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Problem Set 04

Problem Set 04 - Econ 121 – Fall 2010 UC Berkeley...

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Unformatted text preview: Econ 121 – Fall 2010 UC Berkeley Professor Cristian Santesteban Problem Set 4 Due: Thursday, October 7th at the beginning of class Problem 1 Consider a Cournot game with P(q) = a – b*Q. Firm 1’s MC=c1 and firm 2’s MC=c2. a. What is the Nash equilibrium if 0 < ci < a/2 for each firm? b. What if c1 < c2 < a but 2c2 > a+c1? Problem 2 Consider the following sequential move game. Player A can move L or R. Player B moves second and he can move l or r. The set of actions and payoffs is as follows: {(L,l), (8,10) ; (L,r), (7,6) ; (R,l), (6, ‐3) ; (R,r), (10, ‐2)} a) Depict this game in extensive form. What is the Sub‐Game Perfect Nash Equilibrium? b) Suppose now that player 2 cannot observe the choice of player 1 before he has to choose his strategy. Depict this new game in normal form. What are the Nash equilibria of the new game? c) There are now equilibria in the new game that were not an equilibrium in the old game. Does this make sense? Problem 3 There are n firms. First, each firm simultaneously decides whether to enter a market, by incurring a cost C. (If a firm does not enter, its payoff is 0.) Then, knowing which firms entered, each firm i in the market simultaneously produces qi at zero marginal cost, and they sell at price P = max{1 − Q, 0}, where Q is the sum of the qis produced by the firms in the market. The payoff of a firm i in the market is qiP − C. Find all the subgame‐perfect equilibria in pure strategies. Problem 4 Consider the following game: (a) Give a condition on b such that R2 is strictly dominated by R1. (b) Given that (a) holds, find a condition on d such that C1 strictly dominates C2. (c) Given that (a) and (b) hold, find conditions on a and c such that (R1,C1) is a Nash equilibrium. (d) Given that (a)–(c) hold, find conditions on d, e such that (R1,C1) is the unique Nash equilibrium. Problem 5 Consider the following game: (a) Identify the strategies for both players. (b) Derive the normal form for this game and find all of the Nash equilibria. (c) Identify all of the subgames. (d) Find the unique SPNE. (e) Explain why the the SPNE provides a better prediction. What Nash equilibrium depends on a noncredible threat? ...
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