Question

Players 1 and 2 each choose a positive integer up to K. If they choose the same number then player 2 pays 1 pound to player 1, otherwise no payment is made. Each player's preferences are represented by her expected monetary payoff. Show that the game has a mixed strategy Nash equilibrium in which each player chooses each positive integer up to K with probability 1/K.

Answer 1

Answer:

To display the pair of mixed strategies describe in the question is a mixed strategy Nash equilibrium, we have to display that each action of each player yields the same ​​forecast payoff.

Player 1's expected payoff to each pure strategy is 1/K, as with probability 1/K player 2 chooses the same number, and with probability 1 - (1/K) player2 chooses a different number.

Likewise, player 2's expected payoff to each pure strategy is -1/K, because with probability 1/K player 1 chooses the same number,and with probability 1-(1/K) player 2 chooses a different number. hence, the pair of strategies is a mixed strategy Nash equilibrium.