Valence uncertainty and the nature of the candidate pool in elections

1. Introduction

Elections are one of the most defining moments of a democracy. They give citizens the opportunity to have a say in the democratic life of their society by allowing them to select representatives who will ultimately have the power to produce binding policies and laws. When shaping their voting decision, citizens evaluate the candidates’ ideological stances in order to ensure that the policy decisions of their future representatives will be aligned to their best interest. If citizens only care about positional issues, then potential candidates with policy views similar to the median voter’s preferences should be the only ones with any sort of ambition to run for office.

This conclusion does not fit the picture that has emerged in the last few decades in the American political landscape. Among others, Ansolabehere et al. (2001) show how there is consistent evidence of nonconvergence of candidates in congressional races toward the median voter’s preferences. Both in open races and in races with an incumbent, many candidates with relatively extreme positions compete for office. While we know that the voters’ decision to elect extreme candidates further encourages extremism in future electoral races, what is less clear is the reason why extremists enter the race for office in the first place, especially against more moderate opponents (Hall, 2014). If extremists challenge candidates who support policies relatively close to the median voter’s preferences, their chance of winning office should be extremely slim. Why are they willing to sustain the cost of an electoral campaign that will most likely end up with a severe defeat? Under what conditions are they willing to enter the competition? I argue that taking into consideration non-positional issues is crucial in understanding the entry decision of extremists and underdogs in general.

Voters usually tend to lack accurate information about the quality and other personal characteristics of competitors, especially in the absence of an incumbent whose record in office and personal image have been uncovered during his/her tenure. This informational deficiency is usually filled by electoral campaigns. Electoral campaigns often generate new and valuable information that can help the electorate evaluate each competitor’s quality, and as such they can radically change the electoral prospects for each competitor. The emergence of details about the candidates’ pasts, late-breaking scandals, performances in public debates, and particularly unfortunate comments are recurrent elements of campaigns and are all factors that can either improve or worsen the voters’ perception of the candidates’ quality. Additionally, there is substantial evidence documenting how extensively voters use candidates’ images to shape their voting decisions. Candidates and parties that are evaluated positively by the electorate in terms of competence, integrity and other non-policy related domains tend to fare well in the ballot box (e.g. Adams et al., 2011; Andersen et al., 2002; Clark, 2009; Clark and Leiter, 2014; Mondak, 1995; Stone and Simas, 2010). Moreover, the effect of candidates’ valence on the public support they receive is larger in the months immediately before the election (Abney et al., 2013). This confirms that the months leading up to the vote assume great importance in determining the result that ultimately emerges from the election: candidates, who at the beginning of the campaign were widely believed to be comfortably cruising toward office, might all of a sudden find themselves trailing behind.

Given the negative effects that the revelation of additional information about the candidates’ quality might have, potential candidates most likely will include in their strategic calculations the possibility that their bid for office might be severely undermined along the campaign trail. To capture this environment, I present a model where potential candidates decide whether or not to enter the race for office not only on the basis of the attractiveness of their policy stance, but also considering the impact of potential valence shocks to each candidate’s campaign. Once the pool of candidates has emerged, the electoral campaign begins and each candidate’s valence becomes known to all citizens. Depending on the voters’ sensitivity to quality, the information about the candidates’ valence might radically alter the odds of winning office.

I show that extremists can join the competition in equilibrium even though they are clearly behind in terms of initial electoral support. Importantly, extremists are not assumed to have any expressive motivations to run, which would push them to join the race even when their odds of winning are extremely unfavorable. Nor does running in a campaign represent an investment to improve future electoral prospects or other outside options thanks to, for instance, name recognition and visibility. Instead, extremist underdogs rely on the possibility that some campaign shocks might induce some voters (enough to reach the simple majority quota) to change their voting decision and express their preference for the ex ante underdog. The trailing candidate has a chance, albeit small, to win office in the event that his opponents turn out to have low valence, and thus end up being de facto knocked out of the competition. Even if the probability of winning the race is small, there are multiple benefits that a citizen would derive from being in office. In addition to granting ego and material rents and allowing the design of policies congruent with one’s preferences, it winning office also reduces the chances of being represented by a low valence politician. In fact, a contested race offers voters a viable alternative to the frontrunner in case such candidate were to show low quality during the campaign, while such an alternative is obviously unavailable in an uncontested race.

Along with providing a rationale for the entry of extremists, my analysis generates predictions about the conditions under which these potential candidates will actually join the electoral campaign. Although many contested races, especially primaries, feature an extremist candidate challenging a more moderate one (Hall, 2015), a nontrivial share of electoral races in the USA remain uncontested. ¹ I show that when voters place a higher weight on policy considerations than on the candidates’ valence, moderate candidates manage to lead unopposed campaigns. Other citizens with moderate ideological stances are unwilling to enter the race given their ideological proximity to the unique candidate, while citizens with more extreme views, who have a strong interest in preventing the implementation of policies they dislike greatly, do not have any chance of winning the competition. In fact, valence is the only dimension on which they might have an advantage over the other candidate, and in this case voters care mostly about positional issues. On the contrary, when candidate valence has a large impact on the voters’ electoral decisions, running unopposed becomes impossible even for the median citizen himself, who will surely face an underdog opponent.

The possibility that extremists might rationally challenge more moderate candidates raises the question of how extreme these challengers can actually be. More specifically, what are the conditions under which we can observe large degrees of asymmetry between the candidates’ policy positions with respect to the median voter? Analyzing the relative position of each candidate running for office, I find a relationship between the degree of asymmetry in policy position and the importance citizens attach to a candidate’s valence. In particular, large degrees of asymmetry in the polarization of the candidates’ policy stances are possible only when the salience of issues other than policy positions is large, which is when citizens heavily weigh a candidate’s quality when shaping their voting decision. The more voters tend to base their vote choice exclusively on the evaluation of the candidates’ policy positions, the more symmetric the polarization of these policy positions will be.

Finally, I consider an extension of the model in which individual valence levels are private information. Several factors affect candidates’ valence. While some of these factors might not be ex ante known by anyone, even the candidate himself (e.g. suitability for public office in terms of character and image), other aspects are exclusively known by each particular candidate (e.g. honesty, presence of scandalous issues in their past). I show that, in the model with private information about individual valence levels, it is still possible to observe, in equilibrium, electoral races in which a unique candidate ends up on the ballot. Conditions for the existence of equilibria with contested elections are more stringent, but they exhibit the same relation between asymmetry in the candidates’ positions and voters’ distaste for low valence representatives. Additionally, in every contested race there is at least one candidate with high valence. That is, voters will always have the possibility to cast a ballot for a high valence candidate, even though such candidate might stand for relatively extreme and unattractive policies. This stands in contrast to other results in the extant literature, where the valence-advantaged candidate always stands for more moderate policies than valence-disadvantaged candidates. When entry in the competition is endogenous and candidates cannot commit to any policy they want, it is possible to observe in equilibrium contested races in which the valence-advantaged candidate is campaigning on policy stances that are more extreme than the valence-disadvantaged candidate.

The rest of the paper proceeds as follows: in the next section I briefly review the extant literature, in Section 3 I describe the model, in Section 4 I then proceed to present the results, in Section 5 I analyze a modified version of the model where some of the assumptions are relaxed, and in the last section I conclude by discussing the implications of the results. The proofs for all results are in the Supplementary Appendix.

2. Related literature

This paper contributes to a large literature on the increasing polarization in US legislatures in recent times (McCarty et al., 2006). This phenomenon is likely linked to the particular dynamic of candidate entry. More and more extreme candidates are entering primary races and some of them manage to win seats in the US Congress, thus making their party shift toward more extreme positions. Importantly, when a candidate from one party wins office, the other party’s candidate pool in the following primary election becomes more extreme (Hall, 2014). Complementing these results, I show why extremists might enter the race against moderate opponents in the first place. While it has been argued that voters in primary elections tend to slightly prefer extreme candidates (Hall and Snyder, 2014), I focus on cases where extremists enter even when the median voter clearly has a preference for moderate candidates, in terms of policy stances.

Given the emphasis on endogenous entry, this paper is closely related to the political economy literature on the entry of parties and candidates in an electoral competition (Besley and Coate, 1997; Caselli and Morelli, 2004; Messner and Polborn, 2004; Osborne and Slivinski, 1996; Palfrey, 1984). I contribute to this literature by considering the effect that uncertainty with respect to the candidates’ valence has on the features of the candidate pool, using a citizen-candidate model. ²

The main mechanism that drives the entry of candidates in my model is the citizens’ uncertainty about the candidates’ valence. This makes my paper closely related to the literature on valence and the effect of non-positional issues on electoral competition (Aragones and Palfrey, 2002; Aragonès and Xefteris, 2012; Hummel, 2010; Stokes, 1963). A large part of the literature on valence has focused on the relationship between quality and the endogenous policy positioning of two fixed candidates. Groseclose (2001) analyzes an environment where one candidate has a valence advantage and the position of the median voter is uncertain. He shows that the greater the valence advantage of one candidate, the more the candidates diverge, even though the advantaged candidate always chooses a more moderate platform than the disadvantaged one for all values of the valence advantage. Carrillo and Castanheira (2008) focus on how the information revelation about the candidates’ quality during a campaign affect policy positioning and investment in valence. ³ When the probability that voters will learn about the candidates’ quality is not too high and not too low, candidates become polarized. Boleslavsky and Cotton (2015) consider the link between information revelation during electoral campaigns and the policy positioning of two exogenously given candidates. As campaigns become more informative, competition on the policy dimension is softened and hence candidates become more extreme. Aragonès and Xefteris (2014) examine a model where voters differ and have private information about their valuation of the two candidates’ non-policy characteristics, and show a U-shaped relationship between heterogeneity in voters’ valuations and divergence in candidates platforms. Finally, Carter and Patty (2015) analyze a model with two purely office motivated candidates who announce their campaign platforms sequentially. One candidate has a valence advantage which is common knowledge, and candidates might decide not to campaign, even if their name is on the ballot.

In the paper I present, there are at least two key differences from the contributions just described: (1) candidates in my model self-select themselves for the electoral race and (2) the voters’ concern about candidates’ quality has an effect not only on the degree of polarization, but also on the degree of asymmetry of elite polarization. Such asymmetry in policy positions, unlike in Groseclose (2001) and Carter and Patty (2015), does not arise from an assumed and known asymmetry in the quality of the two competitors, and, unlike in Carrillo and Castanheira (2008), is not limited by the existence of only three exogenously given policy positions that parties can take. ⁴

In this paper I focus on a different relationship, namely the one between ex ante valence uncertainty and entry decisions of potential candidates who cannot commit to any policy platform other than their preferred one. I do not assume the existence of two fixed parties competing for office and I allow every citizen in the society to run for office. Along with generating predictions about the degree of asymmetry in the polarization of candidates’ policy positions, this approach also allows me to specify conditions that determine when the race for office will actually feature two competing candidates and when instead some citizens will be able to deter the entry of challengers and lead an uncontested campaign, even if they might end up exhibiting low levels of quality during the campaign.

3. The model

Consider a society S of N citizens, with N being odd. Denote a citizen of this society by i ∈ S = { 1 , … , N } and in particular denote by m the median citizen of the society. The citizens must choose a representative who, in the last stage of the game, has the right to select a policy alternative x ∈ X ⊆ R . Every citizen can run for office, and once the pool of candidates is defined, he can cast a vote for any of the candidates. Running for election is costly, and I denote such cost by c ∈ R + , which can be seen as the average amount of resources that are necessary for a candidate to run an electoral campaign. The citizen elected as a representative receives a benefit b ∈ R + from holding office. In the event that nobody runs for office, a default policy, p 0 ∈ X , is implemented. I assume that b > c , so that a citizen always prefers running unopposed to not running and receiving the payoff derived from the implementation of p 0 . ⁵

Each citizen i is characterized by a level of valence θ i ∈ { θ ¯ , θ ¯ } . He can either have low valence (i.e. θ i = θ ¯ ) or high valence (i.e. θ i = θ ¯ ). For ease of exposition, I fix θ ¯ = 0 . Valence here is meant to capture characteristics of the individual citizens that are of interest to voters/citizens and are different from positional issues. In particular, θ captures features such as charisma, honesty, and perceived image in the eyes of the electorate. While some of these characteristics are known to each individual citizen (e.g. honesty), some of them might not be well understood until after the electoral campaign, during which candidates personal attributes are uncovered and scrutinized. I assume that nobody knows the actual level of valence of any member of the society, that is, each citizen does not know his own valence level let alone every other citizen’s valence. Later on in the paper, I relax this assumption and analyze the case in which each citizen has private information about his own valence level.

In the baseline version of the model, while citizens do not know the valence level of anyone, they have a prior about their own valence and the valence of all the other citizens in the society. In particular, the probability each citizen has low valence is given by Pr ( θ = 0 ) = s , and with complementary probability he has high valence. This probability is assumed to be the same for every single candidate, and each citizen’s valence is independent from other citizens’ valence realizations. Although it is a clear and strong simplification of reality, it serves the purpose of giving us some insights on the strategic calculations of the potential candidates when there is uncertainty about the actual level of the other citizens’ valence. Since θ ¯ = 0 , θ ¯ captures the intrinsic disutility of being ruled by a low valence representative. Moreover, let E [ θ ] = θ * = ( 1 − s ) θ ¯ .

Each citizen i has a distinct ideal policy denoted by x i * ∈ X . Additionally, and purely for simplicity of exposition, I assume that for each citizen i in the society, there does not exist a citizen k who is equidistant from the median citizen, m. ⁶ Let d i , j = | x j * − x i * | be the distance between citizen i’s ideal point x i * and citizen j’s ideal point x j * and let I J take value 1 if citizen j has low valence and 0 otherwise. I denote by u i , j the utility that citizen i receives when citizen j is elected, which depends on both policy and quality in the following way

u i , j = { − d i , j , if I j = 1 − d i , j + θ ¯ , if I j = 0

where the policy component is determined by the distance between i’s ideal point and the policy alternative implemented by the elected representative j (i.e. her ideal point, as will be clear below).

Denote by C the set of citizens running for office (i.e. candidates), and by I C the vector of valence levels when C is the set of candidates. As an example, consider a case with three candidates, i, j, and k. If only candidate i and j turn out to have low valence during the campaign, the vector I C would be I { i , j , k } = ( 1 , 1 , 0 ) . Importantly, I C becomes common knowledge before citizens are called to vote for one of the candidates in C. After citizens cast their ballot, the representative is elected, he then chooses a policy and the game ends. This, along with common knowledge of the structure and parameters of the game, implies that the winner of the election cannot commit to implementing any policy different from his preferred one. ⁷ Citizens anticipate this behavior of the future representative and include this into their calculation when deciding on whether or not to join the race.

Lastly, I assume that when a citizen is indifferent between candidates he votes for the one who has higher valence.

3.1. The timing of the game

Having specified the players and the components of the utility function, I now present the sequence of play in the game.

Nature stage: nature draws independently the valence level of each citizen from the set Θ = { 0 , θ ¯ } . For each citizen i, Pr ( θ i = 0 ) = s . Citizens do not observe their valence level.

Entry stage: knowing the probability with which each citizen has low valence (i.e. s), each citizen i chooses whether or not to enter the competition, and I denote this choice by e i = { 0 , 1 } . The individual entry choices generate a profile of entry decisions e = ( e 1 , … , e N ) that in turn defines the set of citizens running for office, C ∈ 2 S . If citizen i decides to enter the race (i.e. e 1 = 1 ), then he pays a cost c. If no citizen enters the race, a default policy p 0 is implemented and the game ends. If, instead, only one candidate enters the competition, then he is automatically elected, he implements his own ideal policy and the game ends.

Campaign stage: the valence levels of all candidates become common knowledge at this stage. ⁸ Before this stage, each citizen has expected valence equal to θ * .

Voting stage: each citizen casts a vote in favor of one of the candidates running for office and the winner is selected via plurality rule. I denote by v i ∈ C citizen i’s voting decision when the set of candidates running is C. The vector of voting decisions v = ( v 1 , … , v N ) determines the set of winning candidates. In case of a tie between two or more candidates, each of these candidates has an equal probability of winning.

Policy stage: the elected representative chooses a policy alternative x ∈ X and the game ends. Since the game ends with the implementation of the policy alternative chosen by the winning candidate, this policy will be the ideal policy of the winning candidate.

To summarize the different incentives at play, the utility function of a citizen i when citizen j wins the election is given by

U i , j = 1 e i = 1 ( b 1 j = i − c ) − u i , j

where 1 e i = 1 is an indicator function taking value 1 if citizen i enters the race, and 0 otherwise, and 1 j = i is an indicator function taking value 1 if the citizen wining the election is citizen i, and 0 otherwise.

Citizens are able to anticipate, given a certain set of candidates, the equilibrium voting strategies of the other citizens. As a consequence, they are able to anticipate the expected utility they will obtain from each candidate’s victory, and on the basis of this information they will decide whether to enter the competition or not.

I solve for the equilibria of this game, where such an equilibrium is formally characterized by a pair ( e * , v * ) , where e * is an equilibrium of the entry stage given v ( · ) and v * ( C ) is an equilibrium in the voting game ∀ C ∈ 2 S . In particular, I require that equilibria at the voting stage are undominated. Therefore, in equilibrium citizens do not play weakly dominated strategies, which in turn implies that they never vote for their least preferred candidate. The elimination of weakly dominated strategies is typical of voting games and is used to get rid of unintuitive equilibria. As for the entry strategy, I focus on those in which citizens play a pure strategy (i.e. either e = 0 or e = 1 ).

4. Analysis

I now turn to the analysis of the game. After exploring the incentives that can induce extremists to enter electoral races against moderate frontrunners, I will characterize the conditions under which extremists do not enter the race, leaving moderate candidates to lead an uncontested campaign, as well as conditions under which they decide to join the race, giving rise to a multi-candidate competition. Additionally, focusing on two-candidate equilibria, I will show when the two candidates running for office can have ideal policies that are asymmetrically distant from the median’s bliss point.

4.1. Uncontested elections

Many races in the USA and other democracies feature a single candidate running uncontested and thus winning office automatically. Even more surprisingly, long-term incumbents are not the only ones enjoying unopposed electoral campaigns. Several open races at different levels of government and for different offices do not offer voters more than one alternative. Under what conditions can we have uncontested campaigns in equilibrium? What kind of candidates, in terms of valence and ideology, can run without being challenged? In order to answer these questions, I start with introducing some notation and concepts.

Definition 1 V i C . is the set of citizens that are closer to i, in terms of policy preferences, than to any other candidate running for office, where C is the set of candidates. ⁹ Formally, V i C = { k ∈ S : ∀ l ∈ C \ { i } u k , i > u k , l } .

Additionally, let V { i ^ , C \ { i } } be the set of citizens that would vote for i if they were to vote sincerely when the set of candidates is C and only candidate i had low valence.

Definition 2. Let the ‘potential winset of i, W θ ¯ ( i ) , be the set of citizens that, given θ ¯ , beat citizen i in a two-candidate election if i has low valence and they have high valence. Formally, W θ ¯ ( i ) = { j ∈ S : I i = 1 and I j = 0 implies | V j { i ^ , j } | ≥ | V i { i ^ , j } | } .

Given that citizen i is running, the citizens in W θ ¯ ( i ) are the ones who would win the competition against i at least when i has low valence and they do not. Therefore, they are the only citizens who can consider joining the race against i.

We can then partition the citizens with ideal points in W θ ¯ ( i ) into two subsets: (a) citizens that before the Campaign Stage are trailing in terms of electoral support with respect to citizen i and (b) citizens that are ahead in the vote count. Clearly, citizens from the two groups will have different ex ante probabilities of defeating i and thus they will have different incentives to invest in an electoral race. For simplicity, I call a citizen, say k, with | V k { i , k } | > | V i { i , k } | an ex ante frontrunner (or simply frontrunner) in the race against i and a citizen with | V i { i , k } | > | V k { i , k } | an ex ante underdog (or simply underdog) in the race against i.

The only citizen in the society who is an ex ante frontrunner with respect to any other citizen is the median. As a consequence, if a citizen not belonging to the potential winset of the median decides to run, the median will also join the race, since he would win the election even when he has low valence and the challenger does not. This basic logic is summarized in the following result.

Remark 1. If j ∉ W θ ¯ ( m ) , then he cannot run unopposed in equilibrium.

The intuition behind this result is illustrated in Figure 1. The left panel depicts the utility that each citizen derives from electing the median, both in the case he has high valence and in the case he has low valence. Given the drop in utility caused by being represented by a low valence citizen (i.e. θ ¯ ), the right panel shows that the potential winset of m is constituted by the citizens with an ideal policy lying in the interval [ k ¯ , k ¯ ]. In equilibrium, citizens with an ideal policy on the left of k ¯ or on the right of k ¯ cannot run unopposed because the median will always challenge and defeat them, regardless of the valence levels revealed during the campaign.

OPEN IN VIEWER

Figure 1.

Low valence and the size of the potential winset of m.

The immediate implication is that the location of the most extreme citizen who could potentially run unopposed varies depending on how strongly citizens dislike being represented by a low valence candidate. In particular, when θ ¯ is small, k ¯ and k ¯ are close to the median voter’s ideal policy, and so only relatively moderate candidates can possibly run an unopposed campaign; as θ ¯ gets larger, k ¯ and k ¯ move respectively to the left and to the right of x m * , so that the pool of candidates who could potentially win in a two-candidate race against the median becomes increasingly more extreme.

While I have identified the citizens who can potentially run unopposed in equilibrium, whether or not they actually will depends on the parameters of the model. I introduce a result that provides a complete characterization of the equilibria with uncontested elections, depending on θ ¯ , which is the level of disutility citizens derive from the election of a low valence representative as opposed to a high valence representative.

Proposition 1. There exist θ ¯ 0 , θ ¯ ′ , and θ ¯ ″ such that:

a one-candidate equilibrium in which a citizen i different from the median runs unopposed exists iff θ ¯ ∈ [ θ ¯ 0 , θ ¯ ′ ] ;

a one-candidate equilibrium in which i = m runs unopposed exists iff θ ¯ ≤ θ ¯ ″ ;

there does not exist a one-candidate equilibrium iff θ ¯ > θ ¯ ″ .

This result points toward a non-monotonic relationship between the disutility citizens derive from the election of a low valence representative and the set of candidates who can run unopposed in equilibrium. When θ ¯ is small enough (i.e. θ ¯ < θ ¯ 0 ), the winset of the median is empty, and hence no citizen other than the median can lead an unopposed campaign in equilibrium. Moreover, no citizen can challenge the median, not even the citizen closest to the median, since he would never win the race. When citizens value low levels of valence as just barely negative, the only ‘dimension’ that matters in their voting decision is policy. On this dimension, the median has a clear advantage and as a consequence the only possible one-candidate equilibrium is the one in which the unique median runs unopposed. In Figure 2, where I depict the set of ideal points of candidates who can run unopposed in equilibrium, the value of θ ¯ 0 would depend on the location of the citizen closest to the median in terms of ideal policies, and it would be equal to the distance between the ideal point of the median and the ideal point of such citizen.

OPEN IN VIEWER

Figure 2.

Equilibria with uncontested elections.

Past θ ¯ 0 , as θ ¯ gets larger, the cardinality of the set of candidates who can run unopposed first increases starting from the median citizen and then shrinks again about the median. When θ ¯ is relatively small (i.e. θ ¯ ∈ [ θ ¯ 0 , θ ¯ ′ ] ), only a few citizens belong to the potential winset of m, and therefore any citizen able to run unopposed will be relatively close to the median in terms of policy stances. As a consequence, both for potential frontrunners and underdogs, the incentives to run are not strong enough to offset the cost of joining the race. In fact, as long as a citizen running unopposed does not have extreme policy views, the other citizens are better off staying out of the competition. This is true even when citizens have particularly favorable electoral prospects. In short, the prospect of having a low valence representative is not so undesirable to push some citizens to invest in a campaign and insure himself and the rest of the society against the election of an inept representative.

As θ ¯ increases, so does the cardinality of the potential winset of m. This means that more and more citizens can run an unopposed campaign. However, the larger θ ¯ becomes, the riskier it is for the rest of the citizens to let somebody (say i) run unchallenged. Especially for frontrunners with respect to i, who have relatively high chances of winning the race, staying out of the race becomes costlier and costlier, to the point that when θ ¯ = θ ¯ † it is not incentive compatible for the frontrunner whose ideal point is farthest away from i’s ideal point not to run. This implies that the set of citizens who can lead an unopposed campaign shrinks as θ ¯ increases past θ ¯ † .

When θ ¯ = θ ¯ ′ , the costs imposed on the society by a low valence representative are high enough to prevent any citizen other than the median from running an unopposed campaign. The intuition for why the median can still run unopposed in equilibrium is as follows. With respect to the median, all other potential candidates are underdogs: their probability of winning is low, the distance from the median is not as high, and the costs imposed by a low valence office holder are not high enough to offset the immediate cost of entry. As a result, all citizens will stay out of the race, thus allowing the median to win office without facing any competition.

As θ ¯ grows even larger, the stakes of letting the median automatically win the race become larger as well. Eventually, when θ ¯ = θ ¯ ″ , letting any citizen, including the median, run alone in equilibrium is too costly, in expectation. Consider the scenario where the median citizen has entered the competition. The risk that he might turn out to have low valence, coupled with the large disutility generated by having a low valence representative, leads another citizen to enter the race in order to reduce the chances of this undesirable event and to also take advantage of the probability of actually winning the competition. In particular, the citizen with the highest stakes in the race is the one who has the most extreme views among the citizens in the winset of m. For this extreme citizen, letting the median run unopposed generates, in expectation, a large utility loss: in addition to the potential cost associated with being represented by a low quality politician, the extremist strongly dislikes the policy that the median citizen will implement once elected (i.e. the median’s ideal point). As a consequence, the expected benefit of competing for office offsets the fixed cost of running in a campaign, thus pushing the most extreme citizen who still has a positive probability of winning to enter the race.

It is interesting to see how the condition guaranteeing the existence of an equilibrium with an uncontested race is affected by changes in the basic parameters of the game. In particular, how does such a condition respond to a change in the probability of each candidate having low valence? I assess this in the following result.

Proposition 2. The set of parameters for which uncontested elections are not sustainable in equilibrium is largest at s = 1 2 .

To see the intuition behind the result, notice that the variance in candidate valence is equal to ( s − s 2 ) θ ¯ 2 . Therefore, for every value of θ ¯ , ( s − s 2 ) is proportional to the variance. The size of the variance in valence is directly connected to the trade off between valence and policy positions. When the variance is higher, selecting high-valence over low-valence candidates is relatively more important than selecting moderate-ideology over extreme-ideology candidates. As we increase ( s − s 2 ) , valence becomes the more relevant dimension in the voters’ evaluation of the candidates, and thus uncontested elections are impossible in equilibrium for a larger set of parameters. The variance is maximized exactly at s = 1 2 .

4.2. Contested elections: The case of two candidates

Having established the conditions under which it is possible to have uncontested campaigns in equilibrium, the question becomes the following: when can we have contested races in equilibrium? What is the relationship between the voters’ evaluation of the candidates’ quality and the ideological stances of the candidates? I address these questions next, starting with the following result.

Proposition 3. Consider citizens i and k. If c ≤ ( s − s 2 ) ( d i , k + b + θ ¯ ) = c i , k * , i ∈ W θ ¯ ( k ) and k ∈ W θ ¯ ( i ) , then there exists a two-candidate equilibrium in which only i and k run for office.

A citizen k who is within the winset of i (which in Figure 3 is given by the citizens with ideal points in [ l ¯ , l ¯ ] ), knows that he will win the competition if he has high valence and the frontrunner has low valence. This can be seen by the fact that the median m derives a higher utility from k in office rather than i. The cost threshold for a potential challenger k to run against i (i.e. c i , k * ) is equal to the expected benefit of winning office. More precisely, the value of winning the race (and automatically preventing opponent i from holding office) is given by ego and material rents (i.e. b), and by policy-related (i.e. d i , k ) and valence-related gains (i.e. θ ¯ ). These benefits are weighted by the probability of winning the competition, namely ( s − s 2 ). If the cost of running is low enough in comparison to the expected benefit of holding office, citizen k has an incentive to join the race against citizen i.

OPEN IN VIEWER

Figure 3.

Contested elections with i and k running for office.

This choice is optimal even if before the start of the electoral campaign he is trailing by a large number of votes. Hence, in the model I propose, the introduction of uncertainty with respect to the candidates’ valence generates the possibility that clear ex ante underdogs could rationally challenge clear ex ante frontrunners. This behavior is not driven by some particular behavioral assumptions about underdogs, or on positing the presence of an expressive benefit of running irrespective of the final outcome of the election. On the contrary, it is predicated upon the fact that the electoral campaign, by revealing additional information about the candidates’ quality, can alter the voting decisions of several voters, and thus can create a concrete possibility of winning the race.

The result states that whenever the two candidates competing in the election actually have an incentive to run against each other, there is always a voting profile that is part of an equilibrium such that no third candidate wants to join the race. This happens because under such a voting profile, a third candidate’s prospects are not promising enough to offset the investment required to enter the electoral campaign. ¹⁰

After having established the condition for the existence of contested races, let us analyze more in depth the relationship between the relative position in the political space of the two candidates running in equilibrium and the possible quality differential that can be realized during the campaign (i.e. θ ¯ ). In order to do this, I introduce some definitions.

Definition 3. Let θ ¯ i , k be the minimum θ ¯ necessary to sustain a two-candidate equilibrium with i and k running for office.

To have a contested race with two competitors, the first condition that must be satisfied is that the two citizens considered need to have an incentive to run against each other. Consider citizens i and k and assume i is the frontrunner. In order to satisfy the aforementioned condition, it must be true that i ∈ W θ ¯ ( k ) and k ∈ W θ ¯ ( i ) . Now, since i is the frontrunner, he clearly belongs to the potential winset of k. However, if θ ¯ is not large enough, it might well be that k does not belong to the winset of i. The minimum θ ¯ ensuring that each of the two citizens belong to the other’s potential winset is θ ¯ i , k .

Definition 4. Consider a two-candidate equilibrium with i and k running for office. Let γ i , k ∈ X be the cut-off point between citizens in V i { i , k } and citizens in V k { i , k } .

In a two-candidate race, citizens vote sincerely for either of the two contenders. We can interpret γ i , k as the ideal point of a hypothetical citizen who is perfectly indifferent between candidate i and candidate k. All citizens with an ideal point on the left of γ i , k will vote for the ‘leftist’ candidate and all citizens with an ideal point on the right of γ i , k will vote for the ‘rightist’ candidate. We are now ready to introduce the following result, which is directly related to Proposition 3.

Proposition 4. The lowest level of θ ¯ necessary to sustain a two-candidate equilibrium with i and k running is increasing in the degree of asymmetry of i’s and k’s ideal points with respect to the median’s. Formally, consider citizens i, k and j with x i * and x k * lying on opposite sides with respect to the median and x j * lying on the same side as x k * . If d m , γ i , j > d m , γ i , k , then θ ¯ i , j > θ ¯ i , k .

The result stated in Proposition 4 can be interpreted in terms of polarization of the candidates’ ideal policies. Specifically, it offers a prediction about the conditions that lead to the emergence in equilibrium of asymmetry in polarization of the candidates’ policy positions. When the loss in utility derived from being represented by a low valence politician is small, whether or not a candidate has low valence will not make a large difference in the voting calculation of the citizens. This implies that citizens who decide to run for office need to have roughly the same ex ante support in terms of votes, as a potential underdog of the race cannot rely too much on the possibility that the frontrunner’s campaign will be destabilized by the revelation of additional information about the frontrunner’s valence. In order to find running worthwhile, an underdog must have ex ante support almost as large as the frontrunner’s, that is, he has to be barely an underdog. As a consequence, since the two candidates must split the votes of the electorate almost equally, the cut-off point γ i , k will be fairly close to the median. Therefore, only low degrees of asymmetry in polarization can be equilibrium outcomes.

As θ ¯ grows larger, the potential winset of a citizen i also becomes larger. That is, more potential candidates with rather extreme policy stances have a chance of winning office, since now citizens weigh more and more the valence status of each candidate when forming their voting decision. Given this anticipated electoral behavior, potential underdogs ex ante trailing by a large amount of votes have two concrete incentives to run for office: (i) citizens dislike low valence to the point that they might actually have a chance of winning office, and thus of deriving both ego benefits and policy-related benefits; (ii) having a chance, even if slim, of winning office also means reducing the possibility of electing a low valence politician, which would cause a substantial loss in utility given the largeness of θ ¯ . In sum, the more citizens care about their representative’s valence level, the more extreme citizens will be in the potential winset of citizen i, and hence the higher the degree of asymmetry in polarization sustainable in equilibrium. This result points toward a precise empirical implication: we should observe a correlation between the average punishment for low valence politicians and the level of asymmetry in polarization, a correlation that has received empirical support (Burden, 2004).

5. Individual valence as private information

Before discussing the implications of the results presented above, it is worth assessing the role played by one of the main features of the model: symmetric uncertainty with respect to each citizen’s valence level. Assuming this type of uncertainty inevitably limits the applicability of the model and of the results described above to a particular subset of aspects feeding into a citizen’s valence or quality, namely the ones that are likely to be unknown to the citizen himself. However, it is obvious that there are other aspects and features contributing to an individual’s valence that are known to each citizen. Prominent examples are details about the candidates’ past, especially the ones affecting the perception of a candidate’s honesty or moral fiber.

Extending the analysis to consider the impact of personal histories and details about candidates’ past is important for two reasons. First, whenever these details become public during the electoral campaign, they tend to generate political scandals that have been shown to have far reaching consequences on electoral outcomes (Hirano and Snyder, 2012; Puglisi and Snyder, 2011). Prominent examples, among others, would be Gary Hart’s affair emerged during the Democratic presidential nomination in 1988, in which he was initially considered to be the clear frontrunner, and Scott McInnis’ plagiarism scandal, emerged during his bid for the Colorado Gubernatorial seat in 2010. ¹¹ Second, the strategic calculations of potential candidates are now substantially different. When citizens do not know their own valence, even an extremist potential candidate could gamble on the possibility that some valence shocks could reshuffle electoral support in a way that could make him win the competition. When instead each potential candidate knows his own valence, the uncertainty is confined to other candidates’ valence and, as such, they cannot hope (or fear) that their valence might turn out to be higher (or lower) than average. For these reasons, the case in which citizens know their valence levels deserves an attentive analytical exploration.

To this end, I now briefly analyze a version of the model in which during the Nature stage each citizen privately learns his own valence level, which I will refer to as his type, while he remains uncertain about every other citizen’s valence level. Given the presence of private information, it is useful to clearly state how the definition of equilibrium needs to be adapted to the new environment. An equilibrium is still characterized by a pair ( e * , v * ) , where e * is an equilibrium of the entry stage given v ( · ) and v * ( C ) is an equilibrium in the voting game ∀ C ∈ 2 S . However, as to the entry strategy, each citizen now conditions his entry decision on his type, and thus an entry strategy for a citizen specifies an entry decision given each valence level. As to the voting stage, just like in the case of the model with symmetric information, the candidates’ valence levels become common knowledge during the Campaign stage, and as such the citizens’ voting strategies in the model with private information are identical to the ones of the symmetric information case.

As mentioned above, since citizens now know their level of valence before they decide whether or not to enter the election, potential underdogs cannot blindly rely on valence shocks during the electoral campaign to critically affect the electoral result. In a similar fashion, not all potential frontrunners will have to fear the possibility that the information revealed during the campaign might erase the initial advantage they could have in the race for office. Specifically, there is no scenario under which a high valence frontrunner will end up losing the competition. As a consequence, he clearly has an incentive to join and, automatically, win the competition. On the contrary, a low valence underdog facing a frontrunner who always enters, regardless of whether he has high or low quality, has no chance to win the race. Even in the event that the frontrunner turned out to have low valence, the difference in the ideological position will play in favor of the frontrunner.

A high valence underdog instead has a chance of winning the race. If the frontrunner he ends up facing has low quality, the underdog’s disadvantage in terms of policy positions is more than counterbalanced by his advantage in terms of valence, and thus he is able to win the race. ¹² This result points toward an intuitive logic: if an underdog is willing to compete against a frontrunner in an electoral race, there must be a dimension (i.e. valence) on which he might have an advantage over the frontrunner to compensate for the initial disadvantage given by the relative unattractiveness of his policy stances. I summarize this intuitive logic in following result.

Proposition 5. A high valence frontrunner always joins the race against an underdog. A low valence underdog never joins the race against a frontrunner, if both types of the frontrunner enter the race.

This result has a rich set of implications both with respect to the case of uncontested and contested elections. I will discuss first the implications for uncontested elections. To this end, I introduce the following definition.

Definition 5. A one-candidate equilibrium is an equilibrium in which both types of one citizen enter the race, while both types of all other citizens stay out of the race.

Proposition 5 clearly points out that, when valence levels are private information, it is impossible to have a one-candidate equilibrium with a citizen different from the median running unopposed. In fact, the high valence type of the median citizen has an incentive to enter the race against any other citizen since he would win the election for sure. As a consequence, given that the median citizen is a potential frontrunner with respect to each and every citizen in the society, the only possible one-candidate equilibrium is the one where the median citizen runs unopposed. I summarize these arguments in the following result.

Proposition 6. There exists θ ¯ ‡ such that there is a one-candidate equilibrium in which the median citizen runs unopposed iff θ ¯ ≤ θ ¯ ‡ . Moreover, this is the unique one-candidate equilibrium of the game.

The intuition for this result is similar to the one behind Proposition 1. If the disutility from being represented by a low valence citizen is too large, it is simply not possible to have a unique candidate run unopposed in equilibrium, and this is the case even if the candidate in question is the median citizen. The possibility of ending up with a low valence representative and the cost associated with this scenario push other citizens, who stand a chance in winning the election, to join the race and prevent this undesirable outcome.

The impossibility of having a one-candidate equilibrium with a citizen different from the median running unchallenged does not restrict the existence of uncontested races to the case of the median being the unique candidate in the race. That is, there are equilibria of the game that are not one-candidate equilibria and that admit as a possible final outcome an uncontested race with a citizen different from the median being unchallenged. This will become apparent in the analysis of the contested elections case, to which I turn next.

An immediate consequence of Proposition 5 is that in the model with private information about individual valence levels it is impossible to have two-candidate equilibria in which both types of the two candidates run. In fact, in case both types of the frontrunner ran, the low valence type of the underdog has an incentive to stay out of the race. On the contrary, whatever the strategy of the underdog is, the high valence frontrunner will always enter the race and win it. This logic implies the following fact.

Corollary 1. Every contested race has at least one high valence candidate.

This means that we cannot observe a situation in which two candidates with different policy stances both have low valence. That is, in equilibrium, contested election will never offer voters only low quality alternatives. When voters have a meaningful choice to make (i.e. vote for one of the two candidates), they will always have at least one high quality candidate present in the competition. At the same time, the impossibility of having both types of two citizens contesting office implies I have to focus on a particular kind of two-candidate equilibria, which I define below.

Definition 6. An ex ante two-candidate equilibrium is an equilibrium in which at least one type of two citizens enter the race, while both types of all other citizens stay out of the race.

With this definition in mind, it is easy to see how the case in which the individual valence levels are privately known by each citizen significantly differs from the one where they are unknown to everybody. Specifically, it is no longer the case that whenever two citizens i and k have an incentive to run against each other there exists a equilibrium in which only the two citizens i and k compete for office. In the model with symmetric uncertainty (i.e. citizens do not know their own valence), as a consequence of strategic voting, any third candidate’s electoral prospects would not be promising enough to offset the costs he would have to sustain upon entering. In the model with private information, as mentioned above, an ex ante two-candidate equilibrium does not ensure that the competition will actually involve two candidates. On the contrary, we can observe, in equilibrium, uncontested races featuring as a unique candidate a low valence citizen different from the median. If there is only one candidate running for office, the entrance of a third candidate cannot be deterred by any strategic voting logic, since in a race with two alternatives citizens would vote sincerely. As a consequence, it is a lot harder to keep potential third candidates out of the race, especially when they would be frontrunner with respect to both candidates. I characterize the possible ex ante two-candidate equilibria in the following result.

Proposition 7. Consider citizens i and k with i ∈ W θ ¯ ( k ) and k ∈ W θ ¯ ( i ) . There exist thresholds c ¯ , c ¯ , c ¯ ′ , c ¯ ′ , c ¯ ″ , c ¯ ″ such that:

Moreover, these are the only possible ex ante two-candidate equilibria.

A complete characterization of the thresholds on the cost of entry is given in the Supplementary Appendix. From a substantive point of view, the introduction of asymmetric information with respect to the individual levels of valence affects the existence of contested races in a significant way. First, as pointed out already above, identifying ex ante two-candidate equilibria does not automatically ensure the actual presence of two citizens competing for office. Suppose that citizen i is the underdog with respect to citizen k. The entry profile presented in Part 1 of Proposition 7 ensures a contested race only when the underdog citizen i has high valence, that is with probability ( 1 − s ) . As pointed out in Proposition 5, when both types of the frontrunner enter the race, there is no chance of winning for a low valence citizen who is also disadvantaged because of his policy stances. The cost of entry has to be low enough for the frontrunner and underdog to find it worthwhile to run, just like in the case of symmetrically unknown valence levels, thus characterizing the upper bound c ¯ . However, in this case, if citizen i has low valence, the entry profile considered would imply an uncontested race. Thus, to ensure that no other citizen has an incentive to enter and exploit this scenario, the cost of entry has to be high enough, which characterizes the lower bound c ¯ .

In this equilibrium, if the election is a contested one, the underdog always has high valence while the frontrunner might have either high or low valence. This implies that, whenever one candidate has a valence advantage, it has to be the underdog, suggesting in turn that the valence-advantaged candidate will be more extreme than the valence-disadvantaged one. This substantive feature of the equilibrium described in part 1 runs contrary to the prediction of Groseclose (2001) that, for every level of valence advantage, the advantaged candidate always adopts a more moderate position than the disadvantaged one.

The entry profile described in Part 2 of Proposition 7 leads to a contested race only with probability ( 1 − s ) 2 , that is when both citizens have high valence, and it could actually lead to a ‘competition’ with no candidates competing, with probability s 2 . The cost of running must now be higher than in the previous case (i.e. c ¯ < c ¯ ′ ) for two reasons: to dissuade a low valence frontrunner from running, and to dissuade potential third candidates, who would face a unique opponent with probability 2 s ( 1 − s ) and would even run unopposed, thus winning the competition outright, with probability s 2 (that is when both citizens i and k have low valence and therefore stay out of the race). The upper bound on the cost of entry, c ¯ ′ , depends on how unpleasant the default outcome p 0 is for the underdog citizen i. In particular, for this equilibrium to exist, it has to be the case that the default outcome is more appealing to the frontrunner than to the underdog.

Finally, the entry profile in Part 3 of Proposition 7 generates a contested election with probability ( 1 − s ) . In this case, it is the low valence frontrunner who stays out of the race, while the low valence underdog enters the competition. In order to make this profile a mutual best response, the entry cost must be high enough with respect to the potential additional benefits derived from running to dissuade a low valence frontrunner to challenge citizen i. The same has to be true for potential third candidates. At the same time, it has to be incentive compatible for the low valence underdog to run. This is the case when the benefits he derives from entering, namely winning the race in the event that the frontrunner has low valence and hence does not enter, are high enough to offset the campaign investment.

In this equilibrium, when the election ends up being contested and one candidate is advantaged in terms of valence levels, the policy stances of the valence-advantaged candidate are more moderate than the policy stances of the disadvantaged one. The nature of this equilibrium, contrary to the equilibrium described in Part 1, is similar in spirit to the result in Groseclose (2001) in that valence-advantaged candidates tend to be more moderate in terms of policy positions, but it is attained here in the presence of private information about individual valence levels.

Since several of the threshold specified in Proposition 7 depend on the position of the default policy, it is worthwhile to briefly assess the robustness of each equilibrium strategy profile to different specification of the default policy. ¹³ In particular, when the default outcome becomes infinitely distasteful to every player (i.e p 0 → + ∞ ), only the entry profiles described in Part 1 and Part 3 can be part of an equilibrium strategy profile for a set of entry costs of positive measure. In fact, for the entry profile described in Part 2, both the upper and the lower bound depend on the position of the default policy, and thus this entry profile cannot be part of an equilibrium strategy profile when the default policy goes to infinity.

To conclude the analysis of the modified model, it is worth noting that, even with private information about valence levels, there is a positive correlation between the disutility from electing a low valence representative and the degree of asymmetry in the polarization of candidates. In fact, the result stated in Proposition 4 holds unchanged in the asymmetric information version of the model since it does not rely on the assumed information structure of the model. It is instead a consequence of the necessity for a two-candidate equilibrium (or an ex ante two-candidate equilibrium) to exist that both candidates must have an incentive in running against each other. For this to be true, the underdog’s ideal policy has to lie within the winset of citizen k, otherwise even the high valence type of the underdog has no chance of defeating even the low valence type of the frontrunner, thus making entry in the race a strictly dominated strategy.

6. Conclusion

Recent trends in US elections show an increasing number of ideologically extreme candidates participating in electoral competitions at all levels of government. Why do we often observe extremists joining an electoral competition even though they are clearly behind in terms of initial support? Under what conditions can we expect them to enter the race for office? While previous contributions to the literature on candidate entry have shown that it is possible to have two extreme candidates run against each other—one on the left and one on the right of the median voter—what is more challenging to explain is the entry of extreme candidates against moderate ones. To understand this phenomenon, I present a model where the citizens’ entry decision depends not only on the attractiveness of their policy stances, but also on the impact of the revelation of new information about their quality during the campaign.

I show that, when there is uncertainty about the citizens’ quality, we can observe candidates joining the race even though their views are remarkably more extreme with respect to the median voter’s policy stances. I offer an intuitive explanation for the entry of extremists against moderate candidates. That is, extremists rely on the possibility that some campaign shocks might induce some voters (enough to reach the simple majority quota) to change their voting decision and express their preference for them. More importantly, when a moderate candidate has entered the race, the extreme potential candidates are precisely the ones who have the strongest incentive to join the competition: not only do they want to avoid the election of a low valence representative, but they also want to avoid the implementation of moderate policies that they find deeply undesirable. The uncertainty about the the candidates’ valence gives extremists a positive probability of winning the race and, in turn, of preventing a potentially low valence moderate from getting elected.

The analysis generates results about the conditions under which we can expect uncontested or contested races. Only relatively moderate citizens can run unopposed, provided that voters evaluate candidates more on the basis of their policy stances than on the basis of their realized quality. As quality becomes increasingly relevant in the voters’ evaluation of candidates, uncontested elections become harder to sustain in equilibrium, up to a point where not even the median citizen is able to run without competition.

Focusing on contested races, I find an interesting relationship between the degree of asymmetry in the candidates’ policy positions with respect to the median voter and the loss in utility derived from being represented by a low quality candidate. I show how, when the candidates’ quality is uncertain, the sustainability in equilibrium of asymmetric polarization and the degree of asymmetry in policy positions depend positively on how negatively voters evaluate the lack of quality of competing candidates. When the concern for issues other than policy positions is sizable, then very asymmetric polarization can be an equilibrium outcome; the more voters tend to base their voting decision exclusively on the evaluation of the candidates’ policy positions, the more the only kind of polarization that can be sustained in equilibrium is the purely symmetric one.

Finally, I examine the case in which the individual valence levels are privately known by each citizen. This allows me to assess how features contributing to an individual’s valence that are known to each citizen affect the entry strategy of potential candidates. That is, modeling valence as private information can shed light on how the anticipation of possible scandals triggered by the revelation of details about personal histories during the electoral campaign shapes the nature of the candidate pool in the first place. In particular, I show that, while many insights uncovered in the case where there is symmetric uncertainty about valence levels carry over in the new environment, the introduction of private information makes it harder to sustain equilibria in which a candidate different from the median enjoys an uncontested campaign. On the contrary, the relationship between the degree of asymmetry in the candidates’ policy positions and disutility from electing a low quality representative holds unchanged in the new environment. Interestingly, whenever elections are contested, one of the two alternative candidates is a high quality one. Moreover, and contrarily to previous results in the literature, when entry is endogenous and candidates cannot arbitrarily commit to any policy they want, it is possible to observe in equilibrium contested races in which the valence-advantaged candidate is campaigning on policy stances that are more extreme than the valence-disadvantaged candidate, a fact that has been empirically documented in Burden (2004).