Question

1. Consider a gaine where the row player moves first, choosing between T and B. The column player observes the row player's action and must choose between L and R. In the payoff table given in (a), solve for the unique backward induction strategy profile, and compare it to the Nash equilibrium when actions are chosen simultaneously. In the payoff table (b); solve for all pure strategy Nash equilibria. Optional question: Are there any inixed strategy equilibria in (b)? DLR T 1,3 2,1 B 0,5 4,0 T 1, 34,3 B 2,1 0,0

Answer 1

LIR T | 123 1 2,1 B 05 4,0 row. by backward induction method, we can solve for the 1. backwund in duction stategy profile when Row play first move. We can write the entonsive form of the is the que as follows - Row playn T/ (65) (1,3) column Playa Column Playn (0,5) (4,0) (1, 3) (2,4) Scanned with CamScanner

at column Playa " and column Playu in Playu will "" Роуч си < < e совечи Now, by backumd in duction method, play " ľ it Row player choosn B, choose "L it Row Playu Plays T. Now, Row Playn will compare the two outcome the la Playn has chosen and will choose T as th manim. is e-okk . - Haces unique backward induction strategy Profile will Бk :( ті) = (1,3) умеw Ukwң by Now It Row playn choosn T, column Playn will choose - Row P0- %: в со 44-н Роуч %: 4 лақ. in Colии и РИn allo C, Rov Руч ~и Acust 'т . bu olumn P-4 chuvsa R, Rov Piм i close 'р. Huce pme strategy nash equilibrium will be= (,2)= (1,3) then, linque chosm Simultaneonly, Hace when actions are Nash equilibrum outcome backwand induction stalgy profile = *L'. . 4 (1-12) Гь к Now wha Ro~ Рели е и т”, Сокм. Рум се can choose either L ac or R as both gives play, Неодичи едни Su Pra okt < 3. Scanned with CamScanner

at Row playn choosu 'B' Column playn will choose 2: 9k Социт едни слабо, ', 'ow е дни оt u chФse В. 1 When Column Playn choose "P' Huce, pue Stategy Row Plyn will choose T' Nash equilibu' a will be's (TⓇ) = (493) and [ B. 2) =(2, 1) Mined statgy equilibria Suppose that column's mined Strategy probudi assigns Probability weight q to L and probability wilgut (1-9) to R. Now, Rowes in Row playn's expected pay off from T against (2, 1-4)= q11) 7 (1-9) (4) = q +4-42=4-39 and, Row playn's expected pay off from B against - (2, 1-9) = q (2) + (1-2)(0) = 29 But these expected payoffs are to be equal. 4-39 - 29 - Sq = - 4 a) 2-4 csand (l-aw) = (1-) -1 Manscanner

albose that Row bability weight Playn's mined p tot and Strategy assigns probability weight 'B' (1-P) to 3 column's eupected pay off from LE against (P1-P) - P(3) + (1-P) (1) - 3P + l-p = 2P+) Column's enpected pay off from R against (P1-P) = P(3) + (1-P)*o = 3P Mud these expected payoffs are to be equal. Auce 2P+1 = 3P 2 -p= -1 = P=1 (-p) - 0 tuce, mined statigy equilibria : [(P. 1-P), (2, 1-2)]= [(10), (1,77 Scanned with CamScanner