Guessing Numbers Players 1 and 2 each choose a number from the set {1; : : : ;K}. If the players choose the same number, then player 2 pays 1 Euro to player 1; otherwise no payment is made. The...

Guessing Numbers

Players 1 and 2 each choose a number from the set {1; : : : ;K}. If the players choose the same number, then player 2 pays 1 Euro to player 1; otherwise no payment is made. The players can use mixed strategies and each player’s preferences are determined by his or her expected monetary payoff.

(a) Show that each player playing each pure strategy with probability 1 K is a Nash equilibrium.

(b) Show that in every Nash equilibrium, player 1 must choose every number with positive probability.

(c) Show that in every Nash equilibrium, player 2 must choose every number with positive probability

(d) Determine all (mixed strategy) Nash equilibria of the game.

(e) This is a zero-sum game. What are the optimal strategies and the value of the game?