Consider the following simultaneous game between two players: Each player chooses an integer between 1 and 5 (e.g., some number of fingers shown with your hand), and the higher number wins, except when it is just one larger than the lower number, in which case the lower number wins. So 4 beats 2 and 5 but it loses against 3 . Equal numbers are a draw.

(a) Write down the matrix of payoffs to the row player of this zero-sum game. A winning player gets payoff 1 , a draw gives payoff 0 .

(b) Show all pairs of strategies of the row player where one strategy weakly or strictly dominates the other, and indicate the type of domination.

(c) Argue carefully, without calculations, why the value of this game has to be zero.

(d) Find an equilibrium of this game in mixed (including pure) strategies, and explain why it is an equilibrium.