Download A Theory of Voting in Large Elections Richard D. McKelvey Division

A Theory of Voting in Large Elections
Richard D. McKelvey Division of Humanities and Social Sciences California Institute of
Technology Pasadena, CA John W. Patty Department of Government Institute for
Quantitative Social Science Harvard University Cambridge, MA April ,
A previous version of this paper was entitled Quantal Response Voting. The nancial support
of NSF grants SBR and SES to the California Institute of Technology are gratefully
acknowledged. In addition, Patty acknowledges the nancial assistance of the Alfred P. Sloan
Foundation. This paper beneted from helpful comments from seminar participants at
CERGYPontoise Oct , CORE Oct , Tilburg Oct , Caltech Theory Workshop May, , Public
Choice Society New Orleans, March , the Stan and Cal Show Pismo Beach, May, , the APSA
annual meetings Atlanta, , and the Wallis Conference on Political Economy Rochester, NY,
October, . We especially thank John Duggan, Tom Palfrey, Norman Schoeld, Bob Sherman,
two anonymous referees, and an Associate Editor for helpful input. Deceased. Richard
passed away on April , . He is, and will be, terribly missed as both a friend and a scholar.
Corresponding Author jpattygov.harvard.edu
Abstract This paper provides a gametheoretic model of probabilistic voting and then
examines the incentives faced by candidates in a spatial model of elections. In our model,
voters strategies form a Quantal Response Equilibrium QRE, which merges strategic voting
and probabilistic behavior. We rst show that a QRE in the voting game exists for all elections
with a nite number of candidates, and then proceed to show that, with enough voters and the
addition of a regularity condition on voters utilities, a Nash equilibrium prole of platforms
exists when candidates seek to maximize their expected margin of victory. This equilibrium
consists of all candidates converging to the policy that maximizes the expected sum of voters
utilities, exists even when voters can abstain, and is unique when there are only candidates.
Journal of Economic Literature Classication Numbers D, D, D, D.
Keywords Voting, Probabilistic Voting, Quantal Response Equilibrium. Proposed Running
Head Voting in Large Elections
Introduction
Probabilistic voting is central to both theoretical and empirical work in political economy. In
particular, prediction of an individuals voting behavior is inherently imprecise, even with
detailed information about the voter in question. Accordingly, modern empirical studies of
voting behavior universally assume some form of probabilistic voting behavior in the form of,
say, a logit or probit regression model. Theoretical models of probabilistic voting incorporate
this imprecision into the candidates strategic calculations. Probabilistic models of voting yield
predictions more amenable to empirical testing and have been used to examine many topics,
including tax policy, redistribution, and multiparty elections. Theoretically, models of
probabilistic voting are appealing for two reasons. The rst is the link between the model and
the empirical means of evaluating its predictions. In particular, probabilistic voting models
and nearly all statistical analyses of individual vote choice share a common starting point
even the most reliable models of behavior are not guaranteed to predict any given individuals
behavior perfectly. Probabilistic voting simply incorporates this reality into the strategic
calculations of the individuals within the model. This common methodological link has
another advantage as opposed to most models of electoral competition in which voters are
presumed to vote in a deterministic fashion, pure strategy electoral equilibria typically exist
when voters are presumed to vote probabilistically. Indeed, one of the most robust
characteristics of models of probabilistic voting is the stability they induce in models of
electoral competition, as highlighted by Coughlin , Banks and Duggan , and Schoeld . This
paper extends this research in an important way by allowing for strategic behavior within a
general formulation of probabilistic voting. Specically, we investigate the Quantal Response
The probabilistic voting literature began with the work of Hinich , and was initially extended
by Coughlin and Nitzan a,b. An excellent overview of the early work is contained in Coughlin
. Recent work in the area includes Lin, Enelow, and Dorussen , Banks and Duggan , Patty , ,
, and Schoeld . Recent examples include Alvarez, Nagler, and Bowler Lacy and Burden
Quinn, Martin, and Whitford and Schoeld, Martin, Quinn, and Whitford , to name only a few.
For an excellent examination of equilibrium existence in spatial electoral competition, see
Banks, Duggan, and Breton .
Equilibrium see McKelvey and Palfrey , within spatial voting games. Quantal Response
Equilibrium QRE is a theory of behavior in games that assumes that individuals get privately
observed random payoff disturbances for each action available to them. A QRE is then just a
Bayesian equilibrium of this game of incomplete information. In a QRE, although voters
adopt pure strategies, from the point of view of an outside observer who does not know the
payoff disturbance, the players choose between strategies probabilistically, choosing actions
that yield higher utility with higher probability than actions that yield lower utility. The
probability that one action is chosen over another is based on the the utility difference
between the alternatives. The fact that probabilistic voting is generated through a Bayesian
equilibrium implies that voters may vote strategically. This is in contrast to most of the
models studied in previous theoretical work on probabilistic voting, which is reviewed in the
next section. As we discuss further below, our results are very similar in spirit to the results
obtained by previous scholars. However, it must be noted that the approach taken here is
much more gametheoretic than in most previous models of probabilistic voting. In particular,
our model allows for voters to take into account the relative likelihoods of different candidates
winning the election. As opposed to classical models of probabilistic behavior, the expected
utility of casting a particular vote is represented correctly. More precisely, in addition to the
candidates announced platforms and his or her own policy preferences, the notion of QRE
presumes that each voter also considers the other voters strategies when calculating the
expected payoff from each possible ballot he or she may cast. Put another way, each voters
probabilistic voting behavior generated by a QRE is consistent with that voter being aware
that his or her fellow citizens are also voting probabilistically. Providing an equilibrium
derivation of probabilistic voting in large elections is important for three reasons. First, as
described above, the assumptions that underpin our theory are taken as given in many
empirical analyses of voting behavior in economics and political science. Second, our results
indicate that, in equilibrium, public policy outcomes may be governed by voters preferences
even when individual voters probabilities of being pivotal are innitesimal. A somewhat
This stands in contrast to the celebrated median voter theorem Black . it is often but not
necessarily centrist. p. and . . Coughlin and Nitzan a. and more importantly. or the number of
candidates as opposed to both median voter results and many classical probabilistic voting
models. In addition. and he constructed examples in a one dimensional space with equilibria
at other locations. Corollaries . subsequent work see Coughlin . Theorem . this policy is
sensitive to the strengths of individual voters preferences. Coughlin also gives various . there
is an electoral equilibrium in which ofcemotivated candidates within our model offer identical
platforms at the ex ante social welfare optimum.. p. Finally. Hinich showed that the median
voter theorem does not always hold in a setting with probabilistic voting.b see also Coughlin .
. in which the candidate equilibrium is insensitive to these individual strengths. Theorem
.ironic corollary of this fact is that policy outcomes are governed by voters policy preferences
even when any given voters observed behavior is nearly independent of his or her policy
preferences. While this work was not explicitly rooted in a utility maximization framework. and
most importantly. First. he obtained an equilibrium in two candidate elections at the mean
which is the social welfare optimum with those preferences. while the equilibrium we
characterize does require that the electorate be large enough. the equilibrium platform
location that we characterize has several appealing features. In particular. it does not depend
upon restrictions on the dimensionality of the policy space as opposed to most median voter
results. shows how it can be so interpreted. the shape of individuals utility functions as
opposed to median voter results and many classical probabilistic voting models. This policy
is appealing for a number of reasons. Related Literature Many other scholars have studied
probabilistic voting see Coughlin for a review of this literature. with quadratic utility functions.
proved if voters have likelihood of voting functions satisfying the Luce axioms over subsets.
Second. As previous authors have shown in other probabilistic voting frameworks. there is a
local equilibrium at a point maximizing the social log likelihood. .
In his model. In all of the above cited probabilistic voting literature. there is a global
equilibrium. then the expected vote function for each candidate becomes concave.conditions
on voter likelihood functions or on preferences that result in a global equilibrium. He shows
that as long as the density function of the costs of voting is positive at zero. show that one
can also obtain equilibrium for multicandidate elections using probabilistic voting models.
They assume that voters preferences are based on the distance between their own ideal
policy and the candidates announced platform. Ledyard develops a Bayesian model of two
candidate competition that does model the game theoretic considerations for the voter. If the
likelihood functions are concave. there is an equilibrium at the social welfare optimum. Voter
types consist of preferences as well as a cost of voting. Lin et al. . Lin et al. unknown to the
voters. See Banks and Duggan for a summary and generalization many of these results. with
a random utility shock. also nd that if the utility shocks have high enough variance. but they
can abstain as well as vote for one of the two candidates. there is a global equilibrium at the
social utility maximum p. Ledyards model. voters vote deterministically there is no random
utility shock to preferences. require that no voters have negative costs of voting. there is a
global equilibrium in Ledyards model as the number of voters becomes large. Voters are
assumed to vote based on their preferences for the candidate policy positions rather than
based on the effect their vote will have on the outcome of the election. Theorem . . and the
cost of voting is a random variable. In a redistributional model where voters have logarithmic
utility functions for income. He shows that in large elections. if voting costs are nonnegative..
game theoretic considerations for the voter are not modeled. is a global equilibrium. which
under certain restrictive conditions on the distribution of costs. implying the existence of a
global equilibrium. and candidates use a logistic model to estimate the probability that voters
vote for each candidate. Myerson extends Ledyards results in a model where the number of
voters is a Poisson random variable. as well as Myersons generalization of it. and obtain
local equilibria at the social utility maximum. All of the above results are for two candidate
competition.
g. this assumption may appear strange. we consider elections with an arbitrary number of
candidates. Schoeld et al. Tullock . . We nd that for large enough electorates there is a
convergent equilibrium at the alternative that maximizes social welfare. . we work in a
Bayesian framework. In other words. For two candidate contests. the equilibrium is unique. .
The individual experiences a subjective payoff from voting for a specic candidate above and
beyond the instrumental objective benets derived from his or her vote choice. Our equilibrium
is global. as in Ledyard . Lacy and Burden . In addition. Schuessler . . a voter gets some
randomly drawn and privately observed payoff for pulling the lever for candidate j. For
example. the conditions for a global equilibrium are satised by allowing the number of voters
to grow large rather than by assuming the utility shock becomes large.g. We do not require
concavity of preferences. as in Lin et al. We also impose a regularity condition that the ratio
of the variance to expected value of any utility differences is uniformly bounded. and take into
account the game theoretic considerations for the voters. Further. and Schoeld is that our
model assumes that the random utility shocks are independently distributed across the
alternatives. .. Brennan and Lomasky . . the payoff disturbances are candidate specic and
unrelated to the policy position of the candidate or to whether the candidate wins or not. we
assume that voters have privately observed payoff disturbances associated with each action.
Overview of The Model In this paper. it is standardly and uncontroversially adopted in
modern empirical studies of individual vote choice. one interpretation of the payoff
disturbances is as a representation of expressive voting e... but require that preferences are
uniformly bounded. Buchanan . Unlike Ledyard . While this interpretation may create the
appearance that the voters in our model are irrational. In our model. any probit or logit
analysis of individual voting behavior is based on the assumption that voters respond to
random utility shocks that are very similar to those considered here. Alvarez et al. Alvarez
and Nagler and Dow and Endersby discuss considerations surrounding the choice between
different multinomial discrete choice models of vote choice. Interpreted in this way. Our
results basically extend those of the earlier literature. but in our model. Quinn et al. Alvarez
and Nagler . it should be noted The principal difference between our approach and recent
empirical work e. However.
. . under the QRE assumptions. McKelvey and Schoeld . The main difference between our
approach and previous work on probabilistic voting is the way in which we model the
probabilistic voting. by treating the voter decisions as a game. Noting the generic
nonexistence of pure strategy equilibria in multidimensional electoral competition when
voters behaviors are perfectly determined by objective payoffs e. even when voters are
strategic and respond to differences between the candidates proposed platforms.. However.
McKelvey . As in Ledyard . in aggregate. Hence. In other words. in large electorates. Thus.
we explicitly include the pivot probability in the voters expected utility calculations. the main
contribution of this paper to the existing work in the eld is to obtain a global candidate
equilibrium in large electorates with very little in the way of assumptions about voter
preferences. and the logic of strategic voting. the voters choice is determined mainly by the
candidate specic payoff disturbance. the expected utility difference between any two
candidates also goes to zero. there is still enough policy voting at the aggregate level to force
the candidates to the social optimum.that the payoff disturbance structure utilized within QRE
insures that the likelihood that a voter votes in accordance with his or her ex ante
preferences increases as the ex ante utility difference between his or her choices increases.
expressive theories of vote choice. even though individuals become less responsive to policy
differences. voters vote less based on policy.g. because the probability of being pivotal goes
to zero. In addition to providing a link between empirical work on voting. since the total
number of voters is also getting large. a voter is more likely to vote for the candidate whose
platform maximizes his or her instrumental expected payoffs as the expected utility gain from
doing so increases. the results of this paper can be seen as demonstrating a stabilizing role
of expressive voting in large electorates. and more based on candidate attributes as the size
of the electorate grows. In large electorates.
and each We leave both the topology and metric with which T is endowed implicit. and
denote the joint distribution by N . Finally. i .g. as we discuss below. Hence. of type ti i . We
assume that u is uniformly bounded with respect to X and T . lt D. each j K . i. We assume
nothing about T other than that it is a complete separable metric space. all voters policy
preferences are common knowledge. it greatly claries the exposition of the models results.
While this assumption is stronger than we need. is common knowledge. T and K so that Ti T
K for all i N . and Schoeld and Sened . or types of the voter. We write ij to represent the j th
component of i . . Write n N and k K for the total number of each. It should be noted that the
assumption that i is atomless does not rule out the possibility that the policybased
component of voter is type. . We assume that Ti can be represented as the Cartesian
product of two sets. i Ti for the policy x X is ux. ux. Note that the assumption of
independence does not preclude degenerate distributions of i in such a case. respectively.
The space of all type proles is denoted by T iN Ti . Groseclose . and nite sets N and K of
voters and candidates. i . For each i N . the set of all candidates and abstention. T represents
the set of all policybased determinants of preference... there is a space Ti of possible
characteristics. over the Borel sets of Ti . Ansolabehere and Snyder.e. i N . the utility of voter
i N .e. and write K K for the set of alternatives i. u X T . Uniform boundedness would follow
from the traditional assumptions that u is continuous with respect to both of its arguments
and that both X and T are compact. We let indicate abstention. as they play no substantive
role in our analysis. We also assume that the marginal distribution of voter is types is an
atomless probability measure. The rst of these sets. i . there exists a D R such that for all x X
and T .. All of the papers results hold with the weaker assumption that N is absolutely
continuous with respect to the product measure iN i . X . we assume that types are
independently distributed N iN i . Jr. This assumption implies that no candidate possesses a
valence advantage e. where X is bounded. Voters preferences over the policy space are
described by a utility function. The Model m We assume the existence of a nite dimensional
policy space. Aragones and Palfrey . Schoeld . K represents the consumption or expressive
determinants of preference over vote choice. We assume that for each voter. while the
second.
but substantively interpretable loosely put. Lemma and Proposition . it is important to note
that we do not impose any restrictions on the distribution of i across voters or electorates. we
assume that ij is identically distributed for all electorates N . we could relax the assumption
that the distribution of ij is invariant across all electorates N as the discussion indicates. the
assumption that i is atomless does not preclude the possibility that the marginal distribution
of i is degenerate as in the classical spatial model with commonly known ideal points. across
all electorates N . From a gametheoretic standpoint. This assumption is made for three
reasons. given our assumed uniform bound on u across X and N .i T . The third reason is
more technical. our results for candidate competition essentially rely on the existence. each
voter i N . this requirement can be interpreted as requiring that every vote prole can follow
from any prole of announced policy platforms. Specically. We omit this added generality for
clarity of presentation. where D is the bound on u as stated on p. policyindependent payoff
disturbances. Thus. an atom in Ti is a point i . First. each with a cumulative distribution
function that is twice continuously differentiable and hence atomless. Accordingly. the
probability that ij D. Any joint distribution N on T satisfying all of the above conditions is said
to be admissible. and every prole of policy platforms. Put succinctly. Second. Finally. Strictly
speaking. it is in keeping with the notion of Quantal Response Equilibrium. and all i T . for
each voter i and candidate j. n. all of the ij are assumed to be independently distributed
random variables with full support. for example. Thus. It is important to note and briey
discuss our assumption that the distribution of ij is identical across all possible electorates N .
nonatomicity of N is assured by the fact that no such point for any voter i is assigned positive
measure. and each candidate k K. all i N . this assumption ensures that our results do not
depend on an unveriable assumption about the effect of n on the distribution of individuals
idiosyncratic. for each electorate N . We thank a referee for urging us to clarify this
assumption. of some strictly positive uniform lower bound on the probability that voter i will
vote for candidate k in electorate N . all candidates j K. . we could essentially impose a
uniform bound on the upper tail of the distribution of ij specically. as dened by McKelvey and
Palfrey . . many of our results are asymptotic with respect to the size of the electorate. As we
discuss later. since relaxing this assumption would require us to carry further notation
through our arguments. i .
. ti j n Vj y. . by J vJ y.Let be the common mean of ij for j K. . ti Prti Ti W y. and t T n i N si y.
and W y. so that we can dene voter utilities over any nonempty subset of candidates. s. We
assume that a fair lottery is used to select a winner when there is a tie. sn S of the voters.
and then after observing the candidate policy positions. s. and si Si to represent strategy
proles for all voters except voter i. J K. The strategy set Si for voter i N is the set of functions
si Y Ti K . Thus. t j K j argmaxlK Vl y. be the mean of i . . Given a strategy choice y y . . t . jJ .
dene for any j K . and the set of strategy proles for the voters is S iN Si . . write PJ y. and s s
. s. . and the set of strategy proles for the candidates is Y iK Yi . i uyj . i . . to be the set of
winners of the election. with similar notation for candidates. For any subset of candidates J
K. t . to be the proportion of the electorate who chose alternative j. s. and c . We will use the
notation Si ji Sj . the voters vote for a candidate. t J. We now dene a game. the strategy set
Yi for candidate i K is Yi X. . s. to be the probability of a rst place tie among the candidates J.
in which the candidates each simultaneously choose policy positions in X. . yk Y by the
candidates. Then c is the expected cost of voting.
ti J . y. i JK PJ y. ti U y. and si Si .ti . Thus. y. ti K . In other words. ij . si . s Y S is U y. is the
expected utility to voter i of type i of voting for candidate j. s. y. a voter voting for candidate j
si y. voter i changes the outcome from a win for k to a win for j. In particular. s. s uyj .
unconditioned on the payoff disturbance. s JK. ti for the utility that voter i of type ti gets from
voting for strategy j. si . i . i ij . s. given y. ti PJ y. that the voter makes as dictated by the
strategy prole y. si . .The expected payoff to voter i N of type ti i . s. s. ti vJ y. note that the rst
Details about the derivation of pivot probabilities in multicandidate elections can be found in
McKelvey and Ordeshook .kJ J PJ y. s.j. s is the pivot probability for j over k j jk i y. . s.. for all
j K . The pivot probability is the probability that by voting for j rather than abstaining. jk i y.
The difference in the expected utility of voting for j over abstaining can be written in the form
U j. s. i U . ti JK PJ y. si . si y. y. ti receives the expected utility of the policy of the winning
candidate the summation term on the right hand side of Equation . We write U j. s. i isi y. To
understand Equation . i given a strategy prole y. . j. y. follows by reversing the order of
summation in the expression for E j E of the Theorem on p. where U j. y. i uyk . U j. i . ti U j.
j.ti that is associated with the vote. Equation . ti vJ y. plus a payoff disturbance isi y. i kj jk
where i y. j.
A potentially confusing aspect of Equation . then a rational voter should condition on his or
her own type. conditional on si . multiplied by the probability of j being selected from J.g.
which is /J . . .. is /JJ . j. If this assumption is relaxed. s is not necessarily equal to i y. ti is the
probability of a tie between a set of candidates J. when calculating his or her probability of
casting a pivotal vote. ti . the assumption of independence is made for expositional purposes.
Recalling that the pivot probability i is the probability of changing the election winner from k to
j. . rather than the more traditional pivot probability for any single given candidate. . the
probability of breaking Because we are assuming that the types are independent p. Of
course. is it is pairspecic with respect to candidates. si . s due to the fact that it includes the
pos sibility of creating a tie between the two candidates. si . as claimed on p. which includes j
and k. Summing over all potential ties yields Equation . However.. conditional on the other
voters strategies. ti . after voter is vote for candidate j has been included. then it is more likely
that voter i voting for candidate k will create a tie between j and k than it is that voter i voting
for j will create a tie between the two. the probability of having changed the election winner
from k to j. si . which is /J. This probability for a voter i and a candidate j jk is given by kj i . si .
Hence. among many others. The jk probability of victory by candidate k is /N in this case. si .
jk kj Note that i y. the ti argument in PJ y. The second term is the probability of a tie between
the same set of candidates. . note that j appears on the right hand side of Equation . Whether
this potential for asymmetry is realized for any given voter i depends on si if candidate j say
is slightly favored over candidate k by the voters other than some voter i. In particular. ti is
superuous. but k does not. then the probability of changing the election winner from k to j is
/J /J. conditional on voter i voting for candidate j and the probability of candidate j winning
conditional on receiving voter is vote in this case is since we are studying plurality rule. J the
summation occurs over all subsets of candidates including both j and k. introducing the
potential for asymmetry between any two candidates. AustenSmith and Banks and
Feddersen and Pesendorfer .term. This feature of pivot probabilities is related to recent work
on the swing voters curse e. before accounting for voter is vote which is equivalent to the
conditional probability of this tie given abstention by voter i. The probability of is vote having
changed the winner from k to j is the probability of k having won if the set of leading
candidates was J j. PJ y. The pairspecic formulation is necessary because different
candidates platforms may offer different utilities to voter i. conditional on PJ y.
i uyk . s uyk . i uyl . It then follows from Equation . s. s uyj .l i y. s . ti j n E E i N si y. reduces
to jl lj U j. s. we rst dene Vj y. for the case of two candidates. we assume that candidate js
payoff is his or . The key fact is that this is not the only case in which a vote can be pivotal
between the two creating a tie is also potentially important. s. i j E i y.a tie between the two
candidates is independent of the order of the candidates. that the difference in expected
utility of voting for j over l is U j. l. i jk y. ti j n Vj y. i uyl . y. s i y. s. K j. i . s uyj . i uyl . i i y. i . s.
s Et i N si y. y. s. s to be the expected proportion of the electorate choosing alternative j at
the prole y. i . i jl lj i y. i U l. s Et Vj y. To dene the candidate payoff functions. we will nd it
useful to reexpress candidate js expected vote share as Et i N si y. which. i i lk kj. s i y. Using
candidate js expected vote share at y. s uyj . i U l. y. t . i j . i j n E si y. Writing Ez to denote
expectation with respect to a random variable z. s n n iN i iN Vj y. y.
This is important in interpreting the main theorem. Petersburg paradox. since the weights
that individuals are given in the social utility function is determined by this normalization.
such that uxk . Remark Our assumptions about the distribution of voters types. Voter
Equilibrium In this section. s. Thus.her margin of expected victory. Remark The assumption
that the ij are independently and identically distributed with respect to voters can be viewed
as an implicit normalization of utility functions. the voter prefer any policy x over the lottery.
there exists a sequence of voters the limit of whose preferences is unbounded and hence
subject to the paradox. if one constructs the sequence xk kZ such that uxk . . s Vj y.. The
uniformity of the bound on u rules out cases in which. for any xed prole of candidate
positions. The paradox occurs if one can construct a sequence of policies. Such a voter
would not trade the lottery that gives outcome xk with probability k for any policy x the lottery
has an unbounded positive expected payoff. Once y is xed. in which all voter ideal points are
known and common knowledge and models such as Ledyard in which all voter types are
independent and drawn independently and identically from a common distribution on voter
types. s max Vl y. Similarly. Remark Our assumption that u is uniformly bounded rules out
the occurrence of the St. xk kZ . we consider the voter equilibrium to the game dened by
Equation . as the electorate grows larger. Ti K . Petersburg paradox. . y Y . N . Vj . . i lt k . lN j
. encompass both the classical spatial voting framework. the strategy space for the voter
reduces from Si the set of functions si Y Ti K to the set of functions of the form si y. . the
assumption of bounded utility implies that no voter is subject to the St. . i gt k for each k .
which is dened as Vj y.
si y. y. ti max U l. For any xed y Y . Throughout the paper. s is a voter equilibrium for y if for
all i N . j and i . ti j U j. and identically Strictly speaking. ti lK U j. for all i. i . dene si y. after
integrating out i . i j Pri si y. Thus.We write Si y to designate this conditional strategy space.
s. T K . s Sy. Note that the structure of the payoffs is essentially the same as used in
McKelvey and Palfrey in dening the agent quantal response equilibrium AQRE for extensive
form games. For any si y. there exists a voter equilibrium for y. We have assumed that the ij
are independently distributed. and j K . the notation K denotes the K dimensional simplex and
the notation K denotes the K dimensional simplex. y. The proofs for this and all following
numbered results are contained in the appendix. So as long as the distribution of the errors.
we dene a voter equilibrium for y to be a pure strategy Bayesian Nash equilibrium to the
voter game dened by Equation . y. . si y. ij is admissible. s. Si y. i il lK . a Bayesian Nash
equilibrium to the voter game is exactly the same as an AQRE to the game. since we allow
for the distribution of i to have a different mean than the distribution of ij for j K. The following
proposition assures as that a voter equilibrium exists for any prole of policy platforms. Of
particular interest is the average behavior of a voter i of type ti . . i j. i ij max U l. as the
marginal distribution of si with respect to i for any i T and j K . ti Ti . over the strategy space
Sy. s. in which voters always choose an action that maximizes expected utility conditional on
their type. y. This is any prole. s. and Sy to designate the set of proles of conditional
strategies. Proposition For any y Y . our framework is slightly more general than AQRE.
s. y... s. s. y. in which Hj wj exp expwj .. where the density functions of w i c and wj ij for j K
follow a type one extreme value distribution.. i U k. y. y. i . where zl il ij for l K j. i U . i. U j. s.
s. y. applying Equation . This . i Gj U j. y. i U . The independence of ij across i and j implies
Hw j Hj wj . i U . Let H be the cumulative distribution function of i . i U l. y. for j K. s. for all j K.
Gj z Pri ij zj and il ij zl for all l j . and everywhere positive. if s is a Bayesian Nash
equilibrium.distributed for all j K.e. i U j . y. Hw P rij wj for all j K for w of z K K . Thus. And let
Gj be the cumulative distribution function . and zj i ij . s. i il lK Pril ij U j. y. both Hw and Gj z
are twice continuously differentiable and strictly increasing in all arguments.. s. Example One
example of the above is the logit AQRE. y. i ij max U l. s. s. j. y. y. i for all l K j j. Thus. Under
the assumptions we have made on the ij . y. i . s. y. s.. i j PrU j. U j. for any z K . s. i U . si y. i
U j . s.
i and in the case of two candidates.leads to the logistic formula Gj z exp czj P lj expzl . l. i jl a
limn i y. The reason for this result is simple ones vote only matters when it is pivotal. y. As n
grows large. sn jl lj c limn i y. s uyj . s. sn /k y. Thus. i exp c U . Thus. we get si y. k gt .
Proposition Fix y Y . sn /i y. y. i uyl . i uyl . i U j. s. ones vote rarely affects the outcome of the
election. and let sn be any AQRE for the voters. all pivot probabilities go to zero and the
probability of voting for any two candidates in K becomes equal as n . i j . exp c y. Then for
any j. s. for xed . i Ui j. s uyj . y. ones vote only matters when the other voters are either
evenly split between the two top candidates or when the vote difference between the two top
candidates differs by one vote. l K and i. and for any voter equilibrium. y. This implies that
voters effectively become indifferent with respect to which candidate they vote for as n . sn
and jl jl b limn i y. let N be any admissible joint distribution over n Ti . We now show that for
xed candidate positions at y Y . in general. i exp jl y. We formalize this in the following
proposition. i lj . where K j. s. see Myerson and Weber . . i j Gj Uj y. sn For more on the logic
of pivotal voting. this becomes a very low probability event. i jl si y. exp U l. s lj y. s. In this
case. and for each integer n.
jl lj i y. great regularity is imposed asymptotically upon individual pivot probabilities in any
AQRE. N . In addition. sn lt . sn /k y. there is an n such that. jl jl i y. it should be emphasized
that Proposition does not impose any requirements on the marginal distribution of i for any
voter i as the electorate grows. .. i l . if n gt n . k. l. the marginal distribution of voter s type
with respect to e.e.d limn n y. We thank a referee for helping us clarify this discussion. i j sn
y. a voter may vote with higher probability for their a priori secondranked candidate than for
their a priori rstranked candidate if the pivot probability for the rstranked candidate is
sufciently low in relation to that for the secondranked candidate. ij jK are xed across all
electorates N . for all i. si i Further. si Before continuing. While we have assumed that the
marginal distributions of the payoff disturbances i. in all cases. sn lt . j. Thus. y. In other
words. voter s ideal point when there are n voters in the electorate is not necessarily equal to
the marginal distribution of when there are n voters in the electorate. the convergence is
uniform for any gt . We feel that this fact illuminates the strength of the proposition. i l lt . i j sn
y. sn lt .g.. sn /i y. and i n y. Remark Note that the requirement that voters adopt a Bayesian
equilibrium means that voters vote strategically in multicandidate elections. jl i y. sn . the
choice of a marginal distribution of each new voters policybased preference type is similarly
unconstrained.
and lies in the interior of the policy space. we omit these parameters. we establish that for a
large enough electorate. as discussed earlier. sy. is unique. the distribution of the ij is
invariant to the electorate. Making an additional assumption on preferences. We rst show
that in general we cannot expect even a local equilibrium to exist at x N To be technically
correct. To simplify notation. let x N argmaxxX iN Ei u x. since we are considering N and N to
be variables. X. The proofs of our results utilize our assumptions that the ij are i. each
candidate j K is assumed to maximize Vj y Vj y. is a quantal response equilibrium for the
voters. For any admissible type distribution N . N . the expected social optimum. that such a
point exists. Furthermore. s Vj y max Vl y. the global equilibrium is unique for two candidate
elections. we should condition voter and candidate strategies accordingly. Assumption For all
N . and measure N on T iN Ti . . For a xed electorate. y Y . with full support and that. lN j . N
.d. as described in the previous section.i. denote the expected social optimum with respect to
N . let s be any strategy prole for the voters such that for any candidate positions. is unique.
We assume throughout the remainder of the paper that for each N and N . for each j K . i . N
. exists. all candidates adopting the social optimum constitutes a global equilibrium.
Candidate Equilibrium This section examines the incentives of candidates competing for
votes in a world populated by voters who play quantal response equilibrium strategies. Then.
We have not explored the possibility of multiple social optima or the possibility that the social
optimum lies on the boundary of X. x N . and lies in the interior of X.
Using Equations . the expected vote for candidate i is Note this function is symmetric around
zero. with no abstention. the CDF for Gz is Gz / z z . Now assume that in a small
neighborhood of zero. Finally. associated with the following strictly concave utility functions
ux. x/ x . i and i i . To see that is not an equilibrium. suppose that candidate adopts x against
x N . . Then n ux. . i so that the social utility maximizing policy is x N . . let n k. let N be a
measure that puts all mass on the vector such that for i k. Specically. and . and let T . and
letting i i . implying that there exists admissible type distributions that are consistent with this.
For some positive integer k.without some additional restrictions on preferences or the policy
space. i x . x x ux. . X . Example Nonexistence of Equilibrium at x N . . and a one dimensional
policy space. suppose that K . . Consider a simple model of electoral politics two candidates.
There exists an integer n such that. we dene the following condition on the sequence of
preference distributions as the electorate grows. If . it must be the case that . implying that x
is a point of inection. independent of and . . . it must be the case that the third derivative of
candidate s expected vote is negative. . the third derivative of candidate s expected vote at x
is equal to .. To avoid situations analogous to Example . the second derivative of candidate s
expected vote at x is equal to . However. if . for all N with N gt n. Accordingly. x is not an
equilibrium. in order for x to be an equilibrium. dx . Thus. then the rstorder necessary
condition is satised. x x x x x x x x Omitting the straightforward calculations.omitting a few
steps of algebra n V x. Accordingly. Thus. Condition . the rst derivative of candidate s
expected vote at x is equal to dV . Suppose that this is the case. is not satised. then the
rstorder necessary condition for an equilibrium at . Regardless of and . i i G ux. G ux. there
exists a nite . for x to be an equilibrium. but the second derivative of candidate s expected
vote is equal to zero. Gi ux. while the third derivative is nonzero. Equation .
The second corollary notes that. is intended to impose eventually on the sequence of
preference distributions. i ux . Essentially. and uniform upper bounds on x ux N . when the
voters types are independently and identically distributed as in Ledyard . and let Hx denote
the Hessian of with respect to x at x. The preferences in Example do not satisfy Condition . i .
The condition requires that. Finally. Dh ux. x N . be negative denite. and Dh ux.number M
satisfying the following for each N on T E xXx E sup ux .. the requirement of a uniform bound
on the eigenvalues of Hx E with simply requiring that Hx E iN iN ux N . . let x denote the
magnitude of the gradient of with respect to x. The rst lemma establishes that compactness
of X. this requirement is that the sensitivity of individual preferences to policy i. for all
electorates exceeding some xed nite size. can be replaced ux N .. respectively. for all . for
example. and the eigenvalues of Hx E iN ux N . not be arbitrarily larger than the sensitivity of
the sum of individual preferences in a neighborhood of the social welfare optimum. In
addition to providing leverage for the application of this papers results in other settings.
amounts to a regularity condition on the sequences of preference proles we consider for the
remainder of the paper. for any function .. in the direction h. x ux. iN ux. Condition . i iN ux.
We now provide two sufcient conditions for the satisfaction of Condition . The lemmas
require some additional notation.e. are sufcient for the satisfaction of Condition . For any
vector h Dm . denote the rst and second directional derivatives of u. Let Dm be the set of unit
length direction vectors in m and h be an arbitrary element of Dm . i Mgt . Lemma and
Corollary are also intended to illustrate the substantive restriction that Condition ..
smoothness of ux. The distinction here is intimately related to the difference . there exists a
uniform bound across all policies and large electorates on the ratio of the variance and mean
of the voters payoffs.
for all N . Lemma Assume that X is compact and. the required uniformity of the upper bound
is with respect to the set of all feasible electorates. If . and Lemma . B and iN . The next
corollary follows immediately from Lemma . essentially For each electorate N . As illustrated
by the proof of Lemma in the appendix. constructing an example in which the sequence of
type distributions does not satisfy Lemma but does satisfy Condition . for any function. is
satised. be twice continuously differentiable in x. .. ux N . is no greater than B. the maximum
eigenvalue of Hx E then Condition . is satised. then Condition . Before continuing to the main
result. is negative denite at x N .between A strict concavity of a realvalued function and B
negative deniteness of its Hessian matrix. and . x E iN ux N . Corollary Assume that X is
compact and. Quadratic preferences based on Euclidean distance from an ideal point over a
compact space X would satisfy the necessary assumptions. B implies A. B. Hx E iN ux N . for
every T . and all T . Of course. with the exception that it requires that the eigenvalues of the
Hessian of the sum of individuals utility functions have a strictly negative uniform upper
bound at x N . let ux. the Hessian in question is evaluated at the social welfare optimum.
each N is an admissible product measures of the form N iN . demonstrated that strict
concavity of preferences is not enough to guarantee equilibrium. be twice continuously
differentiable in x. . Example . D ISCUSSION OF L EMMA AND C ONDITION . Thus. above.
it is worthwhile to comment further on the relationship between Condition . If there exists a
nite B gt such that. for every T . let ux. Lemma includes Example . but the converse
implication does not hold.
to be violated is the existence of a subsequence of electorates N satisfying the following E E
ux . does not hold for any subsequence of electorates is sufcient to ensure that Condition . .
Accordingly. and any admissible N on T iN Ti . is zero.. Vj y Vj yj . is continuous in x for each
T . which states that the social optimum is a global equilibrium in large enough electorates so
long as preferences are uniformly bounded and the distribution of types across electorates
satises Condition . Corollary If k . iN ux. yj Vj y .requires that one construct a sequence of
type distributions for which the limit of the sequence of determinants of Hx E iN ux N . . then
by the assumption that x N is unique. i . Finally. is satised. Viewed another way. a necessary
condition for . x N constitutes a global equilibrium under the margin of expected victory for
any j K and yj X. is met. There exists an integer n such that for any set of voters N with N n gt
n . For the case of two candidates. and assume that Condition . The Main Result We now
present the main result. Theorem . the uniqueness of the social welfare optimum is
essentially vanishing in such a sequence of electorates. the equilibrium identied in Theorem
is unique. . . verifying that Equation . y x N . if one assumes that X is compact and ux. i iN ux.
with the weak inequality becoming strict whenever yj x N . i xx lim . . Theorem Let u be
uniformly bounded. i ux . . then the equilibrium found in Theorem is unique.
In particular. Does the equilibrium prediction of convergence comport with observed political
platforms While many readers have and undoubtedly will doubt whether policy convergence
in observed in realworld elections. Of course. we have extended equilibrium results of
Coughlin. Ledyard. adequately testing this prediction is difcult for at least three reasons. This
is because the platforms in this paper represent the credible commitments of the parties
regarding which policies they will implement if elected. This is further complicated by the
mechanisms of policymaking in real world democracies. the primary question that any model
must confront is that of empirical validity. the issue of whether the platforms offered by the
major political parties differ from one another is debatable on several levels. Thus.
differences in announced platforms that will not translate into differences in policy outcomes
are not inconsistent with the convergent equilibrium constructed here. In particular. Further
theorizing about electoral competition within richer models of policymaking is necessary
before a denitive conclusion can be reached about the effective amount of divergence
between observed In addition. regardless of the structure of the underlying framework. We
have not explored the impact of multiple social welfare optima on electoral competition within
a quantal response voting framework. party leaders in parliamentary systems generally serve
at the pleasure of their partys MPs. Conclusions and Extensions In this paper we have
provided a general framework for probabilistic spatial voting models in large electorates.
Banks and Duggan. While we do not feel that this assumption is restrictive when the space of
feasible platforms possesses nonempty interior. it becomes much more restrictive if one
allows for nite policy spaces and indifference about certain components of policy by
substantial proportions of the electorate. The President of the United States does not set
policy unilaterally. . or as the result of boundedly rational behavior allows for strategic
behavior by the voters. First. our results are based on the assumption that the social welfare
maximizing policy is unique. In addition. our model incorporates strategic voting within a
probabilistic voting setting. and other researchers to policy spaces of arbitrary nite
dimensionality and elections with both abstention and arbitrary numbers of candidates. while
our model is agnostic as to the cause of probabilistic choice the probabilistic choice in a QRE
model can be assumed to arise either as the result of rational behavior under payoff
disturbances as we have modeled it here. Similarly.
Similarly. Thus. Speaking somewhat loosely. inclusion of different candidate. Groseclose .
even if convergence is not observed. In other words.party platforms. .. Finally. Jr. minor
parties may play outofequilibrium strategies for a variety of reasons. Candidates and/or party
leaders may seek to maximize their own policybased payoff functions e. Wittman . On a
related note. the number of voters required for the social welfare optimum to be an
equilibrium decreases. as expressive motivations become a larger determinant of individuals
vote choices. Calvert . the required size is decreasing in the variance of this distribution.
Furthermore. it is entirely plausible that equilibrium platforms are divergent because of
nonpolicy i. Accordingly. and/or voter motivations represent promising avenues for future
research. Duggan and Fey . the motivations of realworld political parties may include more
than plurality maximization. and Schoeld and Sened . Finally.. the results presented here are
asymptotic. Schoeld . party. our results do not rule out the existence of other equilibria in
races with more than two candidates. as discussed by Wittman . This is in accordance with
the ndings of Lin et al. . however. It should be noted. that the predictions of a valence model
of electoral politics diverge from those presented here in an interesting way only if the
candidates are assumed to have policy preferences as well. properly gauging whether one
should expect convergence in electoral competition requires an estimation of the relative
strength of expressive versus instrumental benets in determining individual vote choice within
a particular polity. Duggan and Fey . valence advantages accruing to one or more of the
parties as examined in Ansolabehere and Snyder. This type of setting is ruled out in our
model by the assumption that voters preferences over the election outcome depend only on
the policy chosen by the winning candidate and not on his or her identity. and others.e.
Secondly. the electorate size that is required to guarantee that the social welfare maximizing
policy is a convergent equilibrium in the electoral competition game depends upon the
distribution of payoff disturbances.g. . Aragones and Palfrey .
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Zi j with probability pij for j K . . Public Choice. Quinn. we can apply Theorem of Milgrom and
Weber to conclude that there exists an equilibrium in distributional strategies. A Proofs of
Numbered Results Proof of Proposition Proof This is a game of incomplete information. .
Donald A. . . with action spaces Ai K and type space Ti for each i N . A Logic of Expressive
Choice. The action spaces are nite. Wittman. and let Z n be the set of sequences Z Z .
Martin. . Andrew D. and the distribution of types is equal to and hence absolutely continuous
with respect to the product distribution of the marginal distributions of types across
individuals. American Political Science Review. Princeton. Lemma Fix gt . Schuessler. is
assumed atomless. Further. . Proof of Proposition To prove Proposition . Alexander A. . .
Western Economic Journal. . since the distribution of player is types. Whitford. it follows from
Theorem in the same paper that the equilibrium can be puried to be in pure strategies. .
Candidate Motivation A Synthesis of Alternative Theories. i . Thus. Princeton University
Press.
. . . the Xin are independent. and for each n. Dene Xin Zin i . . . pi . piK which consists of all
but the rst component of p. piK K . For BJ z K zj zk gt zl for all j. . By the assumption that pij
gt for all j K . and is characterized by a vector p p .where j is the j th unit basis vector in any J
K. which ranges over a compact set. Then for any J K with J . Zn Z n to attain the maximum
in Equation A. The mean of Zi is i pi . dene K . it follows that Vn is strictly positive denite and
hence invertible. . k J. . . . . l K J. Dene Vni to be the variance covariance matrix of Zin and let
Vn variancecovariance matrix of the random variable i n i Vni denote the Xin . Since PrZ BJ
is n continuous as a function of p. and p K n satises pij for all i. .. where pi pi . Let Tn denote
the . Denition . . and dene n J max PrZ BJ n ZZ A. n n Pick Z n Z . p... . pn . where each
random variable Xin has zero mean. Zn Z n consists of independent. . . but not identically
distributed random vectors. Then the Xin form a triangular array see Meerschaert and
Schefer . Write Z n i Zi . . j. . n a limn J n n b limn J /J for any J J Proof An element Z Z . . . . it
follows that such a J and Z n exist.
k J. To prove b. positive denite matrix satisfying Tn Vn . proving a. n. Hence. That is. k J .
and Pr n n i Zij i Zil gt for j J. p that the distribution of Tn n i Xin converges weakly to a
multivariate unit normal distribution. we note that BJ describes a lower dimensional subspace
than BJ . Xin . the random vectors satisfy the i following multivariate Lindeberg condition For
every gt . the PrZ BJ goes to zero faster than PrZ BJ . j. Corollary . the variances and
covariance of Vni are all uniformly bounded away from zero and one. and i Xij Xik Pr n n Xij
Xil gt i pil pij for j J. l K J n Writing Qn for the cumulative distribution function of Xin . Then n
PrZ n BJ n n i Zij i Zik for j. since pij for all i. the same will be true of Vn . which establishes A.
k J. Thus. Further. . we can pick large enough n so that Tn Xi lt of Equation A. establishing
the result. n n lim i Tn Xi gt Tn Xi n dQn X i A. symmetric. The probability that Zij is pij .
converges to with n. It follows by Lindebrghs multivariate version of the central limit theorem
for triangular arrays see Bhattacharya and Rao . the right hand side of Equation A. and Pr A.
goes to zero with n. n n Tn Tn i Xij Xil gt n i pil pij for j J. Thus. n To see this. l K J i n n n Tn i
Xij Xik n Tn i pik pij for j. Hence. So Vn will be invertible... l K J n n i pik pij for j. So each term
in the summation and for any . when J . note that Zin is in the simplex K . an argument
similar to above shows that for all sequences. limn n . Hence the probability it falls in a
subset of any lower dimensional subspace goes to zero.
Reexpressing Prti Ti W y. i U l. sn .y. . i j Gj Uj y. that. . from Equation . s. j. n with i T for all i
gt . . we have. the marginal distribution of ij possesses full support on and is identical for all
voters i N and all electorates N implies that gt . .. sn be the prole where i i voter i votes for
candidate j. ti J jl i y. . ux. . ti A. But. Then from Equation . Then. t J. . Now. s. dene D Ksupx.
i . letting . j. i Gj D . and minjK Gj D. l K .we have D U j. uy. t J as Eti W y. . J PJ y. si . y. . .
with pij sn y. y. using the fact that jl i for all i. j. sn j. we . ti Prti Ti W y. which implies that si y.
given any sequence . PJ y. .. dene the random variable Zni i j if sn y. s. si . s.kJK PJ y. si .
where i denotes the indicator function that is equal to if the condition is true and zero
otherwise.Proof of Proposition . for any J K. i D for all j. Proof To prove a. s. . for each j K . k.
ti J . where . and j. The assumption p. from Equation . sn be the prole where the voter i
abstains. is the unit vector of length K. i j i So Zni i Z n .
jl i y. recall the proof of Lemma and note that for each J K. n where n maxJK J By Lemma .
which proves a. Thus.kJK J n J j. we can write PJ y. the convergence is uniform in all
arguments. si . sn j. j.kJK J n . sj . A. l. t J Eti W y. y. Since n is indepen dent of i. . A similar
argument shows the n second term in Equation A. is less than or equal to J . To show b. ti Etj
lj Znl l BJ . n where the inequality follows from the denition of J in Lemma . Thus. ti J i Eti li
Znl l BJ Znl l BJ li Ei Ei n Ei J n J . . s. ti Eti li Znl l BJ . The right hand sides of these two
expressions differ only by the ith and j th terms. and the corresponding expression for voter j
PJ y. sn . .obtain Prti Ti W y. limn n . by once again applying Lindebrghs multivariate version
of the central limit theorem for triangular arrays .
l. Thus. i U j. i l Prmax ia ij Prmax ia il si i j al Gj Gl . i mN x A. converge. in the rst part of the
proposition we showed all pivot probabilities go to zero uniformly as n gets large. is also. i iN
ux. sn . An analogous argument sufces to establish c. i . sn . i j Prmax U l. . it follows that the
convergence in Equation A. i U j. Corollary . we get that as n . But then we get n lim n y. ti
since both li Znl and lj Znl converge weakly to the multivariate standard normal distribu tion. n
lim PJ y. y. for each J K. it follows that. . i is uniform in all arguments. i uniformly in i. ti n lim
Eti li Znl l BJ Znl l BJ lj n lim Etj n lim PJ y. y. Proof of Lemma Proof Dene the following
function E E ux . Hence. l K. y. sn . Since the convergence of U l. conclusion b follows. y. sn .
i ij . si . . To show d. i ux . for j. A.Bhattacharya and Rao . y. all terms in the sum in A. y. that
sn y. we have from Equation . using Equation . sn . sj . i j sn y.. iN ux. j. U l. sn . . y. i il U j. i lj
Now. p . Thus.
If Condition . i lim supn E lt lim inf n E Dh ux N . we can take any unit length direction vector
h Dm and apply LHopitals rule twice. is satised for any nite M B.To prove the lemma. Thus. i
. iN Dh ux N . we want to show that E E ux. for each N . iN ux. yj . By the fact that ux. is
violated. E ux. will be innite for some N . iN Dh ux N . let y yj . then Equation A. i mh x N lim
mN x N h iN A. and admissible . satisfaction of Equation A. obtaining E E Dh ux N . is
equivalent to the following E lim sup sup E n hDm Since E lim sup sup E n hDm and lim supn
E lim inf n E Dh ux N . i iN it follows that Condition . i iN Dh ux N . for all T we can write
Equation A. as was to be shown. iN Dh ux N . where yl x N for all . i iN iN Dh ux N . as E ux. i
. Accordingly. After normalizing so that ux N . it is sufcient to show that lim supn supxX mN x
lt . i lt . A. mN x for each N . i B B lt . is twice continuously differentiable in x for any . Proof of
Theorem Proof For any set of voters N . iN lim sup sup n xX lt . B iN Dh ux N . i iN Dh ux N .
we can assume utility functions are normalized with ux N . We rst show that for large enough
n. s i y. . i Gj i y. For i N . s . ti k ik ij i y. y. l y. s. the probability of a vote for candidate j is
given by si y. s. i R.l i y. Using Equation .. l j and yj x . i ux . s. . i j s iN A. we have that n Vj y
Ei Gj i y. ti U j. .. s i i i j j. Then. s uyj . s uyj . y. i ux . . i iN Ei Gj i ui iN A. s y. i ux N .
Normalize the i by in the following manner. s . i iN Without loss of generality. from Equation
A. s uyj . s.. i lj where i y. for all l K j. k y. and using Equations . k T i . Given an individual i N .
and .. s i j j i y. i ux N . s uyj . s uyj . s. Vj y Vj yj . i for k K j Pr j and i ij i y. A. we can express
the vote for candidate j as n Vj y Ei i y. i for all i N and i T . i j Pr lK j max U l... Write ui uyj .
the above can be written as n n Vj y Ei Gj i y. let k i i . Then. . and i i y. and j y. yj Vj y .
we can write n n Vj y Vj y Ei Gj D i ui Ei Gj iN Ei D i ui T Gj i ui T Gj zi y i ui iN A. Gj is a k k
symmetric matrix of second partial derivatives of Gj evaluated at .. . and denotes a k
dimensional vector of zeros. where zi y . applying i Taylors theorem. Now. . . for some . . G is
the smallest j . maxiN maxi lt where the inner max is with respect to the components of i . .
where Gj is a k dimensional vector consisting of the gradient of Gj evaluated at .and D . it
follows that for any gt . Using these facts and continuing the derivation of Vj y Vj y . . k It is
easily shown that j gt for all i N and j K. and maxiN maxi lt for all n gt n where the inner max
is with respect to the components of i . . ui . . by Proposition .. for each i N . we can nd a
value n such that max D lt . Vj y Vj y n Ei Gj D i ui i ui T Gj zi y i ui iN Ei T Gj zi y i u i i iN Gj Ei
D i ui n n iN k k Gj Ei ui Gj Ei u i n n iN iN A. . Then. so that i is well dened. .
k k G Gj Ei ui Ei u j i n n iN iN k Gj Ei u i n iN iN lt k Ei ui lt G j n iN A. Thus. Ei u G j i lt . j j
We now show that there exists n such that for all N with n gt n . G must be nite j since Gj is a
cumulative distribution function and hence is bounded and is twice continuously
differentiable. The construction of G is j similarly motivated it is chosen so as to maximize the
impact of Ei u . for i . for D. so that it must be strictly convex on some open interval of . In
order to satisfy Equation A. becomes less than zero for all y x N X. follows from substitution
of kG for Gj .. we must take the supremum of the lefthand side over all y x N . for all y x N .
which follows because y is the unique social welfare maximizer.. It should be noted that G is
positive by the j j assumption that the distribution of ij possesses full support for all i and j.
This supremum is dened to be nite and denoted by M in Condition . and G is positive
because i the marginal distribution of ij possesses full supj port for all i. and G is dened as j G
sup Gj z. ii Gj is bounded between and . j zRk We construct G as the smallest element of G
since the rst term of Equation A. resulting in the following requirement for Equation A. . j for i
. which is positive. and iii Gj is twice continuously differentiable. Of course. is negaj tive. For
a given y x N .element of Gj . k Gj iN Ei ui The inequality in Equation A. and k G for Gj zi y. it
follows that G /G gt . The i nal step in the derivation of Equation A. this is for a given y x N .
we can choose n to exceed n as dened in the statement of Condition . is satised for
sufciently small gt . the righthand side of Equation A. to If necessary.
. s. For z K . i l iN Ei Gj i y. for sufciently small. write Gj z Gl z. y. Thus. s. s. s. where Gl is as
dened in Equation . i U l. Vj y Vj yj . . i A. Equation A.. i uyj . the probability of a vote for
candidate l is given by the following U . As with Equation A. In this case. and i il Pr ij il j y. i U
l. s.be satised for all y x N G j . s i i i j j. lj j y. Next. i uy .j U k. s i y. s. . with l y. s. i U l. s ux N .
and i ik il maxkKl. i for all i N . s i y. and maxkKl. y. y. y. y. k G j M lt A. i where i y. for any yj
Yj . s i jl kj and k y. s is maximized. s. we can assume utility functions are normalized with ux
N . we can express the vote for candidate l as n n Vl y Ei si y. l l i y. yj Vj y with strict
inequality whenever yj x . y. i l Pr U j. i . s ux N .j k y. is satised. s. s ux N . i uyj . Vl yj . i uyj .
iN As above. i . i uyj . i i Gj i y. s i y. j. s y.. i . . s for all k K l.k y. i .l i y. s ux N . i Using
Equation . s i ux N . We pick l K j for which jl y. yj Vl y . and si y. i . l y. s. we show that for
some l j.
Thus. x N gt V y . an analogous argument to that in A. y gt . s. with the exception of the
negative sign. write ui uyj . By Theorem . y lt . This yields a contradiction. Then for at least
one candidate j. y V x N . Consequently. .G. But this cannot be an equilibrium for candidate .
i . Assume W. i iN Ei Gj i ui iN A. Since V y . Note that the above takes exactly the same form
as Equation A. yj Vj y .and i T . V y . the above can be written as n n Vl y Ei Gj i y. and i i y. yj
Vj y and Vl yj . As before. We have shown that Vj yj . yj x N . establishes that we can nd large
enough n so that Vl y Vl y is positive. yj Vl y . above. Proof of Corollary Proof Suppose there
is another equilibrium.L. Then. that j . s uyj . So y is a global equilibrium for the objective
function V . Vl y Vl yj . y . Hence the equilibrium is unique. So Vj yj . y. for any yj Yj . yj Vl y
with strict inequality whenever yj x N . It follows that V y .O.