Should rational voters rely only on candidates characteristics?

Should rational voters rely only on candidates characteristics? Sergio Vicente. IDEA, Universitat Autònoma de Barcelona. February 006. Abstract This paper analyzes the role of information in elections when one candidate enjoys a non-policy advantage. We consider an electorate with preferences over policies and with a minor concern about candidates. A fraction of the electorate is assumed to observe very accurate signals -yet not perfect- of politicians promises on policy. We show that it is optimal for these voters to disregard their information and vote for the candidate they like the most. Hence, we provide a "Downsian" rationale for the personal vote. We also show that if a candidate s personal advantage is large enough, partisan policies will arise. I am extremely grateful to Enriqueta Aragonès, to Humberto Llavador, and to David Pérez-Castrillo for guidance. Special thanks to Micael Castanheira and to Julio González-Díaz for making very detailed comments. I would also like to thank Coralio Ballester, Joan Farré, Rosa Ferrer, Rahmi Ilkiliç, Moshe Justman, Inés Macho, Joan de Martí, Jordi Massó and seminar participants at Universitat Autònoma de Barcelona for their comments. Financial support from Fundación Ramón Areces; from Ministerio de Educación y Ciencia under grant BES-004-5584 and project BEC003-0113; and from Generalitat de Catalunya under project 005SGR-00836 is gratefully acknowledged. All remaining errors are mine. Contact: svicente@idea.uab.es. 1

1 Introduction One of the best documented features in contemporary politics is the ignorance of the American voter. Álvarez and Franklin (1994) conducted a survey in order to provide a direct measure of voters lack of knowledge about elected o cers standpoints on public policy issues 1. Their study indicates that almost 50% of the individuals were "not very certain" about politicians position on taxes. This proportion exceeds 85% when those that reported to be "pretty certain", yet not "very certain", are accounted for. These gures contrast sharply with the high level of certainty in self-placements, as only 10% of respondents declared to be "not very certain" about their own opinion on this issue. Scholars have largely emphasized the ability of imperfectly informed citizens to make use of voting cues, such as candidates personal characteristics, so as to overcome their lack of information (Popkins, 1991). Beyond this, social-psychologists argue that voters make judgements based primarily on what Stokes (1963) dubbed valence factors, such as superior charisma, better name recognition, higher competency or the like. And the fact that quality of candidates, rather than policy issues, explains vote shifts from an election to the following has been recognized for decades (Campbell et al, 1954; Miller and Miller, 1976). However, casting a ballot on the basis of politicians personality has been often considered as irrational in the Downsian tradition: from a rational choice viewpoint, a (perfectly informed) voter facing two candidates making binding commitments on policy should vote for the one whose promise is closer to her bliss point. This paper analyzes the e ects of the lack of information in elections on voters behavior when one candidate enjoys a non-policy advantage on policy outcomes and on electoral competition. We show that the valence advantage, although subsidiary with respect to policy issues, is crucial. A candidate favored by a majority of the electorate can exploit this 1 A National sample of 797 adults from the 48 contiguous states was drawn. Respondents were questioned about their preferences and their perception about the position of a senator from their state on two policy issues (abortion and taxes) and on a liberal-conservative dimension. Groseclose (001) provides a detailed description of the potential sources of a candidate s valence avantage.

advantage to win the elections. Moreover, she can pro t from a su ciently high personal support and implement her favorite platform, even if this policy is not desired by most voters. Our model predicts a very simple strategy for voters: disregard politicians commitments on policy issues and vote for the candidate they like the most. Hence, we provide a rational for employing personal characteristics as a voting cue, even if the electorate mainly care about implemented policies. We propose a spatial model that builds on Downs (1957) and Wittman (1983) 3. We add two novelties to the standard setting. First, we assume that voters have preferences over the competing candidates, although they care primarily about the policy outcome: no elector is willing to trade-o his favorite policy for his preferred candidate. In addition, our model posits two types of voters. Informed electors are assumed to observe candidates policy promises with no error. Hence, they vote for the candidate whose platform choice is closer to their preferred policy. Imperfectly informed voters receive a noisy (informative) signal of platform commitments; this group is intended to capture the role of (the lack of) information in elections. Their choice is based on the inference that they make in equilibrium about candidate s actions, upon observation of the signal. We show that voting according to personality preferences is optimal, even if the advantaged candidate chooses his favorite (extreme) policy. If imperfectly informed voters conditioned their actions on signals, both candidates would choose median platforms. But then signals would not play any role. Since candidates choice would be certain, there would not be room for Bayesian updating upon observation of the signals. Voters should then ignore them, as any signal pointing out at an extreme platform should be attributed to the potential error in the assessment of information that ignorant voters are subject to. Hence, they must cast their ballot for the candidate they like the most. If the advantaged candidate enjoys a su ciently high personal support, she can exploit this voting pattern to implement her favorite policy. Non-median outcomes arise because the disadvantaged candidate cannot credibly signal that she has chosen a moderate policy 3 Our model is similar to Groseclose (001) and Aragonès and Palfrey (005). We add noisy signals to study the role of (the lack of) information in the electorate. 3

to imperfectly informed voters. Even if there is a large share of uninformed voters, elections aggregate information: informed and uninformed voters with the same preferences cast their ballots for the same candidate. This feature holds true even when the winning candidate commits to his favorite policy and this choice is not desired by most voters. Hence, although uninformed voters are responsible for the absence of the disciplining e ects of elections -that politicians go median-, they behave as perfectly informed candidates. The remaining of the paper is organized as follows. Section discusses related literature. In section 3 we describe the model. We characterize the equilibria of the electoral competition game and compare it with a full information benchmark in section 4. Section 5 concludes. Related literature Several recent papers study the e ects of non-policy advantages on electoral competition. Aragonès and Palfrey (00) analyze an electoral competition game between two candidates that maximize the probability of winning in a world where the location of the median voter is uncertain and one of the candidates is preferred by all voters. In their setting, an equilibrium in pure strategies does not exist, as the (o ce-seeker) advantaged candidate would always like to mimic the location of her opponent. Moreover, they predict that the favored candidate will locate more centrally than her adversary. In contrast, an equilibrium in pure strategies always exists in our model. Furthermore, politicians are both policy and o ce motivated in our model, which introduces a centrifugal pressure that is absent in their setting. We show that even if all voters prefer a centrist platform with certainty, the candidate with charisma implements her favorite policy, provided that the share of voters that prefer her is su ciently high. Groseclose (001) also considers policy-motivated candidates. However, the advantage candidate always chooses a more centrist location than her rival. The candidate with charisma tends to moderate because, by doing so, he minimizes the signi cance of policy and hence increases the relative importance of her valence. This centripetal force is absent 4

in our model because our assumption on the value of candidates personal characteristics rules out the possibility that any voter cast his ballot for the favored candidate if he prefers the policy proposed by the disadvantaged politician. Ansolabehere and Snyder (000) elaborate on a multidimensional setting with o ceseeker candidates and certainty about the location of the median voter. The advantaged candidate adopts a centrist position and wins with probability one. Our model also considers that candidates know the preferences of the electorate with certainty, but predicts divergence of platforms due to the facts that candidates have policy motivations and that a fraction of the electorate is not perfectly informed of policy commitments. All the above papers, although very diverse, nd out that favored candidates tend to reinforce their advantage by choosing moderate platforms. In contrast, we nd that a su ciently high support for one candidate translates into partisan policies. Gul and Pesendorfer (005) study an electoral competition game for large elections with a high proportion of uninformed voters and also nd out that a large non-policy advantage may lead to non-median outcomes. Our results di er from theirs in two main respects. First, divergent policies arise in their setting when the favored candidate mainly cares about holding o ce and voters ignore the size of her advantage. Voters expect an o ce-seeker to implement a moderate policy with high probability, although median policies are never chosen whenever the favored candidate enjoys a large personal support. Hence, either when candidates are partisans or when voters know the personal support enjoyed by candidates (or both), central policies emerge as the outcome of the game. In contrast, in our model imperfectly informed voters are able to perfectly forecast the advantage candidate s equilibrium location. Therefore, extreme policies are implemented for any degree of partisanship -either known or unknown by voters- and even if voters know the exact distribution of preferences in the electorate. Second, in their setting ignorant voters cast their ballots for the favored candidate only if he is an o ce-seeker. For a high degree of partisanship of the advantaged candidate, voters switch to the candidate that they dislike the most, because they (incorrectly) expect the candidate with charisma to implement an extreme policy. In our model, imperfectly informed voters cast their ballots according to their personality preference, even if a partisan policy is enacted, because the 5

disadvantaged candidate fails to credibly signal the choice of a moderate platform. 3 The model We consider a Downsian model of electoral competition in which voters have preferences both over policies and over candidates. Our model closely follows Groseclose (001) and Aragonès and Palfrey (005). A fundamental twist of our model is that we consider that some voters do not perfectly observe politicians commitments to policy; instead, they observe a noisy (informative) signal of candidates promises. 3.1 Candidates There are two candidates, indexed by j fa; Dg, who we will refer to as the advantaged candidate and the disadvantaged candidate, respectively. The policy space X consists of three points in the real line f ; 0; +g, 0 < < 1, which are referred to as L (left), C (center), and R (right). Therefore, we assume that platforms are symmetric. This assumption is made because one of our goals is to analyze the e ects of a non-policy advantage on policy choices. If platforms were not symmetric, a voter with bliss point at C would not be indi erent between extreme policies. Hence, we would be conferring a policy advantage to one of the candidates. The assumption of symmetry rules out a potential exogenous source for platform divergence, which arise endogenously in our model due to non-policy factors. Our results generalize to non-symmetric policy spaces, provided that the relative distance from extreme to centrist policies is not very large. We write x A and x D candidates platforms positions. Candidates care both about the policy implemented and about holding o ce. The policy component is characterized by an ideal point bx j, with utility over alternatives in the policy space a strictly decreasing function of the Euclidean distance between the ideal point and the platform position. Candidate A s ideal point is L, whereas D s favorite policy is R, i.e. cx D = cx A =. To reduce the number of cases, we assume that candidate A can choose only between a leftist (L) and a centrist (C) platform, whereas D chooses between a centrist 6

(C) and a rightist (R) policy: x A X A = fl; Cg and x D X D = fc; Rg 4. The weight i i that candidate j places on holding o ce is denoted by j h j ; j, where h j ; j (0; 1). The value that a candidate attaches to holding o ce is private information: only candidate j knows j, which is not observed by any other player. Each j is independently drawn from a distribution with c.d.f. F j with no mass points, which is common knowledge. Formally, if policy x is implemented, candidates get: U j (x; j ) = j I j (1 j ) jx bx j j, (1) where I j is an indicator that takes the value 1 if the winner is candidate j and 0 otherwise. Hence, without loss of generality, we assume that the value of getting o ce is 1. 3.1.1 A disgression on the relative weights between o ce and policy We assume that the weight j that candidates place on holding o ce is private information. This assumption is motivated by the fact that politicians willingness for o ce may well depend on his attachment to a particular party and the weights of di erent factions within it. Voters and other candidates may perhaps ignore the underlying forces shaping the actual value. The analysis of equilibrium leads to an additional consideration. We are aware of the fact that it is hard to provide a convincing argument on how to interpret mixed strategies in a location game. However, as emphasized by van Damme and Hurkens (1997), restricting oneself to pure-strategy equilibria may be potentially misleading. When agents play pure strategies, there is no room for updating beliefs upon observation of any signal. Consequently, the set of (pure-strategy) equilibria of the sequential-move game in which candidates choose locations and voters observe noisy signals coincides with that of a simultaneous-move game in which voters take their actions without observing candidates commitments at all. Hence, one must allow for candidates randomization over the set of policies for pursuing a mean- 4 Proofs take a simpler form with this assumption, but our results do not depend on it. As we shall see below, although candidates were allowed to choose the platform that they dislike the most, they would never make such choice in equilibrium. 7

ingful examination of voters strategies. Harsanyi s (1973) insight on puri cation of mixed strategies allows us to reconcile both concerns. In our game, the weight that each candidate assigns to winning is private information. Hence, candidates pure strategies are mappings from types -the weight that they place on winning- to actions on the space of policies. In equilibrium, di erent types may potentially choose di erent alternatives, inducing a probability measure over the set of actions is through the distribution F j of types. Imperfectly informed voters beliefs about actual actions played by both politicians are given by this probability distribution. 3. Voters We consider a continuum 5 of voters whose payo s depend both on the policy implemented and on the characteristics of the winning candidate. We assume that all voters prefer policy C, the utility over alternatives in the policy space being a strictly decreasing function of the Euclidean distance between C and the platform position. This is an extreme assumption that permits us highlight that platform divergence from moderate outcomes is solely induced by non-policy factors. Our results easily extend to a setting with partisan voters, as long as no group of extreme voters is decisive on its own. More generally, this framework is formally equivalent (in terms of outcomes) to the case in which voters have heterogenous preferences over policy and the median voter s favorite policy is C, with the assumption that voters do not use weakly dominated strategies. Voters di er on their evaluations of candidates personal attributes. The quality advantage of candidate j perceived by voter i is captured by an additive constant i j added to the utility that a voter obtains from the policy enacted. Formally, voter i s utility if candidate j wins the elections is given by V i (x j ) = ij jx j j, () 5 It is easy to show that our results do also hold in a framework with a nite number of voters whose preferences are known by candidates. Moreover, equilibrium outcomes are (asymptotically) equivalent to those of a setting with a large nite number of voters whose preferences are unknown but drawn from the distributions referred to below. 8

We de ne the relative advantage of candidate A over candidate D as perceived by voter i as i = ia id. Hence, voter i prefers candidate A if and only if i > 0. Each i is drawn from a common knowledge c.d.f. F with positive density on [ ; ] and with no mass points. We assume that a majority of voters prefer candidate A, i.e. that F (0) < 1. Our assumptions on voters payo s are such that the median voter s favorite platform is a Condorcet winner: if the favored candidate A chooses L and the disadvantage candidate D s choice is C, then all voters would get higher utility by voting for D. Put di erently, no voter is willing to trade-o his favorite policy for his preferred candidate. Note that if a candidate loses, his utility depends on the policy adopted by the winner. Hence, candidates incentives to moderate are twofold. First, chances to win o ce are higher. Second, moderating can be used as a tool to force one s opponent not to choose an extreme platform. 3.3 Information We consider two groups of voters. A fraction is perfectly informed about candidates commitments to policy. Put di erently, this group regard their information about platforms choice as perfect. The remaining part of the electorate observe a noisy signal of politicians promises. The signal technology works as follows 6. Every voter independently draws a signal from each candidate s platform choice. The probability that signal s j arises when candidate j chooses x j is given by 8 < 1, if s j = x j p(s j jx j ) =, :, if s j 6= x j where 0; 1. We place no other restriction on the value of. For su ciently small values of, the probability that the signal s j = x j is realized is very large. Hence, signals are potentially very informative of actual actions. This modelling is intended to capture the feature highlighted in Álvarez and Franklin (1994) referred to above. A non-negligible share 6 For an extensive discussion of imperfectly observable commitments, see Bagwell (1995), van Damme and Hurkens (1997), and Güth, Kirchsteiger and Ritzberger (1998). 9

of an electorate may be able to place politicians platforms, although they attach positive probability to the fact that the actual choice be another. 3.4 Timing 1. Candidates simultaneously select the ideological position of their platform x = (x A ; x D ); the winner candidate must implement the policy that she commits to.. Second, chance chooses a signal s = (s A ; s D ) X A X D. 3. A fraction of the electorate observe x and the remaining observe s. 4. Voters cast their ballots for one of the candidates (no abstention). 5. The politician with the higher share of votes wins and implements the policy that she has committed to. 4 Equilibrium A (pure) strategy for informed voters is a mapping y i : X A X D! fa; Dg that speci es for whom to vote upon observation of a candidates commitments to policy. We adopt the following convention hereafter: we denote i (x) the probability with which the informed voter i votes for the advantaged candidate A when candidates platforms are given by x. Informed voters face a trivial decision problem. If one of the candidates chooses Center, while her opponent chooses a partisan policy, all voters cast their ballots for the candidate choosing the moderate platform. If either both candidates commit to Center or to Left and Right, respectively, they vote for the candidate whose personal characteristics they prefer the most. For simplicity, we assume that the indi erent electors, i.e. voters with candidates preference parameter i = 0, vote for the the disadvantage candidate. Notice that the assumption that the distribution of voters types F has no mass points guarantees that the set of indi erent informed voters has zero measure. Hence, our results do hold for any 10

strategy that indi erent informed voters may take. To sum up: 8 1 if either x = (C; R) >< i (x A ; x D ) = or x 6= (L; C) and i > 0 >: 0 otherwise. (3) Imperfectly informed voters observe noisy signals about the platforms selected by candidates in the rst stage of the game. A strategy for these voters is a mapping i : X A X D! [0; 1] that speci es the probability with which they will vote for the advantaged candidate upon observation of a signal pair. Let q(x j js j ) denote the probability that the voters attach to the fact that action x j has been taken when the signal corresponding to candidate j is s j. This probability is de ned in equilibrium; voters hold prior beliefs -given by candidates equilibrium strategies- about the choice of candidates, and update these beliefs upon observation of the signal pair using Bayes rule. From (), it is straightforward to see,that voters optimal strategies must satisfy the following condition: 8 < 1 if i > [q(ljs A ) q(rjs D )] i (s A ; s D ) = : 0 if i [q(ljs A ) q(rjs D )]. (4) As before, for any given signal s X A X D, we assume that an indi erent voter b (s) casts his ballot for the disadvantage candidate. This assumption is innocuous for our results under the assumption that F has no mass points. Candidate j s share of the votes is denoted by N j (x), j fa; Dg,, when a platforms pair x = (x A ; x D ) is chosen. Given the choices of candidates, chance will select a signal pair s = (s A ; s D ) and voters will decide for whom to vote, as described above. Therefore j is derived from the primitives of the model for any given and for any set of strategies f i ; i g i that voters may use. In particular, N A takes the following form 7 : N A (x) = R i(x)df + (1 ) X s D X D X s A X A p (s D jx D ) p (s A jx A ) R i(s)df. (5) We denote j (x), j fa; Dg, candidate j s probability of winning, when a platforms pair x = (x A ; x D ) is chosen. We assume that candidate D wins when there is a tie, although 7 Obviously, N D (x) = 1 N A (x). 11

this assumption plays no role in the results. From (5) it easy to derive: 8 < 1 if N A (x) > 1 A (x) =. (6) : 0 otherwise It follows from (1) and (6) that candidate j s (expected) utility, given a pair of platforms x = (x A ; x D ) and a voters strategy pro le (implicit in j ) is given by: u j (x A ; x D ; j ) = j j (x) (1 j ) X k=fa;dg k (x) jx k bx j j. (7) The rst component of this (expected) utility function arises from holding o ce, while the second corresponds to candidates ideology. 4.1 De nition of equilibrium We analyze the sequential equilibria of this electoral competition game. A (sequential) 8 equilibrium of this game is a quadruple fx A ( A) ; x D ( D) ; (x) ; (s)g satisfying the conditions below. A pure-strategy for candidate j speci es an action x j X j for each of her types, which are described by her willingness for holding o ce j. We let = prob( A jx A ( A ) = L) (8) (resp. = prob ( D jx D ( D) = R)) denote the probability with which candidate A (resp. candidate D) chooses her bliss point in equilibrium. We also denote u A (x A ; A j) = u A (x A ; R; A ) + (1 ) u A (x A ; C; A ) (9) candidate A s expected utility, when he chooses x A X A and candidate D s equilibrium choice is summarized by ; analogously, we let u B (x B ; B j) = u B (x B ; L; B ) + (1 ) u B (x B ; C; B ). (10) 8 Since there are no unreached information sets, any Nash equilibrium is sequential (Kreps and Wilson, 198). 1

An equilibrium must satisfy conditions (i) to (iii) below: (i) Given (x) ; (s) and x j ( j ), x j ( j ) arg max xj X j u j x j ; j jprob( j jx j ( j ) = dx j. (ii) For any x X A X B, (x) satis es (3). (iii) Given fx A ( A) ; x D ( D)g, for any s X A X B, (s) satis es (4), where the conditional probabilities q (x j js j ) are derived according to the candidates equilibrium strategy, using Bayes rule, i.e. q (x j js j ) = p(s jjx j )p( j jx j ( j)=x j ) p(s j ). 4. Equilibrium behavior Before characterizing the equilibrium of this game, we describe some properties of optimal strategies. We start by showing that imperfectly informed voters best responses are characterized by a cuto on relative advantage of candidate A over candidate D: if a type i votes for candidate A upon observation of a signal s, all voters with candidate preference 0 i > i also vote for candidate A when observing the same signal; similarly, if a type i casts his ballot for candidate D, all voters with types 0 i < i also vote for this candidate, conditional on having observed the same signal. We then show that imperfectly informed voters optimal strategies place (weakly) higher probability of election to a candidate when his corresponding signal is C, the voters bliss point. As a direct consequence, a candidate has (weakly) higher probability of being elected if she chooses C. We nally draw on the latter result to show that candidates strategies are also of a cuto form: there exists a candidate j s type b j such that all types who are more eager for o ce than her choose the voters bliss point, whereas those with a stronger ideological motive choose their own favorite platform. The following lemma shows that imperfectly informed optimal strategies are characterized by a cuto b (s), that we call the indi erent imperfectly informed voter. Lemma 1 For any signal s X A X D, there is a candidates preference type b (s) such that 8 < 1 if i > b (s) i (s) = : 0 if i b. (s) Proof. Given signal s, de ne the indi erent voter as b (s) = [q(ljs A ) q(rjs D )]. An 13

indi erent voter exists, because F is assumed to have positive density on [ ; ]. The result follows from direct observation of (4). Voter i s payo when candidate A is elected is increasing in the relative advantage i of candidate A over candidate D. Hence, if a voter with candidate preference parameter i votes for candidate A, so do voters with parameters 0 i > i, upon observing the same signal. Notice that informed voters best strategies trivially satisfy the same property. We de ne the indi erent informed voter e (x) analogously, for any given pair x. Hence, we can characterize voters strategies by cuto points and write candidate A s share of the votes N A (x), when a platforms pair x = (x A ; x D ) is chosen, in terms of indi erent voters as: N A (x) = F e (x) + (1 ) X s D X D X s A X A p (s D jx D ) p (s A jx A ) F We shall use this expression extensively in the proof of the main result. b (s). (11) We have shown that voting behavior is monotone in the candidates preference parameter. The following result states that it is also monotone in the signal. If a voter casts his ballot for candidate A upon observation of the signal Left -the partisan policy- and a given signal from candidate D, he should also vote for this candidate when he observes Center and the same signal from candidate D as before. The following lemma proves a slightly stronger result. Lemma For any fx A ( A) ; x D ( D)g, imperfectly informed voters strategies satisfy: (i) (C; s D ) (L; s D ), for all s D X D ; and (ii) (s A ; C) (s A ; R), for all s A X A. Proof. (i) Both and are su cient statistics of candidates equilibrium strategies for voters. Noting that q(ljc) q(ljl) 9 for all 1 and all, the result follows from direct observation of (4). (ii) Analogous, recognizing that q(rjc) q(rjr). The intuition behind this result is better understood when is thought to be very small and (or ) is interior. When an action, say L, is taken, the probability that the signal s j = L is realized is very large. In short, signals are very informative of actions. Consequently, 9 For all < 1, this inequality is strict whenever (0; 1). 14

the probability that voters attach to the fact that action L has been actually taken is much higher when they observe the signal L than when they observe the signal C. Therefore, when the signal C is observed, a candidate must be elected with at least as much probability as when the signal is L (or R). Notice, however, that if a candidate chooses an action almost surely, i.e. when prob( j jx j ( j ) = x j ) f0; 1g 10, signals are totally uninformative. For instance, if = 1, Bayes rules dictates that voters believe that candidate A has played L, no matter what signal they receive. This result shows that candidates face an incentive to moderate, not only to capture informed voters, but even if the whole electorate were not perfectly informed. This is because imperfectly informed voters observe informative signals, and these may be very accurate (for small ). In contrast, Gul and Pesendorfer (005) assume uninformed voters that do not observe any signal about candidates locations. Their assumption on the information of voters is formally equivalent to ours if we take = 1. In their setting, incentives to conform with the electorate arise exclusively from informed voters. A straightforward implication of the previous result is that the share of votes is (weakly) greater when she chooses the policy preferred by voters. Lemma 3 In equilibrium, the probability of winning for candidate A satis es: (i) A (C; x D ) A (L; x D ) for all x D X D ; and (ii) A (x A ; R) A (x A ; C) for all x A X A. Proof. (i) Given a (pure) strategy x D for candidate D, and for any given, candidate A gets at least as much votes from informed voters when she chooses Center. Hence, A (C; x D ) A (L; x D ) is non-decreasing in. Assume = 0. Then, A (C; x D ) A (L; x D ) = P x D X D (1 ) [ (C; x D ) (L; x D )] 0, where the last inequality follows from the previous lemma, for all 1. (ii) The proof is analogous to the preceding one. This result re ects the fact that tensions towards extreme policies arise from purely partisan motives, as no candidate can increase her share of the votes by shifting from the 10 Note that even if prob( j jx j ( j) = x j ) = 0, there could be a zero-measure set of agent j s types that chooses x j. 15

voters bliss point to her own favorite policy. This result is a direct consequence of the assumption we make on the location of the median voter. Aragonès and Palfrey (00), in contrast, assume that the location of the median voter is uncertain and nd out that the disadvantage candidate chooses more extreme locations than the favored candidate for purely o ce-seeking reasons. Our following result establishes that candidates equilibrium strategies can be also characterized by a cuto. Candidates choose their his bliss point depending on their willingness for o ce: the higher the o ce-seeking nature of the candidate, the more likely that the politician chooses policy preferred by the electorate. We shall draw on this lemma in the characterization of the strategy that candidate D chooses in equilibrium. Lemma 4 Any equilibrium in which candidate j is elected with positive probability is such that there exist a unique cutpoint b i j h j ; j that characterize candidate j s strategies in the following way 11 : 8 < bx x j if j < b j j ( A ) = : C if j b j Proof. For notational simplicity, we prove the lemma substituting out the generic agent j by candidate A. To prove that there is a cutpoint c A that characterizes candidate A strategy in this fashion, it su ces to show that u A (C; A) A. A straightforward substitution in (9) yields u A (L; A) is an increasing function of u A (C; A j) u A (L; A j) = A ( [ A (C; R) A (L; R)] + (1 ) [ A (C; C) A (L; C)]) + (1 A ) ( [ A (C; R) A (L; R)] (1 ) A (L; C)), which is a straight line in A. By lemma 3 and the hypothesis that the candidate is elected with positive probability, u A (C; ) u by continuity of u A (C; ) u A (L; ) has a strictly positive slope. Uniqueness follow A (L; ) and the assumption that F A has no mass points. 11 If j = c j, candidate j may randomize in any fashion. However, the assumption that F j has no mass points guarantees that such a situation occurs with zero probability. 16

This result sharply contrasts with that of Gul and Pesendorfer (005). In their setting, a candidate can implement her favorite policy only if her willingness for o ce, known by the electorate, is su ciently high. When casting their ballots, uninformed voters nd it very likely that an o ce-seeker has chosen the moderate policy, whereas they believe that a partisan candidate chooses the extreme policy with high probability. Hence, a partisan candidate is forced to choose a centrist platform in order to capture informed voters, while an o ce-seeker with a large advantage can exploit uninformed voters beliefs to implement the policy she prefers. In our model, the probability of winning is independent of candidates types. Hence, if an o ce-seeker chooses an extreme platform, so will do a partisan. 4.3 The benchmark case Before analyzing the equilibria of the electoral competition game, we study the game where all voters are perfectly informed of candidates choices. Proposition 1 characterizes the (unique) Subgame Perfect Equilibrium of the game with perfect information. Proposition 1 Fix o nf j, j, j, F, and = 1. Let fx A ( A) ; x D ( D) ; (x)g be a (subgame perfect) equilibrium. Then: 8 1 if either x = (C; R) >< (x) = or x 6= (L; C) and i > 0 >: 0 otherwise j, and x A ( A ) = C for all A, Proof. Any x D ( D ) such that < = A A + 1 A. We have already argued that the equilibrium strategies for voters should be as characterized in (see (3). Let (; ) characterize the equilibrium pro le of candidates as de ned in (8). We rst show that = 0. Suppose, for the sake of contradiction, that 17

> 0. Trivially, we have that A (x) = 1 for all x 6= (L; C) and that A (L; C) = 0. Hence, u B (C; Bj) u B (R; Bj) = [ A + (1 A )] > 0. It then follows that candidate B must choose Center with probability 1. But then, since A (L; C) = 0 and A (C; C) = 1, candidate A must choose = 0. Finally, we characterize. The probability with which candidate D chooses Right must be such that no type of candidate A be willing to switch to Left. From lemma 4, we now that it su ces to nd such that this is satis ed by the most partisan type, as candidates strategies are given by a cuto whenever a candidate wins with positive probability. Hence, any pair (; ) is an equilibrium pro le if and only if u A L; Aj < u A = j (1 j ). By lemma 3, we know that u A (x A ; R; A ) u A (x A ; C; A ) for all x A X A and all A h j ; j i, with strict inequality if A (x A ; R) > A (x A ; C). Hence, u A L; Aj is strictly increasing in. It follows that the upper threshold for the probability of choosing Right is given by u A algebra shows that = L; Aj = u A. Since A (L; R) = 0 and A (L; C) = /1, straightforward A A +(1 A). This is the standard Downsian result with the trivial addendum of a valence advantage 1. Even if politicians have strong ideologies, they must choose a median platform. Lemma 3 tells us that this is exactly the case in this game of electoral competition. Incentives to moderate are twofold. On the one hand, the probability of winning cannot decrease by committing to Center. Second, the losing candidate must choose Center with high probability to prevent the winner to implement her favorite policy. The electoral outcomes are independent of the size of the non-policy advantage enjoyed by the favored candidate. The main concern of voters is the policy that will be enacted. Furthermore, the location of the median voter is common knowledge. Hence, valence plays no role in determining the winning platform, since it must be the median preferred. The only impact of non-policy characteristics is on the identity of the winner. As in Ansolabehere and Snyder (000), the winner is the advantaged candidate. Indeed, the perfect information benchmark is a unidimensional version of their model, with policy-motivated candidates (they consider o ce-seekers politicians). 1 Indeed, this is a unidimensional version of Ansolabehere and Snyder (000). 18

4.4 Characterization of equilibria o We x the distribution of candidates types nf j, j, j, the policy space and the proportion of informed voters. We consider two di erent cases, according with voters preferences for candidates. The rst case is such that the personal support enjoyed by candidate A is moderate, while in the second case she enjoys a large valence advantage. We characterize the limiting equilibrium of a sequence of electoral competition games as information becomes better, i.e. as! 0. The following proposition provides such characterization. Proposition Fix a (sequential) equilibrium. Then: 8 j o nf j, j, j, and < 1. Let fx A ( A) ; x D ( D) ; (x) ; (s)g be >< (x) = >: j 1 if either x = (C; R) or x 6= (L; C) and i > 0, and 0 otherwise i) For (1 ) (1 F (0)) 1 8 < 1 if i > q(rjs D ) i (s A ; s D ) = : 0 if i q(rjs D ) for all s A X A, x A ( A ) = C for all A, and ii) For (1 ) (1 F (0)) > 1 Any x D ( D ) such that < = A A + 1 A. 8 < 1 if i > [1 q(rjs D )] i (s A ; s D ) = : 0 if i [1 q(rjs D )] for all s A X A, x A ( A ) = L for all A, and 19

Any x D ( D ) such that > = ( ) (1 ) 1 ( ) (1 ) +, with = F 1 (1 ). Proof. We have already argued that the equilibrium strategies for voters, both informed and imperfectly informed, should be as characterized in (see (3) and (4)). The equilibrium beliefs about the action taken by candidate A are derived from x A ( A) by Bayes rule. Hence, for any s A X A, we have that q(ljs A ) = 0 if (1 ) (1 F (0)) 1 and q(ljs A) = 1 if (1 ) (1 F (0)) > 1. Assume (1 ) (1 F (0)) 1. Let (; ) characterize the equilibrium pro le of can- didates as de ned in (8). First, given (; ), policy space, and distribution preferences over candidates F, observe that pr (lim!0 N j (x) = k) = 1, for some k [0; 1] and a realized x X A X B. Hence, lim!0 j (x) f0; 1g. Now, notice that a type j of candidate A can secure a payo u A = j (1 j ) by choosing Center with probability 1, with independence of candidate D s choice. For = 0, imperfectly informed voters attach probability 1 to x A = C, regardless of the signal that they may observe. Hence, b (s A ; s D ) 0 for all (s A ; s D ) X A X D. Also, since she chooses Center with probability 1, she gets at least a share 1 F (0) of the informed electorate. Therefore, since F (0) < 1, it follows that she wins the elections and implements Center. We rst show that is bounded away from 1 in equilibrium. Suppose, on the contrary, that for all h < 1, there exists > 0, such that > h for all >. This implies that lim!0 A (L; R) = 1; otherwise, from lemma 3, it follows that candidate A would get an (expected) payo of at most (1 ) A (1 A ) < u A, as she could only win if the realized policy is Center. But then, candidate D should play Center with probability 1. Since lim!0 A (L; R) = 1, it follows from lemma 3 that candidate D could not win by choosing Right. Hence, the policy implemented would be either Center or Left. On the contrary, for = 1, it follows that lim!0 b (L; C) =, as q(xd = Rjs D ) = 0 for all s D X D and lim!0 q(x A = Ljs A = L) = 1. Hence, lim!0 N D (L; C) = 1, leading to a higher (expected) payo. It is easy to see that the pair (; ), with lim!0 = 1 and = 0, 0

is not an equilibrium pro le, since lim!0 u A (L; Aj) = (1 A ) < u A. Now, we prove that candidate A chooses Center with probability 1. Suppose, for the sake of contradiction, that > 0. From above, we know that is bounded away from 1. Hence, given, candidate A does not win for certain by choosing Left; otherwise she would choose Left -her preferred policy- for certain. However, the probability of winning by choosing Left is positive; if not, she would choose Center with probability 1, as she would then win for certain and ensure that Right is not implemented. Hence, it follows from lemma 3 that lim!0 A (L; C) = 0 and that lim!0 A (L; R) = 1. Then, by a similar argument as above, it is easy to show that candidate D should choose Center with probability 1. But the pair (; ), with > 0 and = 0, is not a (descriptor of) an equilibrium pro le. For = 0, we have that q(x D = Rjs D ) = 0 for all s D X D. Hence, b (L; s D ) 0 all s D X D. Hence, whenever x A = L, it follows from lemmata 1 and that at least a proportion F (0) of imperfectly informed voters will cast their ballots for candidate D. Also, since the disadvantaged candidate chooses Center with probability 1, she will get all votes from the informed electorate. But since (1 ) (1 F (0)) 1, candidate A loses for certain and get a payo of (1 A ). On the contrary, by choosing Center, she would get a payo of u A. Hence, candidate A chooses Center in equilibrium with probability 1. The characterization of is identical to the case of perfect information, i.e. is such that u A L; Aj = u A. It only remains to show that lim!0 A (L; C) = 0 and that lim!0 A (L; R) = 1. But observe that trivially lim!0 A (L; C) = 0; otherwise, the most partisan type of candidate A would get A > u A by pursuing a deviation. But then 6= 0 in equilibrium. Also, it is easy to show that lim!0 A (L; R) = 1. Since = 0, we have that q(x A = Ljs A ) = 0 for all s A X A. Hence, b (s A ; s D ) 0 for all (s A ; s D ) X A X D. It follows that candidate A gets no less than 1 x = (L; R). Assume now that (1 ) (1 F (0)) > 1 F (0) > 1 for of the votes if the platform pair is and let (; ) describe candidates strategies. We rst show that candidate does not choose Center with probability 1. Suppose, for the sake of contradiction, that = 0. It follows that b (s A ; s D ) 0 for all (s A ; s D ) X A X D. But the, she will secure a share (1 ) (1 F (0)) > 1 of the votes, regardless of her platform 1

choice. Hence, she can opt for Left and win for certain, improving her payo. Now we prove that = 1. Assume, on the contrary, that < 1. This implies that candidate A does not win for certain by choosing Left; otherwise, she would choose Left with probability 1. Hence, from lemma 3, we have that lim!0 A (L; C) = 0. Also, from above, we have that 6= 0. Then, it follows from lemma 3 that lim!0 A (L; R) = 1; if not, candidate A could not win by choosing Left and would then choose Center with probability 1. But then, candidate D loses for certain by choosing Right. In contrast, since 6= 0 and lim!0 A (L; C) = 0, she wins with positive probability by choosing Center. Hence, it must be the case that = 0. However, this would force candidate A to choose Center, as lim!0 A (L; C) = 0. But = 0 cannot constitute part of an equilibrium, as shown above. We end the proof by characterizing. Since = 1, we have that lim!0 A (L; C) = 1. De ne = F 1 1 (1 ) ; notice that the assumptions on F and the fact that < 1 guarantee that is well-de ned and that (0; ). Observe that whenever the pro le x = (L; C) is chosen, candidate D gets all votes from the informed electorate. Hence, for lim!0 A (L; C) = 1 to hold true, candidate D can get at most (1 ) F ( ) share of the uninformed electorate. Now, since = 1, it follows that q(x A = Ljs A ) = 1 for all s A X A. Hence, lim!0 A (L; C) = 1 only if q(x D = Rjs D = C) 1 increasing in, it follows that = ( )(1 ) ( )(1 )+.. Since q(x D = Rjs D ) is When 1, a majority of voters is perfectly informed about candidates s choices. Hence, informed voters are decisive. Consequently, the favored candidate must implement Center in order to win the elections. This is a direct consequence of our assumption on voters preferences over policies, namely that no elector is willing to renounce to her favorite policy for her preferred candidate to win. In order to analyze the e ects of the lack of (perfect) information in electoral outcomes, we focus in the case where imperfectly informed voters do have a bite. Fix the proportion of informed voters < 1. If the advantage candidate enjoys a moder- ate personal support, i.e. if (1 ) (1 F (0)) 1, she must commit to the policy preferred by voters. The intuition is simple. If the favored politician chooses her preferred policy, the disadvantaged candidate can always commit to Center and get the support of at least all

informed electors and those uninformed voters that prefer her personal characteristics. This backing would be enough to defeat her rival. Therefore, the advantaged candidate must choose Center. On the contrary, in contrast with the perfect information benchmark, if the valence advantage is large, the favored candidate wins and implements her favorite policy. The advantaged candidate is able to exploit her non-policy advantage and implement her favorite policy because all imperfectly informed electors that prefer her personal characteristics vote for him, with independence of her policy choice: observe that = 1,! &0 1 and hence b (L; R)! 0, where signals di erent than s = (L; R) arise with vanishing probability. Since the share of such voters is large, this support su ces to win the elections. The main insight of the paper is that even if voters anticipate that the advantaged candidate will choose his own preferred policy, they vote for her whenever they like her identity better than her opponent s. Ignoring very accurate signals of platform choices is rational, even though personal characteristics are less important than policy for all voters. There is a natural explanation for this. If voters conditioned their actions on their information, they would provide too strong of an incentive for candidates to moderate. Since signals are very informative, politicians would be facing an almost pure Downsian game. Therefore, candidates would converge to Center, as in the benchmark case. Then, since both politicians would decide to commit to the median policy for certain, there would not be uncertainty in equilibrium about actual platforms choices. Consequently, the information available would not have any bite to assess the true locations: if an unexpected (extreme policy) signal is observed, it will be attributed to the fact that wrong signals arise with positive probability. Thus, imperfectly informed voters should ignore signals and rely on their preferences for candidates. Gul and Pesendorfer (005) get a similar result. In their setting, however, voters perform a wrong assessments of the policy choice of the advantaged candidate. Partisan outcomes arise only if a candidate enjoys a large valence advantage and voters believe that her advantage is su ciently small so that she will choose a median policy. If ignorant voters were revealed actual choices, they would be willing to switch their votes. On the contrary, our 3

paper shows that relying only on candidates characteristics may be a successful voting cue: if voters were suddenly perfectly informed, they would not be willing to change their ballots, even if they could change the electoral outcome with their action. Another distinctive feature of our analysis is that we consider almost perfectly informed voters. The following proposition shows that our results remain unchanged if uninformed voters are totally ignorant of policy choices. Proposition 3 Fix nf j, j, j o j, and < 1. Let = 1. In any sequential equi- librium, the advantaged candidate wins; she implements her favorite policy if and only if (1 ) (1 F (0)) > 1. Proof. Observe that ignorant voters observe uninformative signals. Hence, their beliefs are merely candidates equilibrium strategies. Furthermore, they cannot condition their actions on signals. Denote ^ the indi erent uninformed voter. Informed voters optimal strategies are as above. Let (; ) characterize the equilibrium pro le of candidates as de ned in (8). The proof is analogous to that of proposition. Consider the case (1 ) (1 F (0)) 1. We rst show that 6= 1. Suppose, on the contrary, that = 1. This implies that A (L; R) = 1; if not, it follows from lemma 3 that candidate A should not choose Left at all. But then, candidate D should choose Center with probability 1, as she can never win by committing to Right, whereas she could win by selecting Center, as ^ 0 for all and (1 ) (1 F (0)) 1. However, given candidate D s strategy, candidate A could improve upon a deviation to Center. We show now that = 0. Suppose, for the sake of contradiction, that > 0. This implies that A (L; R) = 1. Also, since 6= 1 (from above), we know that A (L; C) = 0. But then, it follows that candidate B must choose Center with probability 1. The fact that = 0 together with A (L; C) = 0 forces candidate A to choose Center with probability 1. With this strategy, she gets at least a share F (0) > 1 set of congruent strategies for candidate D is identical to proposition 1. of the votes. The derivation of the Consider now the case (1 ) (1 F (0)) > 1. First, 6= 0. Suppose, on the contrary, that = 0. Then, ^ 0 for all. But then, candidate A could choose Left for certain 4

and get at least a share (1 ) (1 F (0)) > 1. Hence, > 0. We show now that = 1. Suppose that < 1. This implies that A (L; C) = 0. Also, since 6= 0, it follows A (L; R) = 1. But then, candidate D loses for certain by choosing Right. In contrast, since 6= 0 and A (L; C) = 0, she wins with positive probability by choosing Center. Hence, it must be the case that = 0. However, this would force candidate A to choose Center, as A (L; C) = 0. But = 0 cannot constitute part of an equilibrium, as shown above. As Álvarez and Franklin s (1994) paper re ects, the share of uninformed voters does dramatically increase if those who consider themselves to be "pretty certain", yet not "very certain" of candidates standpoints on policy are pool together with truly ignorant voters. This observation may have important empirical implications. 5 Conclusion There is ample evidence in the literature on voting behavior that gathering information about politicians commitments to policy is more di cult than assessing candidates abilities. Also, it has been largely emphasized that voters systematically make use of voting cues in order to overcome their lack of information. Our paper shows that relying only on candidates characteristics is a successful (and rational) one: even if voters were suddenly perfectly informed, they would not be willing to change their ballots. This voting behavior, however, permits a largely favored candidate to implement his own favorite policy, even if it is not median preferred. Elections fail to discipline candidates because in order to provide incentives for candidates to moderate, voters strategies should condition on signals. Such strategies, however, are self-defeating, as they would induce centrist policies and it would then be optimal for voters to ignore them. A fundamental consequence of the Median Voter theorem is that candidates can substantially increase their electoral support by tting with their constituencies interests. Electoral competition is therefore a way to discipline parties, as competition for votes will drive candidates to the median voter s bliss point. Our model shows that a large valence advantage may reduce "Downsian" pressures. Whether median or partisan policies emerge would crucially 5