Robust Bidding in First-Price Auctions:

Transcription

1 Robust Bidding in First-Pric Auctions: How to Bid without Knowing what Othrs ar Doing Brnhard Kasbrgr Karl Schlag Novmbr 8, 2016 Vry Prliminary and Incomplt Abstract Finding optimal bids in first-pric auctions in th classical framwork rquirs dtaild information and spcific assumptions about th othr biddrs valu distributions and bidding bhavior. This papr shows how to bid with lss information. A bidding rul is valuatd by comparing th payoff of th rul to th payoff that could b achivd if on knw th othr biddrs valu distributions and bidding functions. Robust bidding approximats th payoff undr mor information by minimizing th highst payoff diffrnc. W driv robust bidding ruls undr complt uncrtainty and for cass in which on imposs bounds on th bid or valu distributions of th othr biddrs. 1 Introduction W invstigat how to bid in a first-pric auction. Th ruls of th firstpric auction ar simpl and commonly known. 1 Yt, it is vry hard to bid optimally as lots of othr aspcts ar unknown. A biddr might b uncrtain about th numbr of othr biddrs, thir valu distribution, th way Kasbrgr: Vinna Graduat School of Economics and Dpartmnt of Economics, Univrsity of Vinna, brnhard.kasbrgr@univi.ac.at; Schlag: Dpartmnt of Economics, Univrsity of Vinna, karl.schlag@univi.ac.at 1 In a first-pric auction vry biddr submits a sald bid. Th biddr with th highst bid wins th objct and pays his or hr bid. Tis ar rsolvd randomly. 1

2 thy translat valus into bids, and thir blifs about othr biddrs valus and bhavior. First, w propos how to bid if no furthr assumptions on th valu distribution and bidding bhavior ar mad. Thn w introduc rstrictions on th anticipatd bid distribution. Finally, w formulat constraints on othr biddrs valu distributions and on thir bidding bhavior. Th spcific application will idntify whthr it is simplr to incorporat knowldg and prcptions of th nvironmnt by spcifying possibl bid distributions or by spcifying possibl valus and bidding bhavior. This is th first papr that maks thory basd xplicit rcommndations whn th valu distribution is unknown and stratgic uncrtainty is not rsolvd in quilibrium. Each of our rcommndations coms togthr with an rror bound that spcifis an uppr bound on how much mor surplus on could gt if on knw th joint valu distribution and bidding bhavior. Intrstingly, w can mak a rcommndation vn for th cas in which on dos not wish to mak any assumptions about valus and/or bidding bhavior of othrs. In this cas, it is as if w ar comparing th payoff to th cas whr w would know th bids of all othrs. Th rror bound of our proposd randomizd bidding stratgy is 36% of th own valu v. In contrast, th rror bound of a slightly misspcifid Nash quilibrium bidding function can b 100% of th own valu. Our objctiv is to driv bidding functions that minimiz th rror bound for a givn st of concivabl nvironmnts. An nvironmnt is a pair that gnrats th facd bid distribution and consists of th joint valu distribution and th othr biddrs bidding functions, that is thir ruls that transform valus into bids. Th loss for a givn nvironmnt is th diffrnc btwn th hypothtical oracl payoff, that is th highst possibl surplus if th nvironmnt is known, and th surplus gnratd by th bidding function. Th rror bound is th maximal loss; bidding bhavior is chosn to minimiz this maximal loss. W dvlop two approachs to rduc th rror bound obtaind undr complt uncrtainty. A first stp is to think about th driving forcs of th statd rsult. If all nvironmnts ar possibl, thn on has to dal with th cas in which all othrs bid 0 for sur, which gnrats a larg loss for bidding high. An immdiat raction is that this situation is not vry likly. W addrss this in two diffrnt ways, by imposing lowr bounds 2

3 and by imposing variability of bids. First, on can imagin that on has nough information about th contxt that on is willing to rul out that th maximal bid of th othrs will b blow som thrshold L. In this cas w provid a bidding rul that guarants a loss blow 36% of (v L). In particular this mans that if on dos not xpct that th maximal bid is blow 73% of on s own valu thn th rror bound will b at most 10% of v. Howvr, a thrshold L for which on is willing to rul out with crtainty that th maximal bid will not b blow L may b vry small. Clarly, on could choos such a thrshold largr if on is allowd to assign som maximal probability that th maximal bid is blow L. W find that th 10% of v rror bound can b sustaind if on assigns at most 12% to th possibility that th maximal bid will b blow 73% of v. Th rason why th rror bound is smallr is bcaus th constraint on th probability of th maximal bid bing blow L must also hold for th tru nvironmnt. Scond, w considr bidding whn on adds som assumptions about th bids abov L, maintaining th assumption that th maximal bid will not b blow L. Ths assumptions will concrn both indpndnc of bidding and th variability of bids. Indpndntly for ach othr biddr, with probability ε, this biddr is blivd to bid btwn L and v with ach bid bing qually likly. With probability 1 ε that biddr can do anything as long as sh bids abov L. So som biddrs ar blivd to bid indpndntly and uniformly on [L, v], whil no assumptions ar mad on what th othr biddrs do. Hnc, it is not possibl that all biddrs bid L with crtainty. W find, for instanc, if ε is at last 0.15 and thr ar 10 biddrs and L = 0 (or thr ar 5 biddrs and L = v/2) that loss is boundd abov by 10% of valu v. Th paramtr ε masurs th wight on th particular blifs and influncs th variability of bids that ar xpctd. Bounds on th valu distribution and on bidding bhavior can furthr improv th rror bound. A bound ithr on th valu distribution or on bidding bhavior cannot rduc th uncrtainty sufficintly much to rduc maximal loss. Th combination, howvr, can produc sharp rsults. W invstigat th cas in which th valu distribution can b boundd abov so that vry low valus cannot b drawn with crtainty, and th othr biddrs 3

4 bidding functions can b boundd from blow so that not all biddrs bid vry low with crtainty. In this cas, a linar bidding function can bound th rror from abov. Rmarkably, this bidding function only dpnds on th paramtr usd to bound th valu distribution and on th numbr of biddrs. What is mor, if othr biddrs ar prcivd to us th sam linar bidding function, this bidding function rmains to prform wll. Thus, bhavior forms an ɛ-loss-quilibrium, that is similar in spirit to ɛ-nash quilibrium. Th advantag of this solution concpt is that it is applicabl to vry complicatd sttings in which (stratgic) uncrtainty is prsnt. Rlatd Litratur Minimizing th maximal loss is a concpt introducd by Savag (1951) for dcision problms and was subsquntly usd in th litratur on minimax rgrt and robust statistics. To start with th lattr, Hubr (1965, 1981) introducs a loss function to driv robust tst statistics in slightly misspcifid nvironmnts. In this papr, w do not rstrict ourslvs to slightly misspcifid nvironmnts, but considr mor arbitrary sts of concivabl nvironmnts. Th litratur on minimax rgrt (.g. Hayashi, 2008) has, for xampl, lookd at th pricing problm of a monopolist (Brgmann and Schlag, 2008, 2011), and dynamically consistnt robust sarch ruls (Schlag and Zapchlnyuk, 2016). In stratgic sttings, Linhard and Radnr (1989) considr bargaining, and Rnou and Schlag (2011) and Halprn and Pass (2012) dvlop solution concpts for gams whr it is common knowldg all playrs follow minimax rgrt. In th last sction of our papr w also mak rcommndations in an quilibrium styl framwork, but rmain lss prcis in trms of anticipatd blifs and bidding bhavior of othrs. Som of th paprs ar discussd in mor dpth in th txt. W talk about loss and not rgrt bcaus, first, our valuation of prformanc has no bhavioral contxt as th trm rgrt might suggst, and scond, bcaus in th contxt of auctions th trm rgrt is usd diffrntly by Englbrcht-Wiggans (1989). In this litratur (.g. Filiz- Özbay and Özbay, 2007; Englbrcht-Wiggans and Katok, 2008) th valu distributions is known and th objctiv is th maximization of xpctd utility plus additiv rgrt trms. Rgrt ariss from larning othr bids. 4

5 In our stting, th bnchmark of larning th nvironmnt of valu distribution and bidding function is purly hypothtical and thr is no additiv bhavioral motiv. This bnchmark is introducd to masur th compromis that a biddr who blivs to b bttr informd has to undrgo whn following our rcommndations. Our approach should not b confusd with rcnt litratur on robust dcision making (.g. Carroll, 2015) that borrowd th trm robust (Hubr, 1981) for a diffrnt contxt. Common is th objctiv to find a rul for making dcisions without a prior. Howvr, in this altrnativ strand of litratur th bnchmark to b clos to th optimal policy is droppd. Instad, prfrncs ar introducd for how to dal with multipl priors without conncting this to th optimal policy. Th plausibility of th rul dpnds also on how plausibl ths altrnativ prfrncs ar. Stratgic uncrtainty is rsolvd in quilibrium. In th contxt of th first-pric auction, Lo (1998) and Chn t al. (2007) prsnt modls in which symmtric biddrs hav a st of priors ovr th valu distribution. In th first papr, biddrs hav maxmin prfrncs. In th scond papr, biddrs prfrncs ar a gnralization of maximin xpctd utility that allows ambiguity loving. Lvin and Ozdnorn (2004) study auctions with an uncrtain numbr of biddrs who hav maximin prfrncs. Intrstingly, xprimntal rsults by Güth and Ivanova-Stnzl (2003) and Chn t al. (2007) indicat that bidding bhavior is vry similar with and without priors. Thr ar paprs that study rationalizabl bids in first-pric auctions in which biddrs hav a prior ovr th valu distribution. Battigalli and Siniscalchi (2003) show that all nonzro bids blow, and som bids abov quilibrium ar rationalizabl. For sufficintly many biddrs, rationalizabl bids ar vry clos to valu (Dkl and Wolinsky, 2003; Cho, 2005). Whn biddrs bliv that othrs valus ar only slightly lowr than th own valu, thn bidding clos to valu is rationalizabl (Robls and Shimoji, 2012). W nithr assum common knowldg of rationality, nor common knowldg of th blifs about th valu distribution. Othr approachs to complt-information gams with non-rational biddrs ar Eliaz (2002) and d Clippl (2014). Work on robust mchanism dsign, initiatd by Brgmann and Morris (2005), loosns th prior whil maintaining rationality assumptions. Brgmann t al. (2016) analyz 5

6 proprtis of quilibrium play in th first-pric auction that hold for all information structurs for a common prior ovr th joint valu distribution. Th formal mthodology usd for this study is dscribd in th nxt sction. Sction 3 prsnts simpl xampls of th non-robustnss of bidding functions that ar optimal in spcific nvironmnts undr xpctd and maximin utility. Th cor of th papr bgins in Sction 4. Various blifs about th bid distribution ar usd to driv bid rcommndations. In Sction 5 mor spcific blifs about th bidding bhavior and th valu distribution ar mad. 2 Mthodology W considr th bidding bhavior of biddr 1 in a first-pric sal bid auction for a singl indivisibl good. Biddr 1 has a valu v 1 R + for th auctiond objct and quasilinar prfrncs that hav a von Numann Morgnstrn xpctd utility rprsntation. Th utility of losing th auction is normalizd to zro. Winning th auction with bid b 1 yilds utility u 1 (v 1 b 1 ). Biddr 1 mploys th (possibly mixd) bidding function b 1 : R + R + that maps th own valu v 1 into (a distribution of) bids. Likwis, vry othr biddr i > 1 has a valu v i for th objct and bids according to a crtain bidding function. W will b mor spcific about th othr biddrs. Whn choosing thir bids, biddrs in first-pric auctions ar intrstd in th bid distribution of th othr biddrs. Th bid distribution is gnratd from two inputs. First, thr is a tru and xognous joint valu distribution F R n +, with n dnoting th numbr of biddrs. Scond, ach participant s bidding bhavior translats valus into bids. Th bidding bhavior can dpnd on th own valu and on information about othr biddrs. Traditional analysis of bidding bhavior in auctions assums that all biddrs know th valu distribution and othr biddrs risk attituds and that thr is common knowldg of rationality. On thn sarchs for an quilibrium in which ach biddr bst rsponds to th bidding bhavior of th othr biddrs. In short, vry biddr knows th valu distribution and, in quilibrium, th othr biddrs bidding stratgis. W dpart from th classic stting and invstigat how biddr 1 should 6

7 bid if sh is uncrtain both about th valu distribution and th bidding bhavior of th othrs. This uncrtainty is modld in th form of biddr 1 idntifying a st of concivabl nvironmnts E. An nvironmnt E E is a pair (F, b 1 ), whr F is a joint valu distribution and b 1 spcifis th bidding bhavior of othr biddrs. Hnc, an nvironmnt gnrats th bid distribution facd by biddr 1. What follows is th formal dscription of th two componnts of nvironmnts and th st of concivabl nvironmnts. Biddr 1 dos not know th tru F, but sh concivs that F blongs to th class of joint distributions F with F n N Rn +. In particular, on can incorporat uncrtainty ovr th numbr of biddrs n by including distributions with diffrnt numbr of biddrs in F. Complt uncrtainty about th othr s valus can b modld by F bing th st of all joint distributions. Altrnativly, th biddr might assrt that thr ar n biddrs with iid valus. Th st of possibl joint valu distributions F would thn b th st of all joint distributions that ar gnratd from som iid distributions and dnotd by F iid. Th st F plays a rol in th valuation of bidding functions. Biddrs might b uncrtain about th bidding functions of th othrs, just as thy ar uncrtain about th valu distribution. First, w formally introduc th bidding functions of othr biddrs. Thn w introduc sts of bidding functions that ar usd to modl biddr 1 s uncrtainty about th othr participants bidding bhavior. Any biddr i > 1 uss th (possibly mixd) bidding function b i : R k i + R + for som k i N, whr R + dnots th st of probability distributions ovr positiv rals. Dtrministic bidding functions ar b i : R k i + R +. Lt b 1 = (b 2,..., b n ) dnot th othr biddrs profil of bidding functions. For xampl, th othr biddrs can bid indpndntly, in which cas k i = 1 for all biddrs. Anothr possibility is that biddr 1 thinks that th othrs ar communicating or colluding, so k i > 1. Lt B F b th st of bidding functions that biddr 1 concivs that th othr biddrs us undr th joint distribution F. A gnric lmnt of B F is dnotd by b 1. Th biddr may no nothing about th bidding bhavior of othrs, in which cas B F = {(b 2,..., b n ) b i : R k i + R +, for all k i and 1 < i n}, with n givn by F. Sh might conciv that th othr biddrs us idntical linar bidding functions such 7

8 that thy do not submit ngativ bids and nvr bid abov valu, i.. B F = {(b 2,..., b n ) b i (v) = b j (v) = τv, for all τ [0, 1], i, j 1}. On can viw nvironmnts as having a purly xognous part F, and a potntially ndognous part b 1 that might dpnd on th bhavior of othr biddrs. Th biddr concivs that th nvironmnt sh facs blongs to E E, whr E = F F {F } B F is th univrs of all nvironmnts. Th st E is varid throughout th txt. Not that our modl about th biddr s blif about th facd nvironmnt is rich nough to includ th Bays-Nash framwork. Lt F b a joint distribution such that a Bays-Nash quilibrium xits and F = {F }. Lt b = (b 1..., b n ) b th quilibrium stratgy profil, i.. a stratgy profil such that for vry biddr i {1,..., n} th bidding function b i is a bst-rspons to th bidding profil b i. Thn E = {(F, b 1 )}, so th only concivabl nvironmnt within th Bays-Nash quilibrium framwork is (F, b 1 ). Similarly, on can modl th nvironmnt in which som othr biddrs ar in a Bays-Nash framwork. This approach is includd in Sction 5.3. Idally, biddr 1 slcts th bst bid givn th nvironmnt. W considr, howvr, a biddr who dos not know th nvironmnt and hnc cannot prform this task (i.. E > 1). In th following w prsnt our modl of how biddr 1 bids without knowing th nvironmnt. Th prformanc of a bidding function in a givn nvironmnt is masurd using a loss function. Th loss of biddr 1 conditional on hr valu v 1 is dfind as th diffrnc btwn what sh gts and what sh could gt if sh knw th nvironmnt. Formally, loss is givn by l (b 1, F, b 1 v 1 ) = sup {u 1 (v 1 y)q (y, b 1, F )} y u 1 (v 1 x)q (x, b 1, F ) db 1 (x), whr Q is th probability that biddr 1 wins th objct whn biddr i uss bidding function b i and valus ar drawn according to F. Not that sup y {u 1 (v 1 y)q (y, b 1, F )} dscribs th payoff biddr 1 could (approximatly) achiv if sh knw F and th bidding bhavior of th othrs. In gnral, loss is zro if th optimal bidding function for th tru nvironmnt is chosn and boundd abov by u 1 (v 1 ) if no bids abov valu ar placd. What rmains to b spcifid is th dscription of how biddr 1 solvs 8

9 th problm of dciding how to bid without knowing th nvironmnt. A bidding function is valuatd by th maximal loss it can gnrat among th concivabl nvironmnts E. For a givn st of concivabl nvironmnts E, biddr 1 prfrs bidding functions that gnrat smallr maximal loss. Th bst bidding function biddr 1 can choos according to this critrion is th on at which minimax loss is attaind. Minimax loss is always dfind rlativ to a st of concivabl nvironmnts E E. W distinguish minimax loss and dtrministic minimax loss. In th formr all bidding stratgis can b usd to minimiz loss, whras in th lattr only dtrministic (pur) stratgis ar allowd. Dfinition 1. Call ɛ th valu of minimax loss for th concivabl nvironmnts E E if, for all nvironmnts in E, (i) loss is guarantd to b at most ɛ, and (ii) thr is no bidding function that guarants a loss strictly lowr than ɛ. Call ɛ th valu of dtrministic minimax loss for th concivabl nvironmnts E E if, for all nvironmnts in E, (i) thr xists a dtrministic bidding function that guarants loss to b at most ɛ, and (ii) thr is no dtrministic bidding function that guarants a loss strictly lowr than ɛ. 3 Insnsitivity of Expctd and Maximin Utility In this sction w illustrat th notion of loss in a simpl xampl and discuss th robustnss of quilibrium bidding functions that ar drivd undr xpctd utility maximization with a (slightly) misspcifid prior. In particular, w show that a bidding rul that maximizs xpctd utility undr a crtain prior can prform vry badly in anothr stting. Additionally, w commnt on shortcomings of bidding functions that ar optimal undr maximin utility. Suppos thr ar two risk-nutral biddrs participating in a first-pric auction. distribution F δ Thir rspctiv valus ar drawn indpndntly for th valu paramtrizd by δ [0, 1). Blow w comput loss of having a slightly wrong prior blif about δ, but first w dscrib th stup. Th valu distribution puts mass ɛ (0, 1) uniformly on (δ, 1] and mass 1 ɛ on δ. Thus, F δ (x) = 0 for x < δ and F δ (x) = min { 1 ɛ + ɛ x δ 1 δ, 1} 9

10 for x δ. For a givn δ, biddr 2, acting as if both biddrs know F δ, uss th bidding function b δ 2(x) = x x δ F δ ( x) d x F δ (x) = x2 ɛ δ 2 (2 ɛ) + 2δ(1 ɛ) 2(1 δ + xɛ ɛ) that corrsponds to th Bays-Nash quilibrium bidding function. Not biddr 2 has valu δ with probability 1 ɛ, in which cas sh bids hr valu. Lt b δ 1 dnot th bst-rspons to bidding function b δ 2. Sinc b δ 2 is th quilibrium bidding function, b δ 1 b δ 2. Considr biddr 1 with typ v 1 = 1. If biddr 1 (wrongly) blivs that δ = 0, thn th optimal bid is b 0 1(1) = ɛ/2. This bid is nvr winning if th tru δ > ɛ/2. Th loss of th bidding stratgy b 1 (1) = ɛ/2 is maximizd if th tru δ is slightly abov ɛ/2 and qual to sup l(b 1, F δ, b δ 2) = sup 1 b δ (1 δ)(2 ɛ) 1(1) = sup δ>ɛ/2 δ>ɛ/2 δ>ɛ/2 2 = (2 ɛ)2. 4 For ɛ clos to 0, th maximal loss is approximatly qual to th valu, which is is th uppr bound on loss, conditional on not bidding highr than v 1. Hnc, for a slightly misspcifid prior about th nvironmnt, th loss can b as larg as possibl. Undr maximin utility a uncrtainty avrs biddr maximizs xpctd utility givn th worst possibl prior. Stratgic uncrtainty is rsolvd in quilibrium. W show that with ths prfrncs loss can also b as high as 100% of th own valu. This is don by th construction of a simpl two-biddr xampl paramtrizd by γ. Any of th concivabl valu distributions distributs mass γ uniformly on [0, 1 γ) and mass 1 γ uniformly on [1 γ, 1]. Lt 0 < γ 1 < γ 2 < 1, and F = {F γ γ [γ 1, γ 2 ]}, whr F γ (x) = γx/(1 γ) for 0 x 1 γ and F γ (x) = (2γ γx+x 1)/γ for 1 γ < x 1. Lo (1998) shows that biddrs with idntical F and maximin prfrncs slct th worst cas prior F min as th lowr nvlop of concivabl valu distributions in F. In our xampl, this corrsponds to F min = F γ 1. Subsquntly, biddrs bhav as if F min is th tru valu distribution and stratgic uncrtainty is rsolvd in quilibrium, that is th 10

11 bidding function b min (v) = v v 0 F min (x) dx F min (v) v for v < 1 γ 2 1 = (γ 1 1)(2γ 1 +v 2 1) 4γ 1 +2(γ 1 for v 1 γ 1)v+2 1 is usd by both playrs. Not that in particular b min (1) = 1 γ 1. Considr biddr with valu 1. If th tru γ quals γ 2 and γ 2 is larg (clos to 1) thn most biddrs hav valu clos to 0 and biddr 1 should bid vry low to maximiz hr surplus. Howvr, sinc hr bidding follows th maximin solution b min and γ 1 is small (clos to 0), sh bids vry high. Consquntly, hr loss of not bidding as sh would if sh knw th tru γ can b as larg as it can gt, namly it can b approximatly 1, that is hr valu. 4 Conciving th Bids of Othrs Whn many valu distributions and bidding functions ar concivabl thn it can b hard to undrstand how diffrnt rstrictions influnc bidding bhavior. Hnc, it can b simplr and mor insightful to work dirctly with blifs ovr bid distributions, instad of driving ths from valu distributions and bidding functions. In this sction w considr a biddr who narrows down th possibl nvironmnt sh is facing by putting rstrictions on th bids of othrs. First sh imposs a lowr bound on th possibl maximal bid, thn sh allows for som mass blow this thrshold, and finally, w considr a modl with indpndnt bidding and bid disprsion. 4.1 Imposing a Lowr Bound on th Maximal Bid of Othrs W considr th bidding bhavior of risk-nutral biddr 1 in a first-pric auction for a singl good. 2 In this sction, biddr 1 is compltly uncrtain about th othr biddrs typs (valu distributions, risk prfrncs, highrordr blifs, tc.) and thir bidding bhavior. This mans, for xampl, that biddr 1 dos not insist that th othr biddrs bid indpndntly, but also dms colluding bhavior possibl. Mor formally, w allow th st of concivabl nvironmnts to b th st of all nvironmnts E as dfind 2 In Appndix A w considr risk-avrs biddrs. 11

12 in Sction 2. This st is th st of all possibl valu distributions and all possibl bidding functions. Suppos biddr 1 blivs that sh nds to bid at last L 0 so that hr bid bcoms winning. Th valu of L can com from a known rsrv pric, 3 or from th blif that th maximal bid of othr biddrs is at last L. Dfinition 2. Givn L 0, lt E L E b such that E L is th st of nvironmnts blonging to E in which th highst bid is almost surly at last L. In th following w considr th cas whr v 1 > L as bidding undr v1 L is simpl; all bids lss than or qual to v 1 ar optimal. W brifly rpat th considrd problm and solution concpt. Th biddr dos not know th nvironmnt that consists of th joint valu distribution and bidding bhavior of th othr biddrs. All sh knows is that th maximal bid of th othrs will almost surly b abov L. If sh knw th joint valu distribution and thir bidding functions, sh could maximiz xpctd utility. Th objctiv is to gt a payoff that is clos to th on sh would achiv if sh knw th nvironmnt. Th closr hr payoff is to th bst possibl payoff, th smallr th loss. Hnc, th objctiv is to find a bidding function that minimizs th maximal loss. In ordr to maximiz loss w do not nd to considr all concivabl nvironmnts, sinc it suffics to rstrict attntion to a particularly simpl nvironmnts. Loss associatd with bidding function b 1 is th diffrnc btwn th oracl payoff, th maximizd xpctd utility if th nvironmnt is known, and th xpctd utility gnratd by bidding function b 1. First, obsrv that loss is zro if on of th othr biddrs bids highr than v 1. Thrfor, in ordr to maximiz loss w only hav to considr nvironmnts in which th maximal bid of othr biddrs is in th intrval [L, v 1 ]. Scond, notic that for givn nvironmnt E biddr 1 wins if hr bid is highr than th maximal bid among th othr biddrs, dnotd by M. Loss is incrasd if biddr 1 larns this bid M in th oracl. As a rsult, loss is maximizd by simpl nvironmnts in which all othr biddrs bid M 3 Th valu L can also b th ndognously dtrmind rsrv pric in a unit-dmand Anglo-Dutch auction (Binmor and Klmprr, 2002). 12

13 with crtainty. 4 First w considr th prformanc of dtrministic bids. Obsrv that biddr 1 nvr bids abov v 1 bcaus this rsults in a ngativ surplus and nvr blow L, bcaus ths bids ar losing for sur. Biddr 1 wins th auction if hr bid is abov M and loss it othrwis. Winning th auction with bid b 1 yilds utility v 1 b 1 and losing givs zro utility. 5 If biddr 1 bids b, thn loss quals { l(v, M) = sup v b } 1 b>m (v b) = v M 1 b>m (v b), b>m whr 1 b>m = 1 if b > M and 0 othrwis. Th oracl payoff is sup b>m {v b}, sinc biddr 1 knows th bid M sh has to match. 6 Loss can occur from bidding too low and from bidding too high. Biddr 1 bids too low if th bid dos not bcom winning. In this cas, loss is highst if b is slightly outbid and not highr than v b. In th othr cas, th bid b is too high. If b > M L, thn biddr 1 could rais hr xpctd surplus by dcrasing hr bid. Th loss of bidding too high is not mor than v L v +b = b L. Th maximal loss is max {v b, b L}. Maximal loss is minimizd by th bid that quats th two xprssions. This bid is qual to b = v+l and 2 guarants that absolut loss is blow v L. Loss rlativ to th distanc 2 v L is 1. 2 Nxt w show how appropriat randomizd bidding can rduc loss furthr. Assum that biddr 1 uss a mixd stratgy with probability dnsity function (pdf) g(b v) on som support, which is a subst of [L, v]. Bids blow L ar nvr winning and bids abov v yild non-positiv surplus. Th corrsponding cumulativ distribution function (cdf) of th mixd bidding function is dnotd by G(b v). Biddr 1 wins th auction if hr bid is abov M and loss it othrwis. Loss is qual to th following diffrnc whn M 4 Environmnts of this form can b rationalizd by th following bhavior. Suppos thr ar at last two othr biddrs and th valu distribution puts all th mass on M. Lt th n 1 biddrs know this. Thn bidding valu is an quilibrium for thm. 5 W will oftn drop th indx if w think this causs no confusion. 6 Formally, without discrt bids and with a non-dgnrat ti-braking rul, biddr 1 has no bst rspons whn h knows that th maximal bid among th othr biddrs is M < v. Hnc, w considr th suprmum bcaus w ar intrstd in th payoff and not in th spcific bid. 13

14 is known and whn it is not known, i.. l(v, M) = max { { sup v b, 0} } b>m = max {v M, 0} v M v M v b dg(b v) v b dg(b v) (1) If M is known thn biddr 1 gts ithr (approximat) utility of v M by bidding (slightly abov) M, or 0 if M v. All bids abov M ar winning and biddr 1 computs th xpctd utility of using th randomizd bidding function G. Proposition 1. For th st of concivabl nvironmnts E L minimax loss is v L and attaind by th randomizd bidding stratgy with dnsity g(b v) = 1 v b on [ L, v v L ]. (2) Th dtrministic minimax loss is qual to v L 2 and attaind by bidding v+l 2 İn Appndix B, th proof spcifis th dtails how th bidding function is drivd. Th man bid of bidding function (2) is (v + L( 1))/ and lss than th mdian, which is qual to (v( 1) + L)/. Th mdian and th man ar both lss than th dtrministic bid. This shows that on nds to bid rlativly low in ordr to minimiz maximal loss. In th following sctions w show that loss can b mad smallr if vry low bids ar not concivd to b likly. So w find that a biddr who is not willing to narrow down th nvironmnts furthr than E L can only guarant loss to b blow (v L)/. This guarant is achivd by appropriat randomizd bidding and is 74% or lss of th guarant that can b achivd by a dtrministic bidding stratgy. Th mixd stratgy rducs loss bcaus it allows som kind of hdging against unknown bids. Th bidding function in Proposition 1 is indpndnt of th numbr of biddrs. Not that no assumption on th numbr of biddrs is mad. Evn if this numbr was known, th tru valu distributions could assign th sam valu to all othr biddrs, or th bidding function could b such that 14

15 all submit th sam bid, making th numbr of biddrs irrlvant. Brgmann and Schlag (2008) look at a rlatd problm th optimal pricing schm of a monopolist who dos not know th valu distribution of th buyr. Th monopolist minimizs maximal rgrt, whr rgrt is th diffrnc in profit whn th valu is known and whn it is not known. It turns out that th pdf of th monopolist s optimal pricing stratgy rsmbls Equation (2). Bidding in a first-pric auction with no assumptions is lik pricing in markts with no information on dmand. Apart from dirctions whr highr payoffs can b achivd, in auctions on wishs as biddr to hav a low winning bid, in markts as sllr a high sal prics. A mthodological diffrnc is that Brgmann and Schlag (2008) considr x post loss, whil w considr intrim loss. Th diffrnc of thos two concpts is dcision makr s knowldg usd for th computation of th oracl payoff. On th on hand, in Brgmann and Schlag (2008) th monopolist knows th stratgy of th potntial buyr and uss th buyr s valu in th bnchmark. On th othr hand, in this papr th biddr uss only th distribution and bidding function of th othr biddrs whn computing th oracl payoff. Halprn and Pass (2012) introduc itratd limination of stratgis that do not attain minimax rgrt in normal form gams (with a known prior). For this approach it is crucial that all playrs ar known to minimiz maximal rgrt. Thy provid a simpl xampl of a first-pric auction in which thy ssntially look at x post minimax rgrt, limit attntion to dtrministic stratgis, and itration is not ndd. Thy find that th bidding function b(v) = v/2 minimizs maximal rgrt. W do not assum that all biddrs minimax loss and w considr intrim and not x post loss. 4.2 Allowing som Mass Blow th Thrshold In th analysis abov biddr 1 rstrictd hr bids to b abov L, sinc sh dmd that hr bids blow L ar nvr winning. In this subsction w first show that it is bst to ignor possibl bids blow L and to bid as in Proposition 1, as long as th liklihood of bids blow L is sufficintly small. Thn w discuss a rlatd stting in which on knows that th own valu is rlativly small and th implications on loss. 15

16 Dfinition 3. Lt L 0 and p [0, 1]. Dfin E L p E to b th st of all nvironmnts such that th probability that th highst bid among th othr biddrs is blow L is boundd abov by p. Th dfinition dscribs th following. Considr nvironmnts in E L p. Th maximal probability that th highst bid among th othr biddrs is blow L is p. Each nvironmnt spcifis a numbr of biddrs n with n 2. Suppos th othr biddrs bid indpndntly. For vry biddr i > 1 thr is a p i [0, 1] such that at most mass p i of i s bids can b blow L. Thn th maximal probability that th highst bid among th othr biddrs is blow L is 1<i n p i p. Morovr, if p i dos not dpnd on i, thn p n 1 p. In th analysis abov, biddr 1 thought that th minimal maximal bid is L, thus p = 0. In this sction, th maximal bid M can b in [L, 1] with probability 1, but maximal bids blow L can only b inducd by distributions in which th highst bid among th othr biddrs is abov L with probability at last 1 p. Considr biddr 1 having a rlativly high valu v > L and suppos sh uss th randomizd bidding stratgy of Proposition 1 with support [L, v v L ]. Abov w saw that if th highst bid among th othr biddrs is always abov L, loss is at most v L. Thrfor, loss of not bidding blow L can only b mad largr if th highst bid among th othrs is blow L. Potntially, loss can b mad largst by all othr biddrs bidding 0, which can, undr th st of concivabl nvironmnts E L p, only happn with probability p. If biddr 1 larns that all othr biddrs bid 0, thn sh knows that sh bids too high. This insight is associatd with a loss that dpnds on p. Th following proposition stats that if p is sufficintly small, thn maximal loss is minimizd by ignoring potntial bids blow L. Proposition 2. Lt v > L 0 and p v L. For th st of concivabl nvironmnts E L p minimax loss is qual to v L and attaind by th v L+L randomizd bidding stratgy statd in Proposition 1. Not that for any L (0, v) and p i (0, 1), th uppr bound on p in Proposition 2 is satisfid for larg nough n. On can, of cours, also fix th bound on loss for typ v and ask which L and p giv ris to this loss (Exampl 1), or fix L and p and ask for which v th inquality is satisfid (Exampl 2). 16

17 Exampl 1. Lt v = 1 and L = 1 ( 0.73). W nd that p = 10 pn 1 1. Loss is lss than on tnth if n = 2 and p ( 0.12) v L = 1 v L+L 11 or if n = 4 and p ( 1 11 ) 1 3 ( 0.49). 11 Exampl 2. Suppos th tru valu distribution is uniform on [0, 1] and th othr n 1 biddrs ar risk-nutral and play according to th risk nutral Bays-Nash quilibrium β(v) = n 1 v. Biddr 1, howvr, only knows th n mdian bid L = n 1 and p = 1. Th inquality 2n 2 pn 1 v L givs a v L+L bound on v. If v is highr than th uppr bound, thn loss is boundd by v L for th st of concivabl nvironmnts E L, p. If n = 2, thn L = 0.25 and p n 1 = 1 for v (1 + )/4 ( 0.93). For n = 5 th mdian 2 v L v L+L bid is L = 0.4 and v (30 + 2)/75 ( 0.47) is ncssary. If n = 10, thn L = 0.45 and v ( )/10220 ( ) is rquird. So far w hav considrd rlativly larg v, but now w turn attntion to smallr v. In particular, w look at v < L. Th nxt proposition says 1 that biddr 1 can minimiz maximal loss by using th randomizd bidding function of Equation (2) on [0, v v ]. This bidding functions nsurs that all bids ar blow L, sinc v < Proposition 3. Lt v 1 L. 1 v L L and p if L < v. For th st v of concivabl nvironmnts E L p minimax loss is qual to p v and attaind by th randomizd bidding stratgy statd in Proposition 1 valuatd as if L = 0. Th condition p v L implis p v L. Hnc, for v such that v v L+L L < v L Propositions 2 and 3 cannot hold at th sam tim Th ε-uniform Modl W now rturn to our original modl in which th biddr blivs that all bids ar abov L. In our prvious analysis (Subsction 4.1) th biddr thought it is concivabl that all othr biddrs bid th sam valu. This had th consqunc that th optimal bidding function was indpndnt of th numbr of biddrs. Hr w assum that th biddr xpcts a crtain numbr of biddrs and som htrognity among th othr biddrs. W modl this by assuming that th biddr blivs that any givn othr biddr puts a minimal wight of ε on bids abov L. Thus, no rlvant bid can 17

18 b ruld out and, in particular, it cannot b th tru nvironmnt that all othr biddrs bid L for sur. Formally, biddr 1 concivs that th bidding distribution of a givn biddr can b writtn as ε tims th uniform distribution on [L, v] plus 1 ε tims som arbitrary distribution. 7 Dfinition 4. Lt L 0 and ε (0, 1). Dfin E L,ε,n E L to b th st of all nvironmnts such that (i) thr ar n biddrs, (ii) for any biddr i > 1 it is as if th bid is drawn uniformly from th intrval [L, v] with probability at last ε. Within E L,ε,n E L it is again a simpl form of nvironmnts that potntially maximiz loss. Ths nvironmnts gnrat bid distributions such that for vry biddr i > 1 th bid is drawn uniformly from [L, v 1 ] with probability ε and qual to M [L, v 1 ] with probability 1 ε. It is nough to rstrict bids to th intrval [L, v 1 ], sinc w know from abov that loss is mad smallr if bids ar abov valu with positiv probability. Morovr, th mor mass is distributd uniformly, th lowr loss, hnc w considr bid distributions in which th minimum of ε is distributd uniformly indpndntly for vry biddr i > 1. In ths simpl nvironmnts it is as if biddr 1 larns th highst bid among th othr biddrs M whos bid is not drawn uniformly in th bnchmark. Conditional upon bidding abov M, th xpctd utility in th bnchmark is qual to n 1 ( n 1 EU(b b > M) = k k=0 ) ε k (1 ε) n 1 k (v b) ( ) k b L. (3) v L If b > M, thn biddr 1 wins against th n 1 k biddrs who bid M. Th probability that b is th highst among k uniformly distributd bids is ( b L v L) k. W somtim abbrviat EU(b b > M) simply with EU(b). Whn w us EU(M) thn w man that biddr 1 bids slightly abov M. If biddr 1 bids blow M, thn th xpctd utility is EU(b b < M) = ε n 1 (v b) ( ) n 1 b L, v L 7 W assum th uniform distribution for simplicity. With th uniform distribution on gts quit far in trms of closd form solutions. It might b that on has to rly ntirly on numrical calculations for othr continuous distributions. 18

19 sinc biddr 1 only wins if all othr biddrs bids ar drawn uniformly. Th following lmma says that th surplus in th bnchmark is maximizd by bidding (slightly abov) th highst bid among th othr biddrs, or by bidding b, which is indpndnt of M. Whnvr ε 1/n, thn b is smallr than L and thrfor smallr than M. In this cas th bid b cannot bcom winning and biddr 1 nds to bid M to maximiz utility. In th cas of a rlativly larg ε > 1/n, th bid b is largr than L. Hnc, thr can b bids of th othr biddrs M [L, b ) such that biddr 1 ignors this information and bids b, which is indpndnt of M. Lmma 1. Whnvr M < b = v(nε 1)+L nε, xpctd utility EU(b b > M) is maximizd by bidding b. Othrwis M arg sup b EU(b b > M). This lmma has dirct consquncs on th maximization of loss. Suppos ε > 1/n so that b > L and that biddr 1 uss a randomizd bidding stratgy with support [b, b], whr b < b. If it turns out that th highst bid among th othr biddrs is M [L, b ), thn biddr 1 s optimal bid in th bnchmark is b. Loss is non-dcrasing for othr biddrs highst bids M (M, b ). This can b sn by obsrving that for such an M th oracl payoff is unchangd sinc biddr 1 s optimal bid is still b. Th highr th othr biddrs bid, th lss likly it is that a bid drawn from [b, b] is winning. Thrfor, loss cannot b maximizd if th highst bid among th othr biddrs is blow max {L, b }. Th following proposition shows that minimax loss is attaind by a randomizd bidding function that dpnds on ε, th numbr of biddrs n, and th lowst possibl winning bid L. A closd form solution for th uppr bound of th bidding function is not availabl it nds to b computd numrically in applications. As a rsult, also th valu of minimax loss can only b statd implicitly. As ε tnds to zro, th modl and th rsults convrg to th prviously statd 1/ bound. Proposition 4. Lt v > L 0, ε (0, 1), and n an intgr and n 2. For th st of concivabl nvironmnts E L,ε,n minimax loss is attaind by th randomizd bidding stratgy g(b v) = α(b) n 1 β(b)(v(1 εn) + bεn L) (v b) ((ε 1)vβ(b) n + bε (α(b) n β(b) n ) + L (β(b) n εα(b) n )), 19 (4)

20 with α(b) = v(1 ε) + bε L and β(b) = ε(b L), on support [ b, b ], whr b = max {L, b } and b solvs Minimax loss quals EU ( b) ε n 1 b b b b g(b v) db = 1. (5) g(b v) ( ) n 1 b L (v b) db. (6) v L Dtrministic minimax loss is attaind by ˆb such that n 1 ( ) n 1 EU(max{M, b }) EU(ˆb) = ε k (1 ε) n 1 k (v k ˆb)p(ˆb) k k=1 and qual to th valu on ithr sid of th quation. Exampl 3. Tabl 1 provids numrical calculations for v = 1, L = 0 and diffrnt valus of ε and n. For vry ε and n th tabl rports th support of th random bidding function, th man bid, and th uppr bound on loss, all roundd to two dcimals. On intrsting fatur of th modl is that th xpctd bid is incrasing in th numbr of biddrs. As on might xpct, loss is dcrasing in ε and n. Th maximal loss undr random bidding is roughly 74% of th maximal loss undr dtrministic bidding. Figur 1 shows th dnsity of th bidding function of Equation (4) for n = 2 (dottd), n = 5 (dashd), and n = 10 (solid), whr ε = 0.15, L = 0, and v = 1. On can s that for n = 2 and n = 5 th lowr bound of th bidding function is 0, but not for n = 10. For n = 10, b > 0 and thrfor b is th lowst possibl bid. As th numbr of othr biddrs incrass, mor mass is put on highr bids. 5 Bidding with Blifs about Bhavior In Sction 4 w dscribd uncrtainty in trms of th bid distributions th robust biddr thinks sh could b facing. In this sction w focus on whr ths bids com from and formulat bidding in trms of uncrtainty about valu distributions and bidding bhavior of othrs. 20

21 Randomizd Bidding n = 2 n = 5 ε Support Man Bid Loss Support Man Bid Loss 0.10 [0, 0.64] [0, 73] [0, 0.65] [0, 78] [0, 0.65] [0, 0.82] [0, 0.66] [0.2, 0.86] [0, 0.68] [0.5, 0.91] [0, 0.70] [0.6, 0.93] n = 10 Support Man Bid Loss 0.10 [0, 0.83] [0.33, 0.89] [0.5, 0.92] [0.6, 0.93] [0.75, 0.96] [0.80, 0.97] Dtrministic Bidding n = 2 n = 5 n = 10 ε Bid Loss Bid Loss Bid Loss Tabl 1: Loss for diffrnt valus of ε and n, whr L = 0 and v = 1 21

22 Figur 1: Probability dnsity function of Equation (4) for n = 2 (dottd), n = 5 (dashd), and n = 10 (solid), whr ε = 0.15, L = 0, and v = Complt Uncrtainty about Valus In this sction w prsnt two simpl scnarios rgarding blifs about valu distributions and bhavior that imply that th maximal bid of th othrs will crtainly b abov L. This thn mans that w can apply our rsults of Sction 4.1 whr w showd that th maximal loss can b boundd abov by (v L)/. On might think that thr ar at last two othr biddrs, who ar rational, bid indpndntly, and bliv that th valu of th othr is abov L. In this cas, nithr of thm will bid blow L. Similarly, th maximal bid is abov L if on blivs that thr is anothr robust biddr who applis th rsults of this papr and who blivs that th maximal bid of th biddrs h facs is crtainly abov L. Hnc, it is not ncssarily th own blifs that lad to th bound L, but it can b anticipating blifs and bhavior of othrs. On might wondr whthr th rror bound (v L)/ bcoms smallr if on maks mor assumptions on th bidding bhavior of th othrs whil maintaining complt uncrtainty about th valu distributions. W xplor this in th framwork closst to th classic modl of indpndnt privat valus. Assum that all othr biddrs know th valu distribution, that thr is common knowldg of rationality among thm and that thy know th stratgy of th robust biddr. Morovr, assum that th robust 22

23 biddr knows th abov assumption, but dos not know th valu distribution. W show that th rror bound dos not chang if thr ar at last two othr biddrs. To s this, lt th valu distribution F b iid and th limit of ɛ[v 1 ] + (1 ɛ)[m], M L whr ɛ convrgs to zro from abov. 8 Baysian biddrs basically know that all biddrs hav th sam valu M. Consquntly, bidding valu is a bst rspons to ach othr. This bhavior gnrats th sam conditions as in Subsction 4.1 and thrfor Proposition 1 applis. Loss cannot b dcrasd by simply rstricting uncrtainty to uncrtain ovr valus. Howvr, as w show in th nxt sctions, loss can b dcrasd through bounds on th valu distribution. 5.2 Bhavioral Blifs Th fact that th highst bid can b L with probability 1 contributs substantially to th bound on maximal loss providd in Sction 4.1. On can imagin that th biddr s prcption of th nvironmnt lads to constraints that rul out this xtrm cas. Such constraints would nd to apply both to th valu distribution so that not all biddrs can hav valu L with crtainty and to bidding bhavior to imply that not all biddrs bid L irrspctiv of thir typ. In this sction w mak plausibl or intuitiv rstrictions on bidding bhavior. In th nxt sction w mak assumptions on th othr biddrs objctiv. In both sctions w distinguish btwn robust and non-robust biddrs with th undrstanding that all robust biddrs ar facing th sam uncrtainty. If thr ar at last two robust biddrs thn this mans that w no longr hav a dcision problm and consquntly hav to introduc an quilibrium concpt. Th robust biddr concivs th following nvironmnts possibl. Thr ar k othr robust out of n total biddrs, with 0 k n 1. Valus ar distributd indpndntly. All concivabl valu distributions can b boundd from abov. Bidding stratgis of robust and non-robust biddrs can b diffrnt. It is blivd that th bidding stratgis of non-robust biddrs can b boundd from blow. Robust biddrs ar blivd to us a linar bidding function. 9 All biddrs us dtrministic (pur) stratgis 8 Altrnativly, considr an asymmtric valu distribution in which th Baysian biddrs hav som prior ovr v 1, and know that all i > 1 hav th sam valu. 9 A similar modl appars in Eliaz (2002). In his modl, thr ar up to k faulty 23

24 and thir bids ar incrasing in thir valu. Dfinition 5. Givn 0 < α, η and 0 < σ, τ < 1, lt Eαη στ E b such that Eαη στ is th st of nvironmnts blonging to E in which for all concivabl valu distributions F, (i) valus ar distributd indpndntly, (ii) F (v) min {ηv α, 1}, and for all concivabl bidding functions b 1 thr ar k biddrs such that for vry biddr b(v) = τv, and thr ar n k biddrs with b(v) σv. In this sction it is blivd that all robust biddrs us a linar bidding function. This blif is confirmd w solv for th robust bidding stratgy and find that a linar stratgy gnrats suitabl loss. This givs ris to a crtain quilibrium notion. Lt O {1,..., n} b th st of robust biddrs, whr O 1. In this sction thr ar k +1 robust biddrs, so O = k +1. Lt b b th profil of bidding stratgis of th robust biddrs, b : O R R + +. Th profil b is indxd by i O, hnc b (i) dnots th bidding stratgy of biddr i. Loss dpnds on th st of concivabl nvironmnts and w allow biddrs to hav htrognous prcptions E i E for i O. Hnc, for vry biddr i thr xists an individual bound on loss ɛ(i) that might dpnd on v i. This is capturd by dfining ɛ : O R R + +. Dfinition 6. Th stratgy profil b is an ɛ-loss-quilibrium if for all i O, (i) b (i) guarants a loss blow ɛ (i) for nvironmnts in E i, and (ii) if (F, b i ) E i and j O\ {i} thn b i (j) = b (j). Without any quilibrium rasoning, th following proposition shows that a linar bidding functions allows to bound loss substantially. Th blifs about th valu distribution and th bidding functions allow to bound th probability of a winning bid from abov. Th highr α, th lss likly low valus for othr biddrs. Th highr σ and τ, th highr othr biddrs ar xpctd to bid, th lowr th probability of winning with a crtain bid. It turns out that loss can b boundd from abov by nvironmnts, that is, bid distributions, that put as much mass as possibl on a uniqu bid and th rst of th mass abov valu v 1. Thus it is as if playrs who choos any action. W assum that thr ar n k non-robust biddrs bid can b boundd from blow. Also s Gradwohl and Ringold (2014) for fault tolranc in larg gams. 24

25 biddr 1 larns th maximal bid of th othr biddrs, givn this bid is blow valu. Not howvr, that ths distributions ar not always concivd as possibl. If σ > τ and 0 < k < n 2, robust and non-robust biddrs submit diffrnt bids vn if thy hav th sam valu. Sinc valus ar iid, it cannot b th cas that robust and non-robust biddrs hav diffrnt valu distributions. As a rsult, w do not talk about minimizing maximal loss, but rathr that w bound loss. ( Proposition 5. Lt 0 < α, η, 0 < σ, τ < 1 and 0 v 1 σ ) n k n 1. For η 1/α τ th st of concivabl nvironmnts Eαη στ, loss can b guarantd to b blow β β (1 + β) β(1+β) γ ) 1+β v ((1 1+β, (7) + β) β+1 + β β by using th dtrministic bidding function b 1(v 1 ) = (1 + β)1+β v, (8) (1 + β) 1+β + ββ whr γ = η n 1 ( (σ τ ) ) k α τ and β = α (n 1). σ n Bidding function b is rmarkabl in its indpndnc of th blifs about th othr biddrs bidding bhavior (σ and τ) and about th numbr of robust biddr k. Th function only dpnds on th numbr of othr biddrs and th bound on th valu distribution. Whn low typs ar rlativly likly (small α) and thr ar fw biddrs, thn th optimal bid is just abov v/2. For a linar bound (α = 1) and two biddrs, th optimal bid quals 0.8 v. On th othr hand, th valu of loss is dpndnt on all paramtrs of th modl. Th bound on loss incrass in th numbr of robust biddrs k if and only if σ τ, that is, if robust biddrs bid lss aggrssiv than th othr biddrs. Th intuition can b sn from th bound on th winning probability. Loss is incrasing th mor likly low bids ar, that is, th lowr th probability that a low bid is winning. Th winning probability of a bid can b boundd from abov by bids of non-robust biddrs qual to σv. 25

26 Whnvr σ > τ, thn it is th robust biddrs that mak low bids possibl and thrfor loss incrass in thir numbr. Th robust biddr chooss a linar bidding stratgy if sh anticipats that th othr robust biddrs choos a linar stratgy. This givs ris to an ɛ-loss quilibrium if th othr robust biddrs hav th sam prcption about th nvironmnts, i.. us th sam α and n. Th notion of ɛ-loss quilibrium is comparabl to th notion of ɛ Nash quilibrium in th sns that a lowr loss might b possibl, lik a highr xpctd surplus might b possibl in ɛ Nash, but playrs ar contnt with giving up ɛ. Not that th bound on loss is a non-linar function in v. Rmark 1. For k + 1 robust biddrs with concivabl nvironmnts Eαη στ with (1 + β)1+β τ = (1 + β) 1+β + β, β bidding function (8) forms an ɛ-loss quilibrium with ɛ(v) qual to loss givn in Equation (7). Exampl 4. Lt thr b fiv biddrs (n = 5), α = 1, η = 1, and v = 1. 2 Lt b 1(v) = τv = 27/31/, 0.87, and σ = 0.9. For k = 1 loss is guarantd to b blow In th cas of k = 5, loss is boundd by Altrnativly, for σ = 0.8 w nd that η so that th bound on v is satisfid, i.. v 1 ( σ ) n k n 1. η 1/α τ As σ < τ, th robust biddr is mor aggrssiv and loss dcrass in th numbr of robust biddrs. If η = , thn loss is smallr than 0.13 for k = 1 and lss than 0.11 if k = 5, that is, all biddrs bid robustly. 5.3 Explicit Rational Bidding 6 Conclusion On of th major obstacls and challngs to bidding in first-pric auctions is limitd information. It is difficult, in many instancs, to assss othr biddr s valu distributions and bidding functions (i.. th nvironmnt) 26

27 and to spcify blifs and highr-ordr blifs. Misspcification can lad to substantial loss. This is th first papr that drivs robust bidding ruls in first-pric auctions. W dal with th uncrtainty by sarching for a compromis that prforms wll for a wid varity of situations. Th mthodology basd on compromiss is asy to xplain and justify. W valuat bidding functions basd on loss, whr loss compars th payoff in an nvironmnt to th payoff of th bst bidding rul if th tru nvironmnt wr known (that is, in th truly hypothtical and unralistic bnchmark). Our mthod idntifis how important dtaild information is. W show that thr ar instancs in which mor dtaild information is not ndd to guarant loss to b blow 10% of th own valu. In Subsction 4.1, th robust biddr knows vry littl about th othr biddrs. Howvr, loss can b low if sh knows that sh has a rlativly low valu compard to othr biddrs. Mor formally, it is ndd that th robust biddr knows that th highst bid of th othr biddrs is abov th own valu with probability at last Anothr cas is with th likly xistnc of at last on comptitor who bids clos to th own valu. This happns if th maximal rivaling bid is blow 72.8% of th own valu with probability at most In Subsction 4.3, any opn st of bids blow v occurs with x ant positiv probability. This is modld by drawing th bid for vry othr biddr uniformly with probability ε (0, 1) and arbitrarily with probability 1 ε. Loss is dcrasing in th numbr of biddrs n and in th uncrtainty ε. For xampl, if ε = 0.15 and n = 10, thn loss is blow 10% of th valu. In Sction 5, th robust biddr bounds th othr biddrs valu distribution and bidding functions. Among othr things, w solv a modl in which thr ar only robust biddrs. If all robust biddrs us a linar bidding function, thn a linar bidding function is robust. Loss is dcrasing in th numbr of biddrs. Th lss likly low typs ar, th lowr loss. 27