ECON0027 Game Theory
Home assignment 3
1. Two players, Sauron (player 1) and Saruman (player 2), each own a house. Each player values his own house at vi. The value of player i’s house to the other player, i.e. to player j ≠ i, is αvi - c where α > 1. 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 aredistributed uniformly on the interval [0, 1] and are independent across players.
(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 houses. Find a Bayesian Nash equilibirum of this game in pure strategies.
(b) How does 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) Give an intuition about why we should focus on these threshold strategies when looking for an equilibrium.
2. Consider a game of hide and seek, in which agents choose simultaneously and inde- pendently between two locations—A and B. The payofs are
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A
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B
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A
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1 + E1 , -1 + E2
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-1 + E1 , 1
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B
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-1, 1 + E2
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1 -1
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where Ei 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 payof, but does not observe the realization of the opponent’s payof.
(a) Solve for all equlibria when x = 0. Are they mixed or pure?
(b) Let x > 0. Solve for an equilibrium in pure strategies.
(c) Compare the limit of equilibria for x → 0 with the equilibrium in 2a. Give interpretation to mixed strategy NE using your indings.
3. Two people are involved in a dispute. Person 1 does not know whether person 2 is strong or weak; she assigns probability α to person 2’s being strong. Person 2 is fully informed. Each person can either ight or yield. Each person’s preferences are represented by the expected value of a Bernoulli payof function that assigns the payof of 0 if she yields (regardless of the other person’s action) and a payof of 1 if she ights and her opponent yields; if both people ight then their payofs are (-1, 1) if person 2 is strong and (1, -1) if person 2 is weak.
(a) Formulate this situation as a Bayesian game.
(b) Find Nash equilibria of this game if α < 1/2.
(c) Find Nash equilibria of this game if α > 1/2.
4. Two chefs 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 lour from the restaurant at a price pi per kilogram. The restaurant will charge a chef for the lour only if he wins the contest. The price of lourpi 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 lour.
(a) Suppose the realized prices for the lour are publicly observed before the chefs make their choices. Let p1 < p2 and suppose that in the event the chefs bake the cakes of equal size, the chef who can source cheaper lour—i.e., chef 1—wins. Find a Nash equilibrium of this game.
(b) Suppose the 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 lour independently of the contest outcome. Solve for a Bayes-Nash equilibrium of the game.