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PS1solutions - Econ 171 Spring 2010 Problem Set 1 Solutions...

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Econ 171 Spring 2010 Problem Set 1 Solutions to the two-star Problems In some cases I’ve provided more explanation than was asked of you. ** Problem 3 Find the rationalizable strategies for this game. For each iteration, list the eliminated W X Y Z A 5 , 6 4 , 0 2 , 1 1 , 2 B 4 , 3 3 , 4 6 , 0 2 , 1 C 6 , 2 2 , 0 0 , 5 5 , 7 strategy and which strategy dominates it. The unique rationalizable strategy is ( C, Z ). The logic is as follows: All actions are rational for Player 1. A is optimal if she believes 2 will play X with certainty, B is optimal if she believes Y , and C is optimal if she believes W will be played. Only W , X and Z are rational for Player 2 because Y is dominated by Z . Y is eliminated. If Player 1 knows that 2 is rational, she will never play B . She knows Y won’t be played. A is optimal if she thinks X will be played and C is optimal given W . Is B ever optimal? Well, if 1 believes that Pr( Z ) > 1 / 4 then she prefers C to B and if she believes that Pr( Z ) < 1 / 2 then she prefers A to B . This means that B is never optimal, which in this type of game tells us that it is dominated. Any strategy that places probability q on A and 1 - q on C , with q ( 1 2 , 3 4 ) dominates B .
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