Assume that players prefer their second best candidate winning over a tie of all three candidates. In particular, answer the following questions:
"(Voting between three candidates) Suppose there are three candidates, A, B, and C, and no citizen is indifferent between any two of them. A tie for first place is possible in this case; assume that a citizen who prefers a win by x to a win by y ranks a tie between x and y between an outright win for x and an outright win for y. Show that a citizen's only weakly dominated action is a vote for her least preferred candidate. Find a Nash equilibrium in which some citizen does not vote for her favorite candidate, but the action she takes is not weakly dominated. "
Assume the number of voters is n = 3 and n is odd. Argue that a player voting for her most or second-most preferred candidate is not weakly dominated. (Hint: without loss of generality consider player i who prefers A to B to C and argue that B is not weakly dominated by finding some profile of strategy choices by the other players that renders that strategy the unique best response. Do the same for A. In choosing the profile of strategies by the other players seek situations that maximize the impact of i's vote.)
Assume n = 3. Show that it is a weakly dominated strategy for a player to vote for her least preferred candidate. (Hint: without loss of generality, focus on player 1 and assume she prefers A to B to C; then apply the definition of weak dominance.)
Assume n = 3 and that player 1 prefers A to B to C, player 2 prefers B to A to C, and player 3 prefers C to A to B. Find one Nash equilibrium profile such that (i) no player uses a weakly dominated strategy and (ii) one of the players votes for her second best candidate. (Hint: you do not need to check whether deviating to the weakly dominated strategy is profitable.)