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Unformatted text preview: MS&E 246: Game Theory with Engineering Applications Feryal Erhun Winter, 2010 Problem Set # 4 Solutions 1. Induction Hypothesis: If there are exactly k white hats, people would raise their hands for the first time after the observer announces the number k and the people raising their hands would have white hats. Basis: If there had been only one white hat, the person with the white hat would have realized it right after the observer’s announcement that there is at least one white hat, because observing other people’s hats, he would immediately know that the white hat is his. Inductive Step: Now suppose there are exactly k + 1 white hats. Then, a person with a white hat would know of exactly k people with white hats. Since the people with white hats that he knows of did not raise their hands when the observer announced k , he realizes that there must be k +1 white hats. So he can only exactly know that he has a white hat at this point and he raises his hand when the observer announces k + 1. 2. (a) There exists a unique equilibrium where: if player 1 is selected as a proposer, he offers (1 , 0); if player 2 is selected as a proposer, he offers (0 , 1); and any offer is accepted. The resulting expected payoffs are p for player 1 and 1- p for player 2. (b) Since the game is finitely repeated, if they fail to agree at the end of stage 0, they will implement the solution of (a) in stage 1. Thus if they fail to agree in stage 0, they will receive (expected) payoffs ( δp,δ (1- p )). Given this, in the unique SPNE with n = 2, if player 1 is selected as proposer at stage 0 he will propose (1- δ (1- p ) ,δ (1- p )), and player 2 will accept any offer where his share is at least δ (1- p ); if player 2 is selected as the proposer he will propose ( δp, 1- δp ), and player 1 will accept any offer where...
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- Winter '07
- Game Theory, player, payoﬀ