or the left of c = 0.

We conjecture a symmetric equilibrium in which candidates who receive central signals i ∈ C

locate at the conditional median mi,i with probability one. Candidates who receive signals i �∈ C use

mixed strategies that are absolutely continuous with differentiable density gi on convex support

[xi, xi]. We conjecture supports that are non-overlapping, adjacent, and ordered identically to

signals, with xc+1 = mc,c and, for all i > c, xi = xi−1. Our analysis is mainly concerned with

identifying the conditions that must be fulfilled by the distribution Gi used by candidates after

signal realizations i /∈ C. Under the assumptions of this section, we can analyze the cases i < c

and i > c independently, and we therefore focus on the latter. Once we have characterized what

such a strategy pair (G,G) would have to look like were it an equilibrium, we verify that it is an

equilibrium.

Because we assume G has non-overlapping supports, it must be that for every x ∈ (xi, xi),

gj(x) > 0 implies j = i. As a result, for i > c, the first order condition (11) simplifies to

P (i|i)gi(x)(2Fi,i(x) − 1) = −Xj∈C

P (j|i)2

fi,j

�x + mj,j

2

�

−Xj∈I\C

P (j|i)2

�Z x

−∞fi,j

�x + z

2

�gj(z) dz +

Z ∞

xfi,j

�x + z

2

�gj(z) dz

�,

for all x ∈ [xi, xi], unless i = c + 1, in which case the condition holds on the half-open interval

(mc,c, xc+1]. Since the candidate’s expected payoff is constant over the relevant interval, it must, in

particular be linear over this interval, so the second order condition must be satisfied with equality.

Since the disturbance term is uniformly distributed, the second order condition simplifies to

3gi(x)fi,i(x) + g′i(x)(2Fi,i(x) − 1) = 0,

for all x ∈ [xi, xi], with the same qualification as before if i = c + 1. Since the platform mc,c is

chosen with probability zero when the candidate receives signal c + 1, we include it in the interval

as well, yielding a differential equation in gi that is easily solved. We find that

gi(x) = gi(xi)�

1 − 2Fi,i(xi)1 − 2Fi,i(x)

� 32

, (12)

31

for all x ∈ [xi, xi], with associated distribution

Gi(x) = gi(xi)((1/2) − Fi,i(xi))3/2

Z x

xi

1((1/2) − Fi,i(z))3/2

dz. (13)

Substituting for Fi,j(z) = (a−mi,j + z)/2a, equations (12) and (13) become

gi(x) = gi(xi)�

mi,i − ximi,i − x

� 32

(14)

and

Gi(x) = gi(xi)(mi,i − xi)3/2

�2√

mi,i − xi− 2√

mi,i − x

�. (15)

Thus, the second order condition pins down the density gi and distribution Gi up to the location

of xi and the initial condition gi(xi).

These parameters are determined by the first order condition for each signal realization and

by our assumption of non-overlapping adjacent supports. In the stacked-uniform model, the first

order condition can be written as

gi(x) =−Pj∈I:j<iP (j|i) + P (i|i)(1 − 2Gi(x)) +

Pj∈I:i<j P (j|i)

4P (i|i)(mi,i − x).

Evaluated at xi, the first order condition yields

gi(xi) =−Pj∈I:j<iP (j|i) +

Pj∈I:j≥i P (j|i)

4P (i|i)(mi,i − xi),

which is well-defined for xi < mi,i and positive for i > c, as is required. For signal c + 1, by

construction, xc+1 = mc,c. This pins down gc+1(xc+1) through the first order condition. We then

find xc+2 as the solution to

Gc+1(x) = 1,

if such a solution exists. This pins down gc+2(xc+2) through the first order condition, and so on.

Note that for xi < mi,i, a solution to Gi(x) = 1 does indeed exist for all i > c, since 2√mi,i−x goes

to infinity as x increases to mi,i.

It is prohibitively difficult to solve for these variables in the general case, but the next result

establishes several properties of our conjectured equilibrium. We first show that the strategies

defined above do, indeed, form an equilibrium. We then note that the density used following signal

realizations i > c is increasing, and that the support of this density is bounded above by the

conditional median mi,i. Thus, a candidate never chooses positions more extreme than what might

be chosen in the pure strategy equilibrium. In the third part of the theorem, we index the marginal

32

prior on I × I by the elements of a metric space Γ, as in P γ(i, j), and assume that (C10)-(C13)

are satisfied for all γ ∈ Γ. We consider a sequence {γn} of games, where for each n, Gn denotes

the mixed strategy defined above, gni is the corresponding density used after signal realization i,

and xni and xni are the lower and upper bounds on the support of Gni . We show that as the net

probability that the opponent has a more moderate signal goes to zero, the mixed strategy used

by a candidate approaches the point mass on the candidate’s conditional median.

Theorem 10 In the canonical stacked-uniform model with a ≥ bN − b1, assume (C10)-(C13).

1. The above strategies form a mixed strategy Bayesian equilibrium.

2. For i > c, gi is increasing and convex and xi < mi,i.

3. Let {γn} be a sequence with Cγn = C for all n and with

lim infn→∞ P γn(i|i) > 0 for all i ∈ I. For i > c, ifXj∈I:j<i

P γn(j|i) −X

j∈I:j≥iP γn(j|i) → 0,

then xni → mi,i and, for all x < mi,i, we have gni (x) → 0.

Proof: 1. Take any i, k ∈ I with c < k ≤ i. Let X be a pure strategy such that xi = xk, and

let X ′ be a pure strategy such that x′i = x′ ∈ (xk, xk]. Define

ψk(x) = Fi,j

�xk + z

2

�− Fi,j

�x′ + z

2

�=

xk − x

4a,

which is independent of j and z, and note that this quantity is positive. The change in candidate

A’s expected payoff, conditional on signal i, upon moving from xk to x′ is

ΠA(X ′, G|i) − ΠA(X,G|i) =X

j∈C:j<k

P (j|i)ψk(x′) −X

j∈C:j>k

P (j|i)ψk(x′)

+X

j /∈C:j<k

P (j|i)Z xk

−∞ψk(x′)gj(z) dz −

Xj /∈C:k<j

P (j|i)Z ∞

xk

ψk(x′)gk(z) dz

+ P (k|i)"Z x′

xk

ψk(x′)gk(z) dz +Z xk

x′

�Fi,k

�x′ + z

2

�+ Fi,k

�xk + z

2

�− 1

�gk(z) dz

#.

Substituting for Fi,k(z) = 12 + z−mi,k

2a , this simplifies to

ΠA(X ′, G|i) − ΠA(X,G|i) = ψk(x′)

24Xj:j<k

P (j|i) −Xj:k<j

P (j|i)35

+ P (k|i)�ψk(x′)Gk(x′) +

Z xk

x′

xk + x′ + 2z4a

gk(z) dz −�

1 −Gk(x′)a

�mi,k

�

= ψk(x′)

24Xj:j<k

P (j|i) −Xj:k<j

P (j|i)35+ P (k|i)[D − Emi,k],

33

where

D = ψk(x′)Gk(x′) +Z xk

x′

xk + x′ + 2z4a

gk(z) dz

E =1 −Gk(x′)

a.

Note that by construction, ΠA(X ′, G|i) −ΠA(X,G|i) is equal to zero if i = k. Then, the first term

on the right-hand side, namely

ψk(x′)

24Xj:j<k

P (j|k) −Xj:k<j

P (j|k)

35 ,

is positive because c < k, implying that P (k|k)[D − Emk,k] < 0. If k < i, then

Xj:j<k

P (j|i) −Xj:j>k

P (j|i) ≤Xj:j<k

P (j|k) −Xj:j>k

P (j|k)

by (C10), so the first term decreases relative to i = k. The change in the second term,

P (k|i)[D −Emi,k] − P (k|k)[D − Emk,k],

is generally indeterminate, because P (k|k) > P (k|i) and mi,k > mk,k. We now claim that (C11)

and (C13) imply that

P (k|i)[D − Emi,k] ≤ P (k|k)[D − Emk,k],

or equivalently,

E[P (k|k)mk,k − P (k|i)mi,k] ≤ D[P (k|k) − P (k|i)].

To see this, note that D is positive since 0 ∈ [mc,mc] implies that x′ > 0. Clearly, E is also

non-negative. But (C13) implies that the term multiplying E is non-positive, and (C11) imlies that

the term multiplying D is non-negative. Therefore,

ΠA(X ′, G|i) − ΠA(X,G|i) ≤ ΠA(X ′, G|k) − ΠA(X,G|k) = 0,

and we conclude that

ΠA(X ′, G|i) ≤ ΠA(X,G|i)

whenever x′ ∈ (xk, xk]. Indeed, if c + 1 < k, then xk is a continuity point of A’s expected payoff

function, so the inequality also holds at xk.

34

If k = c + 1, then xk = mc,c is not a continuity point, and the last remark no longer holds. In

this case, let X be a pure strategy such that xi = xc+1, and let X ′ be a pure strategy such that

x′i = x′ = mc,c. Let {xn} be a sequence decreasing to mc,c, and let Xn be such that xni = xn. Then

[ΠA(X ′, G|i) − ΠA(X,G|i)] − limn→∞[ΠA(Xn, G|i) − ΠA(X,G|i)]

= P (c|i)�

12−�1 − Fi,c

�xc+1 + mc,c

2

���+ P (c + 1|i)

"Z xc+1

mc,c

�2Fi,c+1

�z + mc,c

2

�− 1

�gc+1(z) dz

#

− P (c|i)�1 − Fi,c+1(mc,c) −

�1 − Fi,c+1

�xc+1 + mc,c

2

���

− P (c + 1|i)�

limn→∞

Z xc+1

xn

�2Fi,c+1

�z + xn

2

�− 1

�gc+1(z) dz

�

= P (c|i)�Fi,c

�xc+1 + mc,c

2

�− 1

2+ Fi,c+1(mc,c) − Fi,c+1

�xc+1 + mc,c

2

��

= P (c|i)��

xc+1 −mi,c

4a

�+�

mc,c − xc+1

4a

��= P (c|i)

�mc,c −mi,c

4a

�≤ 0,

where the inequality follows from c < i and (C4). Therefore,

ΠA(X ′, G|i) − ΠA(X,G|i) ≤ limn→∞[ΠA(Xn, G|i) − ΠA(X,G|i)] ≤ 0,

where the second inequality follows from the arguments of the previous paragraph. We conclude

that, as in the earlier case,

ΠA(X ′, G|i) ≤ ΠA(X,G|i).

Now take any i, k ∈ I with k ≤ i and c < k ≤ c. Let X be a pure strategy such that xi = mk,k,

and let X ′ be a pure strategy such that x′i = x′ ∈ (mk−1,k−1,mk,k]. Then, defining

ψk(x′) = Fi,k(mk,k) − Fi,k((x′ + mk,k)/2) =mk,k − x′

4a,

we have

ΠA(X ′, G|i) − ΠA(X,G|i) = ψk(x′)

24Xj:j<k

P (j|i) −Xj:k<j

P (j|k)

35 + P (k|i)

�Fi,k

�x′ + mk,k

2

�− 1

2

�.

Note that since mk,k ≤ mi,k,

Fi,k

�x′ + mk,k

2

�− 1

2≤ −ψk(x′).

Therefore,

ΠA(X ′, G|i) − ΠA(X,G|i) ≤ ψk(x′)

24Xj:j<k

P (j|i) −Xj:k≤j

P (j|k)

35 ,

35

which is non-positive if i = k, by definition of k ∈ C. If i > k, then the inequality follows from

(C10). We again conclude that

ΠA(X ′, G|i) ≤ ΠA(X,G|i),

whenever x′ ∈ (mk−1,k−1,mk,k]. That the inequality actually holds for x′ = mk−1,k−1 as well follows

from (C4) and an argument similar to that used in the previous paragraph.

Finally, take any i, k ∈ I with k ≤ i and k < c. Let X be a pure strategy such that xi = x ∈[xk, xk), and let X ′ be a pure strategy such that x′

i = xk. Then, defining

ψk(x) = Fi,j

�z + x

2

�− Fi,j

�z + xk

2

�=

x− xk4a

,

which is independent of j and z, we have

ΠA(X ′, G|i) − ΠA(X,G|i) = ψk(x)

24Xj:j<k

P (j|i) −Xj:k<j

P (j|i)35

+ P (k|i)"−ψk(x)(1 −Gk(x)) +

Z x

xk

�Fi,k

�xk + z

2

�+ Fi,k

�x + z

2

�− 1

�gk(z) dz

#.

If i = k, then, by k < c, we know that the first term on the right-hand side is negative. Since

ΠA(X ′, G|k) − ΠA(X,G|k) = 0, by construction, the second term must then be positive. If k < i,

then, by (C10), the first term decreases relative to i = k. Furthermore,

P (k|i)"−ψk(x)(1 −Gk(x)) +

Z x

xk

�Fi,k

�xk + z

2

�+ Fi,k

�x + z

2

�− 1

�gk(z)dz

#

≤ P (k|k)

"−ψk(x)(1 −Gk(x)) +

Z x

xk

�Fk,k

�xk + z

2

�+ Fk,k

�x + z

2− 1

�gk(z) dz

�#,

where the inequality follows from (C11), from the fact that the right-hand side is positive, and from

the fact that Fi,k stochastically dominates Fk,k. Thus, the second term decreases relative to i = k

as well. Once again, we conclude that

ΠA(X ′, G|i) ≤ ΠA(X,G|i),

whenever x ∈ [xk, xk). Indeed, if k < c−1, then xk is a continuity point, so the inequality also holds

for x = xk as well. If k = c − 1, then xk = mc,c, which is not a continuity point of A’s expected

payoffs. In this case, we may argue as above that the inequality remains true for x = mc,c.

We now argue that the candidate has no profitable deviation, conditional on signal i, from

the mixed strategy G. First, consider the case i > c. Let X satisfy xi = xi, and let X ′ satisfy

x′i = x′ < xi. Suppose that x′ does not lie below the supports of all distributions Gj . (That case is

36

easily checked and is omitted.) Then either x′ ∈ [xk, xk] for some k /∈ C, or x′ ∈ [mk−1,mk−1] for

some k ∈ C. Suppose the former. Indeed, suppose k < c, and let the pure strategies Xj satisfy:

For j /∈ C, we have xji = xj , and for j ∈ C, we have xji = mj,j. Our above arguments show that

ΠA(X ′, G|i) − ΠA(X,G|i) = [ΠA(X ′, G|i) − ΠA(Xk+1, G|i)] +X

j:k<j<c

[ΠA(Xj , G|i) − ΠA(Xj+1, G|i)]

+Xc≤j<c

[ΠA(Xj , G|i) − ΠA(Xj+1, G|i)] +X

j:c<j<i

[ΠA(Xj , G|i) − ΠA(Xj+1, G|i)] ≤ 0.

Because xi is a continuity point in the support of Gi, we have ΠA(X,G|i) = ΠA(G,G|i), so that X ′

is not profitable. Other cases are proved the same way, by decomposing a deviation to the left after

signal i into a finite number of moves across the supports for other signal realizations. This proves

that candidate A has no profitable deviation to the left after signal i, and a symmetric argument

shows that there are no profitable deviations to the right. Finally, the same argument for candidate

B establishes that the strategy pair (G,G) is a mixed strategy equilibrium.

2. That gi is increasing and convex is apparent from the functional form of the density. That

xi < mi,i follows from the discussion above, where it was noted that

Gi(x) = gi(xi)(xi −mi,i)3/2�

2√mi,i − x

− 2√mi,i − xi

�

goes to infinity as x increases to mi,i. This observation, with an induction argument starting with

xc+1 = mc,c yields the desired conclusion.

3. Since Cγn = C for all n, xc+1 is constant along the sequence. By part 2, we therefore have

xni < mi−1,i−1 for all i > c. This implies that 1−2Fi,i(xni ) ≥ 1−2Fi,i(mi−1,i−1) > 0 for all n, so the

denominator in the above expression for gi(xi) does not go to zero. We therefore have gni (xni ) → 0.

Then, fixing x < mi,i, we see that gni (x) → 0. And, from the above expression for Gi(x), we see

that Gni (xni ) = 1 for all n implies xni → mi,i.

Words here?

7 Three Signal Realizations

We now consider the canonical stacked-uniform model with three signal realizations, I = {−1, 0, 1}.

It is immediate that for P (1|1) < 1,

F1(m1,1) = P (−1|1)F1,−1(m1,1) + P (0|1)F1,0(m1,1) + P (1|1)F1,1(m1,1) >12,

37

which implies m1 < m1,1. Similarly, m−1,−1 < m−1. We again assume a sufficiently large support

of the disturbance term, i.e., a ≥ bN − b1, but do not restrict the number of disturbances. We also

add the assumption of symmetry about zero to the model of Section 6.

(C14) For all z ∈ X and all i, j ∈ I,

(a) Fi,j(z) = 1 − F−i,−j(−z), (b) P (−j|i) = P (j| − i), and (c)P (1| − 1) < P (1|1) < 1.

Sub-condition (a) implies that mi,j = −m−i,−j, for each i, j ∈ I: Explicitly writing out Fi,j(z) +

F−i,−j(−z) = 1 yields

12

+z −mi,j

2a+

12

+−z −m−i,−j

2a= 1,

which yields the claim. Thus, m0,0 = 0 and m−1,−1 = −m1,1. Furthermore, (C14) combined with

(C2) imply m1,−1 = m−1,1 = 0. Thus, since P (1|1) > 0, we have

F1(0) = P (−1|1)F1,−1(0) + P (0|1)F1,0(0) + P (1|1)F1,1(0) <12.

Combining these observations, since 0 < P (1|1) < 1, it follows that

m−1,−1 < m−1 < 0 < m1 < m1,1.

Thus, the pure strategy equilibrium platform adopted by a candidate “extremizes” the candidate’s

information in the sense that the platform is biased away from the center of the policy space.

For this three signal environment, we now prove that the equilibrium characterized in the previ-

ous section is actually unique among the class of symmetric regular mixed strategy equilibria.7 We

also provide more precise characterizations of when candidates choose extremal positions. Finally,

we examine extremism for several information structures. We say a mixed strategy G is symmetric

about zero if, for all i ∈ I and all z ∈ X, Gi(z) = 1 − G−i(−z). A mixed strategy Bayesian

equilibrium (G,H) is zero-symmetric if G and H are symmetric about zero.

Theorem 11 In the canonical stacked-uniform model with a ≥ bN − b1, assume I = {1, 0,−1},(C8), and (C14). There is a unique regular zero-symmetric equilibrium for P (1|1) �= 1

2 . In it,

each candidate plays x0 = y0 = 0 upon receiving the signal 0. Upon receiving a signal i �= 0, each

candidate plays xi = yi = mi,i if P (1|1) > 1/2. If, instead, P (1|1) < 1/2, the mixed strategy played

upon receiving signal 1 is given by

G1(z) = H1(z) =(1 − 2P (1|1))(√m1,1 −√

m1,1 − z)2P (1|1)

√m1,1 − z

,

for all z ∈ [0, 4m1,1P (1|1)(1 − P (1|1))].7Recall that (G, H) is regular if each Gi and Hj is differentiable at all but at most a countable set of points.

38

Proof: As a preliminary, note that (C9) is satisfied in the stacked-uniform model so that the

assumptions of Theorem 9 are fulfilled. Thus, a candidate’s equilibrium mixed strategy following

signal i has at most one atom, namely, the median conditional on both candidates receiving signal

i. It turns out that with the symmetry imposed in this section, (C10) is not needed for existence.

Condition (C11) is also simplified, as we only use two inequalities from this condition: P (1|1) >

P (1| − 1), and its symmetric counterpart. Condition (C12) is automatically satisfied because

symmetry implies that 0 ∈ C. In fact, C contains only zero unless P (1|1) > 1/2, in which case

C = I. Finally, condition (C13) is vacuously satisfied.

We first show that given any potential regular zero-symmetric equilibrium strategy for one

candidate, a best response for the other candidate when he receives the signal zero, is to locate

at zero. We next use the construction in the previous section to prove the existence of a regular

zero-symmetric equilibrium with support in M . After this, we show that in every regular zero-

symmetric equilibrium, candidates locate at zero following the zero signal. Finally, we pin down

candidates’ strategies following extreme signal realizations.

Step 1: Let H be any strategy symmetric about zero for candidate B with support in M and

with mass points (if any) only at conditional medians, i.e., Hj(y) − Hj(y)− > 0 implies y = mj,j.

Let X be a pure strategy with x0 = 0, let X ′ be a pure strategy with x′0 = x′ > 0, and note that

ΠA(X,H|0) − ΠA(X ′,H|0) =1X

j=−1

P (j|0)

"Z[x′,m1,1]

�F0,j

�z

2

�− F0,j

�x′ + z

2

��Hj(dz)

+Z

(0,x′)

"F0,j

�z

2

�−�1 − F0,j

�x′ + z

2

��Hj(dz) +

Z[−m1,1,0]

�1 − F0,j

�z

2

�−�1 − F0,j

�x′ + z

2

��Hj(dz)

�##.

Recalling that

F0,j (w) − F0,j (z) =w − z

2a, (16)

for all j ∈ I and all w, z ∈ [bN − a, b1 + a], we use the zero-symmetry of H to simplify the above

terms corresponding to j = 1,−1 to:

Z(0,x′)

"F0,j

�z

2

�−�1 − F0,j

�x′ + z

2

��Hj(dz) +

Z(−x′,0]

��1 − F0,j

�z

2

��−�

1 − F0,j

�x′ + z

2

���#Hj(dz).

39

By assumption, F0,−1 (z) = 1 − F0,1 (−z) for all z ∈ �. Using P (1|0) = P (−1|0), it follows that

Xj=−1,1

P (j|0)

"Z(0,x′)

�F0,j

�z

2

�−�1 − F0,j

�x′ + z

2

���Hj(dz)

+Z

(−x′,0]

��1 − F0,j

�z

2

�−�1 − F0,j

�x′ + z

2

��Hj(dz)

��#

= P (1|0)

"Z(0,x′)

�1 − F0,1

�−z

2

�− F0,1

�−x′ + z

2

�+ F0,1

�z

2

�− 1 + F0,1

�x′ + z

2

��Hj(dz)

+Z

(−x′,0]

�F0,1

�−z

2

�− F0,1

�−x′ + z

2

�− F0,1

�z

2

�+ F0,1

�x′ + z

2

��Hj(dz)

#.

To see that each of the integrals on the right-hand side is positive, note that the first integrand is

equal to x′4a , and the second is equal to x′

2a + za , which is positive for all z ∈ [0, x′]. Therefore,

ΠA(X,H|0) − ΠA(X ′,H|0) ≥ P (0|0)

"Z[x′,m1,1]

�F0,0

�z

2

�− F0,0

�x′ + z

2

��H0(dz)

+Z

(0,x′)

�F0,0

�z

2

�−�1 − F0,0

�x′ + z

2

���H0(dz)

+Z

[−m1,1,0]

��1 − F0,0

�z

2

��−�1 − F0,0

�x′ + z

2

��H0(dz)

�#

where the inequality follows from the preceding remarks. Substituting for 1−F0,0

�z2

�= F0,0

�− z2

�and exploiting the functional form from equation (16), the right-hand side simplifies to

P (0|0)2a

Z x′

0(x′ + z)H0(dz) =

P (0|0)2a

"H0(x′) − 1

2+Z x′

0zH0(dz)

#≥ 0,

where the last inequality follows because zero is a median of H0. A symmetric argument holds for

x′ < 0. We conclude that x0 = 0 is a best response to H.

Step 2: All of the conditions of Theorem 10 are fulfilled, except for part of (C11), namely,

P (1|1) ≥ P (1|0) and P (0|0) ≥ P (0|1). For the special case of three signals, the latter inequality is

not used, and the former is only used in Step 1 to prove that x0 = 0 is a best response following

i = 0. It follows that the strategies defined in section 6 form a regular zero-symmetric equilibrium.

Step 3: Let (G,H) be a regular zero-symmetric equilibrium. By Theorem 8, in any equilibrium

(G,H), we must have Gi(m1,1) − Gi(m−1,−1)− = Hi(m1,1) − Hi(m−1,−1)− = 1 for all i ∈ I.

Furthermore, by symmetry and interchangeability, we may assume G = H. To see that H0 puts

probability one on m0,0 = 0, suppose that H0(0) < 1. Since H0 has no mass points other than

m0,0, by Theorem 9 we can choose a strictly increasing sequence {xn} in the support of H0 such

that H0(xn) → 1. Since each xn is a continuity point of A’s expected payoff, conditional on i = 0,

ΠA(Xn,H|0) = ΠA(H,H|0)

40

for all n, where Xn is a pure strategy with xn0 = xn. Take any n such that H0(xn) > 1/2, and

recall that

ΠA(X,H|0) − ΠA(Xn,H|0) ≥ P (0|0)2a

�H0(xn) − 1

2+Z xn

0zH0(dz)

�,

as shown in Step 2. This quantity is positive, however, a contradiction. Therefore, H0(0) = 1. A

symmetric argument shows that H0(0)− = 0. We conclude that H0 puts probability one on zero.

Step 4: To pin down G1, let (G,H) be a regular zero-symmetric equilibrium, which must

have support in M , and again suppose G = H. To see that G1(0) = 0, let X be such that

x1 = x ∈ (m−1,−1, 0), and note that

ΠA(X,H|1) = P (0|1)F0,1

�x

2

�+

Xj=−1,1

P (j|1)

"Z x

m−1,−1

�1 − Fj,1

�x + z

2

�Hj(dz) +

Z m1,1

xFj,1

�x + z

2

�Hj(dz)

�#.

Using Theorem 9, each Hj is continuous at x. Hence, because H is regular, Hj has a density hj

defined at x, and candidate A’s payoff is differentiable at x. Thus,

∂

∂x1ΠA(X,H|1) =

P (0|1)2

f0,1

�x

2

�+

Xj=−1,1

P (j|1)

"Z x

m−1,−1

−12fj,1

�x + z

2

�Hj(dz)

+ (1 − Fj,1(x))hj(x) +Z m1,1

x

12fj,1

�x + z

2

�Hj(dz) − Fj,1(x)hj(x)

#

=P (0|1)

4a+

Xj=−1,1

P (j|1)�(1 − 2Fj,1(x))hj(x) +

14a

(1 − 2Hj(x))�. (17)

Since the median of F1,1 is positive and the median of F−1,1 is zero, the only term in (17) that may

be negative when x < 0 is 1 − 2Hj(x). Now, if G1(0) > 0, then, by Theorem 9, we may take a

strictly decreasing sequence {xn} in the support of G1 such that G1(xn) → 0. For each n, let Xn

be a pure strategy satisfying xn1 = xn. Because each xn is a differentiable point of ΠA, we must

have ∂∂x1

ΠA(Xn,H|1) = 0 for all n. Note, however, that 1 − 2H1(xn) = 1 − 2G1(xn) → 1, while

1 − 2H−1(xn) is bounded below by −1. Therefore, since P (1|1) ≥ P (−1|1), we have

lim infn→∞

∂

∂x1ΠA(Xn,H|1) ≥ P (1|1) − P (−1|1)

4a> 0,

where the last inequality follows from (C14), a contradiction. We conclude that G1(0) = 0. A

symmetric argument shows that G−1(0) = 1.

Suppose P (1|1) < 1/2 and the support of G1 is not an interval with left-hand endpoint zero,

i.e., for some c, d ∈ � with 0 < c < d, we have G1(c) = G1(d) < 1. Let X be a pure strategy with

x1 = d, and let X ′ be a pure strategy with x′1 = d′ = (c + d)/2. We now argue that conditional on

41

i = 1, a deviation to X ′ is profitable for candidate A. Since G is symmetric about zero, we have

ΠA(X ′,H|1) − ΠA(X,H|1) = P (−1|1)

"Z[0,m1,1)

�F1,−1

�d− z

2

�− F1,−1

�d′ − z

2

��G1(dz)

+ (1 −G1(m1,1)−)�F1,−1

�d−m1,1

2

�− F1,−1

�d′ −m1,1

2

��#+ P (0|1)

�F1,0

�d

2

�− F1,0

�d′

2

��

+P (1|1)

"Z[0,c]

�F1,1

�d + z

2

�− F1,1

�d′ + z

2

��G1(dz) +

Z[d,m1,1)

�F1,1

�d′ + z

2

�− F1,1

�d + z

2

��G1(dz)

+�1 −G1(m1,1)−

� �F1,1

�d′ + m1,1

2

�− F1,1

�d + m1,1

2

��#

= P (−1|1)�G1(m1,1)−

�d− d′

4a

�+ (1 −G1(m1,1)−)

�d− d′

4a

��+ P (0|1)

�d− d′

4a

�

+ P (1|1)�G1(c)

�d− d′

4a

�+ (G1(m1,1)− −G1(d)−)

�d′ − d

4a

�+ (1 −G1(m1,1)−)

�d′ − d

4a

��

=�

d− d′

4a

�[P (−1|1) + P (0|1) + G1(c)P (1|1) − (1 −G1(c))P (1|1)]

=�

d− d′

4a

�[P (−1|1) + P (0|1) − P (1|1)] > 0,

where the second equality uses G1(c) = G1(d)−, and the last inequality follows from P (1|1) < 1/2,

a contradiction. Choosing d = sup{d | G1(d) = G1(c)}, which is in the support of G1, the above

inequalities continue to hold. Then we have

ΠA(X ′,H|1) > ΠA(X,H|1) = ΠA(G,H|1),

a contradiction. We conclude that the support of G1 is an interval with left-hand endpoint zero.

Now suppose P (1|1) > 1/2 but the support of G1 is not an interval with right-hand endpoint

m1,1, i.e., for some c, d ∈ � such that c < d < m1,1, we have 0 < G1(c) = G1(d). Without loss of

generality, because G1 is continuous from the right, assume c is in the support of G1. Letting X be

a pure strategy with x1 = c and letting X ′ be a pure strategy with x′1 = c′ = (c + d)/2, the above

arguments again yield ΠA(X ′,H|1) > ΠA(X,H|1), a contradiction.

From Theorem 9, the support of G1 must be, in fact, a non-degenerate interval if P (1|1) < 1/2.

Because candidate A must be indifferent over all platforms in this interval, the first and second

order conditions from the previous section must be satisfied. Then, the analysis preceding Theorem

10 implies that

G1(z) = gs(x1)((1/2) − F1,1(x1))3/2Z x

x1

1((1/2) − F1,1(z))3/2

dz

for all z in the support of G1, where

g1(x1) =−1 + 2P (1|1)

4aP (1|1)(2F1,1(x1) − 1)

42

and x1 = 0. Substituting for F1,1(z) = 12 + z−m1,1

2a , G1(z) simplifies to

G1(z) =(1 − 2P (1|1))(√m1,1 −√

m1,1 − z)2P (1|1)

√m1,1 − z

,

for all z in the support of G1. Solving for G1(z) = 1 yields the upper bound of the support,

4m1,1P (1|1)(1 −P (1|1))]. If P (1|1) > 1/2, then the support of G1 must be degenerate: Otherwise,

the same first and second order conditions must hold with the density of G1 as given above, but

this is negative. From these arguments, it follows that G1 is degenerate on m1,1, as required.

The closed form of G1 in Proposition 11 immediately yields results about the likelihood that

candidates locate more extremely than their information suggests. Let Γ3 index parameterizations

of the canonical three-signal stacked-uniform model satisfying (C8) and (C14). The first corollary

establishes that when the probability the other candidate receives the same signal is close enough

to one half, then with probability arbitrarily close to one, a candidate who receives a left or right

signal (1 or −1) chooses a platform that is more extreme than that suggested by that signal alone.

The corollary also shows that when the other candidate is very unlikely to receive the same signal,

then candidates take very moderate positions.

Corollary 5 Let {γn} be a sequence in Γ3 such that mγn

i,j → mi,j for all i, j ∈ I. Let {(Gn,Hn)}be the corresponding sequence of regular zero-symmetric mixed strategy Bayesian equilibria, and let

zn = 4mγn1,1P

γn(1|1)(1 − P γn(1|1))

denote the upper bound of the support of Gn1 . If Pγn(1|1) ↑ 1/2, then zn → m1,1 and Gn1 (mγn

1 ) → 0.

If P γn(1|1) ↓ 0, then zn → 0 and Gn1 (mγn1 ) → 1.

Inspecting the closed form of G1 in Proposition 11 reveals that fixing z in the support of G1,

the derivative of G1 with respect to P (1|1) at z is

−√m1,1 −√

m1,1 − z

2P (1|1)2√

m1,1 − z< 0,

so that G1(z) is decreasing in P (1|1). This observation yields the next corollary.

Corollary 6 Let γ, γ′ ∈ Γ3 be such that P γ(1|1) < P γ′(1|1) < 1/2 and mγ

1,1 = mγ′1,1, and let

(Gγ ,Hγ) and (Gγ′,Hγ′) be the corresponding regular zero-symmetric mixed strategy Bayesian equi-

libria. Then Gγ′

1 first order stochastically dominates Gγ1 .

Thus, better information leads to more extreme policy platforms. This result implies, in par-

ticular, that there is a unique threshold in P (1|1) that determines whether a candidate is likely

43

to choose a location more extreme than m1, the “non-strategic” choice induced by the candidate’s

private information only.

We end this section by exploring two examples of the three-signal stacked-uniform model, in

which we consider the probability of extremal electoral platforms as a function of model parameters.

The first example highlights how correlation in signals leads to extreme location by candidates.

The second example highlights that the probability candidates locate extremely rises with the

informational content of their signals.

Example: Three Spatial Signals with Conditional Correlation. Consider the three-signal stacked-

uniform model with three equally-likely preliminary cut points, β ∈ {−1, 0, 1}, and three possible

signals, i ∈ {−1, 0, 1}. With probability q, candidates receive conditionally-independent signals

where with probability p the signal is correct, and with probability r the signal is “off by one”:

p = Pr(i = 0|β = 0) = Pr(i = 1|β = 1) = Pr(i = −1|β = −1)

r = Pr(i = 1|β = 0) = Pr(i = −1|β = 0) = Pr(i = 0|β = 1) = Pr(i = 0|β = −1).

We assume that p > r > 1 − p− r, which implies that r < 1/3. With probability 1 − q candidates

receive identical signals drawn from the same distribution.

Given β = k, the joint probability of signals i and j is q Pr(i|k) Pr(j|k)+(1− q) Pr(i|k), if i = j,

and is q Pr(i|k) Pr(j|k), otherwise. Thus, P (1|1)p,q,r = q(p2 +r2 +(1−p−r)2)+(1−q). Solving for

the median conditional only on a candidate’s own signal, and for the median conditional on both

candidates receiving the same signal yields

mp,q,r1 = p− (1 − p− r)

mp,q,r1,1 = q

p2 − (1 − p− r)2

p2 + r2 + (1 − p− r)2+ (1 − q)(p− (1 − p− r)).

We say that a candidate chooses a moderate platform if he locates in [−mp,q,r1 ,mp,q,r

1 ]; and he

chooses an extreme platform if he biases his location further away from the center of the policy

space in the direction of his signal so that |x| > mp,q,r1 . We use Maple to calculate the probability

P(p, q, r) = Gp,q,r1 (mp,q,r1 ) that a candidate chooses a moderate platform after a left or right signal.

We find that P(p, q, r) is maximal for q = 1, p approaching 1/3 from above, and r approaching 1/3

from below, with

limp↓ 1

3,r↑ 1

3,q→1

P(p, q, r) =12

(√

2 − 1) ≈ 0.207.

Thus, the probability that candidates adopt extreme platforms is very high for all model specifica-

tions. Figure 3 illustrates the comparative statics of the model.

[ Figure 3 here. ]

44

In Figure 3a, we fix q = 1 and view P(p, q, r) as a function of p and r alone; in 3b, we fix p = .35

and view P(p, q, r) as a function of q and r; and in 3c, we fix r = .3 and view P(p, q, r) as a function

of p and q. In each, negative values correspond to pairs (p, r) such that G1 is degenerate on m1,1.

Note that increasing the probability of a correct signal, p, makes extreme platforms more likely.

Example: Conditional Independence with an Uninformative Signal. Suppose there are two

equally-likely preliminary cut-points, β ∈ {−1, 1}, and three possible signals, s ∈ {−1, 0, 1}. Can-

didates receive conditionally-independent signals. The key characteristic of this example is that

signal 0, rather than indicating a centrally located cut-point, is completely uninformative. With

probability r a candidate receives an informative signal s ∈ {−1, 1} of quality q, where given

s ∈ {−1, 1} the probability s = β is 1+q2 . Thus, given β, the probability a candidate receives signal

i = β is r(1 + q)/2, while the probability i = −β is r(1 − q)/2. With probability 1 − r, candidates

receive signal s = 0, independently of the value of β, so that this signal is uninformative. Thus, q

captures quality: If q = 1, then a signal of 1 or −1 is completely informative, and if q = 0, then all

signals are equally uninformative. From Theorem 11, a pure strategy Bayesian equilibrium exists

if r > 1/(1 + q2); i.e., fixing the informativeness of signals, a pure strategy equilibrium exists if the

probability of an informative signal is high enough. If r < 1/(1+ q2), then there is a unique regular

zero-symmetric mixed strategy equilibrium. In this mixed strategy equilibrium, candidates choose

m0,0 = 0, upon receiving the uninformative signal 0, and use the mixed strategy distribution

G (z) =(1 − r(1 + q2))

�√2rq −p2qr − r(1 + q2)z

�r(1 + q2)

p2qr − r(1 + q2)z

for all z ∈ (0, 2rq(2 − r(1 + q2)), upon receiving the informative signal 1. We solve

Pr (β = 1|si = 1)Z m1

−a+1

12a

dx+ Pr (β = −1|si = 1)Z m1

−a−1

12a

dx = 1/2,

to obtain the median of µ conditional on signal 1, m1 = q. Figure 4 plots the level sets of P(q, r),

the probability of moderation as a function of the parameters q and r.

[ Figure 4 here. ]

Here, the higher is a level curve, the lower is P(q, r), i.e., the more likely is the candidate to choose

a location that extremizes the informational content of his/her signal. The top level set represents

P(q, r) = 0, and the bottom level set represents P(q, r) = 1. Note that an increase in q (the

informativeness of signals 1 and −1) has an ambiguous effect, depending on r (the probability of

an informative signal). An increase in r, however, reduces the probability of a moderate platform.

45

8 Welfare

In this section we explore how voter welfare is affected by the polling technology. We seek to

glean insights into how preferences over polling technology vary with voter ideology, and into the

determinants of voter preferences over the accuracy of polling technology. We distinguish both the

direct welfare impact of the polling technology due to the candidate’s ability to target the median

voter, and the indirect impact of the polling technology due to the strategic behavior of candidates.

For simplicity, we investigate welfare in the two-signal stacked-uniform model with two equally

likely preliminary cut-points, β ∈ {−1, 1}, and signals i ∈ {−1, 1}. Given β candidates receive the

same signal with probability q, and with probability 1 − q they receive conditionally-independent

signals. A candidate’s signal is accurate with probability p, i.e., Pr(i|β = i) = p, where p ≥ 1/2.

Thus, p captures the precision of the candidates’ signals, while q captures one aspect of signal

correlation. We assume that a > 2, so that the support of α is sufficiently broad, and P (i|i) ≥ 1/2.

Hence, by Theorem 2, the unique pure strategy equilibrium obtains.

The median voter’s preferred platform is µ = α+β, where α is uniformly distributed on [−a, a].

We explicitly model a set of voters so that each voter v’s policy preferences are represented by a

randomly distributed ideal point θv that is correlated with µ in a simple way: θv = µ + δv, where

δv represents the position of v’s ideal point relative to µ. Call δv the “relative ideal point” of the

voter. The distribution of relative ideal points is represented by a distribution function Φ; given µ,

Φ(z − µ) is the proportion of voters with θv = δv + µ ≤ z. Thus, µ simply shifts the distribution

of voter ideal points. Voters have quadratic utility: A voter with ideal point θ receives utility

u(θ, z) = −(θ − z)2

from policy outcome z. Voters vote for the candidate whose platform is closest to their ideal

points, flipping a coin in case of indifference,8 generating probabilities of winning for the candidates

consistent with the stacked-uniform model.

One interpretation of this environment is that at date 1, candidates observe signals about the

median voter’s current location β ∈ {−1, 1}, and then choose platforms. The election is at date 2.

Between dates 1 and 2, the median voter’s preferred platform may change, perhaps due to changes

in the economy, where the support of α captures the degree to this can happen. While p captures

the precision that candidates receive about the median voter’s initial location, and q captures the

initial signal correlation, as support parameter a is increased, the effective ability of candidates

to target α + β falls, while the probability that the candidates both locate on the same side of µ

increases (for example, if α > 1, both candidates’ will locate below µ).8If each candidate receives exactly half of the votes, the tie is resolved by a fair coin toss.

46

In equilibrium, candidates locate at the conditional medians, either

m1,1(p, q) =2p− 1

2p2 + 2q(p − p2) + 1 − 2p

or m−1,−1(p, q) = −m1,1(p, q), corresponding to their signals. The median conditional only on

signal 1 is

m1(p) = 2p− 1,

which is independent of q. To simplify notation, we suppress the arguments of these medians. Since

p−p2 > 0 if p < 1, then as long as signals are not perfectly precise, (a) m1,1 > m1, and (b) m1,1−m1

decreases in the correlation parameter q, implying that candidates locate more moderately. With

probability qp+(1−q)p2, both candidates receive the signal i = β; with probability 2(1−q)p(1−p),

candidates receive distinct signals; and with probability q(1− p) + (1− q)(1− p)2, both candidates

receive the signal i = −β. Thus, platform separation is zero if signals are perfectly precise (p = 1)

or if perfectly correlated (q = 1),and for p < 1, as correlation is reduced, the probability and extent

of platform separation rises.

To calculate the welfare of a voter with relative ideal point δ, suppose without loss of generality

that β = 1. If both candidates receive signal i = 1, then both choose m1,1, yielding voter δ’s

expected utility of (1/2a)R

u(α + 1 + δ,m1,1)dα. If both receive signal i = −1, then δ’s expected

utility is (1/2a)R

u(α + 1 + δ,m−1,−1)dα. If one candidate receives signal i = 1 and the other

receives signal i = −1, then they locate at m1,1 and m−1,−1, and the candidate closest to the

median voter wins the election. That is, the left candidate wins if µ < 0, or equivalently if α < −1;

and the right candidate wins if µ > 0, or equivalently if α > −1. Thus, voter δ’s expected utility is

12a

Z −1

−au(α + 1 + δ,m−1,−1) dα +

12a

Z a

−1u(α + 1 + δ,m1,1) dα.

A similar analysis holds for β = −1. Thus, the voter’s welfare is

W (δ; p, q, a) =12

"(qp + (1 − q)p2)

12a

Z a

−au(α + 1 + δ,m1,1) dα

+�q(1 − p) + (1 − q)(1 − p)2

� 12a

Z a

−au(α + 1 + δ,m−1,−1) dα

+ (2(1 − q)p(1 − p))�

12a

Z −1

−au(α + 1 + δ,m−1,−1) dα +

12a

Z a

−1u(α + 1 + δ,m1,1) dα

�#

+12

"(qp + (1 − q)p2)

12a

Z a

−au(α− 1 + δ,m−1,−1) dα

+�q(1 − p) + (1 − q)(1 − p)2

� 12a

Z a

−au(α− 1 + δ,m1,1) dα

+ (2(1 − q)p(1 − p))�

12a

Z 1

−au(α− 1 + δ,m−1,−1) dα +

12a

Z a

1u(α− 1 + δ,m1,1) dα

�#,

47

where we recall that m1,1 is also a function of p and q.

Gathering the two terms multiplied by qp + (1 − q)p2, corresponding to the candidates both

receiving the correct signal, yields

14a

Z a

−au(α + 1 + δ,m1,1) dα +

14a

Z a

−au(α− 1 + δ,m−1,−1)

= − 14a

Z a

−a(δ − (m1,1 − α− 1))2 dα − 1

4a

Z a

−a(δ − (m−1,−1 − α + 1))2 dα

= −δ2

2+ δ(m1,1 − 1) − 1

4a

Z a

−a(m1,1 − α− 1)2 dα − δ2

2+ δ(m−1,−1 + 1) − 1

4a

Z a

−a(m−1,−1 − α + 1)2 dα

= −δ2 +14a

Z a

−au(α + 1,m1,1) dα +

14a

Z a

−au(α− 1,m−1,−1) dα, (18)

where we use m−1,−1 = −m1,1 and the fact that the expected value of α is zero. Similar manipu-

lations for the terms multiplied by q(1 − p) + (1 − q)(1 − p)2 and 2(1 − q)p(1 − p), establish that

the voter’s welfare is just δ2 less than the welfare of the median voter (δ = 0), independent of the

polling parameters p and q. Thus, all voters agree on the ranking of polling technologies determined

by the expected utility of each (p, q) pair.

As a result, we need only consider the welfare of the median voter and set δ = 0. Exploiting the

inherent symmetry in the environment and using a change of variables yields, we obtain a simple

expression for the median voter’s welfare:

W (p, q, a) = −(1 − q)p(1 − p)�

12a

Z 1

−a(α + 1 + m1,1)2 dα +

12a

Z a

−1(α + 1 −m1,1)2 dα

�

− 14a

Z a

−a[(α + 1 −m1,1)2(qp + (1 − q)p2) + (α + 1 + m1,1)2(q(1 − p) + (1 − q)(1 − p)2)] dα,

where again m1,1 is also a function of (p, q). This, in turn, allows us to analyze the voters’ unanimous

preferences over polling technologies.

Our results identify three major forces that determine welfare. Two of these forces are statistical

properties of polling technology, namely the signal’s precision and correlation. The third is the

strategic effect on candidate location, which compares the equilibrium candidate choices with the

location of “non-strategic” candidates, who condition their choice only on their private information.

It is informative to consider first a simplified version of the model, in which the strategic effect is

not present. Specifically, suppose that a = 0, so that the median ideal point is just β. Then the

signal correlation is

q + (1 − q)(4p2 − 4p + 1),

which rises with p and q. In this benchmark case, strategic considerations do not arise because

m1 = m1,1 = 1 and m−1 = m−1,−1 = −1, so that each candidate locates as if he/she were non-

strategically targeting the median voter given only his/her own information. Because a = 0, a

48

candidate who receives the correct signal successfully targets the median with probability one, a

significant difference from the general model.

The median voter’s utility is therefore zero when at least one candidate receives the correct

signal, and it is equal to −4 if both candidates get the wrong signal, so that welfare reduces to

W (p, q) = −4(q(1 − p) + (1 − q)(1 − p)2).

Differentiating with respect to p and q, it follows immediately that increased precision raises welfare,

but increased correlation, in the sense of higher q, reduces welfare:

∂

∂qW (p, q) = −4p(1 − p) < 0

and

∂

∂pW (p, q) = 4[(2 − q)(1 − p) + qp] > 0.

Though raising either p or q increases the correlation of the candidates’ signals, thereby limiting the

options of the median voter, increasing p does this by increasing the probability that a candidate

receives the correct signal. In the simplified model, this raises the probability that at least one

candidate successfully targeted the median, which is clearly beneficial to the median voter. In

contrast, increasing q has the effect of increasing the probability that both candidates receive the

incorrect signal, which reduces welfare.

To highlight further the adverse effects of correlation, we add a “Blackwell garble” to candidates’

signals. This decreases signal precision, yet, because it also reduces correlation, it actually raises

voter welfare. For simplicity, suppose q = 1, so that signals are perfectly correlated, and introduce

i.i.d. garbles to signals: With probability r a candidate receives his/her original signal, and with

probability 1− r the candidate receives the opposite signal. Then, viewing welfare as a function of

r alone, we have

W (r) = −4[(1 − r)2p + r2(1 − p)].

Differentiating with respect to r reveals that increasing the garble raises welfare if signals are not

too precise: If p < r, then

d

drW (r) = 8(p− r) < 0.

The garble reduces the precision of the signal, and, in particular, it decreases the probability that

both candidates receive the correct signal. But it also reduces the correlation between signals,

raising the probability that at least one candidate receives the correct signal. The latter positive

effect outweighs the former negative effect, resulting in a net increase in voter welfare. The garble

49

would always hurt voters if they had to select a given candidate, because the garble causes any

given candidate to target the median less accurately, but voters can choose between candidates

when they offer distinct platforms, and this option to choose has significant value.

Turning to the full-fledged two-signal stacked-uniform model, we first analyze the strategic

effect. We compare the equilibrium candidate choices with (a) those of non-strategic candidates

who condition their choice only on their private information, and (b) with the optimal location,

m∗, from the standpoint of the voters. That is, m∗ maximizes expected voter welfare, W (p, q, a, δ),

when we treat the position m1,1 as a variable and p, q, and a as parameters. Inspection of equation

(18) reveals that all voters agree on the optimal m∗. Of course, m∗ varies with these parameters,

a dependence that we suppress notationally. We have seen that the strategic interaction between

candidates causes them to locate more extremely than their private information would suggest. We

now show that if the voters could choose the amount by which candidates “biased” their location

following a signal realization, then voters would choose to increase further the bias, as they value

the increased choice.

Proposition 1 The optimal location m∗1 for the voters is increasing in p and decreasing q, and

m∗1 > m1,1 > m1.

The difference m∗ − m1,1 is (1) increasing in a, (2) decreasing in q, and (3) decreasing in p for a

and or q sufficiently large.

Proof: Solving the first order condition for maxW (p, q, a) with respect to “m”, we obtain

m∗ = 2p − 1 + (1 − q)p(1 − p)(a +1a

).

Comparing the optimal m∗ with the actual m1,1 chosen by the candidates, we obtain

m∗ −m1,1 = (2p − 1)�

1 − 1q + (1 − q)(p2 + (1 − p)2)

�+ (1 − q)p(1 − p)(a +

1a

) ≥ 0.

By inspection, this difference is increasing in a, for a > 1. Differentiating this difference with

respect to q for p, q ∈ (0, 1) yields

∂(m∗ −m1,1)∂q

= 2(2p − 1)p(1 − p)

(1 − 2(1 − q)p(1 − p))2−�a +

1a

�p(1 − p)

≤ (2p− 1)p(1 − p)(1 − 2(1 − q)p(1 − p))2

− 2p(1 − p)

= p(1 − p)�

(2p − 1)(1 − 2(1 − q)p(1 − p))2

− 2�≤ p(1 − p)

�(2p − 1)

(1 − 2p(1 − p))2− 2

�< 0.

50

Now differentiate m∗ −m1,1 with respect to p to obtain

∂(m∗ −m1,1)∂p

= p(1 − p)�

2(2p − 1)(1 − 2(1 − q)p(1 − p))2

−�a +

1a

��

which is declining in q and a, and negative for a or q sufficiently large. To see that this derivative

can be increasing, set a + 1a equal to its lower bound of 2.5, and q = 0, in which case

∂(m∗ −m1,1)∂p

= 2p(1 − p)�

(2p− 1)(1 − 2p(1 − p))2

− 1.25�,

which is positive for p ∈ [.728, .856].

Turning our consideration to the statistical properties of the polling technology, it is easy to

show that increased signal correlation, in the sense of higher q, is always bad in terms of welfare,

as it reduces the likelihood that candidates take different locations.

Proposition 2 W (p, q, a) is decreasing in q.

Proof: After substituting m1,1(p, q) into W (p, q, a), Maple yields two solutions, for arbitrary p

and q, to the condition ∂∂a

�∂∂qW

�= 0: These are a = −1 and a = 1. Thus, ∂

∂a

�∂∂qW

�maintains

the same sign on [2,∞). A Maple calculation yields ∂∂a

�∂∂qW

�< 0. We need only show then that

∂∂q

���a=1

W < 0, which we have verified with Maple for all p ∈ [1/2, 1] and q ∈ [0, 1].

The effect of the precision parameter p is ambiguous: It turns out that welfare is increasing

in p over a certain range and then decreasing beyond that. Figure 5, below, illustrates this point.

Here, level sets of W in (p, a) are derived for the case q = 0, i.e., perfectly independent signals, as

well as level sets in (p, q) for the case a = 2.

[ Figure 5 here. ]

We plot the optimal precision, p∗(q, a), as a function of q and a in Figure 6. Note that it is less

than one, decreasing in a, and increasing in q. Furthermore, the optimal precision goes to one as q

goes to one.

[ Figure 6 here. ]

51

These comparative statics contrast with those of the above simplified version of the model, with

a = 0, where an increase in precision leads to an unambiguous increase in voter welfare. An increase

in p still increases the probability that both candidates receive the “correct” signal, in which case

both candidates locate at the corresponding conditional median, but, for general a > 0, that policy

is not necessarily close to the median voter’s ideal point. For concreteness, suppose that β = 1, that

both candidates receive signal 1, and that both, accordingly, locate at m1,1. If the median voter is

indeed to the right of zero, i.e., α+ 1 ≥ 0, then the median prefers m1,1 to m−1,−1, and the median

voter could hope to do no better. However, if the median were to the left of zero, which occurs with

probability (a − 1)/2a > 0, then the median would prefer m−1,−1 and would have been better off

if at least one candidate had received signal −1. Indeed, the ideal situation for the median voter

is that the candidates receive distinct signals, giving the median a choice between the two possible

platforms. An increase in p increases correlation and decreases the probability that the candidates

offer voters a choice, and it can thereby reduce voter welfare.

52

A Appendix

Lemma 1 Let (G,H) be any pair of mixed strategies. For all s ∈ S and all w, z ∈ �, one of threepossibilities obtains: Either

Xt∈T

P (t|s)[Ht(z)−Ht(z)−][Fs,t(w)] =12

Xt∈T

P (t|s)[Ht(z)−Ht(z)−] =Xt∈T

P (t|s)[Ht(z)−Ht(z)−][1−Fs,t(w)],

or

Xt∈T

P (t|s)[Ht(z)−Ht(z)−][Fs,t(w)] <12

Xt∈T

P (t|s)[Ht(z)−Ht(z)−] <Xt∈T

P (t|s)[Ht(z)−Ht(z)−][1−Fs,t(w)],

or the reverse inequalities hold. Likewise for G and all t ∈ T and all w, z ∈ �.

Proof: Given s ∈ S and z ∈ �, the first possibility clearly obtains if P (t|s)[Ht(z)−Ht(z)−] = 0 for

all t ∈ T . Suppose P (t|s)[Ht(z) −Ht(z)−] > 0 for some t ∈ T , and define the distribution function

F ∗ as follows:

F ∗s (w) =

Pt∈T P (t|s)[Ht(z) −Ht(z)−]Fs,t(w)P

t∈T P (t|s)[Ht(z) −Ht(z)−].

Then the three possibilities above correspond to the three possibilities: F ∗s (w) = 1/2, F ∗

s (w) < 1/2,

and F ∗s (w) > 1/2.

Lemma 2 Let (G,H) be a mixed strategy Bayesian equilibrium. For all z ∈ �, if Gs′(z)−Gs′(z)− >

0 for some s′ ∈ S and Ht′(z) −Ht′(z)− > 0 for some t′ ∈ T with P (s′, t′) > 0, then

Xt∈T

P (t|s′)[Ht(z)−Ht(z)−][Fs′,t(z)] =12

Xt∈T

P (t|s′)[Ht(z)−Ht(z)−] =Xt∈T

P (t|s′)[Ht(z)−Ht(z)−][1−Fs′,t(z)].

and

Xs∈S

P (s|t′)[Gs(z)−Gs(z)−][Fs,t′(z)] =12

Xs∈S

P (t′|s)[Gs(z)−Gs(z)−] =Xs∈S

P (s|t′)[Gs(z)−Gs(z)−][1−Fs,t′(z)].

Proof: We prove the first equalities. If they do not hold for some z and some s′ and t′ with

P (s′, t′) > 0, then, by Lemma 1, we may assume that

Xt∈T

P (t|s′)[Ht(z) −Ht(z)−][1 − Fs′,t(z)] >12

Xt∈T

P (t|s′)[Ht(z) −Ht(z)−] (19)

53

orXt∈T

P (t|s′)[Ht(z) −Ht(z)−][Fs′,t(z)] >12

Xt∈T

P (t|s′)[Ht(z) −Ht(z)−].

We focus on the first inequality, as a symmetric proof addresses the second. For each t ∈ T , let λt

denote the measure generated by the distribution Ht, let µt denote the degenerate measure with

probability Ht(z)−Ht(z)− on z, and let νt = λt−µt. Let {xn} be a sequence decreasing to z, and

let Gn be the mixed strategy defined by replacing Gs in G with point mass on xn. Let

πs′,t(w) = πA(z,w|s′, t)

denote A’s probability of winning using z when B receives signal t and chooses platform w, and

πns′,t(w) = πA(xn, w|s′, t)

denote A’s probability of winning using xn when B receives signal t and chooses platform w. Then

ΠA(Gn,H|s′) − ΠA(G,H|s′) =Xt∈T

P (t|s′)Z

[πnt (w) − πt(w)]λt(dw)

=Xt∈T

P (t|s′)[Ht(z) −Ht(z)−]�1 − Fs′,t

�z + xn

2

�− 1

2

�+Xt∈T

P (t|s′)Z

[πnt (w) − πt(w)] νt(dw).

Since πns′,t − πs′,t → 0 almost everywhere (νt), the corresponding integral terms above converge tozero. Thus,

limn→∞ ΠA(Gn, H |s′)−ΠA(G,H |s′) =

Xt∈T

P (t|s′)[Ht(z)−Ht(z)−][1−Fs′,t(z)]−12

Xt∈T

P (t|s′)[Ht(z)−Ht(z)−] > 0,

by (19). It follows that ΠA(Gn,H|s) > ΠA(G,H|s) for high enough n, a contradiction.

We now define a condition that imposes distinct medians, conditional on sets of signal realiza-

tions. It weakens (C4) and, in particular, is stated without the symmetry conditions (C1)-(C3).

(C4′) For all s, s′ ∈ S, not conditionally equivalent, and all T ′ ⊆ T with P (T ′|s) > 0 and P (T ′|s′) >

0, we have ms,T ′ �= ms′,T ′ . For all t, t′ ∈ T , not conditionally equivalent, and all S′ ⊆ S with

P (S′|t) > 0 and P (S′|t′) > 0, we have mS′,t �= mS′,t′ .

(C4′) is a very weak restriction on candidate’s beliefs. Fixing the conditional distributions

{Fs,t} and viewing the marginal P on S × T as an element p of the unit simplex ∆ ⊆ �|S|+|T |,

the next proposition establishes that (C4′) is satisfied generically. Let ms,T ′(p) and mS′,t(p) be the

conditional medians determined by p and the conditional distributions.

54

Proposition 3 Assume that, for all s ∈ S and all z ∈ �, there exist t, t′ ∈ T such that Fs,t(z) �=Fs,t′(z). And assume that, for all t ∈ T and all z ∈ �, there exist s, s′ ∈ S such that Fs,t(z) �=Fs′,t(z). Then the set

P = {p ∈ ∆ : (C4′) is satisfied }

is open and dense in ∆.

Proof: Fix s, s′ ∈ S, not conditionally equivalent, and T ′ ⊆ T . It suffices to show that the

set of p’s such that ms,T ′(p) = ms′,T ′(p) is closed and nowhere dense. That it is closed is obvious,

following from continuity of the conditional distributions. Take any p ∈ ∆, and note that, by

assumption, there exist t, t′ ∈ T such that Fs′,t(ms′,T ′(p)) < Fs′,t′(ms′,T ′(p)). Now perturb p so

that P (·|s) is unchanged, but P (·|s′) moves an arbitrarily small amount of probability from t to t′.

This leads to a decrease in the median conditional on s′ and T ′, as required.

Lemma 3 Assume (C4′). Let (X,Y ) be a pure strategy Bayesian equilibrium. If xs = yt for some

s ∈ S and some t ∈ T with P (s, t) > 0, then xs = yt = ms,t.

Proof: Let S′ = {s ∈ S : xs = z}, and let T ′ = {t ∈ T : yt = z}. Take any s′ ∈ S′ and any

t′ ∈ T ′ with P (s′, t′) > 0. Lemma 2 implies

Xt∈T ′

P (t|s′)P (T ′|s′) [Fs′,t(z)] =

12, (20)

where we use Ht(z) − Ht(z)− = 1. Thus, z = ms′,T ′ . If there exists s ∈ S′ with s′ �= s and

P (s, t′) > 0, then (20) must hold for s as well, implying z = ms,T ′. Since P (T ′|s′) > 0 and

P (T ′|s) > 0, (C4′) implies that s and s′ are conditionally equivalent. The symmetric argument for

candidate B establishes that P (s′, t) > 0 implies that t and t′ are conditionally equivalent. Now

take any t ∈ T such that P (t|s′) > 0. This implies P (s′, t) > 0, so Fs′,t = Fs′,t′ . Therefore, (20)

reduces to Fs′,t′(z) = 1/2, i.e., z = ms′,t′ .

Lemma 4 In the canonical model, assume (C8) and (C9). For each j ∈ I, let αj ∈ [0, 1]. For all

i, i′ ∈ I with i < i′ and for all z ∈ M with

0 < αjP (j|i)P (j|i′)Fi′,j(z) < αjP (j|i)P (j|i′)

55

for at least one j, we havePj∈I αjP (j|i)Fi,j(z)P

j∈I αjP (j|i) >

Pj∈I αjP (j|i′)Fi′,j(z)P

j∈I αjP (j|i′) .

Proof: Take i, i′ ∈ I, z ∈ �, and αj ’s as in the statement of the lemma. By assumption,

αjP (j|i) > 0 and αjP (j|i′) > 0 for some j, so cross multiply and rewrite the desired inequality as

Xj,j′∈I

αjαj′P (j|i)P (j′|i′)Fi,j(z) >Xj,j′∈I

αjαj′P (j|i′)P (j′|i)Fi′,j(z).

We compare the two sides of the inequality one pair {j, j′} at a time. For j = j′, we have

α2jP (j|i)P (j|i′)Fi,j(z) ≥ α2

jP (j|i)P (j|i′)Fi′,j(z)

from (C9). Moreover, there is at least one j such that α2jP (j|i)P (j|i) > 0 and Fi′,j(z) ∈ (0, 1),

which implies Fi,j(z) > Fi′,j(z) and gives us a strict inequality. For distinct j and j′, say j < j′, we

want to show that

αjαj′ [P (j|i)P (j′|i′)Fi,j(z)+P (j′|i)P (j|i′)Fi,j′(z)] ≥ αjαj′ [P (j|i′)P (j′|i)Fi′,j(z)+P (j′|i′)P (j|i)Fi′ ,j′(z)].

Note that by (C9), we have

Fi,j(z) ≥ max{Fi,j′(z), Fi′,j(z)} ≥ min{Fi,j′(z), Fi′,j(z)} ≥ Fi′,j′(z),

and therefore

Fi,j(z) − Fi′,j′(z) > Fi′,j(z) − Fi,j′(z).

Then (C8) implies

P (j|i)P (j′|i′)(Fi,j(z) − Fi′,j′(z)) ≥ P (j|i′)P (j′|i)(Fi′,j(z) − Fi,j′(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 ∈ M , if Gi(z)−Gi(z)− > 0 for some i ∈ I and Hj(z)−Hj(z)− > 0 for some

j ∈ I with P (i, j) > 0, then z = mi,j.

56

Proof: Define the sets

S′ = {i ∈ I : Gi(z) −Gi(z)− > 0}T ′ = {j ∈ I : Hj(z) −Hj(z)− > 0}.

Take any i′ ∈ S′ and j′ ∈ T ′ such that P (i′, j′) > 0. Lemma 2 implies

Xj∈I

P (j|i′)[Hj(z) −Hj(z)−][Fi′,j(w)] =12

Xj∈I

P (j|i′)[Hj(z) −Hj(z)−].

If P (i, j′) > 0 for some i ∈ S′ with i �= i′, then the above equality must hold for i as well. Setting

αj = Hj(z) − Hj(z)−, we see that (C8), (C9), and Lemma 4 imply that i and i′ are conditionally

equivalent. The symmetric argument for candidate B establishes that P (i′, j) > 0 implies that j and

j′ are conditionally equivalent. Now take any j ∈ I such that P (j|i′) > 0. This implies P (i′, j) > 0,

so Fi′,j = Fi′,j′ . Therefore, the above implication of Lemma 2 reduces to Fi′,j′(z) = 1/2, i.e.,

z = mi′,j′.

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