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HW5 - Microeconomic Theory II Assignment 5 Game Theory...

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Microeconomic Theory II Assignment 5: Game Theory Solutions March 10, 2007 Problem 1. [MWG 8.D.3] Consider a first-price sealed-bid auction of an object with two bidders. Each bidder i ’s valuation of the object is v i , which is known to both bidders. Each bidder submits a bid in a sealed envelope. The envelopes are opened, and the bidder who submitted the highest bid gets the object and pays the auctioneer the amount he bid. If both bidders submit the same bid, each gets the object with probability 1 2 . Bids and values are in multiples of dollars. (a) Are any strategies strictly dominated? (b) Are any strategies weakly dominated? (c) Is there a Nash equilibrium? Is it unique? Solution . This is a simultaneous move game with complete information. Let b 1 be the bid of player 1 and b 2 the bid of player 2. The payoffs for each player are u i ( b 1 , b 2 ) = v i b i if b i > b j v i b i 2 if b i = b j 0 if b i < b j i = 1 , 2 j negationslash = i. (a) Let’s assume that for some player i there exists b i which strictly dominates b i , i.e. such that u i ( b i , b j ) > u i ( b i , b j ) for all b j 0. But then, clearly, letting b j = b i + k for some k N we have that b j > b i in which case, u i ( b i , b j ) = 0 = u i ( b i , b j ), which is a contradiction. thus, there are no strictly dominated strategies. * Prepared by ¨ Omer ¨ Ozak ([email protected]), Department of Economics, Brown University. If you find any typos or mistakes please let me know, so that it can be fixed. 1

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Microeconomic Theory II Solutions (b) Let’s show that there exist weakly dominated strategies. If b i > v i , then if bidder i wins, u i ( b i , b j ) < 0 = u i ( v i , b j ), if he looses u i ( b i , b j ) = 0 = u i ( v i , b j ), and if he ties u i ( b i , b j ) < 0 = u i ( v i , b j ). Thus, for any bidder i , all strategies b i > v i are weakly dominated by v i . If v i > v j then for bidder i bidding b i = v j weakly dominates bidding b i = v i , since u i ( v i , b j ) = 0 for all b j , while u i ( v j , b j ) > 0 for all b j v j and u i ( v j , b j ) = 0 for all b j > v j . One can find other ones, but we will not pursue this here. (c) The best-response correspondence for bidder i against bid b j of the other bidder is R i ( b j ) = b j + 1 if b j < v i 2 { b j , b j + 1 } if b j = v i 2 b j if b j = v i 1 { 0 , 1 , . . ., b j } if b j = v i { 0 , 1 , . . ., b j 1 } if b j > v i In equilibrium no bidder will bid b i > v i since that gives him negative expected utility. So b i v i . There are many cases to consider when looking for Nash equilibria, for example: (i) v i = v j 2: Then { ( v i , v i ) , ( v i 1 , v i 1) , ( v i 2 , v i 2) } are NE. (ii) v i = v j + 1 2: Then { ( v j , v j ) , ( v j , v j 1) , ( v j 1 , v j 1) } are NE.
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