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Unformatted text preview: MS&E 246: Game Theory with Engineering Applications Feryal Erhun Winter, 2010 Problem Set # 6 Solutions 1. (All-pay auction) (30 points) (a) (5 points) The bid spaces are B i = [0 , 1] for i = 1 , 2. w ( b 1 ,b 2 ) is 1 if b 1 > b 2 and 0 otherwise. The payments are: p 1 ( b 1 ,b 2 ) = b 1 p 2 ( b 1 ,b 2 ) = b 2 (b) (5 points) Players choose proxy types v 1 , v 2 . The item is allocated to player 1 if s (v 1 ) > s (v 2 ), and to player 2 otherwise. Player i pays s (v i ) regardless of whether he wins or loses. (c) (5 points) The symmetric BNE theorem says that in any symmetric BNE of an auction where the highest bidder wins, the winner always has the highest valuation . Thus the probability that player i wins the auction when his type is v i is: P i ( v i ) = Z v i ( v- i ) dv- i = ( v i ) (d) (5 points) For this auction, S i (0) = 0, since a bidder of zero valuation will never bid anything positive in equilibrium. From the truthtelling lemma and part (c): S i ( v i ) = S i (0) + Z v i P i ( z ) dz = Z v i ( z ) dz (e) (5 points) When player i has type v i , the payment of player i is s ( v i ), and player i gets the object with probability P i ( v i ). Thus, the expected payoff of)....
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