SYMMETRIC AUCTIONS RAHUL DEB AND MALLESH M. PAI

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1 SYMMETRIC AUCTIONS RAHUL DEB AND MALLESH M. PAI ABSTRACT. Symmetrc (sealed bd) auctons such as frst, second, and all-pay auctons are commonly used n practce. One reason for ther popularty s that the rules of these auctons are far n that they are anonymous and nondscrmnatory. In an ndependent prvate value settng wth heterogenous buyers, we characterze the outcomes that can be mplemented when the aucton desgner s restrcted to usng a symmetrc aucton format. We show that symmetrc auctons can yeld a wde varety of dscrmnatory outcomes, such as revenue maxmzaton and affrmatve acton. We also characterze the set of mplementable outcomes when other desderata are mposed n addton to symmetry. These addtonal requrements may prevent the seller from maxmzng revenue. DEPARTMENT OF ECONOMICS, UNIVERSITY OF TORONTO DEPARTMENT OF ECONOMICS, UNIVERSITY OF PENNSYLVANIA E-mal addresses: rahul.deb@utoronto.ca, mallesh@econ.upenn.edu. Date: November 6, 04. We would lke to thank Drk Bergemann, Hesk Bar-Isaac, Amos Fat, Johannes Hörner, René Krkegaard, Bernard Lebrun, Kevn Lleyton-Brown, Steve Matthews, Maher Sad, Larry Samuelson, Xanwen Sh, Andrzej Skrzypacz, Jeroen Swnkels, Juuso Tokka, Gabor Vrag, and numerous semnar audences for ther nsghtful comments. We also thank Yunan L for provdng excellent research assstance. Mallesh Pa gratefully acknowledges the support of NSF Grant CCF-0389.

2 RAHUL DEB AND MALLESH M. PAI An optmal aucton extends the asymmetry of the buyer roles to the allocaton rule tself. The assgnment of the good and the approprate buyer payment wll depend not only on the lst of offers, but also on the denttes of the buyers who submt the bds. In short, an optmal aucton under asymmetrc condtons volates the prncple of buyer anonymty. J. Rley and W. Samuelson (98). Optmal Auctons, Amercan Economc Revew.. INTRODUCTION Symmetrc sealed bd auctons or, smply, symmetrc auctons are wdely used n practce. In these auctons, buyers submt sealed bds, the hghest bdder over the reservaton bd wns and the transfers are determned va an anonymous functon whch maps bds to payments. Standard examples are frst, second, and all-pay auctons. In each of these auctons, the wnner s determned n the same way; nstead, the auctons dffer n terms of the payment rules for the wnnng and losng bdders that n turn affect the outcomes through equlbrum bddng. More complex examples are frst or second prce auctons wth an all pay component (usually an entry fee, as n Levn and Smth (994)), k-prce auctons (Güth and Van Damme, 986), k-prce all-pay auctons (Goeree, Maasland, Onderstal, and Turner, 005), and auctons n whch wnners pay some combnaton of the hghest and second hghest bd (Lebrun, 03), to menton but a few. Symmetrc auctons have the advantage of havng rules that are anonymous and nondscrmnatory. Ths s one of the reasons that they reman popular n the real world despte the fact that they may not acheve the seller s goals such as revenue maxmzaton (when buyers are ex-ante heterogeneous) or affrmatve acton. The partcular symmetrc aucton format chosen by the seller depends on hs objectves and the envronment he faces. For nstance, a revenue maxmzng seller s preference rankng of the frst and second prce aucton s determned by the buyers value dstrbutons (Maskn and Rley, 000; Krkegaard, 0). Motvated by the ubquty of symmetrc auctons, the am of ths paper s to understand the degree of flexblty ths format offers for aucton desgn. Specfcally, we examne the set of outcomes that a seller can acheve when restrcted to usng such a mechansm. As a consequence, our analyss uncovers the extent to whch a seller can dscrmnate amongst buyers (va ther equlbrum bddng) usng a format that appears far on the surface. Mechansm desgn subject to symmetry constrants has a long hstory. In matchng theory, farness s a concern n unversty and publc housng allocatons, parkng space assgnment and student placement n publc schools. Hence, two of the most promnent allocaton mechansms (the random prorty mechansm (Abdulkadroğlu and Sönmez, 998) and the probablstc seral mechansm (Bogomolnaa and Mouln, 00)) are anonymous. More generally, anonymty as a condton has a rch background n the axomatc socal choce theory lterature. Our analyss departs from ths lterature n mposng the symmetry requrement on the ndrect mplementaton as opposed to on the drect mechansm. Addtonally, we dffer from the majorty of socal As we dscuss below, farness s often legally mandated. Often, other notons of farness such as the equal treatment of equals (two agents makng the same reports receve the same allocatons) and envy-freeness (each agent prefers her allocaton to that of any other agent) are mposed nstead.

3 SYMMETRIC AUCTIONS 3 choce theory n consderng the weaker Bayesan (as opposed to domnant strategy) ncentve compatblty crteron commonly employed n aucton theory. 3 It should also be ponted out that symmetry axoms play an mportant role n cooperatve game theory (to characterze the Shapley value) and n both the Nash and Kala-Smorodnsky barganng solutons. We consder a general ndependent prvate value settng wth ex-ante heterogenous buyers. We say that a drect mechansm has a symmetrc mplementaton f there s a symmetrc aucton that has an equlbrum (n undomnated strateges) for whch the outcome yelds the same (ex-post) allocaton rule and the same expected (nterm) payments. 4 Our man result (Theorems and ) s a complete characterzaton of the set of mplementable drect mechansms. Addtonally, we provde a smple qualtatve descrpton of the set of drect mechansms that are not mplementable (Corollary ). We frst argue that a drect mechansm s mplementable only f t s a herarchcal mechansm (Border, 99). In a herarchcal mechansm, there s a (potentally dfferent) nondecreasng ndex assocated wth each buyer s valuaton, and the allocaton rule awards the good to the hghest ndex at each value profle. Ths s an extremely general class of mechansms that ncludes, n partcular, both the effcent (the ndex s the value tself) and the Myerson (98) optmal aucton (the ndex s the roned vrtual value), among many others. We then show that (n a sense we make precse) almost all herarchcal mechansms are mplementable (Corollary ). 5 In other words, symmetrc mplementablty s a generc property of herarchcal mechansms. A strength of our analyss s that t requres very mld assumptons on the dstrbutons of buyer valuatons. In partcular, we do not need to mpose any hazard rate assumptons that are commonly employed n the aucton theory lterature. In ths sense, we vew our man result on the versatlty of symmetrc auctons as akn to (though, of course, not as strong as) the revelaton prncple for drect mechansms. The revelaton prncple states that restrctng the seller to drect mechansms s wthout loss of generalty. Analogously, our characterzaton shows that restrctng the auctoneer to symmetrc aucton formats does not prevent hm from achevng a wde varety of dfferent (and dscrmnatory) goals. In ths regard, our results are also smlar n sprt to the recent work of Manell and Vncent (00) and Gershkov, Goeree, Kushnr, Moldovanu, and Sh (03). These authors show that, n the ndependent prvate values model, any ncentve compatble and ndvdually ratonal outcome that can be acheved n Bayes-Nash equlbrum can also be acheved (n expectaton) n domnant strateges. Thus, as wth the case of symmetry, the requrement of domnant strategy mplementaton s not restrctve n and of tself. A surprsng mplcaton of our man result s that the revenue-optmal outcome can always be acheved va a symmetrc aucton (Corollary 3). Ths result counters what appears to be common ntuton and receved wsdom. Because ts drect mplementaton s asymmetrc, the optmal 3 There s a strand of lterature that uses anonymty n combnaton wth Bayesan ncentve compatblty (an early paper s d Aspremont and Peleg (988)). 4 Of course, changng the drect mechansm or the value dstrbutons wll lead to dfferent symmetrc aucton mplementatons. 5 A consequence of ths s that there s an mplementable herarchcal mechansm that s arbtrarly close to any nonmplementable one.

4 4 RAHUL DEB AND MALLESH M. PAI aucton was beleved to be nonanonymous n the earlest semnal work (see epgraph), and snce then, there have been numerous nstances n the aucton theory lterature where smlar belefs are stated. Some argue that ths observaton justfes the removal of legal hurdles that prevent dscrmnaton. In the context of nternatonal trade, McAfee and McMllan (989) used the theory of optmal auctons to show that explctly dscrmnatng amongst supplers can reduce the costs of procurement. Ther am was to provde an argument aganst the 98 Agreement on Government Procurement (n the General Agreement on Tarffs and Trade), whch set out rules to ensure that domestc and nternatonal supplers were treated equally. 6 Smlarly, Cramton and Ayres (996) suggest that, n government lcense auctons, subsdzng mnorty owned or local busnesses may actually result n more revenue to the government. 7 We show that, at least from a theoretcal perspectve, such goals can be acheved wthout explct dscrmnaton by the auctoneer. That a symmetrc aucton can be used to acheve a broad class of dfferent objectves has mplcatons for government procurement auctons, whch often have dstrbutonal goals n addton to generatng revenue (Athey, Coey, and Levn, 03). Governments often desre to favor certan bdders (small busnesses, women, mnortes, etc.) who are economcally dsadvantaged and hence may be unable to compete wth stronger bdders unless the aucton rules are skewed n ther favor. However, such a preferental polcy s often vewed as unfar. Ths polcy was successfully challenged n the U.S. Supreme Court case 99 (995), and states such as Calforna and Mchgan have explctly changed ther laws (Proposton 09 and Proposal respectvely) to prohbt favored treatment on the bass of race, sex, or ethncty. In Europe, Artcle 87() of the European Commsson Treaty prohbts ad granted by a Member State or through State resources n any form whatsoever whch dstorts or threatens to dstort competton by favorng certan undertakngs... Our results suggest that, n prncple, t s potentally possble to acheve outcomes where partcular classes of bdders are favored wthout havng to resort to explctly basng the aucton. Ths mplcaton can also be nterpreted another way symmetry of the aucton does not mply farness of the outcome. In a sense, ths ntuton s already well known, as ex-ante heterogenous buyers may have dfferent equlbrum strateges even n a symmetrc aucton. For nstance, Maskn and Rley (000) show that stronger bdders often favor second-prce to frst-prce auctons and that the latter format can yeld hgher revenues for the seller. The observaton that a far and transparent aucton can be constructed n a way to mplement dscrmnatory outcomes s mportant n formulatng polcy that prevents favortsm. Ths latter observaton has also been made n the context of affrmatve acton n college admssons. Opponents of affrmatve acton often assert that a ban would lead to hgher-qualty students beng admtted. However, t has been argued that t s theoretcally possble for unverstes to alter the crtera for admssons n response to such a ban n a way that can stll acheve dversty goals wthout explct dscrmnaton (Chan and Eyster, 003; Fryer, Loury, and Yuret, 008). Essentally, ths can be acheved by shftng weght from academc trats that predct performance to socal trats that proxy for race. However, unlke our settng, where bdders are strategc, students cannot choose what to report on ther applcatons. 6 Such an agreement s also currently present n the World Trade Oraganzaton, whch has replaced the GATT. 7 Corns and Schotter (999) test these arguments emprcally by conductng a laboratory experment.

5 SYMMETRIC AUCTIONS 5 Whle the man result of the paper s prmarly theoretcal, we feel that the desgn of symmetrc auctons for real-world applcatons s an mportant aucton desgn problem, and our theoretcal analyss characterzng mplementable outcomes s a necessary step towards ths end. Wth ths n mnd, we consder a handful of addtonal desderata that a seller mght want n an aucton. We solate a number of attractve propertes of frst- and second-prce auctons and then mpose them as addtonal theoretcal requrements on the symmetrc mplementaton. For brevty and tractablty, we focus on the case of two bdders and characterze the set of herarchcal mechansms that are mplementable wth each of these addtonal requrements mposed separately. The key takeaway from these characterzatons s that the optmal aucton s no longer always mplementable when these condtons are requred n addton to symmetry. The frst property we consder s that of nactve losers; that s, the losers n the aucton nether make payments nor receve subsdes. All-pay auctons do not satsfy ths property (snce losers must also pay ther bds), whch s perhaps one of the reasons that they are rarely used n practce. We show that herarchcal mechansms genercally have an nactve losers mplementaton (Proposton ). The second property we examne s contnuty of the payment rule. We provde a necessary and suffcent condton for contnuous mplementaton (Proposton ). Whle ths condton s not genercally satsfed, t s farly unrestrctve. The thrd property we consder s monotoncty of the payment rule n the bds. We separately consder both monotoncty of the payment n the opponent s bd (as n the second-prce aucton) and, monotoncty of the payment n one s own bd (as n the frst-prce aucton). Each of these monotoncty requrements makes the set of mplementable mechansms nongenerc (Propostons 3 and 4, respectvely). The last property we consder s ex-post (as opposed to nterm) ndvdual ratonalty of the equlbrum of the symmetrc aucton mplementaton. Ths property mples that, n equlbrum, losers never have to make payments (but can receve subsdes) and that wnners do not pay more than ther value for the object. Imposng ex-post ndvdual ratonalty can useful to avod stuatons where bdders may not be able to make payments for certan realzed bds due to budget constrants. The set of ex-post ndvdually ratonal mplementable herarchcal mechansms s nongenerc (Proposton 5). The paper s organzed as follows. In Secton, we descrbe the model and set up the notaton. Secton 3 presents an example that hghlghts our approach by demonstratng the mplementaton of a regular optmal aucton wth two bdders. The man results are presented n Secton 4. We present the characterzatons subject to the addtonal desderata n Secton 5. Fnally, concludng remarks are provded n Secton 6. The appendx contans proofs and some results that are not n the text.. THE MODEL We consder an ndependent prvate value aucton settng. A set N = {,,..., n} of rskneutral buyers or bdders (used nterchangeably) compete for a sngle ndvsble object. 8 Buyer N draws a value v V [v, v ] ndependently from a dstrbuton F. We assume that F s 8 Equvalently, our model could be consdered as a procurement settng where a frm or government wants a sngle project to be completed and solcts quotes from contractors, each of whom has an ndependent prvate cost.

6 6 RAHUL DEB AND MALLESH M. PAI twce contnuously dfferentable wth correspondng densty f whch s strctly postve throughout the support [v, v ]. Note that both V and F can be dfferent across, so we allow for ex-ante heterogenous bdders. We denote V j N V j and V j = V j, wth v V and v V denotng typcal elements of these sets. As wth values, we use the notaton F j N F j and F j = F j. We wll use smlar notaton for other vectors and vector-valued functons throughout the paper. A drect mechansm asks bdders to report ther values, and uses these reports to determne allocatons and payments. Allocatons are determned va an ordered lst of functons: a d = a d,..., ad n where a d : V [0, ] and n a d (v). = (Drect Allocaton) Here, a d (v) s the probablty that bdder wns the aucton when the profle of reported types s v. The nequalty above reflects the fact that the seller has a sngle unt to sell, so the probablty of allocatng t cannot exceed at any profle v. Addtonally, ths allows for the possblty that the seller may choose to wthhold the good. Smlarly, payments are determned va an ordered lst of functons: p d = p d,..., pd n where p d : V R. (Drect Payment) Here, p d (v) s the payment made by bdder when the profle of reported types s v. Note that, when t s postve, ths s a transfer to the seller, and, when t s negatve, t s a subsdy from the seller. In addton, the bdder may be requred to make payments even when she does not receve the object. Values are prvate; that s, buyers do not know the realzed valuatons of other bdders. Hence, each bdder s expected utlty from partcpatng n ths mechansm s determned by hs/her expected allocaton and payment. For a gven drect mechansm ( a d, p d), we defne nterm allocatons and payments to be the expected allocatons and payments condtoned on truthful reportng by all the bdders. Formally, these are gven by a d(v ) a d(v, v )df (v ), (Interm Allocaton) V p d(v ) p d(v, v )df (v ). (Interm Payment) V For smplcty, we delberately abuse notaton by denotng nterm allocatons usng the same symbol; the dfference s determned by whether the argument s a sngle value or a value profle. We make the addtonal standard assumpton that the bdders are rsk neutral and that ther utltes are quaslnear n the transfers. Condtonal on truthful reportng by the other bdders, the nterm expected utlty for bdder wth value v who announces a value v s smply v a d (v ) pd (v ). (Bdder Utlty) A mechansm ( a d, p d) s sad to be (Bayesan) ncentve compatble or smply IC f reportng truthfully s a Bayes-Nash equlbrum,.e., v a d (v ) p d (v ) v a d (v ) pd (v ) N, v, v V. (IC)

7 SYMMETRIC AUCTIONS 7 Myerson (98) showed that ncentve compatblty mples that the allocaton rule a d pns down the payments p d up to constants c R; that s, p d (v ) = v a d (v ) v v a d (w)dw + c. (Payoff Equvalence) Addtonally, a mechansm s sad to be ndvdually ratonal or smply IR f truthful reportng leads to a nonnegatve payoff, or v a d (v ) p d (v ) 0 v V. (IR).. Symmetrc Auctons We defne a symmetrc aucton as a game wth three propertes: () buyers smultaneously submt real numbers called bds; () the wnner s the hghest bdder over a gven reservaton bd (tes are broken unformly); and () payments are determned va an anonymous payment functon. Ths s an ndrect sealed bd aucton mechansm wth the addtonal restrcton that allocatons and payments depend only on the profle of bds and not the dentty of the bdders. Formally, n a symmetrc aucton, each bdder chooses a bd b R, and allocatons and payments are determned by functons a s : R n [0, ] and p s : R n R, respectvely. Bdder s allocaton or smply her probablty of wnnng the tem s gven by a s (b, b ) = { #{j N : b j =b } when b max{b, r}, 0 otherwse. (Symmetrc Aucton Allocaton) where r s the reservaton bd. As wth the values, we use b and b to denote the vector of all bds and the vector of all bds except that of bdder, respectvely. Bdder s payment s gven by p s (b, b ), (Symmetrc Aucton Payment) where p s s nvarant to permutatons of b but can depend on the underlyng dstrbuton of values (F,..., F n ). Notce that, snce the allocaton and payment rules do not depend on the dentty of the bdders, we only need a sngle functon, as opposed to lsts of functons, to defne these mechansms. Most commonly used aucton formats, such as frst-prce, second-prce and all-pay auctons are symmetrc n ths sense. In a symmetrc aucton, a pure strategy (henceforth referred to smply as a strategy) for a bdder s a mappng σ : V R, (Buyer Strategy) that specfes the bd correspondng to each possble value. A profle of strateges σ = (σ,..., σ n ) consttutes a (Bayesan Nash) equlbrum of the symmetrc aucton (a s, p s ) f each buyer s strategy s a best response to the strateges of other buyers. Formally, ths requres that, for all N and v V, we have σ (v ) argmax [v a s (b, σ (v )) p s (b, σ (v ))] df (v ). b R V

8 8 RAHUL DEB AND MALLESH M. PAI Symmetrc auctons are useful n stuatons where the seller knows the underlyng value dstrbutons (perhaps from havng conducted smlar auctons n the past) but cannot condton the mechansm on bdder dentty. As we argued n the ntroducton, one reason for ths s that dscrmnaton may be explctly prohbted by law. Alternatvely, the seller could be conductng the aucton n an envronment where t s easy for bdders to conceal ther denttes (such as auctons conducted over the Internet). An advantage of a symmetrc aucton format s that t mantans buyer prvacy by ensurng that they are not forced to reveal ther denttes va ther bds. However, we requre the buyers to know the underlyng value dstrbutons so that they can compute ther equlbrum bd. Admttedly, ths mght be an unrealstc assumpton n certan settngs. That sad, ths requrement s mposed n almost all aucton theory and, n partcular, s necessary for buyers to calculate equlbrum bds even n standard frst-prce auctons. We say that an IC and IR drect mechansm ( a d, p d) s mplemented by a symmetrc aucton (a s, p s ) f there s a pure strategy equlbrum n undomnated strateges of the latter mechansm that yelds the same allocaton and expected payment as the former. Specfcally, we say that a drect mechansm s mplementable f there exsts an undomnated equlbrum strategy profle σ such that, for all v V, 9 a d(v) = as (σ (v ), σ (v )), p d(v ) = p s (σ (v ), σ (v )) df (v ). V In ths noton of mplementablty, we requre the equlbrum allocaton of the symmetrc aucton to be dentcal to the drect mechansm for each profle of values but the payments to be equal n expectaton. Ths s a partal mplementaton crteron as we do not requre the symmetrc aucton to have a unque equlbrum. 0 More generally, we say that an IC and IR drect mechansm ( a d, p d) s mplementable f there exsts a symmetrc aucton (a s, p s ) that mplements t (almost sure and nterm mplementablty are defned analogously). The man goal of ths paper s to characterze the set of IC and IR drect mechansms that are mplementable. To make the exposton cleaner, we have delberately defned mplementaton only n terms of pure strateges for the bdders. Ths restrcton does not affect any of the results n the paper. We show n the appendx that allowng for mxed strateges does not expand the set of mplementable mechansms (or the set of mplementable mechansms subject to the addtonal condtons n Secton 5). We wll also refer to two addtonal weaker mplementaton crtera. The frst s almost sure mplementaton, whch requres () to hold almost surely (over the dstrbuton of buyer values). In other words, accordng to ths crteron, the allocatons and nterm payments are the same (a) (b) 9 We use the addtonal restrcton of undomnated equlbrum strateges to ensure that our symmetrc mplementaton s not based on mplausble buyer behavor. 0 Gven the fact that we allow for very general value dstrbutons, t s perhaps unrealstc to expect a symmetrc aucton mplementaton to have a unque equlbrum. Note that even standard aucton formats lke the frst or second prce aucton can have multple equlbra n our model. Ths s because our settng s more general than even the farly unrestrctve condtons requred for unqueness n frst prce auctons (Lebrun, 006). Snce the addtonal requrement of IR only nvolves changng the payment rules by a constant, our characterzaton results can also be vewed as smply characterzng the set of IC drect mechansms whch are mplementable.

9 SYMMETRIC AUCTIONS 9 except at a measure zero set of values. The second s nterm mplementaton, whch requres the allocaton rule (as wth the payment) to be mplemented n an expected sense or that a d (v ) = V a s (σ (v ), σ (v )) df (v ). The recent work on the equvalence of Bayesan and domnant strategy mplementablty (Manell and Vncent, 00; Gershkov, Goeree, Kushnr, Moldovanu, and Sh, 03) uses an even weaker noton that nstead requres the expected utltes (as opposed to nterm allocatons and payments separately) of the agents to be the same. 3. EXAMPLE: IMPLEMENTING THE OPTIMAL AUCTION WITH TWO BUYERS In ths secton, we explan our approach by descrbng a symmetrc mplementaton of the optmal aucton when there are two buyers. For smplcty, we addtonally assume that the dstrbutons of both buyers satsfy the ncreasng vrtual value property. Formally, ths condton requres that, for each buyer N, the vrtual value φ (v ) = v F (v ) f (v ) (Vrtual Value) s ncreasng n v. An mplcaton s that φ s a sngle valued functon. We denote the allocaton and payment rule of the optmal aucton by (a, p ). Recall that n the optmal aucton, bdders announce ther values and the mechansm awards the good to the bdder who has the hghest postve vrtual value (t s wthout loss to assume that tes are broken equally). Hence, when bdders draw ther values from dfferent dstrbutons, ths drect mechansm s not symmetrc, as the allocaton rule depends on the bdder-specfc value dstrbuton. A natural way to attempt a symmetrc mplementaton of the optmal aucton s to construct a payment rule such that t s an equlbrum for both bdders to bd ther vrtual values. The aucton could then allocate the good to the hgher bd and have a reservaton bd of 0. We denote the set of vrtual values of bdder by B [φ (v ), φ (v )]. The dstrbuton F over V nduces a dstrbuton G over the set B of vrtual values. We clam that the optmal aucton can be mplemented f we can construct a payment rule p s that satsfes p (v ) = p s φ (v ), b j dgj (b j ) for = j and all v V. B j Ths s smply a restatement of the mplementablty requrement where equlbrum strateges of bddng the vrtual value have been substtuted n. Ths clam s easy to see: () Suppose that buyer wth value v bds b B but b = φ (v ). Ths s equvalent to her reportng a value φ (b ) = v n the drect mechansm (a, p ), whch yelds a lower payoff because the optmal aucton s IC. () Suppose that buyer wth value v bds b / B. Ths can be detected wth postve probablty by the auctoneer when the other bdder s bddng truthfully. Ths s because there wll be a postve measure of bds b j such that (b, b j ) / (B B ) (B B ). Such offequlbrum bds can be dscouraged by makng the payments hgh enough at these bds.

10 0 RAHUL DEB AND MALLESH M. PAI We now construct such a symmetrc payment rule. Snce t s easy to dscourage bds that le outsde the support of the vrtual values, the payment rule s delberately defned only for equlbrum bd profles (b, b j ) (B B ) (B B ). We separately construct the payment for bds that le n the supports of only one and both vrtual value dstrbutons respectvely. In equlbrum, bds b B \B j are made only by buyer. Hence, for such bds, we can smply defne the payment rule to be the nterm payment from the optmal aucton or p s (b, b j ) = p (φ (b )) when b B \B j and b j B j. To construct the payments for bds b B B that le n the support of both vrtual value dstrbutons, we frst observe that, for asymmetrc buyers (F = F ), there exsts a ˆb R such that G (ˆb) = G (ˆb). In other words, dfferent value dstrbutons yeld dfferent vrtual values dstrbutons. Consder the payment rule where p s (b, b j ) = { p u (b ) f b j ˆb and b j B B, p l (b ) f b j < ˆb and b j B B, p u (b ) = p (φ (b ))G (ˆb) p (φ (b ))G (ˆb), G (ˆb) G (ˆb) p l (b ) = p (φ (b ))[ G (ˆb)] p (φ (b ))[ G (ˆb)]. G (ˆb) G (ˆb) Accordng to ths payment rule, a bdder who bds b pays an amount p u (b ) when her opponent bds hgher than ˆb and an amount p l (b ) when her opponent s bd s lower than ˆb. Hence, the expected payment of a bdder who bds b B B when bdder j bds φ j (v j ) for all v j V j s p u (b )[ G j (ˆb)] + p l (b )G j (ˆb) = p (φ (b )), (4) whch s precsely the requred payment for mplementaton. Notce also that the above equaton (4) can be used to derve the expressons for p u and p l. An equvalent matrx representaton s the followng system for b B B M [ p u (b ) p l (b ) ] = p p φ (b [ ) φ (b where M = ) G (ˆb) G (ˆb) G (ˆb) G (ˆb) ] () (3). (5) By defnton, G (ˆb) = G (ˆb) mples that M s a full rank matrx. Therefore (5) has a soluton for all b B B, and p u, p l can be obtaned by nvertng M. Note that ths logc bears some resemblance to the ntuton n Crémer and McLean (988) and McAfee and Reny (99), who study the possblty of full surplus extracton n auctons when dfferent buyers values are correlated. In ther settng, n addton to bddng for the object, buyers are forced to make a sde bet on ther opponents reported types (n our constructon, the analogous sde bet s whether your opponent s bd s above or below ˆb). When values are correlated, these sde bets have dfferent expected values for dfferent types of the same buyer, whch allows the payment rule to effectvely dscrmnate between them. They show that a full rank condton (analogous to requrng M to be full rank) on the value dstrbutons s suffcent for full surplus extracton.

11 SYMMETRIC AUCTIONS In summary, the symmetrc payment rule that mplements the optmal aucton n ths example s p u (b ) f b B B, b j ˆb and b j B B, p s p l (b (b, b j ) = ) f b B B, b j < ˆb and b j B B, p (φ (b )) f b B \B and b j B, p (φ (b )) f b B \B and b j B. The followng numercal example llustrates ths constructon. Example. Consder a settng wth two buyers. Buyer has a value that s unformly dstrbuted over [, 4], whle buyer s value s unformly dstrbuted over [, ]. The seller wants to conduct a symmetrc mplementaton of the optmal aucton. In ths settng, the vrtual value of buyer s φ (v ) = v 4, and the vrtual value of buyer s φ (v ) = v. Therefore, buyer s vrtual value (bd) s unformly dstrbuted over B [0, 4], whle buyer s s unformly dstrbuted over B [0, ]. We begn by dervng the nterm payments. These can be determned usng (Payoff Equvalence) as follows: and p (v ) = v a (v ) = { v v p(v ) = v a(v v ) a(w)dw v = v mn{v, } mn{w, }dw for v [, 3] 5 for v (3, 4] = v [ v ] a(w)dw h [ w v Interm payments expressed n terms of bds are p p φ (b ) = φ (b ) = b 6 + b 4 ] dw = v 4 { b 8 + b for b [0, ], 5 for b (, 4], for v [, ]. (6) for b [0, ]. (7) Consder now ˆb =, for whch we have that G (ˆb) = 4 and G (ˆb) =. Ths choce of ˆb yelds p u (b ) = b and p l (b ) = 5b + b 4, from whch we can defne the symmetrc payment rule for equlbrum bds: b f b [0, ] and b j [, 4], p s (b, b j ) = 5b 4 f b [0, ] and b j [0, ), 5 f b (, 4] and b j [0, ]. Our man result n the next secton bulds on the ntuton n ths example. The key dffculty n a symmetrc mplementaton s that the same bd, when made by dfferent bdders, must lead to

12 RAHUL DEB AND MALLESH M. PAI the approprate, potentally dfferent nterm payments. For ths to be the case, the payment rule needs to be desgned n a way that utlzes the dfference n the dstrbuton of the equlbrum bds of each bdder. In ths example, we smply had to charge dfferent amounts dependng on whether the opponent s bd was above or below ˆb. The proof of the man result contans the substantally harder generalzaton of ths constructon to n bdders. 4. CHARACTERIZATION OF MECHANISMS WITH SYMMETRIC IMPLEMENTATIONS In ths secton, we present and dscuss the man result a characterzaton of mplementable IC and IR drect mechansms. A constructve approach to determnng whether a partcular drect mechansm s mplementable would requre frst the desgn of a symmetrc aucton and then a dervaton of ts equlbrum. However, dervng equlbra for a gven symmetrc aucton can be a hard task. For nstance, t s well known that t s dffcult to obtan closed-form solutons for equlbrum bds n the frst-prce aucton for arbtrary dstrbutons. We show that the set of mplementable mechansms s a subset of the set of herarchcal mechansms. Ths smplfes our task. We begn by defnng herarchcal allocaton rules. These are generated by an ordered lst I = (I,..., I n ) of ndex functons that are nondecreasng mappngs I : V R for N. A herarchcal allocaton rule s generated from a gven lst of ndex functons I as follows { a h (v) = when I #{j N : I j (v j )=I (v )} (v ) max{i (v ), 0}, (Herarchcal Allocaton) 0 otherwse. Each buyer s value s transformed nto an ndex va the ndex functon. The good s then allocated to the buyer wth the hghest postve ndex, and tes are broken equally. Restrctng allocatons to buyers wth postve ndces s essentally equvalent to settng a reservaton bd. Choosng a reserve of 0 for the ndex functons s wthout loss of generalty, as all bds can always be moved up or down by a constant. In addton, note that ndex functons can be chosen so that allocatons occur above dfferent reservaton values across the buyers. A herarchcal mechansm (I, p h ) s an IC and IR mechansm that conssts of ndex functons I and payment functons p h. The allocaton a h s determned as shown above from the ndex functons. For the results that follow, we fnd t convenent to denote a herarchcal mechansm n terms of the ndex functons I as opposed to the allocaton rule a h. If two lsts of ndex functons I and I generate the same allocaton rule a h, then t must be that one s a monotone transformaton of the other. Formally, f I and I generate the same allocaton a h, then there exsts a monotone functon Γ : R R such that I (v ) = Γ(I (v )) for all and v. The partcular choce of ndex functons that correspond to a gven allocaton a h does not matter for the statement of any of our results. Snce the ndex functons are nondecreasng, havng a hgher value mples a weakly hgher probablty of wnnng. Ths mples that every herarchcal allocaton rule a h has assocated IC transfers p h (pnned down to constants) that yeld a herarchcal mechansm. All mechansms n appled mechansm desgn that we are aware of fall wthn the class of herarchcal mechansms (we provde examples of nonherarchcal mechansms below). In the effcent Vckrey aucton, Ths term was ntroduced by Border (99).

13 SYMMETRIC AUCTIONS 3 values serve as ndces or I (v ) = v, and n the optmal aucton (wth ncreasng vrtual values) the ndces are gven by the vrtual values or I (v ) = φ (v ). When the vrtual values are not ncreasng, the ndex functons are smply the roned vrtual value functons (Myerson, 98). Alternatvely, suppose an auctoneer wth affrmatve acton concerns wants to subsdze a hstorcally dsadvantaged bdder over a bdder j where the latter has ndex I j (v j ) = v j. The ndex for bdder could reflect ether a flat subsdy I (v ) = v + s (where s > 0) or a percentage subsdy I (v ) = s v (where s > ). We show below that any mplementable mechansm must effectvely be a herarchcal mechansm. Ths allows us to focus on ths smaller class of mechansms, whch n turn smplfes the mplementaton task, as n the prevous secton. Snce the allocaton rule of a symmetrc aucton that mplements a herarchcal mechansm must allocate the good to the bdder wth the hghest ndex, a natural assumpton s to make equlbrum bds correspond to the ndex values. Then, constructng the symmetrc mplementaton essentally bols down to fndng a symmetrc payment rule that yelds the same nterm payments. Gven a herarchcal mechansm (I, p h ), the dstrbuton F on the set of values V nduces a dstrbuton G on the set of ndces or bds B {I (v ) v V }. (Bd Space) At tmes, we wll slghtly abuse notaton and use G as both a dstrbuton and a measure. The meanng wll be clear dependng on whether the argument of G s a real number or a set. The notaton G delberately suppresses the dependence on the ndex functon I ; the meanng wll always be clear from the context. Snce ndex functons I are not necessarly strctly ncreasng, the nduced dstrbutons G may have atoms. Addtonally, notce that the set B need not be an nterval because the ndex functons I may be dscontnuous. A herarchcal allocaton mechansm (I, p h ) can be mplemented f we can fnd a symmetrc payment functon p s such that p h(v ) = p s (I (v ), b ) dg (b ) for all N and v V. ( ) B If such a symmetrc payment functon exsts, t follows that an equlbrum of the symmetrc aucton wth ths payment rule wll nvolve each buyer wth value v bddng ther ndex I (v ). By constructon, such bds generate the requred allocaton. The ntuton s straghtforward and dentcal to that of the example. Suppose that a bdder wth value v makes a bd b B other than her ndex so b = I (v ). Her correspondng allocaton and payment would be dentcal to what she would get by reportng a value v I (b ), resultng n lower utlty as the drect mechansm (I, p h ) s IC. 3 Off-equlbrum bds b / B, whch le outsde the bd space, can be punshed by requrng hgh expected payments at these bds. We now present our man result as two separate theorems. Theorem. Suppose that a drect revelaton mechansm (a d, p d ) s mplementable. Then, there exsts an mplementable herarchcal mechansm (I, p h ) such that ts mplementaton s an almost sure mplementaton of (a d, p d ). 3 Here, I ( ) s the correspondence defned by I (b ) = {v V I (v ) = b }.

14 4 RAHUL DEB AND MALLESH M. PAI Theorem says that t s essentally wthout loss to restrct attenton to herarchcal mechansms. It states that, for any mplementable drect mechansm, there s an mplementable herarchcal mechansm that almost surely has exactly the same allocaton and payments. An mplcaton s that, for any mplementaton of any nontrval objectve, a prncpal can restrct attenton to herarchcal mechansms. Ths result s ntutve. Clearly, a nonherarchcal mechansm cannot have an mplementaton n pure strateges because, f so, the allocaton rule could have been generated by an ndex rule wth ndces equal to the equlbrum bds n the symmetrc aucton. The appendx contans the argument for mxed strateges. Theorem provdes condtons that characterze the set of mplementable herarchcal mechansms. 4 Theorem. A herarchcal mechansm (I, p h ) s mplementable f and only f, for any par of dstnct buyers, j N who have the same dstrbuton of bds (G = G j ), and any par of values for these two buyers v V, v j V j satsfyng I (v ) = I j (v j ), we have that p h (v ) = p h j (v j). We can decompose ths nto two parts: () Whenever bd dstrbutons G dffer across the buyers, the condton of the theorem s vacuously satsfed, and therefore, t s possble to construct a payment rule so that ( ) s satsfed. When there are two bdders, a payment rule lke the one n the prevous secton can be used to construct the mplementaton. The constructon for more that two bdders s consderably more complcated and can be found n the appendx. () When two buyers are such that the two nduced bd dstrbutons are the same, that s G = G j, then the nterm payments must be the same for any two values (one for each buyer) that correspond to the same ndex. Ths s because t s no longer possble to generate dfferent equlbrum expected payments for dstnct buyers who make the same bd. We now present two examples of herarchcal mechansms that cannot be mplemented, thus showng that the theorem s not vacuously true. In the frst example, the good s allocated randomly and n the second, the seller would lke to subsdze one of the buyers. Example. There are two buyers. Buyer has a value unformly dstrbuted on [0, ]. Buyer has a value unformly dstrbuted on [0.5, ]. The seller assgns the good at random (wth equal probablty) to each of the two buyers rrespectve of ther value. Buyer s never asked to pay anythng, whereas buyer s always asked to pay 0.5. Notce that ths mechansm s a herarchcal mechansm where each bdder s ndex functon s a constant nonnegatve functon or I (v ) = I (v ) 0 for all v [0, ] and v [.5, ]. Here, the bd space just conssts of a sngle pont, and dstrbutons G, G are degenerate and therefore satsfy G = G. However, the payments dffer. Therefore ths mechansm volates the condtons of Theorem. It follows that there s no symmetrc mplementaton of ths drect revelaton mechansm. 4 The theorem s actually slghtly stronger. The condtons are also necessary and suffcent for (the weaker crteron of) nterm mplementablty of a herarchcal mechansm.

15 SYMMETRIC AUCTIONS 5 Example 3. Consder an envronment where there are two buyers. Buyer has a value v that s unformly dstrbuted on [0, ]. Buyer has a value v whch s unformly dstrbuted on [, ]. Suppose that the seller would lke to subsdze the bd of buyer by a dollar. Put dfferently, buyer wns the good f and only f hs value exceeds that of buyer by. Therefore, for any v [0, ], the nterm allocaton probabltes are gven by a h (v ) = a h ( + v ). The IC and IR payments are chosen to be such that the lowest type of both buyers for whom there s no probablty of wnnng nether make payments nor are pad. Ths s clearly a herarchcal mechansm wth ndex functons I (v ) = I (v + ), where I ( ) s strctly ncreasng on the nterval [0, ]. Observe that ths mples that the dstrbutons over the bd spaces are dentcal, snce G and G are both U[0, ]. Moreover, ncentve compatblty pns down payments, and therefore we have p h (v + ) = p h (v ) + a h (v ). For all values v (0, ], therefore, the above equaton mples that p h (v + ) = p h (v ). Snce the nterm payments dffer for values that have the same ndex and the bd spaces have dentcal dstrbutons, symmetrc payments cannot be constructed to mplement ths mechansm. However, note that ths mechansm could have been mplemented f buyer s value dstrbuton was anythng even slghtly dfferent that U[, ], as ths would mply that G = G. The condtons n Theorem were on the dstrbutons of the bd space. The followng corollary qualtatvely descrbes the types of herarchcal allocaton rules that cannot be mplemented. Corollary. Suppose that a herarchcal mechansm (I, p h ) s not mplementable. Then there must exst two dstnct buyers j and j such that ther ndex functons can be wrtten as for some non-decreasng functon Γ( ). for = j, j : I (v ) = Γ(F (v )) for almost every v V, The above corollary demonstrates that the only nonmplementable herarchcal mechansms are ones where there are two buyers whose ndces correspondng to a value depend solely on the statstcal rank of that value n the dstrbuton of that buyer s values. Ths s a very specfc and small subset of herarchcal mechansms; n fact the set of mplementable mechansms s generc n a topologcal sense, formalzed below. For each buyer, the dstrbuton F defnes a measure space on V. Consder the space of ndex functons for buyer, as an L p space where p. The space of ndex functons I = (I,..., I n ) s topologzed wth the product topology and denoted I. Snce a fnte product of complete normed vector spaces s a Bare space, standard topologcal notons of genercty are well defned. Recall that a property s sad to be genercally satsfed on a topologcal space f the set that does not satsfy t s a meager set (or conversely, the set that does satsfy t s a resdual

16 6 RAHUL DEB AND MALLESH M. PAI set). Further, recall that a set n a topologcal space s meagre f t can be expressed as the unon of countably many nowhere dense subsets n that space. Corollary. Genercally, on I, G = G j for every par of buyers and j. 5 The ntuton and proof for ths result are straghtforward: two ndex functons that result n the same dstrbuton over bds can be made dfferent by slghtly perturbng them. The fact that a large number of dsparate objectves can be acheved ether exactly or arbtrarly closely va a symmetrc mplementaton s one of the man nsghts of ths paper, and t s worth repeatng ts two man mplcatons. The frst s that symmetry need not mply farness just because an aucton treats the bds of dfferent buyers smlarly, ths does not mply that the resultng outcomes are equal from an ex-ante perspectve. The second s that careful aucton desgn can allow the mechansm desgner to acheve a wde varety of goals n envronments where explct favortsm s mpractcal or prohbted. For nstance, the aucton desgner can choose formats whch favor weaker bdders wthout explctly basng the mechansm. Ths can be helpful for governments strvng to reach dstrbutonal goals (favorng small busnesses, mnortes, etc.) wthout facng legal challenges over favortsm polces. Alternatvely, ths can be useful to encourage competton (and thereby enhance revenue) amongst asymmetrc bdders n settngs such as onlne auctons where the seller may have good knowledge about value dstrbutons (from prevous auctons conducted) but bds are placed anonymously. In fact, the followng corollary ponts out that the revenue-maxmzng aucton can always be mplemented. Snce ths seems to counter prevalng ntuton, we feel that ths s perhaps one of the most surprsng results of the paper. 6 Corollary 3. The optmal aucton can be mplemented symmetrcally. It s worth reteratng that the above corollary requres no hazard rate assumptons on the value dstrbutons. When the dstrbutons satsfy the ncreasng vrtual value property, t s easy to show that f the bdders are asymmetrc, the dstrbuton over vrtual values must also be dfferent. When the vrtual values are not ncreasng then the proof of the Corollary shows that f the dstrbutons over the roned vrtual values are the same then the condton of the Theorem must be satsfed. We end ths secton by pontng out that, f the mplementaton crteron s weakened, the prncpal can acheve the outcomes correspondng to certan nonherarchcal mechansms usng randomzaton. The prncpal can randomze by choosng amongst a set of mechansms va a lottery. After choosng one such mechansm from the set, the prncpal can announce t to the buyer. For nstance, the prncpal could toss a con and choose between a frst- and second-prce aucton. 5 It s possble to restate ths corollary to say nstead that mplementablty s a generc property n the space of herarchcal mechansms (nstead of n the space of ndex functons that do not nclude payments). We have delberately chosen not to do so to avod the dstractng techncaltes nherent n defnng the approprate topology on the space of herarchcal mechansms. The complcatons arse from the fact that the ndex functons restrct the payments (up to constants) va ncentve compatblty, so we cannot smply employ a product topology over ndex functons and payments. 6 For nstance, n an nfluental paper, Cantllon (008) conjectured that bdder asymmetres hurt the auctoneer n any anonymous mechansm after showng that ths s not the case n the optmal aucton. Corollary 3 answers ths conjecture n the negatve by showng that the optmal aucton can be mplemented by an anonymous mechansm.

17 SYMMETRIC AUCTIONS 7 Havng chosen, the buyer s nformed of the aucton format and the game proceeds. Such randomzaton s approprate for a prncpal concerned about expected outcomes (as n Manell and Vncent, 00; Gershkov, Goeree, Kushnr, Moldovanu, and Sh, 03). Randomzaton can be useful n achevng the outcomes of both unmplementable herarchcal allocaton mechansms and nonherarchcal mechansms. A smple two-buyer example of a nonherarchcal mechansm s one where rrespectve of the values, buyer gets the good 5% of the tme and buyer gets t 75%. Clearly, ths s not a herarchcal allocaton snce our defnton of the latter requres the equal breakng of tes. Another example s a mechansm n whch the seller randomly allocates the good 50% of the tme and runs a second prce aucton the remanng 50%. A mechansm (a d, p d ) s defned to be a randomzaton over a set of mechansms M, f there s a measure ζ defned on M such that a d(v ) = a (v )dζ((a, p)) and p d(v ) = p (v )dζ((a, p)). M The lemma below shows that all IC and IR drect mechansms can be obtaned as a randomzaton over herarchcal mechansms. Ths lemma follows from results n Border (99) and Merendorff (0). Lemma. Every IC and IR drect mechansm s a randomzaton over the set of herarchcal mechansms. Clearly, the outcome from any mechansm that s a randomzaton over mplementable herarchcal mechansms can be acheved n such an ex-ante sense. The auctoneer can just randomly choose (usng measure ζ) from the symmetrc auctons that correspond to the mplementable herarchcal mechansms. Note that, strctly speakng, ths s not nterm mplementaton as we defned t. However, for practcal applcatons, t serves the same purpose, as randomzaton s done before the chosen symmetrc aucton s announced to the buyers. The next corollary summarzes ths dscusson, and n t, we use the termnology outcomes are achevable to clarfy the dstncton from nterm mplementaton. Corollary 4. The outcomes from an IC and IR drect mechansm are achevable f t s a randomzaton over mplementable herarchcal mechansms. Fnally, we dscuss the two examples of the unmplementable mechansms and examne whether ther outcomes can be acheved va randomzaton. Example. (Contnued) Recall that, n ths example, the seller assgns the good at random (wth equal probablty), buyer s never asked to pay anythng, whle buyer s always asked to pay 0.5. The outcome from the mechansm can be acheved by randomzng wth equal probablty over two mplementable herarchcal mechansms. In the frst herarchcal mechansm, buyer s always awarded the good rrespectve of value and s not asked to pay anythng. In the second herarchcal mechansm, buyer s always gven the good rrespectve of value and s asked to pay 0.5. Example 3. (Contnued) Recall that, n ths example, buyer wns the good f and only f her value exceeds that of buyer by. The outcome of ths mechansm cannot be acheved usng randomzaton. M

18 8 RAHUL DEB AND MALLESH M. PAI Consder the ndex functon I (v) = I (v + ) = v. By observaton, the allocaton rule a h correspondng to these ndex functons s the unque (almost everywhere) maxmzer of a d j (v j, v j )I j (v j ) f j (v j ) dv, V j {,} amongst all IC drect allocatons a d. Therefore, for any herarchcal allocaton rule ã h = a h that dffers from a h at a postve measure subset of values, t must be that V j {,} ã h j (v j, v j )I j (v j ) f j (v j ) dv < V j {,} a h j (v j, v j )I j (v j ) f j (v j ) dv. Moreover, any allocaton rule that s equal to a h almost everywhere s not mplementable. Therefore, a h s not a randomzaton over mplementable herarchcal allocatons, so ts outcome s not achevable. 5. ADDITIONAL DESIDERATA Our man result from the prevous secton showed that a large class of mechansms can be mplemented symmetrcally. However, symmetry s just one desderatum of a practcal mplementaton. In ths secton, we consder a number of addtonal propertes that one mght want n an aucton mplementaton and dscuss how these propertes, along wth symmetry, restrct the set of mplementable outcomes. Essentally, the goal s to dentfy some propertes n frst- and second-prce auctons whch can be construed to be desrable and mpose them as addtonal restrctons on a symmetrc aucton mplementaton. Throughout the secton, we consder the case of two bdders (n = ), prmarly for the sake of brevty and tractablty. 7 A key takeaway from ths secton s that the optmal aucton s no longer always mplementable under these addtonal requrements. 5.. Inactve Losers An mportant property of frst- and second-prce auctons s that losers nether make nor receve payments. Wth the notable excepton of charty auctons (see, for nstance Goeree, Maasland, Onderstal, and Turner, 005), most auctons conducted n the real world have ths feature. It s often argued that requrng the loser to pay reduces partcpaton, whch s one of the reasons that all-pay auctons are seldom used n practce. Hence, ths mght be construed as a shortcomng of Theorem : the symmetrc mplementaton that we construct there may requre both the wnner and the losers to make payments. A herarchcal mechansm has a symmetrc, nactve losers mplementaton (a s, p s ) f p s (b, b j ) = 0 whenever b < b j. Note that such an mplementaton may sometmes requre the wnner to make payments that are greater than hs value and thus may not be ex-post IR (we consder the ex-post 7 Barrng Propostons 4 and 5, we can extend the results n ths secton to more than two bdders.