onsider 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 ui(d) the utility obtained by player i if alternave 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 a. 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. b. Is (A,A,B) a Nash equilibrium? Comment.
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 ui(d) the utility obtained by player i if alternave 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
a. 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.
b. Is (A,A,B) a Nash equilibrium? Comment.
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