Question

1. (60 marks) Consider a two-person game, in which every player has two pure strategies to play. The payoff matrix of the game is as follows Strategy 2 Player One Player Two Strategy I Strategy II Strategy 1 0,0 1,3 1,1 Find all the Nash equilibria of the game.

Answer 1

2,113 0,0 I I (13) (1) - it player 1 chooses strategy I, then player 2 chooses strateg 2. (Marked by *) - if player 1 chooses strategy II, then player 2 chooses strategy 1. SIMILARLY, - if player 2 chooses strategy 1, ther player 1 chooses strategy I (Marked by ) - it player 2 chooses strategy 2, then player 1 chooses stoategy I. So, pure stogtegy Nash equilibrium : < Stoategy T, Strategy 1 > < Strategy I, Stoategy 2 >. Now, Let Player 1 play strategy I with probability P. . Player 1 play strategy II with prabability (1-P) Player 2 play strategy 1 with probability q. Player 2 play strategy 2 with (1-9). Expected Utility (64) = pxoxq+ 2p(lot 1(1-pq+ I (1-P)(1-2) *= 0+ 20-284+4-p4t 15-pt pa : P-289 +1. Fol 7 de la = 1-29 = 0 729= 1. → a= 0.57' : : : EU2 = 0+ IP(-2)+ 3 (-p)q+1(1-P)(1-9). = 8-pq+ 3q - 3pq+ 1-9 - Pt pq = 29-3P9+ Foc 3€U23 = 2-3p = 0 . . aq IP * = 0.67) Mixed Storategy Naph Eqm . <0.67, 0.5>