On Monotone Strategy Equilibria in Simultaneous Auctions for Complementary Goods

Transcription

1 On Monotone Strategy Equlbra n Smultaneous Auctons for Complementary Goods Matthew Gentry Tatana Komarova Pasquale Schrald Wroy Shn June 20, 2018 Abstract We explore exstence and propertes of equlbrum when N 2 bdders compete for L 2 objects va smultaneous but separate auctons. Bdders have prvate combnatoral valuatons over all sets of objects they could wn, and objects are complements n the sense that these valuatons are supermodular n the set of objects won. We provde a novel partal order on types under whch best reples are monotone, and demonstrate that Bayesan Nash equlbra whch are monotone wth respect to ths partal order exst on any fnte bd lattce. We apply ths result to show exstence of monotone Bayesan Nash equlbra n contnuous bd spaces when a sngle global bdder competes for L objects aganst many local bdders who bd for sngle objects only, hghlghtng the step n ths extenson whch fals wth multple global bdders. We therefore nstead consder an alternatve equlbrum wth endogenous tebreakng buldng on Jackson, Smon, Swnkels and Zame (2002), and demonstrate that ths exsts n general. Fnally, we explore effcency n smultaneous auctons wth symmetrc bdders, establshng novel suffcent condtons under whch neffcency n expectaton approaches zero as the number of bdders ncreases. Ths artcle ncorporates materals from the pror workng papers Smultaneous Auctons for Complementary Goods by W. Shn and On Monotone Strategy Equlbra n Smultaneous Auctons for Complementary Goods by M. Gentry, T. Komarova, and P. Schrald. We are grateful to Edward Green, Vjay Krshna, Roger Myerson and Balazs Szentes for ther comments and nsght. London School of Economcs, m.l.gentry@lse.ac.uk London School of Economcs, t.komarova@lse.ac.uk London School of Economcs and CEPR, p.schrald@lse.ac.uk Korea Insttute for Industral Economcs and Trade, wshn@ket.re.kr M. Gentry, T.Komarova and P. Schrald acknowledge the ESRC for fnancal support. W. Shn receved fnancal support from the Human Captal Foundaton to the Pennsylvana State Un- 1

2 1 Introducton Smultaneous bddng for multple objects s a commonly occurrng phenomenon n many realworld aucton markets, but surprsngly lttle s known about the propertes of equlbra n games nvolvng smultaneous auctons when bdder payoffs are non-addtve. 1 For example, when auctonng drllng rghts n the US Outer Contnental Shelf, the US Mnerals Management Servce typcally offers (and bdders typcally bd on) a large number of drllng tracts smultaneously. Pror emprcal work (e.g. Hendrcks and Porter, 1988, Hendrcks, Pnkse and Porter, 2003) suggests that economcally mportant complementartes may exst between tracts n close proxmty. Yet lttle s presently known ether theoretcally or emprcally about how such synerges mght affect equlbrum behavor n such markets. 2 Ths paper analyzes equlbrum wthn a class of mechansms we refer to as smultaneous standard auctons for complementary goods. In ths settng, a collecton of L 2 objects are offered for sale to a set of N 2 bdders. Bdders have ndependent prvate valuatons over combnatons of objects, where objects are complements n the sense that bdders valuatons are supermodular n sets of objects won. Auctons are smultaneous n the sense that bdders may bd on each object ndvdually but may not submt contngent or combnatoral bds, and standard n the sense that each object l s allocated to a hgh bdder n aucton l and payments n aucton l depend only on bds n aucton l. So long as all auctons are standard, auctons for dfferent objects may have dfferent formats. For smplcty, we frame dscusson n terms of a versty. 1 Examples of markets nvolvng smultaneous bddng nclude hghway procurement n many US states (e.g. Krasnokutskaya [2011], Soman [2013], Groeger [2014] among others), recyclng servces n Japan (Kawa, 2010), cleanng servces n Sweden (Lunander and Lundberg, 2013), ol and drllng rghts n the US Outer Contnental Shelf (Hendrcks and Porter, 1988, Hendrcks, Pnkse and Porter, 2003), and to a lesser extent US Forest Servce tmber harvestng (Athey, Levn and Sera, 2011, among many others). 2 Notable exceptons are Fox and Bajar (2013), who estmate the determnstc component of bdder valuatons n FCC smultaneous ascendng spectrum auctons, and Gentry, Komarova and Schrald (2017), who emprcally study smultaneous bddng n Mchgan Department of Transportaton hghway procurement auctons. 2

3 sngle auctoneer, although ths s nessental for our results. The smultaneous standard aucton game rases a number of sgnfcant theoretcal challenges. Even assumng ndependent prvate types, each bdder s preference structure could n prncple be as complex as a complete (2 L 1)-dmensonal set of valuatons assgned by that bdder to each of the 2 L 1 possble non-empty subsets of objects. Meanwhle, the smultaneous standard aucton permts bdders to submt (at most) L ndvdual bds on the L objects beng sold. Furthermore, as usual n auctons, payoffs n the resultng game may be dscontnuous n bds. The end result s a dscontnuous Bayesan game wth hgh-dmensonal types for whch even basc propertes such as exstence of Bayesan Nash equlbrum are challengng to establsh n general. Secton 2 ntroduces the model and shows that even the (strong) assumpton of supermodular valuatons s nsuffcent to ensure monotoncty of best reples wth respect to the usual coordnatewse order on types n fact, a strct coordnatewse ncrease n type can nduce a strct coordnatewse decrease n best-reply bds. Ths turns out to be because the usual coordnatewse order on types mposes nsuffcent structure on margnal valuatons. Motvated by ths observaton, Secton 3 ntroduces a partal order on bdder types characterzed by a fnte number of lnear nequaltes on margnal valuatons. These nequaltes defne a cone (wth nonempty nteror) strctly contaned n the frst-orthant cone of the (2 L 1)-dmensonal type space.e., the cone descrbng the usual coordnatewse order. Ths stronger partal order turns out to be suffcent for monotoncty n the sense that each bdder has an nterm best reply such that an ncrease n s type wth respect to our partal order wll mply an ncrease n s bds wth respect to the usual coordnatewse order. Equpped wth ths prelmnary result, Secton 4 bulds on the methodology of Athey (2001), McAdams (2003) and Reny (2011) to establsh exstence of pure strategy equlbra on fnte bd spaces whch are monotone n the sense above. Ths result turns on one addtonal condton, 3

4 whch clarfes the relatonshp between the partal order cone and the support of bdder s jont dstrbuton of valuatons. Namely, we show that ths relatonshp s suffcently rch f ths dstrbuton s absolutely contnuous wth respect to the Lebesgue measure n the (2 L 1)- dmensonal space and the support of ths dstrbuton s regular enough. Whle the exstence of a pure strategy Bayes-Nash Equlbrum follows from Mlgrom and Weber (1985), the monotone characterzaton of the equlbra under a sutable partal order on types s novel n ths settng. We then proceed to consder contnuous bddng spaces. Frst, n Secton 5.1, we consder a specal case smlar n sprt to Krshna and Rosenthal (1996), n whch a sngle global bdder bds n smultaneous frst-prce auctons for L objects aganst a collecton of local bdders who bd for sngle objects only. 3 Buldng on proof technques n Reny (2011), we show exstence of a pure strategy Bayes-Nash equlbrum whch s monotone wth respect to our partal order. To the best of our knowledge, both exstence and monotoncty are novel n ths settng. Moreover, monotoncty s here pvotal n establshng exstence; the proof turns on passng from a sequence of monotone equlbra on dscrete spaces to the lmt of ths sequence n a contnuous space, whch s feasble only because the space of monotone strateges s known to be compact n the pontwse convergence topology f the partal order s suffcently rch (Reny, 2011). Whle we beleve that ths fndng s of nterest n ts own rght, ths example also serves to hghlght a subtle challenge arsng n settngs wth more than one global bdder. Specfcally, the nteracton between strategc overbddng by global bdder and dependence across auctons of bds by s global rvals leads to uncertanty regardng a key techncal property better-reply securty of Reny (1999) needed to complete the exstence proof. In Secton 5.2, we therefore turn to an alternatve soluton concept, equlbrum wth endogenous tebreakng, buldng on the work of Jackson, Smon, Swnkels and Zame (2002). Jackson, Smon, Swnkels and Zame 3 In ths secton only, we restrct attenton to smultaneous frst-prce auctons, as our prmary purpose s llustraton. We conjecture, however, that arguments smlar to those n 5.1 could be used to establsh exstence of monotone equlbra n standard auctons more generally. 4

5 (2002, henceforth JSSZ) defne the communcaton extenson G c to a gven game G as the game arsng when, n addton to ther actons under G, players also submt cheap-talk ndcatons of ther types whch the auctoneer may use (only) to resolve tes. A soluton to G c s a strategy profle for bdders plus a tebreakng rule such that strateges are a Bayesan Nash equlbrum gven the tebreakng rule. Startng from a class of dscontnuous games G whch ncludes ours, JSSZ (2002) establsh exstence of solutons to G c n whch bdders play dstrbutonal strateges as defned by Mlgrom and Weber (1985) and communcaton s truthful. In the context of smultaneous auctons for complementary goods, we show that these general conclusons can be sharpened n at least three respects. Frst, rather than permttng bdders to communcate ther full (2 L 1) 1-dmensonal types, we allow bdder to submt (n addton to her bd vector b ) only an L 1 vector of cheap-talk sgnals s ; we refer to the latter as a sgnallng extenson to dstngush t from the communcaton extenson of JSSZ (2002). Second, we show the exstence of a soluton to the sgnallng extenson n whch the auctoneer s tebreakng rule can be characterzed by a set of L weakly monotone tebreakng precedence functons (ρ 1,..., ρ l ), where the auctoneer randomzes object l ndependently among the set of hgh bdders n aucton l wth the hghest tebreakng precedence:.e. among the set of bdders wth b l = max j {b jl } and ρ l (s l ) = max j {ρ l (s jl )}. Ths characterzaton of tebreakng sharpens that n JSSZ (2002), mplyng n partcular exstence of a soluton where allocatons and payments n aucton l depend only on bds and sgnals n aucton l. Furthermore, whereas JSSZ (2002) consder only exstence n dstrbutonal strateges, we obtan exstence n pure strateges whch are addtonally monotone n a sutable partal order sense. Fnally, n Secton 6, we relate our model to the mportant queston of the performance of auctons when the number of bdders becomes large. We consder the case of ex-ante symmetrc bdders playng a Bayes-Nash equlbrum n symmetrc monotone strateges. We obtan a suffcent condton that guarantees that the expected neffcency n a symmetrc monotone 5

6 equlbrum converges to zero. Intutvely, ths condton requres the support of prvate types to contan one type whch domnates all others n a partal order strctly more restrctve than the one ntroduced n Secton 3. Whle admttedly a strong restrcton, ths s to our knowledge one of the frst postve results on effcency n smultaneous auctons for complementary goods. Proofs of all propostons and lemmas are collected n the Appendx. Related lterature The effect of complementartes n smultaneous auctons has been studed, among others and n very specfc setups, by Bkhchandan (1999), by Rosenthal and Wang (1996) and Szentes and Rosenthal (2003) n smultaneous frst-prce auctons, by Krshna and Rosenthal (1996) n smultaneous second-prce aucton, and by Brusco and Lopomo (2002, 2009) and Cramton (1997) n smultaneous ascendng auctons. Szentes and Rosenthal (2003) study the smultaneous frst-prce mechansm n a complete nformaton settng wth two dentcal players who compete va smultaneous frst-prce auctons for three dentcal objects. Ther analyss hghlghts the challenges nvolved n study of the smultaneous frst-prce mechansm even n relatvely smple settngs, equlbrum turns out to have subtle and surprsng propertes. Smlar complexty arses n Krshna and Rosenthal (1996), who study a settng where many dentcal objects are auctoned va smultaneous secondprce auctons to two types of bdders: global bdders, who bd n multple auctons, and local bdders, who bd n one aucton only. Global bdders preferences are characterzed by a onedmensonal prvate type descrbng ther valuaton for each (dentcal) sngle object, wth a determnstc, common knowledge synergy realzed n the event of a multple wn. In contrast, we allow a much rcher type space n whch global bdders to have prvate valuatons for each of the possble (2 L 1) 1 combnatons of auctoned objects. We thereby take a sgnfcant step toward characterzng equlbrum n a broad class of smultaneous aucton games. Games of ncomplete nformaton wth payoffs supermodular n actons have also been stud- 6

7 ed by, among others, Athey (2001), McAdams (2003) and Reny (2011). Vves (1990) establshed the exstence of pure-strategy equlbra n Bayesan games when payoffs are supermodular and upper-semcontnuous n actons. Ths could provde an alternatve path to establshng exstence when the bd space s fnte, but does not speak to monotone equlbra. Meanwhle, Van Zandt and Vves (2007) demonstrate exstence of monotone pure strategy equlbra n games wth supermodular utlty assumng (among other condtons) that utlty s contnuous and exhbts ncreasng dfferences n own and rval actons. The latter condton does not hold even n fnte bd spaces, and therefore cannot be appled n our settng. 4 There s also a substantal lterature analyzng propertes of varous combnatoral aucton mechansms. Notable studes n ths lterature nclude Cantllon and Pesendorfer (2006), Ausubel and Mlgrom (2002), Ausubel and Cramton (2004), Cramton (2006), Krshna and Rosenthal (1996), Klemperer (2008, 2010), Mlgrom (2000a, 2000b), to menton just a few. Detaled surveys of ths lterature are gven n de Vres and Vorha (2003) and Cramton et al. (2006). Whle these studes also consder settngs where bdders have preferences over combnatons, the theoretcal problems generated by smultaneous bddng dffer substantally from those encountered n true combnatoral mechansms. 5 4 As usual n auctons, once one moves from fnte to contnuous bd spaces, utlty s no longer ether contnuous or sem-contnuous n actons. Hence results based on these no longer apply. 5 Though only tangentally related to our problem, there s also a growng lterature on mult-unt dscrmnatory auctons of homogeneous objects. Reny (1999, 2011), Athey (2001), and McAdams (2006) address exstence and propertes of equlbrum n such auctons. Meanwhle, Hortacsu and Puller (2008), Hortacsu and McAdams (2010), and Hortacsu (2011) provde more emprcal perspectves on mult-unt auctons. 7

8 2 Smultaneous standard auctons wth complementartes Consder a settng n whch N rsk-neutral bdders compete for L przes allocated va a class of mechansms we call smultaneous standard auctons, defned as follows: Defnton 1 (Smultaneous standard auctons). We say that objects l = 1,..., L are allocated va smultaneous standard auctons f the bddng mechansm s such that: 1. Bdders may bd for each object l = 1,..., L ndvdually, but may not submt combnaton or contngent bds; 2. Each object l s allocated to a hgh bdder n aucton l, wth payments condtonal on allocaton determned solely by bds n aucton l. Note that whle allocatons are always to a hgh bdder, payment rules need not be the same across l. In what follows, we frame dscusson n terms of a sngle seller, although ths s nessental for our results. For ease of exposton, n analyzng monotoncty and bddng we wll ntally assume that tes are broken randomly and ndependently across objects: Assumpton 1 (Independent te-breakng). Tes are broken ndependently across auctons; te-breakng does not depend on bdders types. We wll mantan ths assumpton through Secton 5.1, whch demonstrates exstence of monotone equlbra n contnuous bd spaces wth one global bdder. It wll be dropped n Secton 5.2, when we consder monotone equlbra wth endogenous tebreakng n contnuous bd spaces wth many global bdders. 8

9 Let an outcome from the perspectve of bdder be an L 1 ndcator vector ω wth a 1 n the lth place f object l s allocated to bdder and a 0 n the lth place otherwse. Smlarly, let the outcome matrx Ω for bdder be the (2 L 1) L matrx whose rows contan (transposes of) each possble outcome ω 0: e.g. f L = 2, Ω T = In what follows, we use the squared Eucldean norm ω 2 to denote the number of objects allocated to bdder n outcome ω. Bdder preferences Let Y ω denote the combnatoral valuaton bdder assgns to outcome ω. We normalze the outcome wn nothng to zero, and assume that valuatons are nondecreasng n the set of objects won: Assumpton 2 (Values Normalzed and Non-decreasng). Y 0 = 0 and Y ω s non-decreasng n the vector of objects won: ω ω mples Y ω Y ω. Let Y be the (2 L 1) 1 vector descrbng the combnatoral valuatons assgns to all possble wnnng outcomes (normalzng Y 0 = 0 as above), wth elements of Y correspondng to rows n Ω. In what follows, we nterpret Y as bdder s prvate type n the bddng game, known to bdder but unknown to rvals at the tme of bddng. We further assume that prvate types Y are..d. across bdders: Assumpton 3 (Independent Prvate Values). Each bdder draws prvate type Y from a contnuous c.d.f. F Y, wth compact support Y R 2L 1, wth F Y, common knowledge and types drawn ndependent across bdders: Y Y j for all, j. 9

10 As our focus s on monotone equlbra, n the bulk of our analyss we wll further assume that objects are complements n the sense that combnatoral valuatons are supermodular n the set of objects won: Defnton 2. We wll say that bdders have supermodular valuatons f for any outcomes ω 1, ω 2, Y ω 1 ω 2 + Y ω 1 ω 2 Y ω 1 + Y ω 2, where ω 1 ω 2 denotes the meet of ω 1, ω 2 and ω 1 ω 2 denotes the jon of ω 1, ω 2. Supermodularty mples that wnnng a larger set of objects ncreases the margnal valuaton assgns to any addtonal object. Actons and strateges Let B l be the set of feasble bds for bdder n aucton l. For each bdder, we assume that B l s a compact subset of R +. 6 The acton space for bdder s the set of L 1 bd vectors b = (b 1,..., b L ) T, wth b B = l B l and B a lattce n R L. As usual, a pure strategy for bdder s a functon σ : Y B. Let σ = (σ 1,..., σ N ) denote a pure strategy profle for all bdders, and σ = (σ 1,..., σ 1, σ +1,..., σ N ) denote a strategy profle for all bdders except. 7 Jont and margnal wnnng probabltes Let P (b; σ ) be the (2 L 1) 1 vector descrbng the probablty dstrbuton over outcomes arsng when submts bd b B l aganst rval strateges σ, wth P ω (b; σ ) the element of P (b; σ ) descrbng the probablty of outcome ω. Smlarly, let Γ (b; σ ) be the L 1 vector descrbng margnal wn probabltes arsng 6 Whle we do not explctly model reserve prces, these can easly be accommodated n our framework by ntroducng a dummy bdder whose acton space s a sngleton ncludng only the relevant reserve prces. 7 Note that although we do not dscuss reserve prces explctly, our framng here n fact mplctly accommodates arbtrary reserve prces. One could, for nstance, smply nclude a dummy bdder whose bd space n each aucton s a sngleton equal to the relevant reserve. All results developed below would then mmedately extend. 10

11 when submts bd vector b B l aganst rval strategy profle σ, wth Γ l (b; σ ) the margnal probablty wns aucton l. Observe that Γ (b; σ ) s related to P (b; σ ) by Γ (b; σ ) = Ω T P (b; σ ). Under Assumpton 1, Γ l (b; σ ) depends only on bd b l. Furthermore, f tes occur wth probablty zero, Γ l (b; σ ) s the c.d.f. of the maxmum rval bd n aucton l. Interm payoffs and expected payments Let π (b ; y, σ ) denote the expected nterm payoff of bdder wth type y Y submttng bd vector b aganst rval strateges σ. Mantanng Assumptons 1-3, we may wrte π (b ; y, σ ) as follows: π (b ; y, σ ) = y T P (b; σ ) L c l (b l ; σ ), (1) l=1 where c l (b l ; σ ) denotes s expected mechansm-determned payment n aucton l as a functon of s bd b l n aucton l gven rval strateges σ. For example, f aucton l s a frst-prce aucton, then we would have c l (b l ; σ ) = b l Γ l (b l ; σ ). Note that the addtvely separable form for payments follows jontly from our hypotheses of standard auctons and ndependent tebreakng; the former mples that payments n aucton l depend only on allocatons and bds n aucton l, whle the latter mples that allocatons n aucton l depend only on bds n aucton l. 11

12 3 Monotone best responses n smultaneous standard auctons wth complementartes A natural frst queston n analyss of smultaneous auctons s whether bddng strateges are monotone n any natural economc sense. As we show n Sectons 4-6, monotoncty s useful n analyzng both techncal questons such as exstence of equlbrum and economc questons such as expected neffcency n large markets. Furthermore, nsofar as we are focusng on a settng wth complementartes between objects, t s natural to expect that hgher valuatons should n some sense translate nto hgher bds. In ths secton, we show that ths s n fact the case: there s a partal order on the space of types Y such that f y y, then has a best reply at type y whch s coordnatewse greater than any best reply at type y. Importantly, however, ths partal order s not the usual coordnatewse partal order; ndeed, even when objects are complements, a strct coordnatewse ncrease n s type can lead to a strct coordnatewse decrease n all elements of s best-response bd. Intutvely, ths s because the usual coordnatewse order on Y mposes nsuffcent structure on the margnal value added by any addtonal object. We begn by llustratng falure of coordnatewse monotoncty n the context of a twoobject smultaneous frst prce example, then proceed to develop the partal order and show that ths s suffcent to restore monotoncty of best responses. Example 1. Consder two bdders competng for two objects va smultaneous frst-prce auctons. Suppose that bdder 2 s fxed strategy s to bd ether n aucton 1 (wth probablty 1) or 2 n aucton 2 (wth probablty 1 ), drawng bds from the unform U[0; 1] dstrbuton n ether 2 case. Consder two types for bdder 1: y 1 = (0, 0, 0, 2) T, y 1 = (0, 1, 1, 5/2) T. 12

13 For b 1, b 2 [0, 1], these types correspond to the followng proft functons: ( 1 π = b 1 2 b ) ( 2 1 b b ) 1 + (2 b 1 b 2 ) b1 + b 2, 2 2 ( 1 π = b 1 2 b ) ( ) ( b 2 2 b ) ( ) b 1 b 2 b1 + b 2, 2 yeldng best response bds b = ( 1, ) 1 T 2 2 and b = ( 1, 1 T 4 4) respectvely. Ignorng the frst component, whch corresponds to the case of wnnng no auctons, we see that y 1 s strctly greater than y 1 n the coordnatewse sense. Thus even when objects are complements n the (strong) sense of supermodular valuatons, a strct coordnatewse ncrease n type (from y 1 to y 1) can generate a strct coordnatewse decrease n s best response bd (from b to b ). As ponted out by Reny (2011) n a substantally dfferent context (mult-unt auctons wth rsk-averse bdders), the fundamental problem s that the coordnatewse partal order on types mposes nsuffcent structure on margnal value added: for nstance, when movng from y 1 to y 1 n Example 1, the value added by object 2 (n events where s already wnnng object 1) falls from 2 to 0.5 even as the value assgns to wnnng objects 1 and 2 together ncreases from 2 to 2.5. The soluton s to seek an alternatve partal order on Y whch mposes addtonal structure on these margnal valuatons. 3.1 Partal order on types We next turn to constructon of an alternatve partal order on Y suffcent to restore monotoncty of best reples. Specfcally, buldng on the ntuton n Example 1, we seek a partal order on Y such that types whch are hgher wth respect to also have hgher margnal valuatons. Toward ths end, we frst make precse the noton of margnal valuatons n ths combnatoral context: Defnton 3 (Margnal valuatons). Let ω and ω be two outcomes such that ω ω n the coordnatewse sense. For bdder, the margnal valuaton of objects correspondng to allocaton 13

14 ω relatve to those n allocaton ω s defned as the dfference Y ω Y ω. Recall that under Assumpton 2 all margnal valuatons are non-negatve. We seek a partal order on the space of types Y such that y y mples that every margnal valuaton s hgher for type y than for type y. Bearng n mnd the combnatoral nature of margnal valuatons, ths leads to the followng defnton for the partal order : Defnton 4 (Partal order). We wll say that ỹ y f and only f for any outcome ω and any object l such that ω l = 0 we have ỹ ω e l ỹ ω y ω e l y ω. (P O) Note that (by constructon) the partal order (P O) s more restrctve than the usual coordnatewse order on Y. In partcular, choosng ω = 0, we fnd that ỹ y mples ỹ e l any object l, whch n turn mples ỹ e l e m y e l e m y e l for for any l m and so forth. Proceedng nductvely n ths way, we ultmately conclude that ỹ y mples ỹ y coordnatewse, as desred n vew of the dscusson above. Whle the economc motvaton for the partal order (P O) s clear, t also has a useful geometrc nterpretaton. Intutvely, ths nterpretaton arses from the observaton that there s a postve cone generatng our partal order. Specfcally, defne the set Z 2 L 1 as follows: Z 2 L 1 = { } z R 2L 1 : takng z 0 = 0, (ω, l such that ω l = 0) z ω e l z ω 0. (2) Then Z 2 L 1 s a sold cone located n the frst orthant of the 2 L 1-dmensonal space R 2L To remnd the readers, a nonempty subset Z of a vector space s sad to be a cone f t satsfes the followng three propertes: () Z + Z Z, () αz Z for all α 0, () Z ( Z) = {0}. Any cone wth a nonempty 14

15 Furthermore, for any y R 2L 1, the set y +Z 2 L 1, whch amounts to the translaton of the cone Z 2 L 1 n R 2L 1 to vertex y, represents all the realzatons of s type n R 2L 1 that domnate y n the sense of partal order (P O). Smlarly, the set y Z 2 L 1, whch amounts to the rotaton of Z 2 L 1 and then ts translaton n R 2L 1 to vertex y, represents all the realzatons of s type n R 2L 1 that are domnated by y under the partal order (P O). 3.2 Monotone best reples To conclude ths secton, we demonstrate that the addtonal structure mposed by the partal order (P O) s n fact suffcent to restore an economcally meanngful noton of monotoncty. Specfcally, gven any set of rval strategy profles σ, we show that f ỹ y n the sense of (P O), then at least one element of s best reply bd correspondence at type ỹ s coordnatewse no smaller than every element of s best reply bd correspondence at type y. We prove ths proposton n three steps. Frst, combnng our hypothess of smultaneous standard auctons wth Assumptons 1 3 above, we show that supermodularty of valuatons n the set of objects won mples supermodularty of nterm payoffs π (b ; y, σ ) as a functon of b : Lemma 1. Suppose that Assumptons 1 3 hold. Fx a rval pure strategy profle σ. Let y be a realzaton of bdder s type. If valuatons are supermodular n the sense of Defnton 2, then the nterm payoff functon π (b ; y, σ ) = y T P (b ; σ ) L c l (b l ; σ ) l=1 s supermodular n b. nteror s a sold cone. In ths case the nteror of Z 2L 1 s clearly nonempty n R 2L 1 ; e.g. for L = 2 we have (z e1, z e2, z e1 e2 ) T = (1, 1, 2) T Z o 3, for L = 3 we have (z e1, z e2, z e3, z e1 e2, z e1 e3, z e2 e3, z e1 e2 e3 ) T = (1, 1, 1, 2, 2, 2, 3) T Z o 7, and so forth for L > 3. Hence Z 2 L 1 s a sold cone n R 2L 1. 15

16 Second, we establsh that bdders nterm payoffs satsfy the followng weak sngle crossng property n y : Lemma 2. Mantanng Assumptons 1 3, fx a pure strategy σ for the rval bdders. Suppose that types ỹ and y are such that ỹ y n the sense of the partal order (P O), and suppose that b b n the coordnatewse sense. Then π ( b ; y, σ ) π (b ; y, σ ) = π ( b ; ỹ, σ ) π (b ; ỹ, σ ). Fnally, we combne these results to establsh the followng weak monotoncty property on the set of s best reples to σ : Proposton 1. Mantanng Assumptons 1 3, suppose that valuatons are supermodular n the sense of Defnton 2. Fx a pure strategy profle σ for the rval bdders. Let b B be a best response to σ when bdder s type s y, and b B be a best response to σ when s type s ỹ, where ỹ and y are such that ỹ y n the sense of the partal order (P O). Then the bd vector b b s also a best response to σ when s type s ỹ. Proposton 1 does not, of course, guarantee exstence of a best reply strategy σ to σ. It does, however, mply that f such a best reply exsts, then there also exsts a best reply strategy σ whch s monotone n that y y n the sense of (P O) mples σ (y ) σ (y ) n the usual coordnatewse order. Ths n turn provdes a foundaton for our analyss of monotone equlbrum below. 4 Monotone equlbrum n fnte bd spaces Buldng on the partal order (P O), we next turn to consder monotone equlbrum n fnte bd spaces. Specfcally, suppose that B s a fnte lattce. Under an addtonal support condton on Y to be defned shortly, we show that there exsts a Bayes-Nash equlbrum n pure strateges (σ1,..., σn ) : Y B wth the property that y y n the sense of (P O) mples σ (y ) σ (y ) n the usual coordnatewse order. 16

17 Toward ths end, we requre one addtonal assumpton, whch guarantees that the support of Y s suffcently rch to permt meanngful comparsons wth respect to the partal order (P O): Assumpton 4. There s a countable subset Y of Y such that every set n F Y - sgma-algebra assgned postve probablty by F Y contans two ponts between whch (here between s understood n the partal order sense) les a pont n Y. For ths assumpton to hold jontly wth atomlessness of the dstrbuton of Y, t s necessary that there exsts a postve F Y -measure of ponts n the support Y that can be compared to each other by means of the partal order (P O). In partcular, (P O) should not reduce to the trval partal order on Y. Fortunately, Assumpton 4 turns out to follow from natural regularty condtons on the dstrbuton of Y : Proposton 2. Suppose that Y s compact and has non-empty nteror wth respect to R 2L 1, and that the boundary of Y has zero Lebesgue measure n R 2L 1. Further suppose that the jont dstrbuton of the (2 L 1)-dmensonal vector Y s absolutely contnuous wth respect to the Lebesgue measure on R 2L 1. Then Assumpton 4 holds. We now turn to ths secton s man result: under Assumptons 1 4, at least one Bayes- Nash equlbrum monotone n the sense of (P O) exsts on any fnte bd lattce. In vew of Proposton 1, the proof of ths statement s relatvely straghtforward. Restrctng bds to a fnte lattce guarantees the contnuty of the nterm payoff functon, whch ensures that each player s nterm best reply correspondence s non-empty. Hence by Proposton 1, we conclude that each player has a best reply whch s monotone and jon closed wth respect to the partal order (P O) on types and the usual coordnatewse order on bds. To guarantee exstence of an equlbrum n monotone pure strateges, we therefore need only verfy condtons G.1-G.6 of Reny (2011), after whch Theorem 4.1 of Reny (2011) delvers the result. Of these, only condton G.3 s potentally problematc, leadng to our Assumpton 4 and suffcent condtons provded n Proposton 2. We thereby conclude: 17

18 Proposton 3. Mantanng Assumptons 1 4, suppose that valuatons are supermodular n the sense of Defnton 2. for each bdder the bd space B R L s a fnte lattce. 9 Then there s an equlbrum n pure strateges whch are monotone wth respect to the partal order (P O) on types and the coordnatewse partal order on bds:.e. such that y y n the sense of (P O) mples b b n the usual coordnatewse sense. As n Reny (2011), Proposton 3 mmedately extends to exstence of symmetrc monotone equlbra when bdders are symmetrc: Corrolary 1. In addton to the hypotheses of Proposton 3, suppose that bdders are symmetrc n the sense that Y = Y j, F = F j, and B = B j for all bdders, j. Then there s an equlbrum n symmetrc pure strateges monotone wth respect to the partal order (P O) on types and the coordnatewse partal order on bds. The proof of ths corollary follows mmedately from the proof of Proposton 3 gven n the Appendx, but nvokng Theorem 4.5 rather than Theorem 4.1 of Reny (2011). We therefore do not provde a separate proof. Note that the man contrbuton of Proposton 3 s not exstence per se, 10 but rather exstence n strateges whch are monotone wth respect to a sutably defned partal order. Snce our analyss n Sectons 5 and 6 pvots on monotoncty, ths addtonal structure turns out to be essental. In Example 2 below we consder a settng wth two objects and two ex-ante symmetrc bdders whose valuaton for the bundle of both objects s the sum of standalone valuatons and a non-negatve determnstc complementarty denoted as k. The bddng set conssts of two ponts and, thus, s dscrete. At the lower bd no loss wll be ncurred ex-post. However, a bdder may potentally ncur a loss when submttng the hgher bd, and whether such a loss happens depends on bdder s type as well as what that bdder won. 9 E.g., B = L l=1 B l where each B l conssts of the fnte number of ponts. 10 Exstence of equlbrum n pure strateges could be obtaned by, for nstance, applyng results n Mlgrom and Weber (1985). 18

19 We consder three dfferent aucton formats (frst-prce, second-prce and all-pay) and llustrate symmetrc BNE n pure monotone strateges. In ths example we fnd that for moderate and hgh levels of complementarty n frst-prce and all-pay auctons exposure happens wth a postve probablty and the probablty of exposure as well as the maxmal degree of exposure s ncreasng n k up to some level, after whch t becomes constant. An analogous result holds for the second-prce aucton wth the dfference that exposure happens for any k 0. We generally fnd that the exposure problem (lookng at the maxmum degree of exposure) n the all-pay aucton s not more severe than that n the frst-prce aucton due to bdders bddng more cautously n the all-pay aucton, and the exposure problem n the frst-prce aucton s not more severe than that n the second-prce aucton. Example 2. Consder the case of two symmetrc global bdders competng for two objects. The valuaton for havng two objects s are two ndependent random varables dstrbuted un- where k 0 s constant, Y e 1 and Y e 2 formly on [0, 1]. Y (1,1) = Y e 1 + Y e 2 + k, = 1, 2, for each object, Suppose the the bdders are only allowed to bd ether b 1 = 0 or b 2 = 1 2 and tes for objects are broken ndependently wth a far lottery. We characterze equlbrum bddng strateges n a BNE n symmetrc pure monotone strateges. We llustrate how BNE changes when complementarty k ncreases and dscuss occurrences of exposure. Frst-prce aucton If k = 0 and, thus, we are n the case of case of no complementartes from wnnng two objects. Each bddes always bddng 0 for each object s a BNE. For k (0, 1), the equlbrum bddng strateges are ( ) β Y (1,0), Y (0,1) = ( 1, ) 1 2 2, f (Y e 1, Y e 2 ( ) A, 1, 0), f (Y e 1 2, Y e 2 ( ) ) B, 0, 1 2, f (Y e 1, Y e 2 ) C, (0, 0), f (Y e 1, Y e 2 ) D, (3) 19

20 where regons A,B,C and D are defned as follows: { A = (y e 1, y e 2 ) [0, 1] 2 : y (1,0) + y (0,1) 2γ(k) k 2, ye 1 γ(k) k 2, ye 2 γ(k) k }, (4) 2 { C = (y (e 1, y e 2 ) [0, 1] 2 : y e 1 < γ(k) k } 2, ye 2 γ(k), { B = (y e 1, y (e 2 ) [0, 1] 2 : y e 1 γ(k), y e 2 < γ(k) k }, 2 D = [0, 1] 2 \(A B C), and k 3 8 γ(k) = + 3k2 k k 2 + k + 1. k (0, 1]. 2 2 Fgure 1 llustrates the BNE n symmetrc monotone pure strateges when k = 0.2 and k = 0.6. For k > 0 small enough, there no exposure for any of the bdders. y e2 y e2 1 1 C A C A D B D B 1 y e1 1 y e1 k = 0.2 k = 0.6 Fgure 1: Illustraton to Example 2 (frst-prce aucton). However, for k (kfp,exp, 1] wth k fp,exp some bdders wth valuatons n regon A become exposed when wnnng one object only. The maxmum possble exposure (ex-post loss) 3 n stuaton s k 4 + k k 2 1, whch ncreases n k from 0 (at k = k k 2 +k+2 fp,exp ) to 0.5 (at k = 1). For k (1, 2], the equlbrum bddng strateges are ( ) { ( 1 β Y (1,0), Y (0,1) =, ) 1 2 2, f (Y e 1, Y e 2 ) A, (0, 0), f (Y e 1, Y e 2 ) D, 20 (5)

21 where regons A and D are descrbed as follows: D = { (y e 1, y e 2 ) [0, 1] 2 : y e 1 + y e 2 γ(k) }, (6) A = [0, 1] 2 \D, wth γ(k) = 2 k 1 + (2 k)(k 1) + 1. Some bdders wth valuatons n regon A are exposed when wnnng one object only. The maxmum possble exposure (ex-post loss) n stuaton s 0.5. When k 2, bdders always submt b 2 = 1 for each object. Agan, some bdders are exposed 2 when wnnng one object only and the maxmum possble exposure (ex-post loss) n stuaton s 0.5. Second-prce aucton In a second-prce aucton, wth k = 0, the equlbrum bddng strateges are such that each bdder submts b 1 n aucton l f y e l < 1 and submts b 3 2 otherwse. Due to the second-prce nature of the aucton, some bdders may be exposed even n the stuaton of k = 0 when wnnng one object and havng to pay b 2 for that object. The maxmum possble degree of exposure n ths stuaton s equal to 1. 3 When the complementarty becomes strctly postve, the equlbrum bddng strateges change. When k (0, ksp), where ksp (to be exact, ksp satsfes (k sp) 3 + (k sp) 2 + k 8 8 sp 1 = 0), then the monotone bddng strateges have the form (3)-(4) wth γ(k) = k k2 + k k k 2 Some bdders wth values n A are exposed when wnnng one object and payng b 2 for that object. The maxmum possble degree of such exposure n ths stuaton s equal to 0.5+ k 4 k 3 8 k k k 2 Moreover, when k [0, ksp,exp1), where ksp,exp , some bdders wth values n A may be exposed even when wnnng both objects and havng to pay b 2 for each object. The maxmum 3k 2 1 k 2 k k 2 + k2 2 possble degree of such exposure s equal to t decreases n k from (at k = ksp,exp1). For k [ksp, 2), the equlbrum bddng strateges are n the form (5)-(6) wth γ(k) = 2 k 1 + (2 k)(k + 1) + 1. (at k = 0) to In ths case some bdders wth values n A are exposed when wnnng one object and payng 21

22 b 2 for that object. The maxmum possble degree of such exposure n ths stuaton s equal to 0.5. Moreover, some bdders wth values n A may be exposed even when wnnng both objects and havng to pay b 2 for each object. The maxmum possble degree of such exposure s equal to 2 k 1 + (2 k)(k + 1) For k 2, bdders always submt b 2 = 1 for each object. The maxmum possble degree of 2 such exposure s equal to 1. The graphcal llustraton of these regons and equlbrum bddng behavor for k = 0.2 and k = 0.6 s gven n Fgure 2. y e2 y e2 1 1 C A C A D B 1 y e1 D B 1 y e1 k = 0.2 k = 0.6 Fgure 2: Illustraton to Example 2 (second-prce aucton). All-pay aucton In an all-pay aucton, wth k = 0, the equlbrum bddng strateges are such that each type submts b 1 = 0 for each object. When k (0, kap), where kap (to be exact, kap satsfes (k ap) 3 1 = 0), the monotone bddng strateges have the form (3)-(4) wth γ(k) = k 3 + k k 2 + k k ap 16 2 For k > 0 small enough, there no exposure for any of the bdders. However, for k (k ap,exp, 4] wth k ap,exp some bdders wth valuatons n regon A become exposed when wnnng one object only. The maxmum possble exposure (ex-post loss) 22

23 k 3 + k2 16 k 2 4 n stuaton s k (k + 1) For k [kap, 4), the equlbrum bddng strateges are n the form (5)-(6) wth γ(k) = 4 k 1 + k(4 k) + 1. In ths case some bdders wth values n A are exposed when wnnng one object and payng b 2 for that object. The maxmum possble degree of such exposure n ths stuaton s equal to 0.5. For k 4, bdders always submt b 2 = 1 for each object. Agan, some bdders are exposed 2 when wnnng one object only and the maxmum possble exposure (ex-post loss) n stuaton s 0.5. The graphcal llustraton of the equlbrum bddng strateges for k = 0.2 and k = 0.6 s gven n Fgure 3. y e2 y e2 1 C A 1 C A D B D B 1 y e1 1 y e1 k = 0.2 k = 0.6 Fgure 3: Illustraton to Example 2 (all-pay aucton). Smlar to the case of sngle-object aucton, for k > 0 (up to a threshold), bdders equlbrum behavor s more aggressve n the second-prce aucton than n the frst prce aucton, and s more cautous n the all-pay aucton than n the frst-prce aucton. 23

24 5 Monotone equlbra n contnuous bd spaces We now turn from fnte to contnuous bd spaces, applyng Proposton 3 to establsh two new results. We begn wth a specal case nspred by Krshna and Rosenthal (1996), n whch a sngle global bdder bds for multple objects aganst many local bdders who bd for sngle objects only. We demonstrate exstence of a monotone pure strategy Bayes-Nash equlbrum n ths context, buldng on lmtng technques n Reny (2011) to extend from dscrete to contnuous bd spaces. Whle nstructve n ts own rght, ths example also hghlghts a subtle techncal challenge: wth more than one global bdder, nteracton between strategc overbddng by global bdders and strategc dependence of bds across auctons renders uncertan a key techncal condton better-reply securty of Reny (1999) needed to complete the extenson proof. Returnng to the full model, we therefore consder nstead a more general soluton concept nspred by the work of JSSZ (2002): equlbrum wth endogenous tebreakng. We defne ths soluton concept n detal n Secton 5.2, and demonstrate that equlbra wth endogenous tebreakng exst for any number of global bdders. 5.1 One global bdder and many local bdders Frst consder the followng specal case of our general model: suppose that one global bdder competes for L objects aganst many local bdders, wth each local bdder competng n exactly one aucton. Let bdder 0 denote the global bdder and let l 1,..., l N denote the auctons n whch local bdders 1,..., N are competng respectvely. We specalze our assumptons to ths envronment as follows: Assumpton 5. The global bdder s rsk-neutral and draws prvate type Y 0 (whch contans combnatoral valuatons for all ω 0) from a contnuous (2 L 1)-varate c.d.f. F 0 wth compact support Y 0 R 2L 1. The global bdder s valuatons are monotone (Y ω Y ω for ω ω) and supermodular n the sense of Defnton 2. 24

25 Assumpton 6. Each local bdder s rsk-neutral and draws prvate type Y for object l from unvarate contnuous c.d.f. F wth compact support Y R, = 1,..., N. Types are ndependently dstrbuted across all bdders. For smplcty, and for ths secton only, further suppose that each aucton l = 1,..., L s a frstprce aucton. We conjecture that smlar results could be shown for other standard auctons, but do not pursue ths further here as our goal s llustraton. For each l = 1,..., L, let B l R be a compact nterval descrbng feasble bds n aucton l. Then a pure a strategy for the global bdder 0 s a map σ 0 : Y 0 B 0 wth B 0 L l=1 B l, whle a pure strategy for local bdder s a map σ : Y B l. It s straghtforward to construct a sequence of fnte lattces { B k 0; B k 1,..., B k L } k=1 such that as k each B k j becomes ncreasngly dense n B j, j = 0,..., L. By Proposton 3, for each k the collecton of such fnte lattces wll nduce a monotone equlbrum strategy profle σ k, where monotoncty of the strategy of the global bdder s understood n the sense of the partal order (P O) on the type space and the coordnatewse order for bd vectors, and monotoncty of the strateges of the local bdders are understood n the usual unvarate sense. By Lemma A.13 n Reny (2011), the space of strateges monotone wth respect to our partal order s compact n the pontwse convergence topology, hence the sequence of strateges { σ k } k=1 wll have a subsequence { σ k j } j=1 whch converges pontwse a.e. to a lmt σ. Ths lmt σ s a monotone pure strategy profle by constructon; we seek to show that t also defnes an equlbrum on the contnuous bd space B {B 0 ; B 1,..., B L }. To acheve ths, we apply the concept of better-reply securty ntroduced n Reny (1999), defned formally as follows: Defnton 5 (Secure a payoff). Player can secure a payoff of α R at strategy profle σ S f there exsts σ S such that Π ( σ ; σ ) α for all σ n some open neghborhood of σ. Defnton 6 (Better-Reply Secure). A game G = (S, Π ) N =1 s better-reply secure f whenever 25

26 ( σ, Π) s n the closure of the graph of the vector payoff functon Π( ) and σ s not an equlbrum, some player can secure a payoff strctly above Π at σ. By Remark 3.1 n Reny (1999) (p. 1038), f a game s better-reply secure, then the lmt of a convergent sequence of ɛ-equlbra, as ɛ tends to zero, are pure strategy equlbra. To establsh exstence of an equlbrum n monotone pure strateges on the contnuous space B, t s therefore suffcent to demonstrate the followng: () There exsts a sequence of fnte lattces { B k 0; B k 1,..., B k L } k=1 such that f σk s a monotone pure strategy equlbrum on { B k 0; B k 1,..., B k L } for each k, then { σk } k=1 s a sequence of ɛ-equlbra on B for whch ɛ 0; () The bddng game s better-reply secure when bds may be submtted on B. We now establsh each of these n turn. Lemma 3. Suppose that Assumptons 1, 4, 5, and 6 hold, and that each aucton l = 1,..., L s a frst-prce aucton. Let { B 0; k B 1, k..., B L k} k=1 be any sequence of fnte lattces such that: 1. Bk 0 B 0, B k l B l for all l = 1,..., L, and H( B k j, B j ) 0 as k, for j = 0,..., N, where H(, ) stands for the Hausdorff dstance. 2. For each l = 1,..., L, B l k s a subset of B 0l k such that mn B 0l k < mn B l k, max B 0l k > max B l k, and for any b l, b l B l k there exsts a pont b l B 0l k such that b l < b l < b l. Let σ k be a monotone pure strategy equlbrum for bd space B k. Then for any sequence {ɛ m } such that ɛ m > 0 and ɛ m 0, there exsts a subsequence {k m } m=1 of k = 1, 2,... such that strategy profle σ km s an ɛ m -equlbrum on the unrestrcted B. Note that the proof of Lemma 3 turns on choosng lattces { B k 0; B k 1,..., B k 1} such that the bd lattce of the global bdder s always fner than the product of the bd lattces of the local bdders. Ths guarantees that at each pont along the sequence of fnte grds consdered the 26

27 global bdder can resolve tes n the drecton she most prefers. Ths n turn allows us to use the global bdder s revealed preference on fnte lattces to bound the her potental gans from devaton n contnuous bd space. Lemma 4. Suppose that Assumptons 1, 4, 5, and 6 hold, and that each aucton l = 1,..., L s a frst-prce aucton. Then the smultaneous frst-prce aucton game wth one global and many local bdders s better-reply secure when consderng monotone strateges played by the bdders. Whle we relegate detals to the Appendx, we emphasze that the proof of Lemma 4 s more complcated here than n standard sngle- or mult-unt auctons. In standard auctons, betterreply securty follows almost automatcally from the fact that bdders almost surely bd below ther margnal valuatons. One can therefore construct payoff-securng devatons by slghtly ncreasng bds at any pont nvolvng potental tes. 11 Here, n contrast, the global bdder may strategcally overbd for a gven object.e. to submt a bd strctly above her margnal valuaton n the hope of wnnng hgher-order combnatons. The proof of Lemma 4 therefore n fact turns on ndependence of rval bds faced by the global bdder. Ths allows us to assert that any ncrease n the margnal probablty of wnnng object l proportonally ncreases the probablty of wnnng all combnatons nvolvng object l, and hence to conclude that the global bdder wll always want to break relevant tes n her favor. Fnally, combnng Lemmas 3 and 4, Remark 3.1 n Reny (1999), and Lemma A.13 of Reny (2011) as descrbed above, we obtan ths subsecton s man result: Proposton 4. Suppose that Assumptons 1, 4, 5, and 6 hold, and that each aucton l = 1,..., L s a frst-prce aucton. In the smultaneous frst-prce aucton game wth one global bdder and many local bdders, a monotone pure strategy equlbrum exsts on the compact convex bd space B. Now consder what may go wrong wth more than one global bdder. Recall that, due to the possblty of strategc overbddng by the global bdder, our proof of Lemma 4 (better-reply 11 see, e.g., Reny (1999, 2011) for examples of ths argument. 27

28 securty) turns crucally on ndependence of bds by local rvals across auctons. Unfortunately, however, f multple global bdders are present, bds by global rvals may n prncple exhbt arbtrary dependence across auctons through strateges σ. When combned wth the possblty of strategc overbddng, ths n turn may be problematc for better-reply securty. For example, consder a settng n whch two global bdders compete for two objects. Imagne a sequence of strateges along whch these bdders converge to a te n aucton 1, such that at each strategy profle n the sequence, any type of bdder 1 bddng just above the te pont n aucton 1 also submts a bd for object 2 whch wns aucton 2 wth certanty. Then any devaton by bdder 2 to a pont just above the te wll produce a strct ncrease n the probablty that bdder 2 wns aucton 1, wthout ncreasng the probablty that bdder 2 wns objects 1 and 2 together. If bdder 2 also engages n strategc overbddng for object 1, ths could n turn mply a strct decrease n bdder 2 s expected payoff. Hence even f the lmt profle (wth tes) s not an equlbrum, bdder 2 may not have a proftable devaton. Ths n turn undermnes any proof of better-reply securty parallelng Lemma 4, and thereby any proof of exstence parallelng Proposton Monotone equlbrum wth endogenous te-breakng Recall that the techncal challenge n establshng better-reply securty wth many global bdders s not overbddng per se; rather, t s the fact that by slghtly ncreasng her bd n aucton l, bdder may wn aucton l only aganst types of global rvals aganst whch s lkely to lose n other auctons. There may therefore n prncple exst a sequence of strateges {σ k } k=1, convergng to a lmt σ nvolvng tes, such that no type tyng at the lmt wshes to devate at any pont along the sequence, but a postve measure of tyng types wsh to devate at the lmtng profle σ. The fundamental problem n such a case s that ndependent tebreakng at the lmtng profle σ may lose nformaton regardng the order n whch near-tes are broken 28

29 along the sequence types tyng at the lmt could submt dfferent bds n aucton l (and hence have near-tes resolved wth dfferng precedences) at every strategy profle σ k n the sequence. If ths te-breakng order could be preserved n the lmt, then σ would n fact represent a monotone pure strategy equlbrum. Motvated by ths observaton, n analyzng smultaneous auctons wth many global bdders, we focus on a soluton concept whch generalzes Bayes-Nash equlbrum along the lnes proposed by Jackson, Smon, Swnkels and Zame (2002, henceforth JSSZ). In what follows, we refer to ths soluton concept as an equlbrum wth endogenous tebreakng n the sgnallng extenson of the smultaneous bddng game. We defne ths soluton concept formally as follows. Let G be a smultaneous aucton game satsfyng Assumptons 2 4 on prmtves above. In what follows, we no longer presume that tes are resolved ndependently across actons accordng to some pre-specfed tebreakng rule.e. we no longer mantan Assumpton 1 above. Rather, from ths pont forward, we nterpret G as a game of ndetermnate outcomes n the language of JSSZ (2002): that s, a game requrng only that each object l s awarded to a hgh bdder n aucton l, wthout specfyng whch hgh bdder wll receve the object n the event of a te. We defne the sgnallng extenson G s to the game of ndetermnate outcomes G by augmentng each bdder s strategy space as follows. For each aucton l, we allow bdder to submt, n addton to her bd vector b, a vector of cheap-talk sgnal s [0, 1] L ndcatng her desred tebreakng precedence n each aucton l. These sgnals are rrelevant for allocatons and payoffs except n case of tes, n whch case the auctoneer may consder (s 1,..., s N ) n determnng how to break tes. A pure strategy for bdder n the sgnallng extenson G s s therefore a functon σ s : Y B S, where σ : Y B denotes s bddng strategy as above and τ : Y S denotes s tebreakng strategy n the sgnallng extenson. 12 As above, let Y = Y, B = B, and 12 Snce we focus on pure strateges, wrtng bdder s strategy as the cross-product of her bddng and tebreakng strateges nvolves no loss of generalty. 29