Consideraciones en torno a las poblaciones y las muestras

Equilibria with multiple candidates, and sincere-strategic equilibria with multiple candidates are possible in the model. In this subsection I show that under some restrictions, the sincere-strategic refinement eliminates all equilibria in which weak candidates run with little support.

Models without uncertainty can distinguish between “winning candidates” (those who win with positive probability) and “spoilers” (those who run just to affect the outcome indirectly, but with no probability of actually winning). With uncertainty, every candidate with at least one supporter has some positive probability of victory, albeit possibly a very small one. Nevertheless, we can still describe some candidates as spoilers: Those who trail in support and run mostly to influence who

wins when they lose.

just on their probability of winning, without taking into account the effect that their candidacy would have on the electoral outcome if they lose. Competitiveness is defined relative to a specific

joint strategy of all citizens at the support stage; thus a candidate can be competitive for some profile of support and not competitive given a different support profile.

Definition 7 Given a joint support strategy profileσ, a candidate i_∈C is competitive if

{b+vi(pi)−

X

k∈C\i

vi(pk) Pr[Wk = 1|C\i]}Pr[Wi= 1|C]> c,

and a spoiler otherwise.

Competitive candidates run to win. Spoiler candidates run motivated by the advantages that running for a defeat entails for them. Equilibria with three competitive candidates, which do not exist under mild assumptions in Besley and Coate’s [9] model, exist in this model with uncertainty (an example is available from the author). Equilibria with four candidates may have one or two competitive candidates, as in Besley and Coate [9], or three or four, as in Osborne and Slivinski [44]. However, with the refinement of the sincere-strategic equilibrium, in all equilibria in which

citizens use pure strategies at the entry stage, spoilers must all be in between two competitive candidates. Therefore, pure equilibria with a single competitive candidate and several spoilers will not be sincere-strategic.

Lemma 9 In any pure sincere-strategic equilibrium with multiple candidates , ifi_∈C∗_{is a spoiler,}

then there existj, h_∈C∗ such thatpi∈(pj, ph).

Corollary 10 In any pure sincere-strategic equilibrium with two or more candidates, there exist at least two competitive candidates.

Entry by spoiler candidates who deprive stronger candidates from crucial support can effectively affect the outcome of an election. However, the entry decision by a candidate who draws no support

from the electorate is intuitively irrelevant to the outcome of an election. If a candidate with no support drops out of the race, the most intuitive reaction by all other citizens is to keep their support decisions unaltered.

I say that a support strategy profile is consistent if whenever a candidate who receives zero

support drops out of the race, the support for all the other candidates remains unaltered.

Definition 8 A support strategy profile σ is consistent if for any C and i _∈C, Si(C, σ) = 0 =⇒

σj(C) =σj(C\i)for all j∈N.

Next I show that if preferences are Euclidean, support strategies areconsistent,and the electorate is large enough, then in equilibrium all candidates must have a similar expected share of support. For afixedµ,I consider a sequence of societies of increasing sizeN and I show that the result holds for

all societies larger than some sizen.To prove this I add a technical assumption on the distribution of ideal policies to guarantee that no agent is indifferent between two distinct ideal policies.

LetQ_⊂[0,1]be a set composed offinitely many points such that for anyx, y_∈Q, x+₂y _∈/Q.

Given a positive uncertainty µand a setQcomposed offinitely many staggered points, if pref-

erences are Euclidean, the ideal policy of every agent lies in the set Q, and the electorate is large enough, then in any pure sincere-strategic equilibrium withconsistent strategy profiles all candidates

receive a similar expected share of support. Proposition 11 states this result formally.

Proposition 11 Given µ >0 and Q, for any >0 there exists a positive integer n such that if preferences are Euclidean, pi ∈ Q for all i ∈ N, and N > n, then in any pure sincere-strategic

equilibrium with consistent strategy profiles

¯ ¯ ¯ ¯ E[Si] N − E[Sj] N ¯ ¯ ¯ ¯< for alli, j∈C∗.

The intuition for the proof is as follows: If the electorate is large enough, a candidate whose expected share of support is less than that of the strongest candidates will lose with probability

approaching one. Then, by lemma 9 and corollary 10, it must be that this weak candidate is a spoiler, between two competitive candidates. But then, given two competitive candidates and a trailing weak candidate, votes for the spoiler are wasted votes, and only citizens who are indifferent among the top two candidates will support the spoiler. Indifference between two candidates with distinct ideal policies is ruled out by assumption and as a result the spoiler will get no support, then the spoiler not only cannot win, but with zero support she cannot even influence the election if the

support strategy profile isconsistent. Then, she prefers to drop out.

A complete characterization of multi-candidate equilibria is beyond the scope of this thesis.