Question

1. A game with two players, Player 1 and Player 2, is represented in the matrix below. Player 1 has two possible actions, U, M and D, and Player 2 has three possible actions, L, C and R.

Player 2

   L C R

U 1,0 4,2 7,1

Player 1 M 3,1 4,2 3,0

D 4,3 5,4 0,2

(a) Which of player 1’s actions are best responses when player 2 chooses action R?

(b) Which of player 1’s actions are best responses when player 2 uses the mixed strategy (σ2(L), σ2(C), σ2(R)) = (2/3, 0, 1/3)?

(c) Which of player 1’s actions are best responses when player 2 uses the mixed strategy (σ2(L), σ2(C), σ2(R)) = (1/2, 1/4, 1/4)?

(d) Can you construct a strategy (possibly mixed) for player 1 that makes L a best response for player 2? If yes, do so. If no, explain why not.

Answer 1

a) When player 2 chooses action R, action U gives player 1 the highest payoff = 7.

b) When player 2 uses the mixed strategy (σ2(L), σ2(C), σ2(R)) = (2/3, 0, 1/3), we see that actions U and M can both be best responses for Player 1 as both of them give expected payoff 3.

c) When player 2 uses the mixed strategy (σ2(L), σ2(C), σ2(R)) = (1/2, 1/4, 1/4), all the actions U,M and D provide player 1 with equal payoffs = 13/4