Problem_set_14_Solutions

Problem_set_14_Solutions - Kaushik Basu Spring 2008 Econ...

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Kaushik Basu Spring 2008 Econ 367: Game-Theoretic Methods Problem Set 14 [This was the final examination in 2006.] 1. Consider the two-player normal-form game described below. L R T 4, 4 0, 0 B 0, 0 2, 2 (a) Locate all the pure and mixed-strategy Nash equilibria of this game. Answer: There are three NE – (T,L), (B,R), {(T with prob. 1/3, B with prob. 2/3); (L with prob. 1/3, R with prob. 2/3)}. (b) Suppose we use p to denote the probability of player 1 playing T and q to denote the probability of player 2 playing L . Using a diagram where the horizontal axis represents q and the vertical axis, p , draw the ‘reaction function’ of player 1, that is, a graph showing the optimal choice of p for every value of q . Answer: Expected pay-off for choosing T = 4q, expected pay-off for choosing B = 2(1 – q). Therefore, BR 1 (L,q) = 1 if 4q > 2(1 – q), = p є [0,1] if 4q = 2(1 – q), = 0 if 4q < 2(1 – q). Or, BR 1 (L,q) = 1 if q > 1/3, = p є [0,1] if q = 1/3, = 0 if q < 1/3.
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2. There is a city in which people live around a circle, as shown, below. M To go from any point on the circle to another point on the circle, people have to travel along the circle (of course, taking the shorter direction). In other words, think of the circle as a ring road along which people live and assume there is water in the middle. Two vendors, A and B, have to choose where to locate each store on the ring road. Given their location, citizens go to the nearest store and when a person is indifferent between stores she chooses a store randomly by assigning equal probability to each store. (a) Assuming that people are uniformly distributed along the circle, describe all possible Nash equilibrium locations of the two vendors. Anywhere on the circle is a Nash equilibrium; each vender will always get ½. (b) Suppose a tall residential block comes up at point M so that some extra people now live at M and everywhere else population is uniformly distributed. Does this game have a Nash equilibrium? If not, why not? If yes describe it. BR 1 2 1/3 1/3 1 1 p q
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Both firms at M. There is no incentive to move, because they would get less than half the population elsewhere. This is the only one equilibrium: If one is at M and the other not, the one that is not at M will move to M. If neither is at M, the one who is farther away than M will move toward M so that he is closer to M than the other. (Even if they are the same distance away from M, they will want to move to be closer.) (b) Now suppose that when both stores are equally close to a person’s residence he or she goes to store A . Otherwise, he or she goes, as before, to the nearest store. Does this game have a Nash equilibrium? If not, why not? If yes describe it. There is no Nash.
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Problem_set_14_Solutions - Kaushik Basu Spring 2008 Econ...

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