Lecture notes: Auction models 11 Theory (primer)An auction is a game of in complete information. Assume that there are N players, orbidders, indexed by i = 1, . . . , N. There are two fu ndamental random elements in anyauction model.• Bidders’ private signals X1, . . . , XN.• Bidders’ utilities: ui(Xi, X−i), where X−i≡ {X1, . . . , Xi−1, Xi+1, . . . , XN}, the vec-tor of signals excluding bidder i’s signal. Since signals are private, Vi≡ ui(Xi, X−i) isa random variable from all bidders’ point of view. In what follows, we will also referto bidder i’s (random) utility as her valuation.Differing assumptions on the f orm of bidders’ utility function lead to an important distinc-tion:• Private value mo del: Vi= Xi, ∀i. Each bidd er knows his own valuation, but notthat of his rivals.1• (Pure) common value model: Vi= V, ∀i, where V is in tur n a rand om variablefrom all bidders’ point of view, and bidders’ signals are to be interpreted as their noisyestimates of the true but known common value V . Therefore, signals will generallynot be independent when common values are involved.• More generally a common value model arises when ui(Xi, X−i) is functionally depen-dent on X−i. In CV models, each bidder does not know his valuation with certainty,but only has a noisy signal of it.Examples:• Symmetric independent private values (IPV) mo del: Xi∼ F , i.i.d. across all bid-ders i, and Vi= Xi. Therefore, F (X1, . . . , XN) = F (X1) ∗ F (X2) · · · ∗ F (XN), andF (V1, X1, . . . , VN, XN) =Qi[F (Xi)]2.• Conditional independent model: signals are independent, conditional on a commoncomponent V . Vi= V, ∀i, but F (V, X1, . . . , XN) = F (V )QiF (Xi|V ).1More generally, in a private value model, ui(Xi, X−i) is restricted to be a function only of Xi.1Lecture notes: Auction models 2Models also differ depending on the auction rules:• First-price auction: the object is awarded to the highest bidder, at her bid.• Second-price auction: awards the object to the highest bidder, b ut she pays a priceequal to the bid of the second-highest bidder. (Sometimes second-price auctions arealso called “Vickrey” auctions, after the late Nob el laureate William Vickrey.)• In an English or ascending auction, the price the raised continuously by the auc-tioneer, and the winner is the last bidder to remain, and he pays an amount equ al tothe price at which all of his rivals have dropped out of the auction.• In a Dutch auction, the price is lowered continuously by the auctioneer, and thewinner is the first bidder to agree to pay any price.1.1 Equilibrium biddingIn discussing equilibrium bidding in the different auction models, we will focus on the generalsymmetric affiliated model, used in the seminal p aper of Milgrom and Weber (1982). Theassumptions made in this model are:• Vi= ui(Xi, X−i)• Symmetry: the joint distribution function F (V1, X1, . . . , VN, XN) is symmetric (i.e.,exchangeable) in the indices i so that, for example, F (VN, XN, . . . , V1, X1) = F (V1, X1, . . . , VN, XN).• The random variables V1, . . . , VN, X1, . . . , XNare affiliated. Consider a joint distri-bution function F (Z1, . . . , ZM), and let~Z ≡ (Z1, . . . , ZM) and~Z∗≡ (Z∗1, . . . , Z∗M)denote two independent draws from this distribution. Let¯Z and Zdenote, respec-tively, the component-wise maximum and m inimum. Then we say that Z1, . . . , ZMare affiliated if F (¯Z)F (Z) ≥ F (Z1, . . . , ZM)F (Z∗1, . . . , Z∗M). In other words, largevalues for some of the variables make large values for the other variables most likely.Some useful implications of affiliation:• E[Z1|Z2] ≥ 0.• Let Yi≡ maxj6=iXj, the highest of the signals observed by bidder i’s rivals. Givenaffiliation, the conditional expectation E[Vi|Xi, Yi] is increasing in both Xiand Yi.2Lecture notes: Auction models 3• Rules out negative correlation between bidders’ valuations.Winner’s curse Another consequence of affiliation is the winner’s curse, which is justthe fact thatE[Vi|Xi] ≥ E[Vi|Xi> Yi]where the conditioning event in the second expectation (Xi> Yi) is the event of winningthe auction.To see this, note that2E[Vi|Xi] = EX−i|XiE [Vi|Xi; X−i] =Z· · ·Z|{z }N−1E [Vi|Xi; X−i] F (dX1, . . . , dXi−1, dXi+1, . . . , dXN|Xi)≥ZXi· · ·ZXi|{z }N−1E [Vi|Xi; X−i] F (dX1dXi−1, dXi+1, . . . , dXN|Xi)= E [Vi|Xi> Xj, j 6= i] = E [Vi|Xi> Yi] .In other words, if bidder i “naively” bids E [Vi|Xi], her expected payoff from a first-priceauction is negative for every Xi.2Law of iterated expectation:Ex =Zxf(x)dx=Zx»Zf(x, y)dy–dx=Zx»Zf(x|y)f(y)dy–dx=Z»Zxf(x|y)dx–f(y)dy=ZE[x|y]f(y)dy = EyE[x|y].3Lecture notes: Auction models 40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.10.20.30.40.50.60.70.80.91valuationbid naivesophisticatedIn equilibrium, therefore, rational bidders should “shade down” their bids by a factor toaccount for the winner’s curse. This winner’s cur se intuition arises in many non-auctionsettings also. For example, in two-sided markets where traders have private signals aboutunknown fundamental value of the asset, the ability to consummate a trade is “good news”for sellers, but “bad news” for buyers, imp lying that, without ex-ante gains from trade,traders may not be able to settle on a market-clearing price. The result is the famous“lemons” result by Akerlof (1970), as well as a version of the “no-trade” theorem in Milgromand Stokey (1982). Glosten and Milgrom (1985) apply the same intuition to explain bid-askspreads in financial markets.Next, we cover the fir st- and second-pr ice auctions in some detail.4Lecture notes: Auction models 51.2 Second-price auctionsLet b∗(x) denote the equilibrium bidding strategy (which maps each bidder’s private infor-mation to his bid). Assume it is monotonic. Next we derive the functional form of thisequilibrium strategy.Given monotonicity, the price that bidder i will pay (if he wins) is b∗(Yi): the bid su bmittedby his closest rivals. He only wins when his bid b < b∗(Yi). Th erefore, his expected profitfrom participating in the auction w ith a bid b and a signal Xi= x is:EYi[(Vi− b∗(Yi)) 1 (b∗(Yi) < b) |Xi= x]=EYi[(Vi− b∗(Yi)) 1 (Yi< Xi) |Xi= x]=EYi|XiE [(Vi− b∗(Yi)) 1 (Yi< Xi) |Xi= x, Yi]=EYi|Xi[(E(Vi|Xi, Yi) − b∗(Yi)) 1 (Yi< Xi)]≡EYi|Xi[(v(Xi, Yi) − b∗(Yi)) 1 (Yi< Xi)]=ZXi−∞(v(x, Yi) − b∗(Yi)) f (Yi|Xi= x) .(1)In equilibrium, bidder i also