Candidate Citizen Models

Candidate Citizen Models

General setup Number of candidates is endogenous Candidates are unable to make binding campaign promises whoever wins office implements her ideal policy Citizens preferences are common knowledge voters infer policy choices from candidates identity Citizens perform two roles: each voter may enter the competition as a candidate voters also vote over those candidates that entered

General setup Policy space is real number line R, generic policy x Utility of voter i with ideal policy x i when policy x is implemented is Euclidean u i (x, x i ) = x x i Median voter has ideal policy x m Benefit from holding the office v 0 Cost of running is δ > 0 Both parameters (v, δ) are common for all citizens

Two Versions of the Candidate Citizen Model Osborne and Slivinski (1996) consider sincere voting sincere voting (naive voting) occurs when a voter always supports a candidate who if elected to office gives him higher expected utility Besley and Coate (1997) consider strategic voting when more than two viable candidate, strategic voting (tactical voting or sophisticated voting) occurs when a voter may support a candidate other than his or her sincere preference in order to prevent an undesirable outcome

Timing of the Events Entry all citizens simultaneously and independently decide whether to become candidates Voting all citizens vote over the set of candidates the winner is decided by plurality rule, with ties settled using an equal-probability rule single-candidate elections are won with certainty by the sole candidate Policy choice the winning candidate implements her ideal policy x R in the event that no candidate has entered the race, a status quo policy x R is implemented

One-candidate equilibria Consider the following parameterization default policy is x = 0 cost of entry δ = 1 4 median voter ideal policy is x m = 1 2 exogenous payoff from winning v = 0 Derive the set of one-candidate equilibria Find the set of citizens such that if one such citizen enters all other citizens prefer to stay our each citizen in this set indeed prefers to enter if noone enters

One-candidate equilibria Intuitively, candidates with ideal points close to x m are invulnerable to entry find how close to x m candidates must be Consider entry by x i < 1 2 no citizen who would lose against i would enter, since δ > 0 no citizen with the same ideal as i would enter, since v = 0 how about x j = 1 x i? if enters, then wins with probability 50% to deter his entrance, we need x i x j 1 2 ( xi x j ) + 1 2 ( xj x j ) 1 4 x i 1 4 how about x i < x j < 1 x i? if enters, then wins for sure among those citizens, those with ideal points furthest from x i have the greatest incentive to deviate by entering consider x j = 1 x i ɛ for ɛ > 0 very small to deter this citizen, we need x i x j ( x j x j ) 1 4 x i 3 8 ɛ for all ɛ > 0 2

One-candidate equilibria Among the two conditions, the second is binding Thus, among all i such that x i < 1 2, only those citizens i such that x i [ 3 8, 1 2) are invulnerable to entry should they enter themselves Similarly, only those i with x i ( 1 2, 5 8] are invulnerable to entry should they enter themselves The set of one-candidate equilibria is [ 3 8, 5 ] 8

Two-candidate equilibria Citizens have ideal points x i U[0, 1] with x m = 1 2 Cost of entry is δ > 0 and payoff from winning is v 0 Voting is sincere if v 2 δ v 2 + and < 1 3 then there exists an equilibrium in which exactly two candidates x L and x R enter and they are located symmetrically around median voter x L = x m and x R = x m + Voting is strategic it is possible to support equilibria in which the two extreme candidates enter: x L = 0 and x R = 1 even though plurality prefers a hypothetical centrist candidate, a coordination failure among centrist voters prevents that individual from achieving a plurality should she enter

Three-candidate equilibria In the Hotelling-Downs model, Nash equilibria with three parties do not generally exist In contrast, three-candidate equilibria are possible with the citizen-candidate model We will derive here the conditions for existence of a particular three-candidate equilibrium when ideal points of voters are distributed uniformly in this equilibrium the candidates have different ideal points each candidate wins with probability 1 3

Three-candidate equilibria Citizens vote sincerely and have ideal points distributed x i U[0, 1] Equilibrium with three candidates ( 1 6, 1 2, 5 6) exists if and only if v 3δ Check that candidate 1 6 prefers to enter: if he deviates by not entering the race, then 1 2 wins elections for sure, thus, δ + v 3 + 1 ( 16 3 12 ) + 1 ( 16 3 56 ) 1 6 1 2 v 3δ Similarly for candidate 5 6 we get the same condition

Three-candidate equilibria Check that candidate 1 2 prefers to enter: if he deviates then the two extreme candidates win with equal probability δ + v 3 + 1 ( 16 3 12 ) + 1 ( 12 3 56 ) 1 2 1 6 1 2 1 2 1 2 5 6 v 3δ 1 3 this condition is weaker than v 3δ

Three-candidate equilibria Check that no other citizen prefers to enter the race, given the presence of ( 1 6, 1 2, 5 ) 6 no citizen with x i < 6 1 prefers to enter, because then 2 1 and 1 6 would each win with prob 50%, which is worse expected policy payoff for citizens with x i < 6 1 and in addition requires paying costs of entry δ similarly, x i > 5 6 wouldn t ( ) want to enter the race no citizen with x i 16, 2 1 would enter the race, because then candidate 5 6 wins for sure, which is a worse expected policy payoff and ( requires ) paying cost of entry δ similarly, x i 12, 5 6 wouldn t want to enter no citizen with x i = 1 2 would enter, because then 1 6 and 5 6 each win with prob 50%, producing a worse expected policy payoff (though the same expected policy - the difference is that now x m is implemented with probability zero) for citizen i and requiring that citizen i incur the cost of entry δ.

Additional Predictions of Candidate Citizens Model Model so far is deterministic candidate positions map deterministically onto voter choices This implies some odd predictions of the model 1. for certain parameter values, one-candidate equilibria exist and its existence does not depend on the number of voters at odds with empirical reality: most candidates do not run unopposed in large electorates, though they may do so when the electorate is small (e.g., in elections for department chair) 2. if finite and odd # of voters, then no two-candidate equilibria in two-candidate equilibria the candidates must have ideal points equidistant from the median ideal point to make x m indifferent, very special case with odd number of voters, two candidates can t have equal # of supporters, thus, loser should withdraw from the race

Citizen Candidates under Uncertainty Solve these problems by assuming that candidate positions (identity) map only probabilistically onto voter choices Eguia (2007) model makes a distinction between supporting a candidate and voting for that candidate with positive probability any citizen fails to cast her vote work or family obligation prevents a citizen from voting There is no uncertainty in voters preferences, only in whether they actually vote

Citizen Candidates under Uncertainty, framework Society consists of N 3 citizens with ideal points x i U[0, 1] with Euclidean utility u i (p) = x i p Median voter is m Elected politician enjoys utility of v and any citizen that enters the race pays the cost δ > 0, where δ < v 2 At the voting stage, each citizen i who supports candidate j casts a vote for j with probability (1 µ) with probability µ [0, 1) citizen i s support is lost this probability is the same for each citizen and it is uncorrelated across citizens Candidates cannot anticipate the outcome of the election, because they do not know how many voters will show up

Citizen Candidates under Uncertainty, results One-candidate equilibrium Lemma 1: there exists a one-candidate equilibrium if and only if N is odd, the median is unique and j N\{m} δ + λ (v x j x j ) + (1 λ) ( x j m ) x j m where λ = Pr[j wins {j, m}] which means λ (v + x j m ) δ This equilibrium is unique among one-candidate equilibria and m is the single candidate The intuition of this result is that the median would enter a run against any other citizen who was running alone, so only a unique median can run unopposed Moreover, the median can run unopposed only if any challenger would have a small enough probability of victory, which is precisely the condition above

Citizen Candidates under Uncertainty, results One-candidate equilibrium Theorem 1: Given µ > 0, there exists some n such that if N > n, there is no single-candidate equilibrium. The intuition of this result: candidate that has small number of supporters almost certainly loses if the electorate is also small but as the electorate gets larger, the number of lost votes will increase and a candidate with the same small number of supporters will have a better chance of victory for instance, in an electorate with 5 citizens, a 3-2 split of support will give the weaker candidate a very small chance of victory however, in an electorate with millions of citizens a split of support in which the stronger candidate has only one more supporter is a virtual tie, and both candidate have an almost equal probability of victory

Citizen Candidates under Uncertainty, results Two-candidates equilibrium Theorem 2: given µ > 0, there exists some n such that if every citizen has a distinct ideal position and N > n, then a two-candidate equilibrium exists. The proof is by construction: if there is a unique median, in the absence of uncertainty, two-candidate equilibria did not exist unless the median citizen is indifferent between the two candidates, a rare event however, as the electorate grows, a positive uncertainty raises the probability that a candidate trailing by one supporter wins the election as the electorate grows, the probability that a weaker candidate trailing by a given number of supporters wins the election converges to 1 2 and given large benefits from holding office δ < v 2 this weaker candidate would want to run, and the equilibria with two-candidate exists.