Question

) Consider a game where player 1’s possible three strategies are a, b and c and player 2’s possible strategies are A, B and C. The players choose their strategies simultaneously. The payoff matrix of the game is as follows:

                                          Player 2

A B C

   a 8,5 9,7 10,8

player 1 b 6,1 10,3 7,9

c 5,4 8,6 6,4

1. (5 pts) Is there a dominated strategy for player 1? For player 2? Justify your answer.
2. (5 pts) Is there a dominant strategy for any player? Justify your answer.
3. (5 pts) Determine the Nash equilibrium of the game. Explain your answer.

Answer 1

a) Dominated strategy: Player 1: c; player 2: A

reason: When Player 2 moves A, Player 1 will choose a (8>6>5). When Player 2 moves B, Player 1 will choose b (10>9>8). When Player 2 moves C, Player 1 will choose a (10>7>6). Thus Player 1 will never choose c. It is dominated strategy for Player 1.

For Player 2: When Player 1 moves a, Player 2 will choose C (8>7>5). When Player 1 moves b, Player 2 will go B (9>3>1). When Player 1 moves c, Player 2 will choose B (6>4=4). Thus Player 2 will never choose A. A is dominated strategy for Player 2.

b) There is no dominant strategy for either player. Because either player has no strategy that he will aways choose no matter what the other player does. Player 1 chooses between a and b; Player 2 chooses between B and C.

c) Nash equilibrium is (a, C) with payoff (10, 8).

reason: When Player 2 moves A, Player 1 will choose a (8>6>5). When Player 2 moves B, Player 1 will choose b (10>9>8). When Player 2 moves C, Player 1 will choose a (10>7>6).

For Player 2: When Player 1 moves a, Player 2 will choose C (8>7>5). When Player 1 moves b, Player 2 will go B (9>3>1). When Player 1 moves c, Player 2 will choose B (6>4=4).

Thus we see that Player 1's strategy a and Player 2's strategy C coincide. This combination is threfore Nash equiibrium.