Assume the following modified version of the citizen-candidate model. As in the model we studied in lectures, all citizens have ideal points in [1, 1] and a citizen with ideal point gains a utility of | | if policy is implemented by the winner in the election. In addition, the winner in the election is chosen by a simple winner-takes-all rule, i.e., the candidate that garnered the highest vote share wins the election and implements their ideal policy. The modification we study in this question is in how the voters' ideal points are distributed. In particular, we assume that half of the population of voters have ideal points uniformly distributed in the interval [0, 1] and the other half have ideal points uniformly distributed on [0.5, 0]. As in the model studied in lectures, besides policy preferences mentioned above, citizens also derive utility from becoming candidates and winning elections. A citizen that chooses to become a candidate incurs a cost of = 0.01 and if they win the election, they will receive an additional payoff of = 100.

(a) Is there an equilibrium in which two citizens, both with ideal points at zero, are the only citizens that run for office? Prove your answer. [8 marks]

(b) Consider the profile of strategies in which one citizen with ideal point 0.25 and another citizen with ideal point at 0.25 are the only ones running for office. Suppose that a citizen with an ideal point at 0.05 is considering a deviation to run for office. If he runs for office, will he win the election? Prove your answer. [9 marks]

(c) Consider the profile of strategies in which one citizen with ideal point 0.25 and another citizen with ideal point at 0.25 are the only ones running for office. Show that this is not an equilibrium by showing that the citizen at 0.05 wants to deviate. [9 marks]

(d) Find an equilibrium with only two candidates running. [9 marks]