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Unformatted text preview: ECE 496 SOLUTIONS TO HOMEWORK ASSIGNMENT V Spring 2007 1. Suppose the players play a total of T stages. If you are Player 1, and Player 2 plays the proposed strategy, can you do better by playing a different strategy? Any such different strategy would result in a payoff to you of 1 on any stage(s) on which you deviate from the proposed strategy. If you stuck with the proposed strategy, you would collect a payoff of either 2 or 3 on every stage. Accordingly, if Player 2 plays as proposed, then if you deviate from the proposed strategy youll definitely collect a lower payoff than if you stay with the proposed strategy. Similar reasoning applies if youre Player 2. It doesnt matter, by the way, whether T is odd or even. The bottom line: the strategy profile in the problem statement is a Nash equilibrium. 2. (a) For each i ( i = 1 or 2) u i ( D,C ) = 5 > 3 = u i ( C,C ) and u i ( D,D ) = 1 > 0 = u i ( C,D ) . so playing D is strictly dominant for each player. (b) Suppose ( a 1 , a 2 ,..., a n ) is a PSNE profile in some n-player game. Suppose, for some i , action a i is strictly dominated by some other action a i for Player i . Then u i ( a 1 ,..., a i- 1 ,a i , a i +1 ,... a n ) > u i ( a 1 ,..., a i- 1 , a i , a i +1 ,... a n ) , so Player i could do better than to play a i against the other Players hatted actions. Conclusion: ( a 1 , a 2 ,..., a n ) cant be a PSNE profile. (c) Without loss of generality, suppose Player 1 has some strictly dominant strategy a 1 . Then no matter what Player 2 plays, a 1 is a best reply for Player 1. So if Player 2 plays a best reply to a 1 call that best reply a 2 then ( a 1 , a 2 ) will be a PSNE profile. The idea: under that profile each player is playing a best reply to what the other player is playing, and thats the definition of a Nash equilibrium profile. 3. (a) Its pretty clear that no PSNE exists. To see this, suppose Player 1 plays S . Then Player 2s only best reply is R ; but S is not a best reply for Player 1 against R , so there exists no PSNE wherein Player 1 plays S . Similarly, if Player 1 plays P , then Player 2s only best reply is S , and P is not a best reply for Player 1 against S , so no PSNE exists wherein Player 1 plays P . If Player 1 plays R , then Player 2s only best reply is P , and R is not a best reply for Player 1 against P , so no PSNE exists wherein Player 1 plays R . Accordingly, any Nash equilibrium profile will feature a mixed strategy for at least one of the players. Suppose ( 1 , 2 ) is a Nash-equilibrium profile where at least one of the strategies is mixed. If 2 = q 1 S + q 2 P where q 1 + q 2 = 1 and both q 1 and q 2 are nonzero, then P is not a best reply for Player 1, and it follows that 1 = p 1 S + p 3 R 1 where p 1 + p 3 = 1. The only way to make P a best reply for Player 2, which it must be since 2 has P 2 in its support, is for p 3 = 0, which means Player 1 is playing...
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- Spring '07