Question

Consider the following game: - 2 LR 2

a.) Find all pure-strategy Nash equilibria.

b.) *Find all mixed-strategy Nash equilibria.

c.) Explain why, in any mixed-strategy equilibrium, each player must be indifferent between the pure strategies that she randomizes over.

Answer 1

(1-9) l pulsu (1) D4?s ? (as Pure Strategy Nash equilibrium No fure strategy hash equilibrium. (b) der Player 1 Choose loo sanctomize its Strategies with Probability b for u and (1-6) Jor D. and Player-2 sandomize with Probability q and (1.2l for Land R Resfectively. In order to find out mixed strategy hash hayer-1 must be indifferent ble his strategy u and o given Player-2 is randomizing among his strategies. Exlected Payoff from o = expected Payoff from o 59 + 4(19) = 49 + 5 (19) sqru-uq = uqt 5-59 4+4 = –q+5

29 = 5.4 = q = 1 2 (1-9) = 1/2 Foo Player-2 expected Payoff from a = expected Payoff from R 2p+ 3(1-P) = 3b + 2(1-6) 26+ 3-3p = 3p+2-2p - 6+3 = pt2 - 3-2 = btb = = 26 - p= 1 / 2 uns ese n=» (1- - 1 Player - I will choose mixed to strategy with b = 1 Player-2 will choose wixced with q=1,

(e, consider hayer-1. exlected Payoff from u = 9+4 11 D= 5-9 so if Elul > E(0) (that is he is not indifJerent ) heb 9ч » S-4 - atq> 5-4 - 29 > 1 » q> hayer- I will choose o if expected Payoff from u is more than expected layoyy from D. which gives that q> since hayer-2 is better fanciomizing blw Land R with Possitive Probability. Player-I docs not know wheather a is less than or equal to 1 / 2 if ELUILE (O) qtu 5-9 - 29 al = q 4 1 / 2 flayer-I will choose o if , q< 1 / But Player - I doesn't know ah 4 or q> 1 / 2