Auction Theory

1 Auction TheoryJonathan LevinOctober 2004 Our next topic is auctions. Our objective will be to cover a few of themain ideas and highlights. Auction Theory can be approached from differentangles from the perspective of game Theory (auctions are bayesian gamesof incomplete information), contract or mechanism design Theory (auctionsare allocation mechanisms), market microstructure (auctions are models ofprice formation), as well as in the context of different applications (procure-ment, patent licensing, publicfinance, etc.). We re going to take a relativelygame-theoretic approach, but some of this richness should be The Independent Private Value (IPV) A ModelThe basic Auction environment consists of: Biddersi=1,..,n Oneobjecttobesold Bidderiobserves a signal Si F( ), with typical realizationsi [s,s], and assumeFis continuous. Bidders signalsS1,..,Snare independent.

2 Bidderi s valuevi(si)= this basic set-up, specifying a set of Auction rules will give riseto a game between the bidders. Before going on, observe two features ofthe model that turn out to be important. First, bidderi s information (hersignal) isindependentof bidderj s information. Second, bidderi s value isindependent of bidderj s information so bidderj s information isprivatein the sense that it doesn t affect anyone else s Vickrey (Second-Price) AuctionIn a Vickrey, or second price, Auction , bidders are asked to submit sealedbidsb1,..,bn. The bidder who submits the highest bid is awarded the object,and pays the amount of the second highest 1In a second price Auction , it is a weakly dominant strategyto bid one s value,bi(si)= s value issi, and she considers biddingbi> bdenote the highest bid of the other biddersj6=i(fromi s perspective this isa random variable).

3 There are three possible outcomes fromi s perspective:(i) b>bi,si;(ii)bi> b>si; or (iii)bi,si> b. In the event of thefirst orthird outcome,iwould have done equally well to bidsirather thanbi> (i) she won t win regardless, and in (ii) she will win, and will pay bregardless. However, in case (ii),iwill win and pay more than her value ifshe bids b, something that won t happen if she ,idoes betterto bidsithanbi>si. A similar argument shows thatialso does better tobidsithan to bidbi< each bidder will bid their value, the seller s revenue (the amountpaid in equilibrium) will be equal to the second highest value. LetSi:ndenote theith highest ofndraws from distributionF(soSi:nis a randomvariable with typical realizationsi:n). Then the seller s expected revenue isE S2:n .The truthful equilibrium described in Proposition 1 is the unique sym-metric Bayesian Nash equilibrium of the second price Auction .

4 There are alsoasymmetric equilibria that involve players using weakly dominated strate-gies. One such equilibrium is for some playerito bidbi(si)=vand all theother players to bidbj(sj)= Vickrey auctions are not used very often in practice, open as-cending (or English) auctions are used frequently. One way to model suchauctions is to assume that the price rises continuously from zero and playerseach can push a button to drop out of the bidding. In an independent pri-vate values setting, the Nash equilibria of the English Auction are the sameas the Nash equilibria of the Vickrey Auction . In particular, the unique sym-metric equilibrium (or unique sequential equilibrium) of the English auctionhas each bidder drop out when the price reaches his value. In equilibrium,the Auction ends when the bidder with the second-highest value drops out,so the winner pays an amount equal to the second highest Sealed Bid (First-Price) AuctionIn a sealed bid, orfirst price, Auction , bidders submit sealed bidsb1.

5 , bidders who submits the highest bid is awarded the object, and payshis these rules, it should be clear that bidders will not want to bidtheir true values. By doing so, they would ensure a zero profit. By biddingsomewhat below their values, they can potentially make a profitsomeofthetime. We now consider two approaches to solving for symmetric equlibriumbidding The First Order Conditions ApproachWe will look for an equilibrium where each bidder uses a bid strategythat is a strictly increasing, continuous, and differentiable function of do this, suppose that biddersj6=iuse identical bidding strategiesbj=b(sj) with these properties and consider the problem facing s expected payoff, as a function of his bidbiand signalsiis:U(bi,si)=(si bi) Pr [bj=b(Sj) bi, j6=i]Thus, bidderichoosesbto solve:maxbi(si bi)Fn 1 b 1(bi) .Thefirst order condition is:(si bi)(n 1)Fn 2 b 1(bi) f b 1(bi) 1b0(b 1(bi)) Fn 1 b 1(bi) =0At a symmetric equilibrium,bi=b(si), so thefirst order condition reducesto a differential equation (here I ll drop theisubscript):b0(s)=(s b(s)) (n 1)f(s)F(s).

6 This can be solved, using the boundary condition thatb(s)=s, to obtain:b(s)=s RsisFn 1( s)d sFn 1(s).1In fact, it is possible to prove that in any symmetric equilibrium each biddermustusea continuous and strictly increasing strategy. To prove this, one shows that in equilbriumthere cannot be a gap in the range of bids offered in equilibrium (because then it wouldbe sub-optimal to offer the bid just above the gap) and there cannot be an atom in theequilibrium distribution of bids (because then no bidder would make an offer just belowthe atom, leading to a gap). I ll skip the details is easy to check thatb(s) is increasing and differentiable. So any symmetricequilibrium with these properties must involve bidders using the strategyb(s).B. The Envelope Theorem ApproachA closely related, and often convenient, approach to identify necessaryconditions for a symmetric equilibrium is to exploit the envelope this end, supposeb(s) is a symmetric equilibrium in increasing dif-ferentiable strategies.

7 Theni s equilibrium payoffgiven signalsiisU(si)=(si b(si))Fn 1(si).(1)Alternatively, becauseiis playing a best-response in equilibrium:U(si)=maxbi(si bi)Fn 1(b 1(bi)).Applying the envelope theorem (Milgrom and Segal, 2002), we have:ddsU(s) s=si=Fn 1(b 1(b(si)) =Fn 1(si)and also,U(si)=U(s)+ZsisFn 1( s)d s.(2)Asb(s) is increasing, a bidder with signalswill never win the Auction therefore,U(s)= (1) and (2), we solve for the equilibrium strategy (again drop-ping theisubscript):b(s)=s RsisFn 1( s)d sFn 1(s).Again, we have showed necessary conditions for an equilibrium ( anyincreasing differentiable symmetric equilibrium must involve the strategyb(s)). To check sufficiency (thatb(s) actually is an equilibrium), we canexploit the fact thatb(s) is increasing and satisfies the envelope formulato show that it must be a selection fromi s best response given the otherbidder s use the strategyb(s).)

8 (For details, see Milgrom 2004, Theorems ).4 Remark 1In most Auction models, both thefirst order conditions and theenvelope approach can be used to characterize an equilibrium. The trick istofigure out which is more is the revenue from thefirst price Auction ? It is the expectedwinning bid, or the expected bid of the bidder with the highest signal,E b(S1:n) .Tosharpenthis,defineG(s)=Fn 1(s). ThenGis the proba-bility that if you taken 1draws fromF, all will be belows( it is thecdf ofS1:n 1). Then,b(s)=s RssFn 1( s)d sFn 1(s)=1Fn 1(s)Zss sdFn 1( s)=E S1:n 1|S1:n 1 s .That is, if a bidder has signals, he sets his bid equal to the expectation ofthehighestoftheothern 1 values, conditional on all those values beingless than his this fact, the expected revenue is:E b S1:n =E S1:n 1|S1:n 1 S1:n =E S2:n ,equal to the expectation of the second highest value. We have shown:Proposition 2 Thefirst and second price Auction yield the same revenuein Revenue EquivalenceThe result above is a special case of the celebrated revenue equivalence the-orem due to Vickrey (1961), Myerson (1981), Riley and Samuelson (1981)and Harris and Raviv (1981).

9 Theorem 1(Revenue Equivalence) Supposenbidders have valuess1,..,snidentically and independently distributed with cdfF( ). Then all auctionmechanisms that (i) always award the object to the bidder with highest valuein equilibrium, and (ii) give a bidder with valuationszero profits, generatesthe same revenue in consider the general class of auctions where bidders submit bidsb1,..,bn. An Auction rule specifies for alli,xi:B1 .. Bn [0,1]ti:B1 .. Bn R,5wherexi( ) gives the probabilityiwill get the object andti( )givesi srequired payment as a function of the bids (b1,..,bn).2 Given the Auction rule, bidderi s expected payoffas a function of hissignal and bid is:Ui(si,bi)=siEb i[xi(bi,b i)] Eb i[ti(bi,b i)] .Letbi( ),b i( ) denote an equilibrium of the Auction game. Bidderi sequi-libriumpayoffis:Ui(si)=Ui(si,b(si) ) =siFn 1(si) Es i[ti(bi(si),b i(s i)] ,whereweuse(i)to writeEs i[xi(b(si),b(s i))] =Fn 1(si).)

10 Using the fact thatb(si) must maximizei s payoffgivensiand opponentstrategiesb i( ), the envelope theorem implies that:ddsUi(s) s=si=Eb i[xi(bi(si),b i(s i))] =Fn 1(si),and alsoUi(si)=Ui(s)+ZsisFn 1( s)d s=ZsisFn 1( s)d s,whereweuse(ii)to writeUi(s)= our expressions forUi(si), we get bidderi s expected paymentgiven his signal:Es i[ti(bi,b i)] =siFn 1(si) ZsisFn 1( s)d s=Zsis sdFn 1( s),where the last equality is from integration by parts. Sincexi( )doesnotenter into this expression, bidderi s expected equilibrium payment given hissignal is thesameunder all Auction rules that satisfy(i)and(ii). Indeed,i s expected payment givensiis equal to:E S1:n 1|S1:n 1<si =E S2:n|S1:n=si .So the seller s revenue is:E[Revenue] =nEsi[i s expected payment|si]=E S2:n ,2So in afirst price Auction ,x1(b1,..,bn)equals1ifb1is the highest bid, and otherwisezero. Meanwhilet1(b1.)