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 <1) 111,X(( we? m,x(( webg) =max( if}? j) By taking the first order condition, we get: 0 = Vi C which is equivalent to bi = Vi, meaning that by setting c = ; so that each bidder bids , times its own valuation, we achieve a Bayesian Nash Equilibrium! 2 Cournot Duopoly with Incomplete Information Now for a market game consider a Cournot duopoly operating in a market with inverse demand P(Q) = a - Q, where Q = q1 + 92 is the aggregate quantity on the market. Both firms have total costs ci(qi) = cqi, but the cost is uncertain: it is high (c = CH) with probability y and low (c = CL) with probability 1 -7. Furthermore, information is asymmetric: firm 1 knows whether the cost is high or low, but firm 2 does not. All of this is common knowledge. The two firms simultaneously choose quantities. Let's calculate the pure-strategy Bayesian Nash equilibrium for this game of incomplete information! The payoff for player 1 is w1(q1, 92, CH) = (a - (q1 + q2) - CH)q1 if c = CH and U1 (91, 92, CL) = (a - (q1 + 92) - CL)q1 if c = CL. If c = CH and for a fixed q2, it is maximized by 91 (CH) = =(a - q2 - CH). Analogously, 91 (CL) = = (a - q2 - CL). Player 2 has expected payoff Y(a - (91 + 92) - CH)q2 + (1 - 7)(a - (91 + 92) - CL)q2 = (a - (91 + 92) - 7CH - (1 - 7)CL) q2, which is maximized for fixed qf = vqi (CH) + (1 -v)qi(CL) by 92 = (a - YCH - (1 - 7)CL - q1) = 5(a - YCH - (1 - V)cp) - 2 ((a - 92 - ca) + 1,2(a- q2 -CL) ) = a - YCH - (1 - 7)CL + 92 4 This implies 92 = a - YCH - (1 - V)CL _ a - YCH - (1 - 7)CL (1) 3 2qi (CH) a _ a - YCH - (1 - 7)CL 2a + (7 - 3)CH + (1 - 7)CL 3 CH) (2 ) 6 and qi(CL.) a a - YCH - (1 - 7)CL - CL ) = 2a + YCH + (-2 - 7)CL (3) 3 6 92, q1(CH), and q1(CL) from (1), (2), and (3) constitute the unique pure-strategy Bayesian Nash equilibrium of this game. Next class, more on auctions! Stay tuned!2. Consider an nplayer rst-price sealed-bid auction in which the bidders' valuations on a good are independently and uniformly distributed on [0,1]. The bids bi 2 0 with 1 S i S n are submitted simultaneously. Bidder 2' has valuation 1),; for the good, that is, if bidder 3' Wins the auction she gets the good but pas her own bid bi, earning 1).; bi as payoff While the other bids get 0 as payoff. In case of a tie, the Winner is distributed uniformly at random. Give a Bayesian Nash Equilibrium linear in the valuation. [5 points]