Question

find any dominant strategies and any nash equilibrium

Player 2 X Y 10,10 15,5 Player 5, 15 12, 12

Answer 1

Player 1/2	X	Y
A	10,10	15,5
B	5,15	12,2

The numbers in bold represent the player's best response given other player's strategy. For example, if player 1 plays A, player 2 has to choose between playing X and Y. Since his payoff from playing X is greater than his payoff from playing Y, he will play X. Similarly, when player 1 plays B, player 2 plays X to maximize his payoff. Since player 2 plays X irrespective of player 1's strategy, X is the dominant strategy for player 2.

If player 2 plays X, player 1 has a choice between playing A and B. Player 1 plays A to maximize his payoff. Similarly, when player 2 plays Y, player 1 plays A to maximize his payoff. Since player 1 plays A irrespective of player 2's strategy, A is the dominant strategy for player 1.

Since both players will always play their dominant strategies, the strategy (A,X) is the Nash Equilibrium and payoffs at Nash Equilibrium are (10,10).