Consider a two-player game where A1 = A2 = {1, . . . , k}, so each player chooses a number between 1 and k; player 1 wins if the two numbers match, and player 2 wins if they do not:

(See Above)

(a) Find all the Nash equilibria of the mixed extension of this game. Justify your answer. (b) Consider a modified version of the game above where player 1s utility increases exponentially as a function of the number that is matched:

(See Above)

Player 2s utility is not changed. Find all the Nash equilibria of the mixed extension of this game.

Consider a two-player game where A1=A2={1,,k}, so each player chooses a number between 1 and k; player 1 wins if the two numbers match, and player 2 wins if they do not: u1(x,y)={10ifx=yotherwise and u2(x,y)={10ifx=yotherwise. (a) Find all the Nash equilibria of the mixed extension of this game. Justify your answer. (b) Consider a modified version of the game above where player 1's utility increases exponentially as a function of the number that is matched: u1(x,y)={2x10ifx=yotherwise