Consider a game of hide and seek, in which agents choose simultaneously and inde-pendently between two locations—A and B. The payoffs areABA1 +1,−1 +2−1 +1,1B−1,1 +21,−1whereiis a random variable distributed uniformly on [−x,x] fori= 1,2.Thisrandom variables are independent across players. A player knows the realization ofhis payoff, but does not observe the realization of the opponent’s payoff.(a) Solve for all equlibria whenx= 0.Are they mixed or pure?Answer:This game has a unique equilibrium in which each action is playedwith probability 1/2 by both players (it is mixed strategy equilibrium).(b) Letx >0. Solve for an equilibrium in pure strategies.Answer:Suppose the player 1 believes that the actionAwill be played bythe opponent with probabilityp. ThenAis a best response ifp−(1−p) +1≥−p+ (1−p)or if1≥2−4p.Similarly if player 2 believes that player 1 playsAwith probabilityq,Ais abest response if−q+ (1−q) +2≥q−(1−q)ECON0027 Game Theory, HA