Consider the following voting game. There are three players, 1, 2, and 3. And there are three alternatives: A, B, and C. Players vote simultaneously for an alternative. Abstaining is not allowed. Thus, the strategy space for each player is {A, B, C}. The alternative with the most votes wins. If no alternative receives a majority, then alternative A is selected. Denote u_i(d) the utility obtained by player i if alternative d {A, B, C} is selected. The payoff functions are,

u₁(A) = u₂(B) = u₃(C) = 2

u₁(B) = u₂(C) = u₃(A) = 1

u₁(C) = u₂(A) = u₃(B) = 0

1) Let us denote by (i, j, k) a profile of pure strategies where player 1's strategy is (to vote for) i, player 2's strategy is j, and player 3's strategy is k. Show that the pure strategy profiles (A, A, A) and (A,B,A) are both Nash equilibria.

2) Is (A, A, B) a Nash equilibrium? Comment.