Question

3. For the game represented below 1) Modify the game matrix by eliminating dominated strategies 2) Find a Nash Equilibrium in pure strategies. 3) Find NE in mixed strategies Y 3; 4 2; 2 3,5 D 2;3 1; 5 Z 5; 2 2;1 3:3 0:4 3; 0

Answer 1

U 5,2 3,4 4, 4 2,2 33 3,5 44 04 D 2,3 1,5 30 At First comparing the payoffs for Player 1 under all action A, B, C o. Now For Player 2, consider that payoffs for From X & Z. The set of payoff from X for Player = {2,4,5,33 is greater than from y = { 1 3 4 of for each action of player 1 hongannen). i Action 't' dominates 'z' for players . We can eliminate the dominated strategy z. This reduces our game to X Y For Player 1: 4 42,2 Band I are donimated è 3,5 44 Bis dominated by s and Dis darinated by A, B, C C - We can climate these dominated strategies this reduces our game to la O 5,2 3,4 In this redived game, J JC 3,5 y, y no dominated strategy exist This reduers our you, A 52 3 4 А.

(2) NE in pure stotlegy A 45,2 3,44 C 3,54 44 Best Response of Player 1 to Playa & playing X:BR (X) " BR, (X) = {A} broz 573 diversise BR, CY) = {c} 11 423 ther player 2 BR2 (A): Best Response of Playa 2 to BR₂ (A) = {43 g Prying a BR₂ CC) = {x} (soul is imply that there exist no mar Nash Essen in pure strategy. (3) Mixed strategy NE X Y Let up be the probs Cup c 395 44. of a playing A (-p) prob of player 1 playing C "s be the nob of player 2 playing X ing' be the pros of player & playing y. Expected Payoff of layer 1 from playing A. E (A) = 59 + 3(1-9) = 3+29

We note that the expected payot haver I depends on the pros of playan 2 I choosing his actions (adig! E, (C) = 39 +401-q) = 4-9. Randomization & E,CA) E (C) 2 3+2 = 48 39-1 n go Y3 1-9-43 for player 2 : E2(X) = 2p+ 56+p) = 5-3p Ez (Y) = 40 +421-p) = 4 Randomisation by Player 2 = E(X)=E(Y). - 5-3p = 4 2 3p = 1 (1-P). 2/3 p=73 Mine M,+)30), (734, 43773