Configuration of Strategies Form equilibrium of Voters

Question

Assuming a citizen candidate model with a runoff system. If a candidate receives more than 50% of the vote in the first round, he wins the election. If no candidate receives more than 50% of the votes then there is a runoff election between the two candidates with the highest vote shares (there is a lottery with equal probability if several candidates have the same vote share). We are checking whether the following configuration of strategies is equilibrium. Three citizens run, one at the median and two symmetrically on both sides of him at (2/3) and -(2/3).

(a) Show that all three candidates receive the same vote share in the first round.

(b) What is the probability that the candidate on the left (with ideal at -(2/3)) wins the election (both   rounds). What is the probability that the central candidate wins the election?

(c) Under what conditions on b and c do the three candidates indeed want to run?

(d) Under what conditions, if at all, does this configuration of strategies form equilibrium?

Summary: This question belongs to political economy and discusses about a citizen candidate model with a runoff system and configuration of strategies is equilibrium.

Total word count: 108