ECON0027 Game TheoryHome assignment 31. Two players, Sauron (player 1) and Saruman (player 2), each own a house. Eachplayer values his own house atvi. The value of playeri’s house to the other player,i.e. to playerj6=i, isαvi−cwhereα >1. Each player knows the valueviof hisown house to himself, but not the value of the opponent’s house. Both players knowα. The valuesviare distributed uniformly on the interval [0,1] and are independentacross players.(a) Suppose players announce simultaneously whether they want to exchange theirhouses (without paying each other). If both players agree to an exchange,the exchange takes place. Otherwise, they stay in their own houses. Find aBayesian Nash equilibirum of this game in pure strategies.Answer:Let’s look for an equilibrium in threshold strategies: playeriagreesto exchange if his valuation is below a thresholdxi. If playeri’s opponentfollows this threshold strategy, the player will gain nothing over his currenthouseviif he keeps his house and will gainE[αvj|vj≤xj]−c−viif hedecides to trade and the opponent agrees to trade as well. At the threshold—i.e., whenvi=xi, the player is indifferent. ThusE[αvj|vj≤xj]−c=xiorαxj2−c=xiFrom symmetry, we get thatx1=x2=2cα−2. Note that if2cα−26∈[0,1] we getan equilibrium in which either players always exchange the houses or playernever do so.(b) How does this equilibrium depend onα? In particular how does the probabilityof exchage depends onαin this equilibrium? Is the equilibrium outcome alwaysefficient?Answer:The probability of exchange isp=x2=(2cα−2)2.It is increasing incand decreasing inα. In equilibrium, ifαis small and the opponent wants toexchange, it is good news for the player because he thinks that the house hasECON0027 Game Theory, HA31TURN OVER