Consider three-party competition in an election governed by a runoff rule: if no party receives a majority of votes in the first round of voting, then there is a second (runoff) round of voting. In the second round, voters choose among the top two vote-winners in the first round. If all three parties receive the same share of votes in the first round, then two parties are chosen using an equal-probability rule to compete in the second round (so that the probability that any party enters the second round equals two-thirds). If two parties tie for second place in the first round, then one of the tying parties is chosen by a fair coin toss to compete in the second round. Assume that parties cannot change their positions between the first and second rounds. Voters have single-peaked preferences over policies on the real number line, with ideal points distributed uniformly on [0,1]. Parties maximize their probability of winning.
a) Find the Nash Equilibrium or Equilibria of this game: that is, find the policies proposed by each of the three parties such that no one has incentives to deviate and show that no other set of policies can constitute an equilibrium.
b) Can the same equilibrium/equilibria exist if the election rule was a plurality rule rather than a runoff? (Under the plurality rule, the winner of the election is the one obtaining the most votes; ties are broken by assigning an equal probability of victory to all parties that receive the most votes).