Consider three individuals with the following preferences:
Suppose the following dynamic voting protocol is in effect: first, there is a vote between a and b; then the winner goes against c; and the winner of this contest will be implemented. Focus on SPE where voters do not use "weakly dominated" strategies at any stage.
(a) Show that these preferences are single peaked, but sincere voting is not equilibrium behavior.
(b) Characterize the SPE of this game under strategic voting by all players.
(c) Consider a generalization in which the society H consists of H individuals and there are finite number of policies, R = {p1, p2,...,pM}. For simplicity, suppose that H is an odd number. Voting takes M - 1 stages. In the first stage, there is a vote between p1 and p2. In the second stage, there is a vote between the winner of the first stage and p3, until we have a final vote against pM. The winner of the final vote is the policy choice of the society. Prove that if preferences of all agents are single peaked, then the unique SPE implements the bliss point of the median voter.