A_Course_in_Game_Theory_-_Martin_J._Osborne 33

A_Course_in_Game_Theory_-_Martin_J._Osborne 33 - Page 18...

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Page 18 The notion of a strategic game encompasses situations much more complex than those described in the last five examples. The following are representatives of three families of games that have been studied extensively: auctions, games of timing, and location games. • Example 18.1 ( An auction ) An object is to be assigned to a player in the set {1,. .., n } in exchange for a payment. Player i 's valuation of the object is v i , and v 1 > v 2 > . .. > v n > 0. The mechanism used to assign the object is a (sealed-bid) auction: the players simultaneously submit bids (nonnegative numbers), and the object is given to the player with the lowest index among those who submit the highest bid, in exchange for a payment. In a first price auction the payment that the winner makes is the price that he bids. • Exercise 18.2. Formulate a first price auction as a strategic game and analyze its Nash equilibria. In particular, show that in all equilibria player I obtains the object. In a
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