Mixed Strategy Equilibrium and DeepCovering in Multidimensional Electoral

Competition

John Duggan∗ Matthew O. Jackson†

Preliminary: January 13, 2004This Draft: December 12, 2005

Abstract

We prove existence of mixed strategy equilibrium in a generalmodel of elections that includes multicandidate Downsian models andprobabilistic voting models as special cases. We do so by modellingvoters explicitly as players, enabling us to resolve discontinuities inthe game between the candidates, which have proved a barrier toexistence. We then give a partial characterization: the supports ofequilibrium mixed strategies in two-candidate Downsian games mustlie in the deep uncovered set.

Journal of Economic Literature Classification Numbers: C72, D72.Key words: Electoral Competition, Colonel Blotto, elections, mixed

strategy equilibrium, undominated strategies, uncovered set

∗Department of Political Science and Department of Economics, University ofRochester, Rochester, NY 14627, http://www.rochester.edu/College/PSC/duggan, email:dugg@troi.cc.rochester.edu. Financial support under NSF grant SES-0213738 is gratefullyacknowledged.

†Division of Humanities and Social Sciences, California Institute of Technology,Pasadena, CA 91125, http://www.hss.caltech.edu/∼jacksonm/Jackson.html, email: jack-sonm@hss.caltech.edu. Financial support from the Center for Advanced Studies in theBehavioral Sciences, the Guggenheim Foundation and from the NSF under grant SES-0316493 is gratefully acknowledged.

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1 Introduction

The formal theory of elections, originated by Hotelling (1929), Downs (1957),and Black (1958), has developed despite a well-known and unresolved founda-tional issue arising in the basic model: the existence of equilibrium in mul-tidimensional policy spaces. Plott (1967) showed that a multidimensionalmodel typically does not admit a pure strategy equilibrium, and in mostcases it has remained unclear even whether mixed strategy equilibria exist.As a consequence, much research has either been limited to unidimensionalpolicy spaces, or has been confined to special variations of the basic modelthat circumvent the existence problems; such as probabilistic voting, policy-motivated candidates, citizen-candidates, candidates who wish to maximizevote share from a heterogeneous continuum of voters, and repeated elections.

Clearly multidimensional policy spaces are fundamental to the under-standing of political interaction, and it is essential to be able to model them.Moreover, it would be especially useful to incorporate multidimensional poli-cies in classical models. The hurdle that such models face, and which makesthe existence questions so hard to answer, is that such models exhibit discon-tinuities that could potentially preclude the existence of equilibrium (evenwhen mixed strategies are allowed). To see this, consider the most basicDownsian model of competition between two candidates in a multidimen-sional space. Each candidate simultaneously chooses a policy from somemulti-dimensional space. Voters then vote for their most preferred candi-date. A candidate receives a payoff of 1 if her policy position is preferredby a majority of voters to the opposing candidate’s position and a payoff of−1 if this situation is reversed (and a payoff of 0 otherwise). There are dis-continuities in candidates’ payoffs, since when they choose the same policies(or, more generally, any time they choose policies that render some votersindifferent), the voter behavior changes from preferring one of the candidatesoutright and voting for that candidate, to being indifferent. Thus, a candi-date can have a majority of voters all along a sequence of policy choices, andnot have a majority in the limit; or vice versa. This means that standardresults on existence of mixed strategy equilibria do not apply. This problemis not only endemic to Downsian models, but to a very wide variety of othermodels as well.

In this paper, we establish existence of equilibria in a general class of

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electoral games, including the Downsian model and others where such dis-continuities are present. Our starting point is the observation that the dis-continuities of the classical Downsian game, as well as many other votinggames, arise from the implicitly fixed behavior of indifferent voters. It isgenerally assumed that a voter who is indifferent between two candidatesflips a fair coin. The central observation that allows us to overcome thesedifficulties is that by including the voters as players in the game, so that anindifferent voter might flip a biased coin or simply vote for one candidate orthe other when indifferent, existence can be established in a very wide classof games, including all of the classical models.

Our proof technique builds on an insightful theorem by Simon and Zame(1990) which shows that existence of equilibrium in games with discontinu-ities can be overcome if outcomes at discontinuity points can be properlyselected. We show that by including voters as players in the game, the neces-sary selections of election outcomes can in fact be rationalized by equilibriumplay on the part of voters.1

Our results apply to a general electoral model, which includes completeinformation as a special case, but also allows voters to have uncertain pref-erences. Candidates can care about policies, or winning, or vote-shares, orelaborate combinations of these outcomes. Similarly, voters can care aboutwho wins, or which policy is chosen, or some combination of the two. Thecandidates strategies are general sets of campaign strategies, which we inter-pret as campaign platforms. The strategy sets are sufficiently general thata strategy could be a plan for running a campaign in the reduced form ofa highly complex election: we are not bound by the traditional assumptionof simultaneous chosen policy platforms. We eliminate dominated votingstrategies, but in contrast to the classical approach, we model the votes ofindifferent voters as endogenous. We prove the existence of equilibria in

1It remains an open question to what extent equilibria exist when voter behavior isfixed (to say flipping fair coins when indifferent). Theorems on equilibrium existence indiscontinuous games have proved difficult to apply because they employ complex sufficientconditions, e.g., Dasgupta and Maskin’s (1986) weak upper semicontinuity or, more gen-erally, Reny’s (1999) better reply security. Duggan (2005) provides a sufficient conditionfor better reply security and verifies that it holds in the classical Downsian model withthree voters, but his approach does not extend to multiple voters. Another approach fromthe auctions literature (e.g., Lebrun (200?) and Jackson and Swinkels (2005)) that mightbe fruitful is to prove existence with endogenous outcomes, and then show that ties neveroccur and that changing voter behavior at ties would not matter.

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the multicandidate model – indeed, we establish existence of Duvergerianequilibria in which only two candidates receive votes.

Beyond the results on existence, we also show that candidates only choosestrategies that lie in a refinement of the top cycle of the majority voting rela-tion. In particular, we show that the support of winning candidates’ strate-gies lie in a variation of the uncovered set. This shows that results found tohold in finite policy settings have proper analogs in continuous multidimen-sional policy settings. In particular, Laffond, Laslier, and Le Breton (1993)showed that if policies must be chosen from a finite set and there are nomajority indifferences, then the mixed strategy equilibrium of a Downsiangame is unique and that the support of the equilibrium strategy lies in theuncovered set.2 McKelvey (1986), in the context of the multidimensionalspatial model, conjectured that the support of equilibrium mixed strategies- if any exist – must lie in the uncovered set, and this was confirmed byBanks, Duggan, and Le Breton (2002). These results leave open the issue ofexistence in the classical model, and also whether the equilibrium strategiesmust have this property outside of a particular spatial model. Here we proveexistence and then also show that the support of equilibrium strategies is inthe uncovered set in a general class of elections.

The paper proceeds as follows. Section 2 presents an preview of thegeneral electoral model and the results of the paper. Section 3 provides a fulldescription of the setting and the details omitted in the overview. Section4 contains equilibrium existence results that “black box” the behavior ofvoters as a correspondence satisfying several technical properties. In Section5, we apply these results to establish existence in the multicandidate completeinformation model. In Section 6, we develop the deep uncovered set and showthat this set bounds the equilibrium outcomes of two-candidate Downsianelections. Section 7 applies our results on endogenous voting equilibria to ageneral probabilistic voting model. Our final section discusses an extensionto a world where public randomization is possible. The proofs appear inan appendix, and another appendix contains an example illustrating thetightness of our bounds on equilibrium strategies.

2In the presence of majority indifferences, the literature assumed that indifferent votersrandomize over candidates with equal probabilities. Assuming a finite policy space, Duttaand Laslier (1999) show that supports of equilibrium strategies still lie in the uncoveredset.

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2 A Preview of the Model and Results

In this section, we provide statements of special cases of our electoral modeland results. This section lacks the full generality of the model and resultsthat follow in later sections, but allows for a simpler presentation that coversmany of the standard cases from the literature and should be more easilyaccessed by the non-specialist.

We consider an election among a set of candidates C = {A,B, . . . , M} tobe decided by an electorate of voters V = {1, 2, . . . , n} using plurality rule.

The candidates C ∈ C simultaneously choose policy positions, xC , in acompact subset X of IR`. Let x = (xA, . . . , xM) denote a profile of policyplatforms for the candidates, and let X denote the set of platform profiles.

Voters observe the policy positions and then cast votes. Voter i, havingobserved the candidates’ platforms, casts a ballot bi in C for one of thecandidates, with no abstention allowed. Let b = (b1, . . . , bn) ∈ Cn denotea profile of ballots, i.e., an election return which indicates which candidateeach voter voted for. The winner is the candidate with the greatest numberof votes, with ties broken by an even lottery among the tied candidates. Let

W (b) =

{C ∈ C : #{i : bi = C} ≥ #{i : bi = C ′} for all C ′ ∈ C

}

denote the set of potential winners (before any ties are broken), for a givenelection return (b.

Candidate C has a continuous utility function uC : Cn ×X → IR, wherethe payoff to candidate C from an outcome (b, x) is uC(b, x). For our existenceresults, we impose a restriction on the ways in which uC is allowed to dependon election returns. Because that restriction is mainly technical, here wewill simply note that it is satisfied in the following three special cases, allprominent in the literature, and refer the interested reader to later sectionsfor the fuller model.

• win-motivation:

uC(b, x) =

{1

#W (b)if C ∈ W (b)

0 else.

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• vote-motivation:

uC(b, x) = #{i : bi = C}.

• policy-motivation:

uC(b, x) =∑

C′∈W (b)

uC(xC′)

#W (b).

Voter i with a continuous utility function ui : C×X → IR, and the payoffto voter i from outcome (b, x) is

ui(b, x) =∑

C′∈W (b)

ui(C′, xC′)

#W (b).

An important special case is where voters care only about policy outcomes:we say that voter i is policy-oriented if ui(C, x) does not depend on C.

We allow candidates and voters to use mixed strategies in equilibrium,as in many applications this is actually necessary for equilibrium existence.Some of our formal results focus on electoral equilibria in which candidatemixed strategies are symmetric, i.e., the candidates mix over policy platformswith the same probability distribution. We consider voting equilibria inwhich no voter puts positive probability on a “dominated” strategy. Fortechnical reasons (discussed in the next section), our refinement is weakerthan the usual notion of undominated strategies: we say a ballot bi for voteri is undominated* if bi is not the strictly worst candidate for voter i. Ourdefinition of undominated strategies coincides with the usual one when thereare only two candidates.

We say that an electoral equilibrium is Duvergerian if there are two can-didates such that, for all profiles of platforms, only those two candidatesreceive votes. That is, there are only two “viable” candidates. In this case,we say a voting strategy for i is symmetric if i’s vote between the two viablecandidates depends not on the name of the candidate, but only on the candi-dates’ platforms. So, for example, if i votes for candidate A with probability1/3 and candidate B with probability 2/3, and then we consider a platformprofile in which the candidates’ platforms are interchanged, then the votermust now vote for A with probability 2/3 and for B with probability 1/3.

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Our first result establishes the general existence of subgame perfect equi-libria in the electoral model. (Theorems are numbered as they are presentedin later sections.) The key step in the proof of Theorem 3 is to convert theelectoral game to a game with an “endogenous sharing rule,” allowing us toapply results of Simon and Zame (1990) and Jackson, Simon, Swinkels, andZame (2002) to specify the behavior of indifferent voters appropriately.

Theorem 3 Consider the multi-candidate complete information electoral model,where there are at least four voters or there are only two candidates. Thereexists a subgame perfect equilibrium in undominated* voting strategies. More-over, there exists such an equilibrium that is Duvergerian, in which two ar-bitrary candidates A and B receive votes. If all voters are policy-orientedand candidates A and B are both win-motivated or both vote-motivated, thenthere exists such an equilibrium that is symmetric on {A, B}.

We next give a partial characterization of equilibrium policy platformsin the Downsian model, where we assume candidates seek to maximize theirprobabilities of winning. We show that policy platforms lie in the deepuncovered set, a centrally located region of the policy space that has receivedconsiderable attention in the political science literature. It is one of manypossible extensions of the uncovered set to settings where ties are possible inthe majority voting relationship.

The deep uncovered set consists of the policies x such that, for every otherpolicy y, either x is weakly majority-preferred to y or there is a policy z suchthat x is weakly majority-preferred to z and z is weakly majority-preferredto y.

Theorem 5 In the two-candidate Downsian model (where candidates arewin motivated and voters are policy oriented), in every subgame perfect equi-librium that has symmetric and undominated* voting strategies, the supportof each candidate’s strategy lies in the deep uncovered set.

Theorem 4 tells us that every policy in the support of an equilibriummixed strategy is weakly majority-preferred to every other policy in at mosttwo steps.

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We also provide an existence result for a general model of probabilisticvoting in two-candidate elections. Each voter’s policy preferences now de-pend on a type ti, unobserved by the candidates, and we extend voter utilitiesto include this as an argument, as in ui(C, x, ti). We do not restrict voterpreferences, thereby capturing two special cases prominent in the electoralmodelling literature.

• stochastic candidate bias:

ui(C, x, ti) = ui(x) + ti,C .

• stochastic policy preference:

ui(C, x, ti) = ui(x, ti).

Although most applications of the probabilistic voting model impose somestructure on the distribution of voter types (requiring, for instance, thatvoters are indifferent with probability 0), we do not restrict the distributionof voter types in any way.

Theorem 6 Consider the multi-candidate probabilistic voting model, wherethere are at least four voters or there are only two candidates. There existsa perfect Bayesian equilibrium in undominated* voting strategies. Moreover,there exists such an equilibrium that is Duvergerian, in which two arbitrarycandidates A and B receive votes. If all voters are policy-oriented and can-didates A and B are both win-motivated or both vote-motivated, then thereexists such an equilibrium that is symmetric on {A,B}.

Note that the complete information model is a special case of the proba-bilistic voting model.

Thus, we prove existence of mixed strategy equilibria the canonical modelsof elections, and we give a partial characterization in terms of the uncoveredset for the basic two-candidate Downsian model. In the sequel, we also giveconvergence results as we vary voter preferences and (in the probabilisticvoting model) the distribution of voter types.

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3 A General Model of Elections

We now present the full model.

Candidates and Policies Each candidate C ∈ C chooses a policyposition, xC , in a second countable, locally compact Hausdorff space XC .Let x = (xA, . . . , xM) denote a profile of policy platforms for the candidates,and let X =

∏C∈C XC denote the set of platform profiles. Let X =

⋃XC

denote the space of all possible policies. A mixed platform for a candidateC ∈ C is a Borel probability measure ξC on XC , with the product measureξ = ξA×· · ·×ξM representing a profile of mixed platforms for the candidates.

Voting Correspondences We explicitly model voters’ behavior in sub-sequent sections, but here we analyze the game among just among the can-didates, black-boxing voter behavior. For now, we capture voter behaviorthrough a correspondence, Ψ: XM ⇒ ∆(Cn), from platform profiles to prob-ability distributions over election returns. The correspondence Ψ representspossible (equilibrium) voter behavior contingent on the candidates’ positions.A measurable selection ψ from Ψ determines a well-defined (but generallydiscontinuous) game among the candidates.

Election Returns and Statistics We wish to admit applications wherecandidates have preferences over more than simply whether they win and/orwhich policy is enacted. For instance, we wish to allow for applications wherecandidates care about their vote totals or vote shares. In order to definecandidate payoffs in such a general manner, we let candidate preferencesover election returns be dependent on a statistic σ : Cn → S, where S isa nonempty, compact, and convex subset of a finite-dimensional Euclideanspace.

We impose two conditions on this statistic, both of which only apply to

situations where at most two candidates receive votes. Let BAB

denote theset of election returns in which only candidates A and B receive votes; i.e.,

BAB

={b ∈ Cn | for all i ∈ V, bi ∈ {A,B}. }

The first condition that we impose on the statistic is that whenever bal-lots are restricted to any pair of candidates, the range of statistic is one-

dimensional ({σ(b)|b ∈ BAB} is one-dimensional for any A and B).

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The second condition that we impose on the statistic is that wheneverballots are restricted to any pair of candidates, the statistic is nondecreasingin the set of votes cast for one of the candidates (and hence nonincreasing inthe votes cast for the other). Given two candidates A and B and two election

returns b, b′ ∈ Cn, write b ≥A b

′if everyone who voted for A in b also votes

for A in b′, i.e., for all i ∈ V, b′i = A implies bi = A. Then the monotonicity

condition is for all A,B ∈ C, either

for all b, b′ ∈ B

AB, b ≥A b

′implies σ(b) ≥ σ(b

′),

or

for all b, b′ ∈ B

AB, b ≥A b

′implies σ(b) ≤ σ(b

′).

Candidate Preferences Each candidate C has a utility function uC :S × X → IR, which is jointly continuous. Moreover, uC(s, x) is affine ins. The payoff to candidate C from an outcome (b, x) is uC(σ(b), x). Letu denote a profile of utility functions, one for each candidate. An electoralgame is then a triple (X, Ψ, u) satisfying the maintained assumptions statedabove.

The main approaches to modeling candidates in the literature all satisfyour assumptions. Some prominent special cases of the model are:

• win-motivation: let S be the unit simplex in IRM , let

σC(b) =

{1

#W (b)if C ∈ W (b)

0 else,

and define uC(s, x) = sC . That is, candidate C’s payoff is just thecandidate’s probability of winning.

• policy-motivation: let S and σ be as above, but let

uC(s, x) =∑

C′∈C

sC uC(xC′),

where uC : X → IR is continuous. That is, the candidate’s payoff isthe expected utility generated by a utility function defined over policyoutcomes.

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• vote-motivation: let S be the cube [0, n]M in IRM , let

σC(b) = #{i : bi = C} − maxC′ 6=C

#{i : bi = C ′},

and define uC(s, x) = sC . That is, the candidate’s payoff is just thenumber of votes received in the election.3

To verify the dimensionality assumption for win-motivated and policy-motivated candidates, just note that if two candidates A and B receive allof the votes in an election return b, then σA(b) + σB(b) = 1; and undervote-motivation, σA(b) + σB(b) = n. Our monotonicity assumption is clearlysatisfied by all of the above specifications.

Note that the distinction between win-motivation and vote-motivationis often overlooked, as these two objectives lead to the same pure strategyequilibria. When we consider equilibria in mixed strategies, however, the dis-tinction becomes potentially significant. The same observation holds whenvoting behavior is probabilistic, and the distinction between the two objec-tives is, accordingly, well-known in that literature.

We extend payoffs to lotteries over outcomes in the standard expectedutility manner, thereby inducing preferences over strategy profiles for candi-dates. Given a measurable selection ψ of voting behavior, preferences of acandidate C over platform profiles are represented by the measurable payofffunction uψ

C : X → IR, defined by

uψC(x) =

∑

b∈Cn

uC(σ(b), x) ψ(x)(b).

Given the selection ψ, the expected payoff of candidate C from the profile ξof mixed platforms is then

UψC (ξ) =

∫uψ

C(x) ξ(dx).

Equilibrium An endogenous voting equilibrium is a measurable selectionψ from Ψ, and a profile of mixed strategies ξ, such that ξ forms a Nashequilibrium under the payoff functions Uψ

C .

3It could also be that the candidate cares about vote share, or vote differences. Forinstance, the condition is met if σC(b) = #{i : bi = C} − maxC′ 6=C #{i : bi = C ′}, orσC(b) = #{i : bi = C}/ maxC′ 6=C #{i : bi = C ′}, or σC(b) = #{i : bi = C}/n, etc.

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To define symmetric equilibrium, we use the following notational conven-tions. Given two candidates A and B and platform profile x, let xAB denotethe result of interchanging the two candidates’ platforms, leaving the othercandidates unchanged, i.e.,

xABC =

xB if C = AxA if C = BxC else.

Similarly, let bAB

denote the result of interchanging ballots for the two can-didates, i.e., for all i ∈ V,

bABi =

B if bi = AA if bi = Bbi else.

If XC = X for all candidates C, then we say the selection ψ is symmetricon C′ ⊆ C if

• for all x ∈ X, ψ(x) has support in C′, i.e., ψ(x)(b) > 0 implies that forall i ∈ V, bi ∈ C′,

• ψ is symmetric in the positions of candidates in C′, i.e., for all A,B ∈ C′,all x ∈ X, and all b ∈ Cn, we have ψ(x)(b) = ψ(xAB)(b

AB).

We say ξ is symmetric on C′ if for all A,B ∈ C′, ξA = ξB. Finally, we sayan endogenous voting equilibrium (ξ, ψ) is symmetric on C′ if ξ and ψ areboth symmetric on C′. That is, all candidates in C′ use the same mixedstrategy, only those candidates receive votes, and voters’ ballots depend onthe platforms of those candidates in a neutral way. When C′ = C, we use theabove terminology without reference to the subset C′.

The electoral game (X, Ψ, u) is symmetric on C′ ⊆ C if there is a measur-able selection ψ that is symmetric on C′ and for all A,B ∈ C′ and all x ∈ X,we have uψ

A(x) = uψB(xAB). That is, the game is symmetric on a subset of

candidates if there is a selection that is symmetric on that subset and, giventhat selection, the payoffs of those candidates are symmetric.

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4 Endogenous Voting Equilibria

We now present results on existence and continuity of endogenous votingequilibria that only rely on the structure provided above. In the followingsections, we then use these results to derive further results for electoral modelswhere voters are players.

Our first result establishes the existence of an endogenous voting equilib-rium.

Given a distribution ν ∈ ∆(Cn), let σΨ: X ⇒ S denote the correspon-dence defined by

{∫σ(b)ν(db) | ν ∈ Ψ(x)

}.

That is, σΨ(x) consists of the mean of the statistic generated by all possiblevoting behavior at platform vector x.

Theorem 1 Suppose that X is a compact metric space and that σΨ is up-per hemi-continuous with nonempty, compact, and convex values. Then thereexists a Borel measurable selection ψ from Ψ and a profile ξ of mixed strate-gies, such that (ξ, ψ) forms an endogenous voting equilibrium. Moreover, ifthe electoral game is symmetric on some subset of candidates C′, then thereexists an endogenous voting equilibrium (ξ, ψ) that is symmetric on C′.

The theorem follows from a direct application of theorems by Simon andZame (1990) to establish existence, and by Jackson, Simon, Swinkels andZame (2002) to obtain existence of symmetric equilibria. The proof (whichappears in Appendix A) consists of verifying that the conditions required bythose theorems are satisfied in our setting.

We remark that one can choose the strategies of the candidates who areoutside of the support of ψ arbitrarily, and so the candidates’ strategies canbe selected to be fully symmetric, even when ψ has support only on somesubset of candidates.

Another useful result is a convergence result across electoral games. Thisis useful for working with finite approximations, as well as deriving results

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about existence of perfect equilibrium, among other things. Fix the set ofcandidates. We say that a sequence of electoral games (X

r, Ψr, ur) converges

to (X0, Ψ0, u0) if:

• for each C ∈ C, each XrC and X0

C are compact metric spaces that aresubsets of some common compact metric space XC , and such that Xr

C

converges to X0C in the Hausdorff metric,

• the graph of Ψr converges to the graph of Ψ0 in the Hausdorff metric,

• for every ε > 0 there exists δ > 0 and r such that, for all r′ > r, all

C, C ′ ∈ C, and all xr ∈ Xr

and x0 ∈ X0

such that d(xr, x0) < δ, wehave |ur

C(C ′, xr)− u0C(C ′, x0)| < ε.

Our next result shows that limits of equilibria corresponding to such a se-quence of electoral games are indeed equilibria of the limiting game. The re-sult follows directly from Theorem 2 in Jackson, Simon, Swinkels, and Zame(2002) (noting from the proof of Theorem 1 that the necessary conditions toapply their Theorem 2 are satisfied).

Theorem 2 Consider a sequence of electoral games (Xr, Ψr, ur) that con-

verge to (X0, Ψ0, u0), and a corresponding sequence of endogenous voting

equilibria (ξr, ψr). Then there exist an endogenous voting equilibrium (ξ0, ψ0)of (X0, Ψ0, u0) and a subsequence (ξr′ , ψr′) such that

• ξr′C → ξ0

C weak*,

• ψr′ξr′ → ψ0ξ0 weak*,4 and

• for each C ∈ C, U r′,ψr′

C (ξr′) → U0,ψ0

C (ξ0).

4ψξ denotes the set function defined by

ψξ(Y ) =∫

Y

ψ(x)ξ(dx),

which measures the ballots cast when candidates use platform profiles in Y .

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5 Equilibrium with Voters as Players

We now explicitly model voters as part of the game, and show that we canstill establish existence of equilibrium. As mentioned in the introduction,the key to establishing this is to account for how voters vote when they areindifferent, and noting that this may have to differ from the flip of a fair coinin order to guarantee existence.

Voters’ Preferences and Strategies Now let X be a compact metricspace, and let each voter i have a continuous utility function ui : C×X → IR.The payoff to voter i from outcome (b, x) is

ui(b, x) =∑

C′∈W (b)

ui(C′, xC′)

#W (b).

Thus, voters care about who wins the election and/or what the policy out-come is. They voters do not care directly for the specific ballot they castexcept to the extent that it determines the outcome of the election.

An important special case of the model is that in which voters are policy-oriented, i.e., for each i, ui(C, x) is independent of the first argument anddepends only on the policy.

A strategy for voter i is a behavioral strategy βi, where βi(·|x) is a proba-bility distribution on C, expressing the probability that voter i votes for eachcandidate given that the candidates have chosen policy positions x. Viewingβi(·|x) as a vector in the simplex, it is required to be Borel measurable asa function of x. Let the product measure β(·|x) = β1(·|x) × · · · × βn(·|x)represent a list of behavioral strategies for the voters, and let uβ

C(x) denotethe expected payoff to candidate C when the candidates take positions x andthe voters employ strategies β. That is,

uβC(x) =

∑

b∈Cn

uC(σ(b), x)β(b|x).

Let UβC(ξ) denote the expected payoff to candidate C when voting behavior

is given by β and candidates use

the mixed platforms ξ. That is,

UβC(x) =

∫uβ

C(x)ξ(dx).

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We seek a profile (ξ, β) of candidate and voter strategies that form a subgameequilibrium of the electoral game, after suitable refinements are imposed.

Refining to Undominated* Strategies A known drawback of Nash-based equilibrium concepts in voting games is that a wide variety of (de-generate) outcomes can be supported by a Nash equilibrium of the votinggame by specifying voting strategies that make no voters pivotal. A stan-dard approach to dealing with this problem is to require that voters notuse dominated strategies. Say that βi is undominated if for all platformprofiles x, βi(C|x) > 0 implies that either there exists C ′ 6= C such thatui(C

′, xC′) < ui(C, xC) or for all C ′ we have ui(C, xC) = ui(C′, xC′).

In discontinuous games with continuum action spaces, however, it is gen-erally natural (in order to guarantee existence) only to require that votersuse strategies that are in the closure of the set of undominated strategies.5

We use a slightly weaker definition, saying that a behavioral strategy βi isundominated* if for all platforms x, βi(C|x) > 0 implies that either thereexists C ′ 6= C such that ui(C, xC) ≥ ui(C

′, xC′).6 Note that when there are

just two candidates, the undominated* voting strategies coincide with thevoting strategies that are undominated in the conventional sense.

When voters are policy-oriented, it is natural to look for equilibria wherevoters treat the set of viable candidates symmetrically. Given a subset C′

of candidates, we say that β is symmetric on C′ if for all A,B ∈ C′ and allx ∈ X,

βi(A| x) = βi(B| xAB).

In particular, if there are just two viable candidates, say A and B, andthese candidates take the same policy position x, then all voters emplying

5For instance, see Jackson and Swinkels (2005) and the discussion there.6Undominated strategies are those where a voter only votes for a least preferred candi-

date in the case where the voter is totally indifferent. Here, undominated* strategies havea voter only voting for a least preferred candidate in the case where the voter is indifferentbetween that candidate and at least one other. The idea here is that a vote vi is undomi-nated* at x if there is some sequence xk → x such that the vote vi is undominated for eachxk. Provided that preferences are such that whenever a voter finds some candidate pairtied at a given policy profile there exists arbitrarily small perturbations in policies thatcan lead the voter to prefer either candidate, and that either there are at least four votersor there are just two candidates, then this notion of closure leads to our formal definitionof undominated* strategies. In order not to add such additional preference restrictions,we simply define undominated* directly, rather than through the closure.

15

symmetric strategies would flip fair coins to decide their votes: βi(A|x, x) =βi(B|x, x) = 1

2.

We now consider existence in the general multi-candidate model, allowingfor an arbitrary number of candidates, for arbitrary candidate motivations,and for voters who care not just about policy but also for the particularcandidates elected. We establish existence of a subgame perfect equilib-rium in undominated* voting strategies. Moreover, we prove existence of anequilibrium in which there are just two candidates who receives votes in allsubgames. Formally, we say β is Duvergerian if there are two candidatesA,B ∈ C such that for all x ∈ X and all voters i, βi({A,B}|x) = 1. Whenvoters are policy-oriented, we can find an equilibrium where voting strategiesare symmetric on those two candidates.7

Theorem 3 Consider the multi-candidate complete information electoral model,where there are at least four voters or there are only two candidates. Thereexists a subgame perfect equilibrium in undominated* voting strategies. More-over, there exists such an equilibrium that is Duvergerian, in which two ar-bitrary candidates A and B receive votes. If all voters are policy-orientedand candidates A and B are both win-motivated or both vote-motivated, thenthere exists such an equilibrium that is symmetric on {A, B}.

6 Downsian Elections and the Deep Uncov-

ered Set

In this section, we consider a special case of the multicandidate completeinformation model. That special case, the Downsian model, is one wherecandidates are win-motivated and voters are policy-oriented.

Theorem 3 directly implies the existence of a subgame perfect equilib-rium in which only two candidates, say A and B, receive votes, candidatestrategies are symmetric, and voting strategies are symmetric on {A,B} and

7Note that this is not the same as saying that voters always flip a coin when they areindifferent. It only requires that they break ties in symmetric ways more generally (e.g.,if they are indifferent between x and y and vote for A when A proposes x and B proposesY , then they do the reverse when the candidates’ platforms are reversed).

16

undominated*. In such equilibria, the positions of candidates other than Aand B are immaterial and so, without loss of generality, we focus attentionon two-candidate elections. That is, in this section let C = {A,B}. Hereundominated and undominated* voting strategies coincide.

Corollary 1 In the two-candidate Downsian model, there exists a sym-metric subgame perfect equilibrium in undominated voting strategies.

6.1 The Deep Uncovered Set

While existence of equilibrium follows from Theorem 3, we have not yetaddressed the characterization of equilibrium platforms. We do so now bydemonstrating that in the equilibrium of Corollary 1, the candidates adoptplatforms in the “uncovered set” (a prominent choice set in the extensiveliterature on voting rules) with probability one.

In settings with a continuum of outcomes, there are various definitionsof the uncovered set that differ with respect to treatments of ties under themajority relation. So, we need to be explicit in defining the set that weconsider.

Define strict and weak majority preferences, respectively, as follows: forall x, y ∈ X,

xPy ⇔ |{i | ui(x) > ui(y)}| > n

2

xRy ⇔ |{i | ui(x) ≥ ui(y)}| ≥ n

2.

The relations P and R are dual, in the sense that xPy if and only if notyRx (and xRy if and only if not yPx). Clearly, P is asymmetric and R iscomplete, and our continuity assumption on ui implies that P is open and Ris closed. We use the standard notation,

P (x) = {y ∈ X | yPx} and R(x) = {y ∈ X | yRx},

for the upper sections of these relations. As is customary, let xIy denotemajority indifference, where neither xPy nor yPx (or equivalently, xRy andyRx).

17

We bound equilibrium policy positions by a weak version of the uncoveredset, derived as the maximal elements of a strong version of the usual coveringrelation. Following Duggan (2006), we say that x deeply covers y, writtenxDC y, if R(x) ⊆ P (y). That is, x deeply covers y if every alternative weaklymajority-preferred to x is strictly majority-preferred to y. Equivalently, xdeeply covers y if whenever y weakly defeats some alternative z, then xstrictly defeats z. Note that this implies that if xDC y implies xPy, i.e.,deep covering is a subrelation of P . Obviously, DC is transitive. Moreover,Duggan (2006) shows that the deep covering relation, DC , is open under ourmaintained assumptions. The deep uncovered set is defined as the maximalelements of the deep covering relation:

UDC = {x ∈ X | there is no y ∈ X such that y DC x}.Assuming X is compact, the fact that DC is transitive and open immediatelyyields non-emptiness of the deep uncovered set, and in fact UDC is compact.Note that the deep uncovered set is equivalently defined by the following“two-step” principle:

x ∈ UDC ⇔ for all y ∈ X, there exists z ∈ X such that xRzRy.

Thus, a policy position x is in the deep uncovered set if and only if, forevery other position y, x is either weakly majority-preferred to y directly orindirectly, via some other alternative.

The next theorem shows that, given any equilibrium in which voters usesymmetric, undominated strategies, the candidates locate in the deep uncov-ered set with probability one. Note that the result does not assume candidatestrategies are symmetric, so the result holds even for non-symmetric equilib-ria.

Theorem 4 In the two-candidate Downsian model, let (ξ, β) be a subgameperfect equilibrium in symmetric, undominated voting strategies. Then thesupports of both candidates’ strategies lie in the deep uncovered set; that is,ξA(UDC) = ξB(UDC) = 1.

Although Theorem 4 applies to equilibria in which candidates may usenon-symmetric strategies, the focus on equilibria where voters are using sym-metric strategies is needed for the result. For instance, if voters always fa-vored one candidate in the case of ties, it is easy to construct examples where

18

that candidate wins and the other candidate chooses arbitrary policies. InAppendix B, we present a (finite) example showing that if voters favor onecandidate over the other, then there exist equilibria in which one candidatelocates outside the weak uncovered and that candidate wins with positiveprobability.

7 Probabilistic Voting

We now address another prominent model from the literature, namely wherevoters’ preferences are unknown to the candidates. The candidates havebeliefs about voter preferences, so that given any pair of campaign plat-forms, any voter’s behavior is a random variable from the perspective ofthe candidates. Here we examine a general setting that captures models of“probabilistic voting” from the political science literature as special cases.We return to the general case in terms of candidate and voter preferences.

Define the probabilistic voting model as follows. The policy space, X, isa compact metric space. For each voter i let Ti be a space of types, with ageneric type for voter i denoted ti. Let T = ×Ti be the space of type profiles,with generic element t = (t1, . . . , tn). We let (T, T, µ) be a probability space,with each t ∈ T representing a state of the world and µ being a probabilitymeasure.8 Each voter i has a utility function ui : C × X × Ti → IR, whereui(C, x, ti) is continuous in x and measurable in ti. Note that the voters’policy preferences are completely determined by their types, so that this isa setting of “private values,” in the sense that each voter knows his or herpreference when voting. Neverthless, the general form of the probabilitymeasure µ allows the model to handle arbitrary correlation structures invoters preferences, and so to admit cases where there could be substantialcommonality in voters’ preferences.

The payoff to voter i from outcome (b, x) is

ui(b, x) =∑

C′∈W (b)

ui(C, xC′ , ti)

#W (b).

8As is clear in the proof, it is not necessary that µ be a probability measure - it couldbe a more general measure.

19

We say that voter i is policy-oriented if ui(C, x, ti) is constant in its firstargument.

Two prominent electoral models that are captured are:

• stochastic candidate bias: Ti = IRM and

ui(C, x, ti) = ui(x) + ti,C

• stochastic policy preference:

ui(C, x, ti) = ui(x, ti).

In the stochastic candidate bias model, the policy preferences of the votersare fixed, t is a n×M matrix, and the only unknown component in a voteri’s utility is the term ti,C which embodies the incremental utility received bythe voter if candidate C wins. It is commonly understood that this utility isderived from fixed characteristics of the candidates, or alternatively that theannounced policies are not binding and by knowing which candidate is electedthe voters can predict the policy that will ensue. In the stochastic policypreference model, a voter’s utility depends not on the particular candidatewho wins, but instead on that candidate’s announced policy position.

Let us emphasize that our model is more general that the usual one,along a number of dimensions. For instance, in both of the cases above, it istypically assumed that there is a zero probability that voters are indifferentbetween the two candidates (provided the adopt distinct platforms in thestochastic policy preference model). We impose no restriction on the distri-bution of types. We also admit general forms of preferences for voters andcandidates, in terms of how they value policy-candidate combinations.

A strategy for voter i is a behavioral strategy βi : X × Ti → [0, 1], whereβi(xA, xB, ti) represents the probability that voter i votes for candidate Agiven policy platforms xA and xB and type ti. As usual, we require that βi ismeasurable. We consider perfect Bayesian equilibrium of the electoral gamein which voters use undominated∗ voting strategies.9

9Note that because candidates are uninformed at the time that they adopt platforms,and because voters know their own preferences and thus have well-defined undominated∗

voting strategies (under the same definition as with complete information), the updatingof beliefs does not play a role in the equilibrium analysis. Hence it is inconsequentialwhether we examine perfect Bayesian equilibrium or some stronger refinement.

20

Theorem 5 Consider the multi-candidate probabilistic voting model, wherethere are at least four voters or there are only two candidates. There existsa perfect Bayesian equilibrium in undominated* voting strategies. Moreover,there exists such an equilibrium that is Duvergerian, in which two arbitrarycandidates A and B receive votes. If all voters are policy-oriented and can-didates A and B are both win-motivated or both vote-motivated, then thereexists such an equilibrium that is symmetric on {A,B}.

If candidates are win-motivated or office-motivated, and if the probabil-ity of indifference between the candidates when they adopt distinct positionsis zero, then it can be shown that discontinuities in candidate payoffs arerestricted to the diagonal. Thus, for an interesting subclass of models, dis-continuities in payoffs take a very restricted form. In fact, candidate payoffsare fully continuous in the stochastic candidate bias model as long as votertypes are continuously distributed. The aspect of Theorem 5 that admitsdiscontinuities and requires the proof supplied here is that the distributionof voter preferences is arbitrary. This allows them to be indifferent in termsof how they vote, or to allow for correlation in their types.

Note also that the endogenous voting equilibrium approach can still befruitful, even in the case where voters are (almost) never indifferent. If oneconsiders limits of equilibria of probabilistic voting models as candidate un-certainty regarding voter preferences goes to zero; the limiting equilibria arethemselves equilibria of the limiting complete information game. This canbe proven either using Theorem 2 or Theorem 2 of Jackson, Simon, Swinkelsand Zame (2002) (and noting that incentive compatibility is not an issue atany tie-breaking point). Thus, this offers another proof for existence of equi-librium in the two-candidate Downsian model: (i) examine a probabilisticvoting version of the model, (ii) consider a sequence where the uncertaintyin the stochastic candidate bias goes to zero, (iii) to select a correspond-ing sequence of equilibria in those models, and (iv) to take the limit of aconvergent subsequence.

8 Public Randomization and Voting

We close with an extension of our results.

21

The importance of Duvergeian equilibrium in our analysis deals with con-vexity of the outcome correspondence. In particular, suppose that there areseveral possible candidates who could be elected given their platforms. Ifall voters are voting for just two candidates, then it must be that there aresome indifferent voters. By varying the probabilities that they vote for ei-ther candidate, we can get any probability that one or the other wins. Thisallows us to convexify the outcome correspondence, which is important in es-tablishing existence. When there are more than two candidates who can winin different equilibrium configurations of voters’ behavior, it is possible thatthe outcome correspondence is not convex. This happens because voters areindependently choosing their strategies, and full convexity over more thantwo candidates can require some correlation in how voters behave. Thus,to establish existence of equilibrium (in undominated strategies) where morethan two candidates get votes, we need to have some means of correlatingvoters behavior. It is enough to have them observe a randomly generatedpublic signal.

The definition of equilibrium with a publicly observed signal is againperfect Bayesian equilibrium.

Theorem 6 In the general probabilistic voting model, there exists a per-fect Bayesian equilibrium with public randomization in undominated* votingstrategies. Furthermore, if for all voters i, all distinct policies x and y, andall candidates A and B, µ({t ∈ T | ui(A, x, ti) = ui(B, y, ti)}) = 0, then thereexists a perfect Bayesian equilibrium with public randomization in undomi-nated voting strategies. If, in addition, for all voters i, all policies x ∈ X,and all candidates A,B ∈ C, µ({t ∈ T | ui(A, x, ti) = ui(B, x, ti)}) = 0, thenthere exists a perfect Bayesian equilibrium in undominated voting strategies.

The proof is a straightforward extension of the proofs of Theorems 3 and5, and so we just discuss the critical step of showing convexity. Given thefinite set of candidates, the convex hull of the set of possible equilibria ofvoters for any given policy choices of the candidates is defined by a finiteset (say with cardinality K) of extreme points, with corresponding equilib-ria γ1, . . . , γK . A convex combination with weights (α1, . . . , αK) of theseis obtained by having K different signals, each generated with these corre-sponding progabilities. Voters then play the equilibrium corresponding to

22

the observed signal. Note that here since we are not starting with two a pri-ori specified candidates A and B, we can start with equilibria in the closureof the set of undominated strategies. If types are almost never indifferent,then this equilibrium can be found so that voters play dominated strategieswith probability zero.

Three features of Theorem 6 are worth noting. First, the general prob-abilistic voting model allows for a degenerate probability measure µ, and ittherefore encompasses the multicandidate model with “deterministic” voting.Second, when the probability that voters are indifferent between distinct al-ternatives is zero, which precludes the deterministic model, we find equilibriain which voters eliminate undominated strategies in the standard sense. Thisis because subgames in which the undominated and undominated* strategiesdiffer are probability zero from the perspective of candidates, allowing usto “patch up” our selection from equilibria in voting subgames. Third, ourstrongest assumption is satisfied in the stochastic candidate bias model whentypes are continuously distributed, but it is not satisfied in the stochastic pol-icy preference model: there, all voters are indifferent between two candidateswho adopt the same policy platform. Candidate payoffs are continuous inthe candidate bias model when there are just two candidates, and we havenoted that equilibrium existence is not an issue in that case. But with threeor more candidates, it is not generally possible to take a continuous selectionfrom equilibria in voting subgames, so the result of Theorem 6 is non-trivialeven in that relatively tractable model.

9 Appendix A: Proofs

Proof of Theorem 1: Let Eν(σ(b)) denote the integral∫

σ(b)ν(db). Thefirst part of Theorem 1 follows upon verifying the conditions of the mainresult in Simon and Zame (1990).10 We claim that the payoff correspondencedefined by

{(uψA(x), . . . , uψ

M(x)) | ψ is a measurable section from Ψ }10Simon and Zame (1990) work with a correspondence of utilities rather than outcomes.

Theorem 1 in Jackson, Simon, Swinkels and Zame (2002) works with outcome correspon-dences and directly implies the first statement in Theorem 1.

23

is upper hemi-continuous with nonempty, compact, convex values. Nonempti-ness is clear. Now take a sequence {xr} in X converging to x and a se-quence ψr of measurable selections from Ψ such that {uψr

C (xr)} convergesfor all C ∈ C. Since ∆(Cn) is compact, there is a subsequence of {ψr(xr)},which we continue to index by r for simplicity, that converges to some dis-tribution ν ∈ ∆(Cn). Since Ψ has closed graph, it follows that ν ∈ Ψ(x).Letting ψ be any measurable selection of Ψ such that ψ(x) = ν, we haveuψ

C(x) = lim uψr

C (xr) for all C ∈ C, and (uψA(x), . . . , uψ

M(x)) ∈ Ψ(x), estab-lishing closed graph. To establish convex values, take any x, any α ∈ (0, 1),and any measurable selections ψ and ψ′ from Ψ. Let ν, ν ′ ∈ Ψ(x) satisfyν = ψ(x) and ν ′ = ψ′(x). For all C ∈ C, linearity of uC in s implies

αuψC(x) + (1− α)uψ′

C (x) = uC(αEν(σ(b)) + (1− α)Eν′(σ(b)), x).

By the assumption that σΨ(x) is convex, we can choose ν ∈ Ψ(x) indepen-dent of C such that

E ν(σ(b)) = αEν(σ(b)) + (1− α)Eν′(σ(b)).

Let ψ be a measurable selection of Ψ satisfying ψ(x) = ν. Then

αuψC(x) + (1− α)uψ′

C (x) = uψC(x),

which establishes convex values, as required. The second part then followsfrom Theorem 2 in Jackson, Simon, Swinkels, and Zame (2002, remark (ii)).

Proof of Theorem 3: Pick two candidates A and B from C. Consider arestricted game where voters can only vote for one of these two candidates,and we model only these two candidates’ strategies. We prove existence of anequilibrium in this modified game. Note that if there are more candidates,and at least four voters, then the following is a subgame perfect equilibriumin undominated* voting strategies of the original game: have voters follow thestrategies from the restricted game independent of the policies of the remain-ing candidates, and have the remaining candidates play arbitrary strategies.Note that the role of having at least four voters is that any deviation by asingle voter cannot result in the election of any other candidate. Thus, weneed only prove that there exists an equilibrium in the two candidate game.

Construct a voting correspondence Ψ as follows. Let Ψ(xA, xB) consist ofall distributions ν ∈ ∆(Cn) such that for all voters i, there exists γi ∈ [0, 1]satisfying

24

(1) for all b ∈ Cn,

ν(b) =

∫

T

( ∏

i:bi=A

γi

)( ∏

i:bi=B

(1− γi)

)µ(dt)

(2) for all i ∈ V,

γi = 1 if ui(A, xA) > ui(B, xB)

γi = 0 if ui(A, xA) < ui(B, xB).

Let γ = (γ1, . . . , γn). Here, we interpret γi as the probability that voter ivotes for candidate A as a function of the voter’s type, where we require thevoter to put probability zero on dominated votes. This then determines theprobability that any particular ballot vector is realized.

Clearly, Ψ has nonempty values. We claim that, for all x ∈ X, Ψ(x)is connected. Given ν generated by γ and ν ′ generated by γ′, let να be thedistribution generated by αγ+(1−α)γ′. Then the path swept out by να as αvaries over [0, 1] connects ν and ν ′, establishing the claim. Note that Eν(σ(b))is continuous in ν, and therefore σΨ(x), as the image of a connected set undera continuous function, is connected. By our dimensionality assumption, therange of Eν(σ(b)) is one-dimensional, and it follows that σΨ(x) is indeedconvex. Therefore, σΨ has convex values.

To verify that σΨ has closed graph, take a sequence (xkA, xk

B, sk) such that(xk

A, xkB, sk) → (xA, xB, s) and, for all k, sk ∈ σΨ(xk

A, xkB). By the latter, for

each k, there exists νk ∈ Ψ(xkA, xk

B) such that sk = Eνk(σ(b)). Thus, for each

k, there exists γk satisfying (1) and (2) above with xkA and xk

B in condition(2). By compactness, there is a convergent subsequence of {γk}, still indexedby k for simplicity. Let γ denote the limit of this subsequence, and let ν bedefined from γ as in condition (1), so that s = Eν(σ(b)). Then continuity ofui ensures that condition (2) also holds, and therefore s ∈ Ψ(xA, xB). Thus,σΨ has closed graph.

By Theorem 1, there is an endogenous voting equilibrium (ξ, ψ), whereψ is a selection from Ψ. Define the behavioral strategy βi for voter i asfollows. For all xA, xB ∈ X, since ψ selects from Ψ, there exists γ satisfying(1) and (2) with ν = ψ(xA, xB). Then let βi(xA, xB) = γi), completing theequilibrium construction and proving the first part of the theorem.

25

For the second part of the theorem, note that if voters are policy-orientedand A and B are both win-motivated or both vote-motivated, then the elec-toral game (X, Ψ, u) is symmetric on {A,B}, and Theorem 1 yields an en-dogenous voting equilibrium that is symmetric on {A,B}. We then use theassociated selection to define symmetric voting strategies, as required.

We now turn to the proof of Theorem 4. First we present a useful lemma.Our equilibrium analysis uses the following dominance relation on the can-didates’ strategies. We say x stage dominates y, written x SD y, if, for everysymmetric, undominated β, we have uβ

A(x, z) ≥ uβA(y, z) for all z ∈ X, with

strict inequality for at least one z ∈ X (where the first entry in uβA(x, z) is

A’s chosen policy). Note that, given undominated β, xPy implies that a ma-jority of voters must vote for the candidate, say A, with position x, implyinguβ

A(x, y) = 1 and uβB(x, y) = −1 (noting that in the Downsian setting the

election is a zero sum game and assigning a value of 1 to winning the elec-tion). The next result establishes a tight connection between deep coveringand stage dominance of the electoral model.

Lemma 1 In the two-candidate Downsian model, xDC y implies x SD y.Assuming n is odd, the converse holds as well.

Proof of Lemma 1: Assume xDC y, and take any z ∈ X. If xPz, thenuβ

A(x, z) = 1, and the weak inequality in the definition of stage dominance isfulfilled. If zRx, then, by deep covering, zPy. This implies uβ

A(y, z) = −1,fulfilling the weak inequality in stage dominance. Finally, when z = x, wehave uβ

A(x, z) = 0 by symmetry of β, and, by xPy, we have uβA(y, z) = −1,

fulfilling the strict inequality in stage dominance. Therefore, x SD y. Nowassume n is odd and x SD y, and take any z ∈ R(x). Suppose z /∈ P (y), i.e.,yRz. Since n is odd, we have

|{i | ui(z) ≥ ui(x)}| > n

2and |{i | ui(y) ≥ ui(z)}| > n

2.

Therefore, there exists a symmetric, undominated β such that uβA(x, z) = −1

and uβA(y, z) = 1, a contradiction. Therefore, z ∈ P (y), and we conclude

that xDC y.

An immediate corollary, using the above observations on the deep uncov-ered set, is that when n is odd the set of stage undominated policy positions

26

is non-empty and compact. When n is even, it need not be the case thatstage dominance implies deep covering: in this case, we may have x SD y yetxIz and yIz for some z, if no voters are indifferent between x and z andnone indifferent between y and z – so majority indifference is the result ofan equal split among the voters, with all voters having strict preferences. Inthis case, the latitude in specifying the behavior of indifferent voters doesnot play a role, and no contradiction need arise.

Proof of Theorem 4: Let (ξA, ξB, β) be a subgame perfect equilibrium insymmetric, undominated voting strategies, and suppose that the support ofξA is not contained in UDC. Letting V denote the set of positions that aredeeply covered, this means ξA(V ) > 0. Let ξA = ξB = ξA. Since the gamebetween the candidates induced by β is symmetric and zero-sum, (ξA, ξB, β) isa subgame perfect equilibrium in symmetric, undominated voting strategies.For each y ∈ V , there exists z ∈ X such that z DC y. And since the deepcovering relation DC is open, there exists an open set Gy ⊆ X such thaty ∈ Gy and z DC Gy. Giving V the relative topology, then {Gy∩V | y ∈ V } isan open cover of V . Since X is second countable, V is as well, and it thereforepossesses the Lindelof property: so there is a countable open subcover givenby, say, {Hk}. Since ξA(V ) > 0, it follows that ξA(Hk) > 0 for some k. Letzk ∈ X satisfy zk DC Hk, and transform ξA to ξ′A by “replacing” any policyin Hk with zk. Specifically, define the mapping φ : X → X as follows:

φ(x) =

{zk if x ∈ Hk

x else,

for all x. Define ξ′A = ξA ◦ φ−1, where (ξA ◦ φ−1)(Y ) = ξA(φ−1(Y )), for allBorel measurable Y ⊆ X. Then

UβA(ξ′A, ξB, β)

=

∫uβ

A(x) (ξ′A × ξB)(dx)

=

∫uβ

A(φ(xA), xB) ξ(dx)

=

∫

Hk×Hk

uβA(φ(xA), xB) ξ(dx)

+

∫

(X×X)\(Hk×Hk))

uβA(φ(xA), xB) ξ(dx).

27

Note that zk = φ(xA)DC Hk for all xA ∈ Hk, so uβA(φ(xA), xB) = 1 for all

xA, xB ∈ Hk, which implies∫

Hk×Hk

uβA(φ(xA), xB) ξ(dx) = ξA(Hk)ξB(Hk) > 0.

In contrast, symmetry implies∫

Hk×Hk

uβA(xA, xB) ξ(dx) = 0.

¿From Lemma 1, we have zk = φ(xA) SD Hk for all xA ∈ Hk, which implies∫

(X×X)\(Hk×Hk))

uβA(φ(xA), xB) ξ(dx) ≥

∫

(X×X)\(Hk×Hk))

uβA(xA, xB) ξ(dx).

Combining these observations yields

UβA(ξ′A, ξB, β) >

∫

Hk×Hk

uβA(xA, xB) ξ(dx)

+

∫

(X×X)\(Hk×Hk)

uβA(xA, xB) ξ(dx)

= UβA(ξA, ξB, β),

contradicting the assumption that (ξA, ξB, β) is an equilibrium.

Proof of Theorem 5: Just as in Theorem 3, we pick two candidates Aand B from C. Consider a restricted game where voters can only vote for oneof these two candidates, and we model only these two candidates’ strategies.We prove existence of an equilibrium in this modified game. Note that if thereare more candidates, and at least four voters, then the following is a perfectBayesian equilibrium in undominated* voting strategies of the original game:have voters follow the strategies from the restricted game independent of thepolicies of the remaining candidates, and have the remaining candidates playarbitrary strategies. Note that the role of having at least four voters is thatany deviation by a single voter cannot result in the election of any othercandidate. Thus, we need only prove that there exists an equilibrium in thetwo candidate game.

Construct a voting correspondence Ψ as follows. Let Ψ(xA, xB) consist ofall distributions ν ∈ ∆(Cn) such that for all voters i, there exists a measurablemapping γi : Ti → [0, 1] satisfying

28

(1) for all b ∈ Cn,

ν(b) =

∫

T

( ∏

i:bi=A

γi(ti)

)( ∏

i:bi=B

(1− γi(ti))

)µ(dt)

(2) for all i ∈ V and all ti ∈ Ti,

γi(ti) = 1 if ui(A, xA, ti) > ui(B, xB, ti)

= 0 if ui(A, xA, ti) < ui(B, xB, ti).

Let γ = (γ1, . . . , γn). Here, we interpret γi as the probability that voter ivotes for candidate A as a function of the voter’s type, where we require thevoter to put probability zero on dominated votes. This then determines theprobability that any particular ballot vector is realized.

Because ui is measurable, Ψ has nonempty values. We claim that, forall x ∈ X, Ψ(x) is connected. Given ν generated by γ and ν ′ generated byγ′, let να be the distribution generated by αγ + (1 − α)γ′. Then the pathswept out by να as α varies over [0, 1] connects ν and ν ′, establishing theclaim. Note that Eν(σ(b)) is continuous in ν, and therefore σΨ(x), as theimage of a connected set under a continuous function, is connected. By ourdimensionality assumption, the range of Eν(σ(b)) is one-dimensional, and itfollows that σΨ(x) is indeed convex. Therefore, σΨ has convex values.

To verify that σΨ has closed graph, take a sequence (xkA, xk

B, sk) such that(xk

A, xkB, sk) → (xA, xB, s) and, for all k, sk ∈ σΨ(xk

A, xkB). By the latter, for

each k, there exists νk ∈ Ψ(xkA, xk

B) such that sk = Eνk(σ(b)). Thus, for each

k, there exists γk satisfying (1) and (2) above with xkA and xk

B in condition (2).We must construct ν and γ satisfying (1) and (2) for the limiting platformsxA and xB and such that s = Eν(σ(b)).

For each i, let T>i denote the set of types for which i prefers candidate A

with platform xA to B with platform xB, let T<i denote the set of types for

which i prefers B, and let T=(i) be the set of types such that i is indifferent:

T>i = {ti ∈ Ti | ui(A, xA, ti) > ui(B, xB, ti)}

T<i = {ti ∈ Ti | ui(A, xA, ti) < ui(B, xB, ti)}

T=i = {ti ∈ Ti | ui(A, xA, ti) = ui(B, xB, ti)},

29

and let T= =⋃

i∈V(T=i × T−i) denote the set of type profiles at which some

voter is indifferent. Take any t ∈ T>i . By continuity of ui, voter i prefers

candidate A for sufficiently high k, and therefore γki (ti) = 1; similarly, if

ti ∈ T<i , then γk

i (ti) = 0 for sufficiently high k. Given α ∈ [0, 1]n, defineγα

i : Ti → [0, 1] by

γαi (ti) =

1 if ti ∈ T>i

αi if t ∈ T=i

0 if t ∈ T<i ,

which is clearly measurable. Letting γα = (γα1 , . . . , γα

n ), we see that γα fulfillscondition (2). Let να be the distribution determined by γα through condition(1).

We have left to establish existence of α ∈ [0, 1]n such that Eνα(σ(b)) = s.

For convenience, define

Γk(b|t) =

( ∏

i:bi=A

γki (ti)

)( ∏

i:bi=B

(1− γki (ti))

),

and define Γα(t) similarly using γα. Note that

Eνk

(σ(b)) =

∫

T

∑

Cn

σ(b)Γk(b|t)µ(dt)

Eνα

(σ(b)) =

∫

T

∑

Cn

σ(b)Γα(b|t)µ(dt).

We have argued that each γki converges pointwise to γα

i on T \T=, and there-fore Γk(·|b) also converges pointwise to Γα(·|b) on T \ T=. As a consequence,we have

∫

T\T=

∑

Cn

σ(b)Γα(b|t)µ(dt) = limk→∞

∫

T\T=

∑

Cn

σ(b)Γk(b|t)µ(dt).

If µ(T=) = 0, then the above equality yields Eνα(σ(b)) = s, as desired.

Otherwise, if µ(T=) > 0, define

θ = limk→∞

1

µ(T=)

∫

T=

∑

Cn

σ(b)Γk(b|t)µ(dt).

30

By our dimensionality and monotonicity assumptions, we may assume thatb ≥A b

′implies σ(b) ≥ σ(b

′). We therefore have

1

µ(T=)

∫

T=

∑

Cn

σ(b)Γ0(b|t)µ(dt) ≤ θ ≤ 1

µ(T=)

∫

T=

∑

Cn

σ(b)Γ1(b|t)µ(dt).

Thus, by continuity, we may choose α ∈ [0, 1] such that

θ =1

µ(T=)

∫

T=

∑

Cn

σ(b)Γα(b|t)µ(dt),

yielding α such that Eνα(σ(b)) = s, completing the proof of closed graph.

By Theorem 1, there is an endogenous voting equilibrium (ξ, ψ), whereψ is a selection from Ψ. Define the behavioral strategy βi for voter i asfollows. For all xA, xB ∈ X, since ψ selects from Ψ, there exists γ satisfying(1) and (2) with ν = ψ(xA, xB). Then let βi(xA, xB, ti) = γi(ti), completingthe equilibrium construction and proving the first part of the theorem.

For the second part of the theorem, note that if voters are policy-orientedand A and B are both win-motivated or both vote-motivated, then the elec-toral game (X, Ψ, u) is symmetric, and Theorem 1 yields a symmetric en-dogenous voting equilibrium. We then use the associated selection to definesymmetric voting strategies, as required.

10 Appendix B: An Asymmetric Voting Ex-

ample

It is relatively easy to construct examples that violate Theorem 4, once therestriction of symmetric voting is removed. For example, suppose that allvoters have the same ideal point and vote for candidate A if A proposes thatpolicy, and otherwise specify any undominated votes; let candidate A choosethe voters’ ideal point, and let candidate B choose any policy position. Thisis a subgame perfect equilibrium in undominated voting strategies, but itexhibits two features that make it uninteresting. First, all voters vote for A,even if both candidates adopt the voters’ ideal point, an extreme violation ofsymmetric voting. Second, the position chosen by candidate B, which may

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a b c d e f g

a 0 0 0 −1 −1 1 0(−1)

b 0 0 0 −1 0(−1) −1 1

c 0 0 0 1 1 0(−1) −1

d 1 1 −1 0 1 −1 −1

e 1 0(1) −1 −1 0 −1 −1

f −1 1 0(1) 1 1 0(1) 0(1)

g 0(1) −1 1 1 1 0(1) 0(1)

Table 1: Covered equilibrium platform

be covered, does not arise as a policy outcome, since A wins the election withprobability one.

The example in Table 1 shows that, even if we maintain the assump-tion that voters flip a fair coin to decide their vote in case both candidatesadopt the same position, it is possible that one candidate adopts a coveredpolicy position. In fact, in the example, that candidate wins with positiveprobability after adopting that position.

Here, let rows denote the strategies of candidate A, columns the strate-gies of B, and let entries denote A’s payoffs: “0” denotes majority indiffer-

32

ence, and a superscript indicates the resolutions of indifferent voters. Forexample, “0(1)” indicates that a sufficient number of indifferent voters votefor A with probability one that A wins the election with certainty. Spec-ify mixed strategies for the candidates so that ξA = (0, 0, 0, 0, 1

3, 1

3, 1

3) and

ξB = (13, 1

3, 1

3, 0, 0, 0, 0). It is straightforward to check that these strategies,

with the voters’ strategies implicit in the candidates’ payoffs, form a subgameperfect equilibrium (and we may assume that voters use undominated strate-gies). Note, however, that the voters’ strategies do not treat the candidatessymmetrically: when A chooses g and B chooses f , voters resolve indifferencein favor of A, and they also do so when A chooses f and B chooses g.

In this example, candidate A chooses policy e with probability one third,despite the fact that it is covered by d: driving this is the fact that the votervotes for A with probability one when A chooses e and B chooses b, giving Aan expected payoff of 1/3 from choosing e (as with f or g). It then becomescritical to limit B’s payoff to −1/3 if that candidate deviates to e, f , or g,necessitating the asymmetric voting strategies employed in the example.

Note that A’s expected payoff upon choosing e, as a function of B’s choiceof a, b, or c, is actually identical to the payoff the candidate would receiveupon choosing d. In fact, letting A choose d with probability one third insteadof e, we have an “equivalent” equilibrium in which the candidates use onlyuncovered strategies. Theorem 4 can be generally extended in this way toallow for asymmetric voting, when appropriately restated: for every subgameperfect equilibrium in undominated voting strategies, say (ξ, β), there existsanother such equilibrium, say (ξ′, β), such that ξ′A and ξ′B have support onthe deep uncovered set and the distribution of each candidate’s payoffs areequivalent to that under (ξ, β).

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