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PS4_212Fal10

PS4_212Fal10 - ECON 212 Game Theory Fall 2010 Prof Andrew...

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ECON 212 Game Theory Prof. Andrew Postlewaite Fall 2010 University of Pennsylvania Suggested Solution for Problem Set 4 1. a. There are essentially two states: G in which ( B, B ) is expected to be played and B in which ( C, C ) is expected. Let V i , i = G, B be the expected payoff in a repeated game each player expects from the prescribed strategy in state i (Note that both players get the same expected payoff according to the strategy profile). Then V G = (1 - δ )4 + δV G = 4 V B = (1 - δ )( - 1) + δV G = 5 δ - 1 . For the given strategy profile to be a SPNE, no player should have an incentive to deviate. The strategy profile is incentive compatible in state G iff V G (1 - δ )5 + δV B ( deviation to A ) V G (1 - δ )1 + δV B ( deviation to C ) Solving these inequalities, we end up with δ 1 5 . The strategy profile is incentive compatible in state B iff V B (1 - δ )0 + δV B ( deviation to either A or B ) δ 1 5 We conclude that the strategy profile is a SPNE if δ 1 5 . b. ( A, A ) is the unique strategy-game NE. Since this is a finitely repeated game, always playing ( A, A ) is the unique SPNE. c. Consider the following strategy profile: each player plays A in the first stage. If ( A, A ) was played in the first stage, then play A . Otherwise, play C . Ob- viously, this is not a SPNE because ( C, C ) is not a stage-game NE. Incentive compatibility in both periods are immediate.
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