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Lecture 16 1 Two-Player First-Price Auction Now, it is time to get back to auctions! Auctions are much more fun with the notion of types, and that's why auctions show up as main actors from the Bayesian model on! Let's now consider rst-price auctions, where the highest bidder gains the item but pays her own bid, so that the winner's payoff equals her valuation for the item minus her bid, while the others get 0. [In case of a match, the winner is randomly decided.) Note that paying her own bid instead of the secondhighest one as in the secondprice auction (whose Nash Equilibrium is also a Bayesian Nash Equilibrium if all valuations get private) changes a lot the game dynamics, since bidding her own valuation is not anymore interesting in this scenario. Consider a two-player rst-price sealed-bid auction in which the bidders' valuations on a good are independently and uniformly distributed on [0,1]. The bids 5.,- 2 U with 1 S 3' S 2 are submitted simultaneously. Bidder 3' has valuation 1),; for the good, that is, if bidder 2' wins the auction she gets the good but pays her own bid 33,-, earning 1),; b, as payoff while the other bids get 0 as payoff. In case of a tie, the winner is distributed uniformly at random. Would that be possible to nd a symmetric Bayesian Nash Equilibrium linear on the valuation? Let's suppose that each bidder 3' bids 5,- : vic where c is a constant. Note that for 1 S i,j S 2 and 2' 7E 3', PH), 2 533-] = 0 given a value of 1),, and hence, player i's optimal bid solves = mbaxvg- 7 delbi > \"0ch =1,3X(( .mb) [w