Question

QUESTION 9 Player 11 DE F 3,-1 1,1 6,1 4,-1 0,0 6,5 -1,-2 -2,-2 7,-1 Player B C Consider the game in normal form above and select all that apply. a. The strategy profile (B,D) is a Nash Equilibrium. Ub. There is a unique Nash Equilibrium in pure strategies. C. There is an equilibrium in mixed strategies. d. The strategy profile (C,F) is a Nash Equilibrium.

Please explain why the answer is what it is!

Answer 1

Question a 1 Plays II (1-21 - 22 요 뜸 D E 도 Playu IP, A 3-1 1 61 P2 B 4-1 oo 65 (1-P-P2) e -1,-2 -2,2 2-1 If Playa I sclects A, Playa 2 will select lithu E or F It Playn I sclects B, Playn 2 will select It playn I selects a Playn 2 will schlect f. It plays II Sclects D pla Yu 1 will Select B, It Playn I sclect E, playn 1 will select, It plays It sclects of playn 1 will select e. Huce nash equilibria (A,E) = (1,1) and (e. F) = (2-1) AN Scanned with CamScanner

Now suppose Probabilitis Playu II plays D, E, F wo q, q2 and it 2 -22) with probabilites and Playa I plays A, B, C P, P2 and [I-P - P2) Now, for playa 1 E (A) = 3q1 + 1 (24) + 6(1-91-92) = 391 +92 + 6-691-692 = 24 sat6 - 341-59276 ECB) = 491 + 0 (21) + 6(1-21-22) = 491 tot 6-621-692 - -221-692 +6 and E (c) = -1 (21) - 2 (22) +7 (1-91-92) = -21-2921 7- 72,- 792 = -69 - qqzt7 Nou Now, E (A) = E(e) implis Coltell - 39,- 582 +6=-621-92277 => 321+ 4 22 = 1 -(1) and E (B) = E(C) un plisi- - -29 - 622+6= -691-9qzt7 => 4q1+34 = 1 --(2 ) Scanned with CamScanner

- Solving equation (1) and (2) we get! 3414 422 = 1 t) x4 421+ 322 = 1 - 2/1 X3 12q + 1692 = 4 /2qt qq2 = 3 Wed 75% 92 = 5 Huce 341+ 4 q2 = 1 => 39 + 4 = 1 => 341 = 1 - 21 ts Now q = 4 2 2 1 - 1 and (1- q - 2) = (1 - 15 h - soi-3 (1) IS15 3 = 4 Scanned with CamScanner

Similarly for Playn 2? 2 - 1 (Pilt (-1) P2 + (-2 (1-P.-P2) =- 2 +24+ 222 PitP2 - 2 110.) + 0(P2) + (-2) (1-P1-P) - - 2 + 2pit 2P2 E (E) + E(F) 2 11 p.) + 5 (R) + (-1) (1-03-27 10 P1 + 58 ltp 1+PL -= 2P1+6P2=1 Now equating E (D) = E(F) => pit P2 = 2 = 2P1+6P2 = 1 n at - iPi - 5 P2 = 1 =) Pit SP2 = -1 - (3) E quating ECE) = E(F) Bosna i po => 3 P1+2P2-2 = 2P1+6P2-1 - - p ,- 4 PL = 1 cannes ¢ + ੫ : - - -(੫ ) CamScanner

Solving equation (3) and (4) PTSP2 =- 1 1111 Pi +42E-1 p. +5 P2 = -1 P + 5/2 = -1 - L +4 B 2-1 P2 = 0 Pi + 5 P2 = -1 =) PE and (I-P1-P2) = ( 14 14-0) = 2 As probabilities Cannot be negative, there is no equilibuum in mined strategy. Hunce auswu is! d) The strategy Profile (e, F) is a Nash Il equilibrium! Ics Bcanned with