ps6(2010)

ps6(2010) - MS&E 246 Game Theory with Engineering...

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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 Φ on [0 , 1] that has a continuous density φ on [0 , 1] that is nonzero everywhere. (In the notation of lecture, φ 1 = φ 2 = φ .) 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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ps6(2010) - MS&E 246 Game Theory with Engineering...

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