Question

Determine ALL of the Nash equilibria (pure-strategy and mixed-strategy equilibria) of the following 3 games:

Player 1 H T Player 2 HT (1, -1) (-1,1) | (-1,1) (1, -1) | Н Player 1 H D Player 2 D (2, 2) (3,1) | (3,1) |(2,2) | Player 2 A (2, 2) (0,0) Player 1 A B B (0,0) | (3,4)

Answer 1

P2 # It P1 H11, -1/-1,1 There is no pune pure strategy nash equin TH11,-1 in this game, I players have incentives to deviate from each & part. suppose pl chooses it with probability P and Twith probability (1-P), P2 chooses it ... with probability q and a with probability (1-9). Eq (H) = 1.q+(-1)(1-9) expecked payoff of a) Lpl by choosing it] = 2q-1 EI (T) = -1.q+ 1 (1-9) - 1-20 EICH) = E(1) in mixed strategy an aq -1 = 1-2a on 4q = 2 aq = 2/4 = 1/2 and, E2 (H) = -1 p + 1 (1-P) [ Expected pay of of P2 by choosing = 1 - 2 Ez (T) = lip+(-1)(1-P) = 2p-1 had sholegy Ez (H) = E2(T) in mixe 4 1-2p = 2p-1 H mixed on 4P = 2 . P =/ so player 1 chooses it with probability / and T with probability 2 player 2 chooses, It with probability and I with probability /

P2 Then is no pne strategy 2,2 3,1 hash equilibrium in D 13,1 12,2 this game. Mixed shakeqyt suppone player 1 chooses it with probability p and b with probability (1-1) player 2 chooses it with probabiliny q and D with probabtlity (1-9). EI/H) = 2.9+3(1-9) - 3-9 Ei (1) = 3,9 + 2(1-9) 2 O + 2 E, (H) = E, (D) an 3-q = q + 2 an 29 = 1 2 q = 1 and 1-q = 1/2 Ez(t) = ap+ 1. (1-P) = P + E2 (A) = lip + 2(1-P) - 2-P E₂(H) = 62(D) an p+1 = 2-P on 2 = 1 and l-p= . P = / So, Pn mixed strategy player 1 chooses t. with probability Y and with prabability 1/2 Player 2 chooses it with probability Y and with probability /

P2 B A 0,0 P2 A when player 1 chooses p4 A 2,2 10,0 A, Baill chart2 will choose B A as B0,0 3,4 270, and when R2 will choose A, P1 will choose A as 270. So, (A, A) is a pure strategy nash equilibrium. when player 1 chooses B, P2 will choose B as 400, when p2 chooses B, Pl will choose B as 3 0 . so, (B, B) is a puu strategy nach equm Mixed strategy nash: Suppas pl chooses A wop. P and B wip. (1-1) P2 chooses A cup. q and B wip. (1-9). E,LA) = 2q +0.(1-9) EI (B) = 0. q + 3(1-9) à E(A) = E1(B) on 29 = 3-39 or 5% = 3 q=²5 and (1-9) = 2 E2 (A) = 219. 2p + (1-9), 0 = 2p B2 (B) = 0.P+ 4(1-9) = u-uq an 20 = 4-4 q . an 6pg = 4 u. app= 2 and 1-p= 1 / 2 so In mixed strategy, pl chooses A with probability 2 and B with probability /3. P2 chooses A with probability 3/5 and B with probability 25.