Question

Ann and Bill play the following game in which 4 gold coins are to be won. Ann selects a word from the set {up, out, over} and Bill selects a word from the same set (it could be the same word). If the words selected by the two players differ, then Ann wins one gold coin, and the remainder go to Bill. If the words selected by the two players are the same, then the number of gold coins won by Ann equals the number of letters in the chosen word, and the remainder go to Bill.

(a) Represent this scenario as a zero-sum 3 × 3 game.

(b) Calculate the equilibrium strategies for the game, and explain how the players should play the game using these strategies.

(c) Use the answer you deduced in part (b) to determine the game value v and show explicitly that Ann’s payoff is bounded above by this game value.

Answer 1

Therefore, For Ann - P(A) = 3/7; P(B) = 3/7; P(C) = 1/7, A,B,C are Up, Out and Over for Ann respectively.

For Bill - P(P) = 6/11; P(Q) = 3/11; P(R) = 2/11, P,Q,R are Up, Out and Over for Bill respectively.

Therefore, any change in the value of P(A), P(B) and P(C), will not change the expected value of Ann's Payoff if Bill chooses to play at the equilibrium strategy. And that value is equal to the game value for Ann.

This means that for any strategy that Ann chooses to play, Ann's payoff is bounded above by the game value because the expected value of Ann's payoff is constant if Bill chooses to play his equilibrium strategy.

Do comment for any doubts.