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Econ171_H1 - Econ 171 Introduction to Game Theory Homework...

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Econ 171 °Introduction to Game Theory Homework Problems #1 These problems are intended to supplement the in-class lectures (as well as give you exposure to the types of problems you will see on the midterms and ±nal exams). They will not be collected or graded. However, it is very di¢ cult to learn game theory without working problems. Therefore, I strongly encourage you to work through them. 1. Static Games: Consider the following 3x3 game. Player 2 West Midwest East West -2, 3 -3, 1 -2, 2 Player 1 Midwest 2, 4 -2, 4 -2, -1 East 0, -1 0, -1 0, 0 (a) Name the strategies that are eliminated by Iterated Elimination of Strictly Dom- inated Strategies. (b) Find all Pure-Strategy Nash Equilibria. (c) Find any Mixed-Strategy Nash Equilibria. 2. Static Games: Players 1 and 2 are bargaining over how to split one dollar. Both players simultaneously name shares they would like to have, s 1 and s 2 , where 0 ° s 1 ; s 2 ° 1 . If s 1 + s 2 ° 1 , then the players receive the shares they named; if s 1 + s 2 > 1 , then both players receive zero. What are the pure strategy Nash Equilibria of this game? 3. Cournot Revisited: Assume there is a 2-°rm duopoly in the Widget market. The price of Widgets will be determined by the total amount of Widgets brought to the market by the two °rms, speci°cally, P ( Q ) = 100 ± Q , where Q = q 1 + q 2 . Assume further that the output of the two °rms is homogenous, and that they have symmetric cost functions with constant marginal cost, i.e. C 1 ( q 1 ) = C 2 ( q 2 ) = 10 q i for i = 1 ; 2 . In class, we found that the Nash Equilibrium in this game is ( q ° 1 ; q ° 2 ) = (30 ; 30) , and the Nash Equilibrium payo/s are ( ° ° 1 ; ° ° 2 ) = ($900 ; $900) (you may want to derive this result again for practice).
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