ECON0027 Game Theory Home assignment 3

1.Two players, Sauron (player 1) and Saruman (player 2), each own a house.

Each playervalues his own house at vi. The value of player i’s house to the other player,i.e.to player j = i, is αvic where α > Each player knows the value vi of his own house to himself, but not the value of the opponent’s house. Both players know α. The values vi are distributed uniformly on the interval [0, 1] and are independent across players.  代写博弈论作业

(a)Suppose players announce simultaneously whether they want to exchange their houses (without paying each other). If both players agree to an exchange, the exchange takes place. Otherwise, they stay in their own Find a Bayesian Nash equilibirum of this game in purestrategies.

(b)Howdoes this equilibrium depend on α? In particular how does the probability of exchage depends on α in this equilibrium? Is the equilibrium outcome always efficient?

(c)Givean intuition about why we should focus on these threshold strategies when looking for an equilibrium.

2.Considera game of hide and seek, in which agents choose simultaneously and inde- pendently between two locations—A and B. The payoffs are  代写博弈论作业

 A B A 1 + s1, −1 + s2 −1 + s1, 1 B −1, 1 + s2 1, −1

where si is a random variable distributed uniformly on [ x, x] for i = 1, 2. This random variables are independent across players. A player knows the realization of his payoff, but does not observe the realization of the opponent’s payoff.

(a)Solve for all equlibria when x = 0. Are they mixed orpure?  代写博弈论作业

(b)Let x > Solve for an equilibrium in purestrategies.

(c)Compare the limit of equilibria for x 0 with the equilibrium in 2a. Give interpretation to mixed strategy NE using your fifindings.

3.Two people are involved in a dispute.  代写博弈论作业

Person 1 does not know whether person 2 isstrong or weak; she assigns probability α to person 2’s being  Person 2 is fully informed. Each person can either fight or yield. Each person’s preferences are represented by the expected value of a Bernoulli payoff function that assigns the payoff of 0 if she yields (regardless of the other person’s action) and a payoff of 1 if she fights and her opponent yields; if both people fight then their payoffs are (-1, 1) if person 2 is strong and (1, -1) if person 2 is weak.

(a)Formulatethis situation as a Bayesian game.

(b)FindNash equilibria of this game if α < 1/2.

(c)FindNash equilibria of this game if α > 1/2.

4.Twochefs are competing for a position at a restaurant called “Food for Thought”.

The value of the position to each chef is equal to v. The competition takes a form of a contest in which one of the two chefs who bakes a bigger cake wins. In order to bake a cake of size x > 0, chef i has to procure x kilograms of flour from the restaurant at a price pi per kilogram.

The restaurant will charge a chef for the flour only if he wins the contest. The price of flour pi for each chef i is drawn randomly from a uniform distribution on [1, 2]. The prices for the two chefs are independent. The chefs choose the sizes of their cakes simultaneously to maximize the expected value of the position net of the expenses for the flour.  代写博弈论作业

(a)Supposethe realized prices for the flour are publicly observed before the chefs make their  Let p1 < p2 and suppose that in the event the chefs bake the cakes of equal size, the chef who can source cheaper flour—i.e., chef 1—wins. Find a Nash equilibrium of this game.

(b)Supposethe chefs privately observe the realization of their prices: chef 1 ob- serves only p1 and chef 2—only p2. Solve for a Bayes-Nash equilibrium of the game.

(c)Continue assuming that the chefs privately observe the realization of their prices. In addition, suppose that the restaurant charges chefs for the flour independentlyof the contest  Solve for a Bayes-Nash equilibrium of the game.