PS4F07

# PS4F07 - Fall 2007 GAME THEORY IN THE SOCIAL SCIENCES...

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Fall 2007 G AME T HEORY IN THE S OCIAL S CIENCES M ETHODS Problem Set 4 (Due in Lecture on Thursday, November 15th) 1. Three players, 1, 2, and 3, have to agree how to divide \$300. The bargaining proceeds as follows: 1 gets to announce how much it is demanding for itself. Players 2 and 3 hear 1’s announcement and then each secretly writes down how much it’s demand on a slip of paper and seals it in an envelope. The envelopes are collected and opened. If the sum of the demands is \$300 or less, each player receives its demand. If the sum of the demands is larger than \$300, each player that demanded more than zero pays \$10. (a) Sketch the tree for this game. (b) Find a subgame perfect equilibrium for this game. (c) Find a Nash equilibrium for this game which is not subgame perfect. (d) Explain why the Nash equilibrium in (c) fails to be subgame perfect. 2. This question examines how much two actors can cooperate. Insay and Peak are two high- tech firms that are thinking about forming a strategic alliance between them by sharing research and development expenses and information. The firms have to decide how “deep” to make the alliance. The deeper it is, the larger the gains from cooperation but the more vulnerable each firm to the other. The matrix below illustrates the situation where the firms have to decide between participating in the alliance and not; the payoff to the alliance is d (for depth); and the gain from exploiting the other firm is . d 2 Insay Participate Not Participate Participate Peak d , d -5, d 2 Not Participate d 2 ,-5 0,0 1

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Assume that this game is repeated infinitely many times; each firm’s discount factor is .8; and the firms follow the strategies: Peak : Participate in round 1. Participate in the current round if Insay has always participated in the past; otherwise do not participate . Insay : Participate in round 1. Participate in the current round if Peak has always participated in the past; otherwise do not participate . a. Assuming that the depth of the alliance is 4, i.e. d = 4 , is the outcome (participate, participate) individually rational? b. Continuing to assume , are the strategies above a Nash equilibrium of the repeated game? d = 4 c. Now consider a deeper alliance. Suppose, in particular, that d = 8 . Are the strategies above a Nash equilibrium of the game? d. How deep of an alliance can the firms create if they follow the strategies above? That is, what is the largest value of d for which the strategies above are a Nash equilibrium? e. As we know, there are many equilibria of a repeated game. Can the firms participate is a deeper alliance than what you found in (d) if they play according to some other equilibrium strategies? 2
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## This note was uploaded on 12/25/2010 for the course PO 137 taught by Professor Power during the Fall '10 term at Berkeley.

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PS4F07 - Fall 2007 GAME THEORY IN THE SOCIAL SCIENCES...

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