Suppose two candidates A and B compete for office in an election with 5 voters.

Suppose two candidates A and B compete for office in an election with 5 voters. They simultaneously making binding announcements of policies they will choose if elected, and then all voters simultaneously vote for either A or B after observing both announcements (no one abstains). There are 5 possible policies to choose 1 from numbered 1 to 5 (candidates cannot locate in between these policies; e.g. 2.5 is not a possible choice). Voter 1’s ideal point is policy 1, 2’s ideal point is 2, and so forth. Each voter obtains 0 jollies if her ideal policy is enacted, and loses 1 jolly for each unit of difference between the policy chosen by the winning candidate and her favorite policy. Assume that if any voter likes both candidates equally well, s/he flips a coin to determine his/her vote. Candidates care only about winning the election: if a candidate wins, she gets 1 jolly; if she loses, she gets 0 jollies; if they tie, each candidate gets 0.5 jollies.
(a) Suppose candidates win if they get at least a majority of votes (at least 3). What policies do they announce in equilibrium? Does either party win for sure?
(b) Suppose candidate A is the incumbent, who has manipulated electoral rules to his advantage: candidate B must receive at least a majority of votes to win, and otherwise A wins. What policies do they announce in equilibrium? Who wins?
(c) Consider the same problem as above, but now B must receive at least 2 3 of the votes in order to win; otherwise A wins. Which combinations of policy proposals are equilibria? (Hint: there may be more than one.) Who wins in all equilibria?

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