Whenever we have modeled political candidates who stand for election, we have assumed that they care only about winning and are perfectly content to change their position in whatever way maximizes the probability of winning. Now consider a different way of thinking about political candidates: Suppose that the citizens again have uniformly distributed ideal points on the Hotel ling line [0, 1]. Before any election is held, each citizen has to decide whether to pay the cost c > 0 to run as a candidate—with the payoff from probability π of winning the election equal to (π−c).
A: Assume candidates cannot change their position from their ideal point, and citizens who do not become candidates get payoff equal to minus the distance of the winning candidate position x∗ to their own on [0,1]. The highest attainable payoff for a non-candidate is therefore 0. (Candidates that lose get the same payoff as citizens who do not run—except that they also incur cost c from having run.)
(a) For what range of c is the following an equilibrium: A citizen with the median position 0.5 is the only candidate to enter the race and thus wins.
(b) How high does c have to be in order for the following to be a possible equilibrium: A citizen with position 0.25 enters the race as the only candidate and therefore wins? How high must c be for an equilibrium to have a citizen with position 0 be the only candidate to run (and thus win)?
(c) For what range of c will it be an equilibrium for 2 candidates with position 0.5 to compete in the election?
(d) For what range of c is it an equilibrium for two candidates with positions 0.25 and 0.75 to compete?
(e) For what range of c is it an equilibrium for two candidates with positions 0 and 1 to compete?
B: Consider the same set-up as in part A.
(a) Let x ∈ [0, 0.5). For what range of c is it an equilibrium for a citizen with position x to be the only candidate to run for office? Is your answer consistent with what you derived for A (b)?
(b) For what range of c is it an equilibrium for two candidates to compete—one taking position x and the other taking the position (1− x)? Is your answer consistent with your answers to A (d) and A (e)?
(c) Let ǫ be arbitrarily close to zero. For what range of c will two candidates with positions (0.5−ǫ) and (0.5+ǫ) be able to run against one another in equilibrium. What does this range converge to as ǫ converges to zero?
(d) How does the range you calculated in (c) compare to the range of c that makes it possible for two candidates with position 0.5 to run against one another in equilibrium (as derived in A(c))?