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Unformatted text preview: MS&E 246: Game Theory with Engineering Applications Feryal Erhun Winter, 2010 Problem Set # 6 Due: March 11, 2010, 5:00 PM outside Terman 305 Reading. Gibbons, Section 3.2 and 3.3; and Mas-Colell, Whinston, and Green, Section 8.E. 1. (All-pay auction) Consider an all-pay auction where the types of players 1 and 2 are identically distributed, with a distribution on [0 , 1] that has a continuous density on [0 , 1] that is nonzero everywhere. (In the notation of lecture, 1 = 2 = .) In an all-pay auction, each player selects a bid from [0 , 1]. The highest bid still wins the item, but both players must pay exactly what they bid even if they do not win the item. (In the event that both players submit the same bid, the item is awarded to player 2.) (a) Define the auction: give the bid spaces of both players, and define the functions w ( b 1 ,b 2 ), p 1 ( b 1 ,b 2 ), and p 2 ( b 1 ,b 2 ). (b) We will search for a symmetric Bayes-Nash equilibrium, i.e., a function s : [0 , 1] [0 , ) such that s 1 = s 2 = s is a Bayes-Nash equilibrium. Assume we are given such a symmetric Bayes-Nash equilibrium. Using the revelation principle, construct a direct revelation mechanism (DRM) for which truthtelling is a Bayes-Nash equilibrium, and the outcomes are the same as those of the all-pay auction at the original symmetric Bayes-Nash equilibrium....
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