problemset2_2012_solutions

problemset2_2012_solutions - Game Theory and Strategy ECON...

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Game Theory and Strategy ECON UA - 216 Solutions to Problem Set 2 1 Penalty shots revisited l m r L 4,6 7,3 9,1 M 6,4 3,7 6,4 R 9,1 7,3 4,6 Part A No, no strategy is dominated by any other (pure) strategy. Part B For what beliefs about Player 1’s strategy is m a best response for Player 2? We can answer this question at two different levels. The intuitive one: in pure strategies, m is a best response strategy for Player 2 when he believes Player 1 is picking M . More rigorously, we look at mixed strategies and let p, q and (1 - p - q ) respectively be the probabilities that Player 2 attaches to Player 1 playing L, M, R. For m to be a best response, it must ensure to Player 2 a payoff at least as high as any other strategy. In other words, it must (weakly) dominate both l and r. To find the conditions under which m (weakly) dominates both l and r, we have to solve the following system of inequalities: 3 p + 7 q + 3(1 - p - q ) 6 p + 4 q + (1 - p - q ) [m dominates l] 3 p + 7 q + 3(1 - p - q ) p + 4 q + 6(1 - p - q ) [m dominates r] With a bit of algebra, this yields q + 2 5 p [m dominates l] 5 p 3 - 6 q [m dominates r] Under any belief satisfying both conditions, m a best response for Player 2. We’ve already notice that in pure strategies, m is a best response strategy for Player 2 when he believes Player 1 is picking M . Note that this belief, corresponding to q = 1 , p = 0 satisfies both conditions: 1 + 2 0 and 0 3 - 6 · 1 . 1
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Game Theory Solutions to P. Set 2 For what beliefs about Player 2’s strategy is M a best response for Player 1?
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