Question: Consider a voting game in which three players, 1, 2, and 3, are deciding among three alternatives, A, B, and C. Alternative B is the "status quo" and alternatives A and C are "challengers." At the first stage, players choose which of the two challengers should be considered by casting votes for either A or C, with the majority choice being the winner and abstentions not allowed. At the second stage, players vote between the status quo B and whichever alternative was victorious in the first round, with majority rule again determining the winner. Players vote simulta-neously in each round. The players care only about the alternative that is finally selected, and are indifferent as to the sequence of votes that leads to a given selection. The payoff functions are u1 (A) = 2, u1(B) = 0, u1(C) = 1: u2(A) = 1, u2(B) = 2, u2(C) = 0; u3(A) = 0, u3(B) = 1, u3(C) = 2.
(a) What would happen if at each stage the players voted for the alternative they would most prefer as the final outcome?
(b) Find the subgame-perfect equilibrium outcome that satisfies the additional condition that no strategy can be eliminated by iterated weak dominance. Indicate what happens if dominated strategies are allowed.
(c) Discuss whether different "agendas" for arriving at a final decision by voting between two alternatives at a time would lead to a different equilibrium outcome.
(This exercise is based on Eckel and Holt 1989, in which the play of this game in experiments is reported)