A Equilibria of Greedy Combinatorial Auctions

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1 A Equlbra of Greedy Combnatoral Auctons BRENDAN LUCIER, Mcrosoft Research ALLAN BORODIN, Unversty of Toronto We consder auctons n whch greedy algorthms, pared wth frst-prce or crtcal-prce payment rules, are used to resolve mult-parameter combnatoral allocaton problems. We study the prce of anarchy for socal welfare n such auctons. We show for a varety of equlbrum concepts, ncludng Bayes-Nash equlbrum, low-regret bddng sequences, and asynchronous best-response dynamcs, the resultng prce of anarchy bound s close to the approxmaton factor of the underlyng greedy algorthm. 1. INTRODUCTION The feld of algorthmc mechansm desgn studes systems that depend upon nteracton wth partcpants whose behavour s motvated by ther own goals, rather than those of a desgner. Relevant solutons must therefore merge the computatonal consderatons of computer scence wth the game-theoretc nsghts of economcs. The focus of ths paper s the mult-parameter doman of combnatoral allocaton problems when the goal s to assgn m objects to n agents n order to maxmze the socal welfare, subject to arbtrary downward-closed feasblty constrants. Ths class ncludes all combnatoral aucton problems that allow sngle-mnded declaratons ncludng mult-unt combnatoral auctons, unsplttable flow problems, and many others. For the goal of optmzng socal welfare, the celebrated Vckrey-Clarke-Groves (VCG) mechansm addresses game-theoretc ssues n a strong sense. In the absense of colluson, t nduces full cooperaton (e. truthtellng) as a domnant strategy. However, the VCG mechansm requres that the underlyng welfare-maxmzaton problem be solved exactly. For all but the smplest settngs, ths optmalty requrement s undesrable: exact maxmzaton may be computatonally ntractble, t may requre an unrealstc amount of communcaton from the buyers, and the resultng wnner determnaton rules may be dffcult to explan to a typcal partcpant. One way to bypass these complexty ssues s to desgn new, specally-talored mechansms for specfc assgnment problems. Indeed, there has been sgnfcant progress n desgnng domnant strategy ncentve compatble (DSIC) alternatves to the VCG mechansm. Whle ths venture has been largely successful n settngs where agent preferences are sngle-dmensonal [Archer and Tardos 2001; Brest et al. 2005; Lehmann et al. 1999; Mu alem and Nsan 2008], general settngs have proven more dffcult. It has been shown that the approxmaton ratos acheveable by DSIC mechansms and ther nonncentve compatble counterparts exhbt a large asymptotc gap for some problems [Papadmtrou et al. 2008; Dobznsk 2011; Dughm and Vondrák 2011; Dobznsk and Vondrak 2012]. Author s addresses: B. Lucer, Mcrosoft Research, brlucer@mcrosoft.com; A. Borodn, Department of Computer Scence, Unversty of Toronto, bor@cs.toronto.edu. Permsson to make dgtal or hard copes of part or all of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes show ths notce on the frst page or ntal screen of a dsplay along wth the full ctaton. Copyrghts for components of ths work owned by others than ACM must be honored. Abstractng wth credt s permtted. To copy otherwse, to republsh, to post on servers, to redstrbute to lsts, or to use any component of ths work n other works requres pror specfc permsson and/or a fee. Permssons may be requested from Publcatons Dept., ACM, Inc., 2 Penn Plaza, Sute 701, New York, NY USA, fax +1 (212) , or permssons@acm.org. c YYYY ACM /YYYY/01-ARTA $15.00 DOI: Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

2 A:2 A. Borodn and B. Lucer Alternatvely, one mght study classes of natural allocaton algorthms, that appear ntutve as aucton allocaton rules, wth the hope that they have desreable ncentve propertes when mplemented as mechansms. As t turns out, for many combnatoral allocaton problems, conceptually smple determnstc algorthms (e.g. greedy algorthms) meet or approach the best-known approxmaton factors subject to computatonal constrants [Lehmann et al. 1999; Mu alem and Nsan 2008; Brest et al. 2005; Babaoff and Blumrosen 2004]. These natural methods tend to be computatonally effcent and easy for bdders to understand, whch are desrable propertes n auctons. Unfortunately, such algorthms are not, n general, DSIC [Lehmann et al. 1999; Borodn and Lucer 2010]. Rather than abandonng these methods n favor of other, potentally more complex, mechansms, we are pursung an alternatve approach. Namely, rather than strvng for domnant strategy truthfulness, t may be acceptable for a system to admt strategc manpulaton, so long as the desgner s objectves are met after such manpulaton occurs. To ths end, we explore the performance of mechansms at equlbra of bdder behavor, gven an approprate model of belefs. Broadly speakng, our motvatng queston s: When can an algorthm be mplemented as a mechansm that acheves hgh socal welfare at every equlbrum? 1 And how robust are the resultng mechansms to varatons of the equlbrum concept? We demonstrate that for combnatoral allocaton problems, any greedy-lke approxmaton algorthm can be converted nto a mechansm that acheves nearly the same approxmaton factor at every equlbrum of bdder behavour. Our analyss s very general, and apples to a range of dfferent equlbrum concepts, ncludng pure and mxed Nash equlbra, Bayes-Nash (correlated) equlbra, no-regret equlbra, and terated myopc best-response. We are thus able to decouple computatonal ssues from ncentves ssues for ths class of algorthms, as one can desgn a greedy algorthm wthout consderng ts economc mplcatons, and then apply a straghtforward prcng scheme n order to acheve good performance at equlbrum. Performance of games at equlbrum has been studed extensvely n the algorthmc game theory lterature as the prce of anarchy (POA) of a gven game 2 : the rato between the optmal outcome and the worst-case outcome at any equlbrum [Papadmtrou 2001]. Put nto these terms, our goal s to convert an algorthm wth approxmaton factor c 1 nto a mechansm whose prce of anarchy s not much larger than c. Ths paper s a synthess and revson of results n [Lucer and Borodn 2010], [Lucer 2010], and results n the frst author s thess [Lucer 2011]. The paper s organzed as follows. The remander of ths secton outlnes our results and relates our work to recent papers n ths area. Secton 2 defnes the necessary concepts and applcatons for our results. Secton 3 ntroduces the concept of strongly loser-ndependence (generalzng the loser-ndependence concept from [Chekur and Gamzu 2009]) whch becomes the key property of greedy algorthms that we wll explot. Sectons 4 and 5 analyze (respectvely) prce of anarchy results for frst-prce and crtcal-prce mechansms. In Sectons 6 and 7 we consder soluton concepts for repeated games: under regret mnmzaton and best-response dynamcs, respectvely. Secton 8 concludes wth some open problems. 1 Domnant strategy truthfulness of an approxmaton mechansm s conceptually stronger as a soluton concept than that of a mechansm that approxmates the optmal socal welfare at every equlbrum. However, as noted elsewhere [Chrstodoulou et al. 2008], Bayesan Nash equlbrum s not, strctly speakng, a relaxaton of domnant strategy truthfulness. There exst truthful mechansms whose approxmaton ratos are not preserved at all Nash equlbra, such as the famous Vckrey aucton. 2 For the purpose of ths paper, we shall not consder cost mnmzaton problems. We note that the prce of anarchy concept was ntroduced n terms of cost mnmzaton games but to the best of our knowledge the only prce of anarchy results for mechansm nduced games apply to maxmzaton problems. Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

3 Equlbra of Greedy Combnatoral Auctons A: Our Results The basc queston of algorthmc mechansm desgn s ths: when can computatonally effcent algorthms be converted nto mechansms that preserve approxmaton bounds wth respect to truthfulness or POA results? We address the prce of anarchy consderatons wth respect to socal welfare maxmzaton for a broad class of allocaton problems. In the full nformaton and Bayesan settng, we study the prce of anarchy for frst prce and crtcal prce mechansms derved from greedy algorthms. Roughly speakng (and n contrast to results regardng approxmaton and truthful mechansms) we are able to show that there s often lttle or no loss from the approxmaton rato of a greedy algorthm to the correspondng mechansm prce of anarchy. We also study the long term behavor of the use of these mechansms when used n repeated games. One-Shot Auctons.. We frst consder one-shot auctons, n whch the allocaton problem s resolved only once. Followng Chrstodoulou et al. [Chrstodoulou et al. 2008], we focus our attenton on the standard (n economcs) ncomplete nformaton settng, where the approprate equlbrum concept s Bayes-Nash equlbrum. That s, we assume that agents preferences are prvate, but drawn ndependently from commonlyknown pror dstrbutons, and that players apply strateges at equlbrum gven ths partal knowledge. We pose the queston: can a gven black-box approxmaton algorthm be converted nto a mechansm that approxmately preserves ts approxmaton rato at every Bayes-Nash Equlbrum? We show that for a broad class of greedy algorthms, the answer s yes. Theorem (nformal): Suppose A s a greedy c-approxmate allocaton rule for a combnatoral allocaton problem. Then the aucton that uses A to choose allocatons, and uses a pay-your-bd payment scheme, has a Bayes-Nash Prce of Anarchy of at most c + O(c 2 /e c ). We also show that the small (and exponentally decreasng) loss n our prce of anarchy bound s necessary, by gvng an example (for every c 2) where the resultng prce of anarchy s at least c + Ω( c e ). We note that the mechansms we 4c consder are all pror-free. Thus, as n the fullnformaton case, whle we assume the exstence of type dstrbutons n order to model ratonal agent behavour, our mechansm need not be aware of these dstrbutons. In the specal case that each player s type dstrbuton s a pont mass, Bayes-Nash equlbrum reduces to standard Nash equlbrum. Our mechansms therefore also preserve approxmaton ratos at every (mxed or pure) Nash equlbrum of the full nformaton game. Our analyss also extends to the more general class of coarse correlated equlbra. For the case of pure Nash equlbrum, our prce of anarchy bound mproves to c. As s standard, our bounds on the Bayesan prce of anarchy wll assume that agent types are dstrbuted ndependently. However, we show that a weaker bound of O(c) holds when agent types are drawn from an arbtrary dstrbuton over the space of all type profles. Ths result apples to greedy algorthms that are non-adaptve, as descrbed n Secton 2.4. Thus, even f agent types are arbtrarly correlated, our mechansms yeld performance at equlbrum asymptotcally matchng that of the underlyng allocaton algorthm. A smlar bound also apples to mechansms that use the crtcal-prce payment scheme, whch s a natural extenson of second-prce payments n sngle-tem auctons. Such a payment scheme charges each bdder the mnmum bd at whch he would have mantaned hs allocaton. These bounds requre a standard no-overbddng assump- Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

4 A:4 A. Borodn and B. Lucer ton, whch s that agents are avod bddng more than ther value for any gven subset of tems. Theorem (nformal): Suppose A s a greedy c-approxmate allocaton rule for a combnatoral allocaton problem. Then, under the assumpton that agents do not overbd, the aucton that uses A to choose allocatons, and uses a crtcal-prce payment scheme, has a Bayes-Nash Prce of Anarchy of at most c + 1. We also show that the extra +1 term s necessary, by gvng an example for every c 2 n whch the resultng prce of anarchy s exactly c + 1. As wth the frst-prce results, our bounds extend to coarse correlated equlbra, and a bound of O(c) holds f agent valuatons can be correlated. Furthermore, we show that a slght modfcaton to the mechansm allows us to replace the no-overbddng assumpton wth the (conceptually weaker) assumpton that bdders avod weakly domnated strateges. Repeated Auctons.. Our bounds on effcency at equlbrum do not explctly model the manner by whch agents reach equlbrum, or mpose upon the agents any computatonal constrants whatsoever. We smply post that equlbra (or approxmate equlbra 3 ) are predctve of the behavor of ratonal agents n hgh-stakes auctons. However, n settngs where auctons are explctly repeated, one mght naturally model the dynamcs under whch bdder behavor evolves. We wll therefore also consder a repeated-game varant of combnatoral allocaton problems, n whch an aucton problem s resolved multple tmes wth the same objects and bdders. Perhaps the most well-studed modern examples of repeated auctons are auctons for advertsng spaces or slots [Edelman et al. 2005], but ths model apples also to bandwdth auctons (such as the FCC spectrum aucton), arlne landng rghts auctons [Cramton et al. 2005], etc. In these settngs a mechansm for the (one-shot) aucton problem corresponds to a repeated game to be played by the agents. Rather than vew a repeated aucton as an extensve-form game, we consder models of lmted ratonalty that attempt to capture natural bddng behavour. We we study two such models: external regret mnmzaton and asynchronous best-response dynamcs. In the frst model, agents can play arbtrary sequences of strateges for the repeated aucton, under the assumpton that they obtan low regret relatve to the best fxed strategy n hndsght. More precsely, for each bdder, the dfference between the average utlty obtaned by the bdder and the average utlty that would have been obtaned by the best sngle declaraton n hndsght must tend to 0 as the number of aucton rounds ncreases. Under the assumpton that bdders are able to mnmze external regret, our goal s to desgn an aucton mechansm that acheves an approxmaton to the optmal socal welfare on average over suffcently many rounds of the repeated aucton. Ths s precsely the problem of desgnng a mechansm wth bounded prce of total anarchy, as ntroduced by Blum et al [Blum et al. 2008]. As observed by Blum and Mansour [Blum and Mansour 2007] and Roughgarden [Roughgarden 2009; 2012], n the full nformaton settng, prce of anarchy wth respect to coarse correlated equlbrum s equal to the total prce of anarchy. Hence, the bounds stated above for coarse correlated equlbrum apply also to the total prce of anarchy. We further show that for the greedy mechansms we consder, regret-mnmzng strateges can be computed effcently, assumng a natural representaton of the bdders valuaton functons. 3 All of our bounds on socal effcency degrade gracefully when agents apply strateges n approxmate equlbrum. Namely, whenever we convert a c approxmate allocaton algorthm nto a mechansm achevng say f(c) prce of anarchy, the same proof shows that the mechansm acheves at least (and often better) an f(c + γ) approxmaton at every (1 + γ) approxmate equlbrum. Notably, f c 1 + γ, the c-approxmaton for pure equlbra of the frst prce mechansm remans a c approxmaton at every approxmate equlbrum. Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

5 Equlbra of Greedy Combnatoral Auctons A:5 Theorem (nformal): Suppose A s a greedy c-approxmate allocaton rule for a combnatoral allocaton problem. Then the aucton that uses A to choose allocatons, and charges crtcal payments, acheves a (c + 1) approxmaton to the optmal welfare when agents apply regret-mnmzng strateges. Moreover, a regret-mnmzng strategy can be mplemented n tme polynomal n the number of XOR bds 4 used to represent an agent s valuaton. In the second model, we assume that agents choose strateges that are myopc bestresponses to the current strateges of the other agents. We model ths behavour as follows: on each aucton round, an agent s chosen unformly at random, and that agent s gven the opportunty to change hs strategy to the current myopc best-response. As n the regret-mnmzaton model, our goal s to desgn aucton mechansms that acheve approxmatons to the best possble socal welfare on average over suffcently many aucton rounds, wth hgh probablty over the random choces of bdders to update. Ths s the concept of the prce of (myopc) snkng, as ntroduced by Goemans et al [Goemans et al. 2005]. We conjecture that any greedy c-approxmate allocaton rule can be mplemented as a mechansm wth prce of snkng O(c). As partal progress toward ths conjecture, we desgn mechansms talored to two partcular combnatoral allocaton problems: the unrestrcted combnatoral aucton problem and the cardnalty-restrcted combnatoral aucton. Each mechansm has prce of snkng O(c), where c s the approxmaton factor of the best-known algorthm. We recall that one method for boundng the prce of snkng s to prove that there exsts an equlbrum state that s reachable from any declaraton profle by some polynomal-length sequence of best-response steps. Ths would mply that an equlbrum state would be reached wth hgh probablty after exponentally many steps. We do not take ths approach, but rather prove that the average socal welfare obtaned after a polynomal number of steps wll approxmate the optmal welfare wth hgh probablty Related Work The semnal paper n algorthmc game theory and more specfcally algorthmc mechansm desgn s that of Nsan and Ronen [Nsan and Ronen 1999]. The basc ssue ntroduced n [Nsan and Ronen 1999] s to reconcle the competng demands for revenue and socal welfare optmzaton wth the need for computatonal effcency n the context of self nterested (.e. selfsh) agents. The two most studed soluton concepts n algorthmc game theory are truthfulness (.e. ncentve compatablty) and behavor at (all) equlbra (.e. the prce of anarchy (POA) concept). Intal POA results for games were frst ntroduced to algorthmc game theory n the semnal papers by Papdmtrou [Papadmtrou 2001] and Roughgarden and Tardos [Roughgarden and Tardos 2000]. Chrstodoulou et al. [Chrstodoulou et al. 2008] ntated the study of the prce of the prce of anarchy n the Bayesan settng. Whereas the emphass of algorthmc mechansm desgn has been to consder the approxmatons acheveable by truthful mechansms, to the best of our knowledge, our conference paper [Lucer and Borodn 2010] was the frst to consder ths constructve aspect of mechansm desgn and prce of anarchy. Snce the ntal conference verson of ths work there has been sgnfcant progress on the understandng of the prce of anarchy of mechansms n varous aucton settngs. Some examples nclude the Generalzed Second Prce aucton for sponsored 4 Equvalently, the mnmum number of (subset, value) pars (S, w ) needed so that valuaton v satsfes v(t ) = max : S T {w }. Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

6 A:6 A. Borodn and B. Lucer search ads [Caraganns et al. 2012], smultaneous sngle-tem auctons [Chrstodoulou et al. 2008; Hassdm et al. 2011; Bhawalkar and Roughgarden 2011; Feldman et al. 2013], and mult-unt auctons [Markaks and Telels 2012; de Kejzer et al. 2013]. A framework unfyng much of ths work was proposed by Syrgkans and Tardos [Syrgkans and Tardos 2013]. Chekur and Gamzu [Chekur and Gamzu 2009] defned loser-ndependent algorthms, and n the conference verson of our paper [Lucer and Borodn 2010] we argued that the basc property of greedy algorthms that we were explotng was a multparameter verson of loser-ndependence. In the frst author s thess [Lucer 2011], a strengthenng of loser-ndependence, called strong loser-ndependence, was ntroduced to smplfy the proofs and that wll be the basc property of greedy algorthms we wll use n ths paper. Loser-ndependence s conceptually related to the concept of smoothness, whch was ntroduced by Roughgarden [Roughgarden 2009] as a general way to derve prce of anarchy results for one shot and repeated games (wthout reference to mechansms that derve games). Loser-ndependence has been shown to be dfferent from ths orgnal noton of smoothness [Lucer 2011]. However, alternatve notons of smoothness defned by Lucer and Paes Leme [Lucer and Leme 2011] and Syrgkans and Tardos [Syrgkans and Tardos 2013] can also be used to derve results smlar to our results. In partcular, Syrgkans and Tardos use ther smoothness condton to derve many prce of anarchy results for allocaton mechansms, ncludng those derved from greedy c-approxmaton algorthms. Ther result for the (non correlated) mxed Bayesan and coarse correlated equlbrum mproved upon our conference results: as n our current paper, they show that the resultng prce of anarchy approaches c wth a term exponentally decreasng n c. In partcular, they show that the prce of anarchy s never worse than c As we wll show n Secton 3.1, our applcaton of strong loser-ndependence can be nterpreted as a proof of smoothness. 2. PRELIMINARIES 2.1. Feasble Allocaton Problems We consder a settng n whch there are n agents and a set M of m objects. An allocaton to agent s a subset x M. A valuaton functon v : 2 M R assgns a value to each allocaton. We assume that valuaton functons are monotone, meanng v(s) v(t ) for all S T M, and normalzed so that v( ) = 0. A valuaton functon v s sngle-mnded f there exsts a set S M and a value y 0 such that for all T M, v(t ) = y f S T and 0 otherwse. A valuaton profle v s a vector of n valuaton functons, one for each agent. In general we wll use boldface to represent vectors, subscrpt to denote the th component, and subscrpt to denote all components except, so that v = (v, v ). An allocaton profle x s a vector of n allocatons. The goal n our socal welfare maxmzaton problems s to choose an allocaton for each agent n order to maxmze the sum of agent values. A combnatoral allocaton problem s defned by a set of feasble allocatons, whch s the set of permtted allocaton profles. We further assume n combnatoral allocaton problems that ths feasblty constrant s separable, meanng that f x s feasble then (, x ) s also feasble 5 for all. Note that separablty s a weaker assumpton than the standard downward-closure property of packng problems, whch would stpulate that f x s feasble then (y, x ) s also feasble for all y x. An allocaton rule A assgns to each valuaton profle v a feasble outcome A(v); we wrte A (v) for the allocaton to agent. An allocaton rule s component-wse monotone f t satsfes the 5 We note that the combnatoral publc projects problem (CPPP) [Papadmtrou et al. 2008] s not separable. Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

7 Equlbra of Greedy Combnatoral Auctons A:7 followng property for every agent : If v (S) < ṽ (S), v (T ) = ṽ (T ) T S, and A (v, v ) = S, then A (ṽ, v ) = S We wll tend to wrte A for both an allocaton rule and an algorthm that mplements t. We wll sometmes abuse notaton and use x for an allocaton rule, rather than a specfc allocaton. Each agent [n] has a prvate valuaton functon v, hs type, whch defnes the value he attrbutes to each allocaton. The socal welfare obtaned by allocaton profle x, gven type profle v, s SW (x, v) = v (x ). We wrte SW opt (v) for max x {SW (x, v)} and say that algorthm A s a c approxmaton algorthm 6 f SW (A(v), v) 1 c SW opt(v) for all v. A type profle v and an allocaton rule A for a combnatoral allocaton problem defne crtcal values, θ (S, v ), for any agent and set S M. The value θ (S, v ) s the mnmum amount that agent could bd on set S and stll wn S, assumng the other agents have profle v. That s, θ (S, v ) = nf{z : x (z, v ) = S}. We note that ths defnton of crtcal values holds even f t s not the case that ncreasng one s value for a set necessarly ncreases the probablty of obtanng that set. However, most of the mechansms we consder n ths work do satsfy ths monotoncty property, whch motvates the termnology of a crtcal prce Mechansms A drect revelaton mechansm M(A, P ) s composed of an allocaton rule A and a payment rule P that assgns a vector of n payments to each declared valuaton profle. The mechansm proceeds by elctng a valuaton profle d from each of the agents, called the declaraton profle. It then apples the allocaton and payment rules to d to obtan an allocaton and payment for each agent. Crucally, we do not assume that d s equal to v. We wll wrte SW (d) for SW (A(d), v) when the allocaton rule and type profle are clear from context. We wll be concerned wth two dfferent payment rules, frst prce and crtcal prce. In a frst prce mechansm, an agent s charged ther declared bd d (S) for any allocated set S. For notatonal convenence, we let M 1 (A) denote the mechansm usng allocaton rule A and the frst prce payment rule. In the crtcal prce payment rule, an agent s charged hs crtcal value θ (S, d ) for any allocated set S. We wll let M 2 (A) denote the mechansm usng allocaton rule A and the crtcal prce payment rule Equlbra of One-shot Auctons The utlty of agent n mechansm M = (A, P ), gven declaraton profle d and type profle v, s u v (d) = v (A (d)) P (d). We wll often omt the dependence on v when t s clear from context, and wrte smply u (d). We say that declaraton d weakly domnates d f for all d, u (d, d ) u (d, d ), and that there exst at least one d for whch the nequalty s strct. We consder a Bayesan settng n whch the true types of the agents are not fxed, but are rather drawn from a known probablty dstrbuton F over the set of valuaton profles. We frst assume that F = F 1... F n s the product of ndependent dstrbutons, where F (v ) s the probablty that agent has type v. (Later we wll also consder correlated dstrbutons over type profles.) We wrte SW opt (F) for E v F [SW opt (v)]. A bddng strategy for agent s a functon b that maps a type v to a dstrbuton over declaratons for agent. We thnk of b (v ) as the (randomzed) bddng strategy 6 Our conventon wll be to have approxmaton ratos c 1. Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

8 A:8 A. Borodn and B. Lucer employed by agent gven that hs true type s v. We wll abuse notaton slghtly and also wrte b (v ) for the random varable representng a declaraton chosen from the correspondng dstrbuton. We wrte b(v) = b 1 (v 1 )... b n (v n ) for the (dstrbuton over) declaraton profles resultng from applyng the bd functons n b to type profle v. The strategy profle b forms a (mxed) Bayesan Nash Equlbrum (BNE) f, for every [n] and every v n the support of F, agent maxmzes hs expected utlty by makng a declaraton drawn from dstrbuton b (v ). That s, for each agent, each possble type v, and every dstrbuton ω over declaratons, E v F [u (b(v))] E v F,d ω [u (d, b (v ))]. For a mechansm M = (A, P ), we wll wrte SW M (F, b) to mean E v F [ v (A (b(v)))], the expected socal welfare gven type dstrbuton F and strategy profle b. The (mxed) Bayesan prce of anarchy (BPoA) of mechansm M s defned as sup F,b SW opt (F) SW M (F, b) where the supremum s over all type dstrbutons F and mxed BNE b for F. In other words, the BPoA of M s the worst-case rato between the expected welfare at BNE and the expected optmal welfare. We can further extend the defnton of BNE to allow a correlated dstrbuton over type profles. The defnton for correlated BNE and correlated Bayesan prce of anarchy s then the same as the above defntons, where we would no longer assume that F s a product of ndependent dstrbutons. Returnng to the case n whch F s a product dstrbuton, a number of specal cases deserve menton. When all type dstrbutons are pont masses (.e., each agent s type s determned), a BNE s referred to as a (mxed) Nash Equlbrum (NE). The Prce of Anarchy (PoA) of a mechansm M s defned analogously to the BPoA, but wth respect to fxed type profles and mxed Nash equlbra. It follows that the BPoA s always at least the PoA for a gven mechansm. A BNE (or NE) s called pure f ts consttuent bddng strateges are determnstc. In general a pure Nash equlbrum may not exst for a gven mechansm and type profle; see Appendx A. One can generalze mxed NE by relaxng the assumpton that the declaraton dstrbutons are ndependent. That s, one mght allow b(v) to be an arbtrary dstrbuton over declaratons, rather than a product dstrbuton. A dstrbuton ω over declaraton profles s a coarse correlated equlbrum (CCE) for type profle v f, for all and all declaraton dstrbutons ω, E d ω [u (d)] E d (ω,ω )[u (d)]. (1) Note that when the agent declaraton dstrbutons are ndependent, CCE s equvalent to mxed NE. We defne the analogous prce of anarchy concepts; t follows that the pure prce of anarchy s at most the mxed prce of anarchy whch n turn s at most the coarse correlated prce of anarchy Greedy Allocaton Rules We descrbe a specal type of allocaton rule, whch we wll refer to as a greedy allocaton rule. These are motvated by the prorty framework n Borodn, Nelsen and Rackoff [Borodn et al. 2002] and the monotone greedy algorthms of Mu alem and Nsan [Mu alem and Nsan 2008], extended to be adaptve as n Borodn and Lucer [Borodn and Lucer 2010]. We begn wth some defntons. A partal allocaton profle Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

9 Equlbra of Greedy Combnatoral Auctons A:9 Prorty Algorthm Input: Declaraton profle d = d 1,..., d n. 1. Fx a monotone prorty functon r. Let N =. 2. Repeat untl N = [n]: 3. Choose / N and feasble allocaton S M for gven N that maxmzes r(, S, d (S)) 4. Set x = S; add player to N 5. return x = (x 1,..., x n ) Fg. 1. The framework for a non-adaptve prorty algorthm. for agents N [n] s an allocaton profle x wth x = for all N. A partal allocaton profle s feasble f there s some feasble allocaton profle that extends t. Gven a partal allocaton profle x for subset N, some N, and allocaton y M, we say y s a feasble allocaton for gven N f the partal allocaton remans feasble when we set x = y. A prorty functon s a functon r : [n] 2 M R R. We thnk of r(, S, v) as the prorty of allocatng S M to player when v (S) = v. We say that r s monotone f t s non-decreasng n v and monotone non-ncreasng n S wth respect to set ncluson. We consder two types of greedy allocaton algorthms. A non-adaptve greedy allocaton algorthm A s an allocaton algorthm as defned n Fgure 1. We say that A s monotone when the prorty functon r s monotone. We assume that tes n step 3 are broken n an arbtrary but fxed manner (.e. we assume that the prorty functon s a 1-1 functon nducng a total orderng). A non-adaptve algorthm fxes a sngle prorty functon that s used throughout ts executon. By constrast, an adaptve greedy allocaton algorthm can change ts prorty functon on each teraton, dependng on the partal allocaton formed on the prevous teratons Applcatons We now descrbe some applcatons of greedy algorthms for partcular combnatoral allocaton problems. Combnatoral Auctons. The general combnatoral aucton problem s defned by the feasblty constrant that no two allocatons can ntersect. Lehmann et al [Lehmann et al. 1999] show that the (non-adaptve) greedy allocaton rule wth r(, S, v) = v acheves a 2m approxmaton rato for CAs. S Cardnalty-restrcted Combnatoral Auctons. In the specal case that players desres are restrcted to sets of sze at most k, the non-adaptve greedy algorthm wth r(, S, v) = v s k-approxmate assumng sngle-mnded agents. Ths translates to a (k + 1) approxmate algorthm for general (.e. mult-mnded) agents. Multple-Demand Unsplttable Flow Problem. In the unsplttable flow problem (UFP), we are gven an undrected graph wth edge capactes. The objects are the edges, and each valuaton functon s such that agent has some value v(s, t) for beng gven a path from s to t. Each agent addtonally specfes a fractonal demand d [0, 1] correspondng to a desred amount of flow to send along the gven path. An allocaton s feasble f the total allocated flow along each edge s no more than ts capacty. Let B be the mnmum edge capacty. A prmal-dual algorthm, whch s an Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

10 A:10 A. Borodn and B. Lucer adaptve greedy allocaton rule, obtans an O(m 1/(B 1) ) approxmaton for any B > 1 [Brest et al. 2005]. Convex Bundle Auctons. In a convex bundle aucton, M s the plane R 2, and allocatons must be non-ntersectng compact convex sets. We suppose that agents declare valuaton functons by makng bds for such sets. Gven such a collecton of bds, the aspect-rato, R, s defned to be the maxmum dameter of a set dvded by the mnmum wdth of a set. A non-adaptve greedy allocaton rule usng a geometrcallymotvated prorty functon yelds an O(R 4/3 ) approxmaton [Babaoff and Blumrosen 2004]. Alternatve greedy algorthms yeld better approxmaton ratos for specal cases, such as rectangles. Max-proft Unt Job Schedulng. In ths problem, each bdder has a job of unt tme to schedule on one of multple machnes. A bdder has varous wndows of tme of the form (release tme, deadlne, machne) n whch hs job could be scheduled, wth a potentally dfferent proft resultng from each wndow. The profts and wndows are prvate nformaton to each bdder. The goal of the mechansm s to schedule the jobs to maxmze the total proft. The greedy algorthm that orders bds by value obtans a 3-approxmaton, and s symmetrc wth respect to agents and objects. Unlke the prevous examples, for the case of sngle-mnded bdders, there s an optmal dynamc programmng algorthm that runs n tme O(n 7 ) [Baptste 1999]. Snce ths algorthm solves the problem optmally, t s ncentve compatble. In ths case, the resultng prce of anarchy for the greedy algorthm s appealng prmarly due to ts lnear runtme and smple allocaton rule. 3. STRONG LOSER-INDEPENDENCE Chekur and Gamzu [Chekur and Gamzu 2009] ntroduced a property known as loser-ndependence for combnatoral allocaton algorthms n sngle-parameter domans. They defne an algorthm for a combnatoral allocaton problem to be loserndependent f, whenever A (d, d ) = A (d, d ) = for some, d, d, and d, then t must be that A(d, d ) = A(d, d ). That s, f a losng agent (.e. an agent who s allocated no tems) modfes hs declaraton n such a way that he stll receves no tems, ths cannot affect the outcome of algorthm A. Note that loser-ndependence s a condton on declaraton profles, rather than on bddng functons, snce the loserndependence noton s purely algorthmc and s not a condton on equlbra. In our results we wll make use of a stronger property of greedy algorthms, whch we call strong loser-ndependence. DEFINITION 3.1. An allocaton rule A s strongly loser-ndependent f, whenever d and d satsfy A(d) A(d ), there exsts an agent and set S such that d (S) d (S) and ether A (d) = S or A (d, d ) = S. Roughly speakng, f A s a strongly loser-ndependent algorthm, then whenever a valuaton profle changes from d to d va modfcatons to losng bds (.e. an agent s declared value for sets that are not allocated to hm, when others bd accordng to d ), algorthm A wll return the same outcome on nputs d and d. We note that our defnton requres that ether A (d) = S or A (d, d ) = S, rather than A (d ) = S. The ntuton s that we thnk of losng bds as beng losers wth respect to the orgnal declaraton profle d. The property of strong loser-ndependence strengthens the defnton of loserndependence due to Chekur and Gamzu n two ways. Frst, we extend from sngleparameter settngs to multple-parameter settngs by consderng losng bds rather Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

11 Equlbra of Greedy Combnatoral Auctons A:11 than losng agents. Second, we requre that the algorthm outcome be unaffected f multple agents smultaneously modfy losng bds. It s clear from the defntons that all strongly loser-ndependent algorthms are loser-ndependent (.e. by consderng the case when d and d dffer only on the declaraton of a sngle agent). However, not all loser-ndependent algorthms are strongly loser-ndependent, even n sngle-mnded domans. For example, consder the combnatoral aucton problem and suppose that A s an algorthm that optmzes socal welfare exactly and breaks tes consstently. Then A s loser-ndependent, snce a losng agent s bd does not affect the optmal allocaton. However, A s not strongly loser-ndependent, as the followng nstance shows. Consder an aucton of two tems {a, b} to three bdders. If the (sngle-mnded) bdder declaratons are d 1 ({a, b}) = 10, d 2 ({a}) = 3, and d 3 ({b}) = 3, then the outcome s that agent 1 wns hs desred set. On the other hand, f the bdder declaratons are gven by d where d 1 ({a, b}) = 10, d 2 ({a}) = 6, and d 3 ({b}) = 6, then the outcome changes: agents 2 and 3 wn ther desred sets. However, d and d do not dffer n ther declaratons for any sets allocated by A(d), A(d 1, d ), A(d 2, d ), or A(d 3, d ), as agent 1 wns hs desred set n each of these four cases. Ths contradcts the defnton of strongly loser-ndependence. As we now show, all greedy algorthms satsfy the strong loser-ndependence property. LEMMA 3.2. Every (monotone) adaptve greedy algorthm s (component-wse monotone) and strongly loser-ndependent. PROOF. The monotoncty property follows mmedately when the prorty functon n the greedy algorthm s a monotone functon. Let A be an adaptve greedy allocaton rule, and choose any d and d such that A(d) A(d ). We wll show that there exsts some and S such that d (S) d (S) and ether A (d) = S or A (d, d ) = S. Recall the defnton of an adaptve greedy algorthm, and consder the teratons of A on nputs d and d. Let k be the frst teraton n whch the allocaton of A dffers on these two nputs. Suppose that A allocates set U to agent l on teraton k when the nput s d, and allocates T to agent j on teraton k when the nput s d. For each teraton q < k, wrte q for the agent allocated to by A (on ether nput profle) and S q for the set allocated to q. Note that f d q (S q ) d q (S q ) for any q < k then we have the desred result wth = q and S = S q. We can therefore assume that d q (S q ) = d q (S q ) for all q < k. Ths mples that the bds resolved by A are dentcal on all teratons preceedng k on nputs d and d, and therefore the values of rankng functons used n each teraton up to k must be dentcal for nputs d and d. Wrte r q for the rankng functon used n teraton q for each q k. Thus, snce the allocaton on teraton k changed from choosng set U for agent l to choosng set T for agent j, t must be that ether r k (l, U, d l (U)) r k (l, U, d l (U)) or r k (j, T, d j (T )) r k (j, T, d j (T )). Ths mples that ether d l (U) d l (U) or d j (T ) d j (T ). If d l (U) d l (U) then we have the desred result wth = l and S = U, snce A l (d) = U. We can therefore assume that d l (U) = d l (U) and d j (T ) d j (T ). Consder now the behavour of algorthm A on nput (d j, d ). We clam that A j (d j, d ) = T. Note that ths mples the desred result wth = j and S = T. To prove the clam, recall that d q (S q ) = d q (S q ) for all q < k. Thus, for each q < k and each feasble set S that could be allocated to agent j on teraton q, r q ( q, S q, d q (S q )) = r q ( q, S q, d q (S q )) > r q (j, S, d j (S)) snce A allocates S q to q on nput d. We conclude that, on nput (d j, d ), A allocates S q to agent q on each teraton q < k. On teraton k, we have r k (j, T, d j (T )) > Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

12 A:12 A. Borodn and B. Lucer r k (j, T, d j (T )) for any feasble T T (snce A allocates T to j on nput d ) and r k (j, T, d j (T )) > r k (l, U, d l (U)) = r k (l, U, d l (U)) r k (, S, d (S)) for any feasble j, due to our assumpton that d l (U) = d l (U) and the fact that A allocates U to l on teraton k for nput d. We therefore conclude that A j (d j, d ) = T as requred. We next explore an mplcaton of a strongly loser-ndependent algorthm A beng a worst-case c-approxmaton. If A s a c-approxmate algorthm, then (on any nput) the sum of the declared values for ts output profle approxmates the sum of the declared values for the optmal allocaton. We now show that t also approxmates the sum of the crtcal values of the optmal allocaton profle. LEMMA 3.3. If A s a c-approxmate strongly loser-ndependent algorthm, then for any type profle v and allocaton profle y, [n] v (A (v)) 1 c [n] θ (y, v ). PROOF. Choose any ɛ > 0. For all, let v be the sngle-mnded declaraton for set y at value θ (y, v ) ɛ. Let v be the pontwse maxmum of v and v. That s, for all S M, v (S) = max{v (S), v (S)}. By defnton of crtcal prces, we have that A (v, v ) = A (v) for all, and furthermore v (A (v)) = v (A (v)). Snce A s strongly loser-ndependent, we must therefore have A(v) = A(v ). Snce A s a c-approxmaton, we conclude that SW (x(v), v) = SW (x(v ), v ) 1 c SW (y, v ) 1 c [n] θ (y, v ) nɛ. The result follows by takng the lmt as ɛ 0. For brevty, for the remander of ths paper we wll say monotone strongly loserndependent to mean both strongly loser-ndependent and component-wse monotone Applyng Strong Loser Independence Strong loser-ndependence s a strctly algorthmc concept, devod of game theoretc consderatons. Our general approach wll be to derve prce of anarchy results for any mechansm that uses a strongly loser-ndependent c-approxmaton A as ts allocaton algorthm. To do so, we wll be usng Lemma 3.3 n conjucton wth the assumpton that a gven bd profle s an equlbrum. At a hgh level, our argument wll be as follows. For each prcng rule and equlbrum concept, equlbrum wll mply an nequalty of the form v (y ) λ θ (y, d ) + µ v (x (d)) where y s an optmal allocaton. (For Bayesan equlbrum, these terms wll be expectatons.) Ths allows us to charge the optmal gan for each agent to ts crtcal value and ts welfare from the algorthm. We then explot Lemma 3.3 to convert ths bound nto a relatonshp between the optmal welfare and the welfare at equlbrum. To make ths more specfc, n our pure Nash equlbrum result for a frst prce mechansm (Theorem 4.3), we show the followng (somewhat stronger) nequalty: v (y ) θ (y, d ) + v (x (d)) d (x (d)) 7 For pure Nash equlbrum prce of anarchy results, f we assume no over-bddng, we do not need monotoncty but t s necessary for all of our other results. Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

13 Equlbra of Greedy Combnatoral Auctons A:13 It wll then follow (gnorng an nɛ term whch dsappears as ɛ 0) that: v (y ) θ (y, d ) + v (x (d)) d (x (d)) (c 1) d (x (d)) + v (x (d)) In other words, the hgh-level approach s to charge an agent s welfare n the optmal outcome aganst hs welfare at equlbrum plus the welfare of other prce-settng agents. Ths approach s smlar to the smoothness argument as formulated by Syrgkans and Tardos [Syrgkans and Tardos 2013]. However, there s a dfference n our approaches. The smoothness condton n [Syrgkans and Tardos 2013] s talored to allocaton mechansms and asserts the exstence of some d (for each player ) satsfyng such an nequalty whereas we are assumng that d s an equlbrum. The beneft of ther mmedate reducton to smoothness s that ther prce of anarchy results for pure equlbra carry over mmedately to Bayesan prce of anarchy. However, ths prohbts establshng certan tght bounds; for example, n the frst prce mechansm we show that the pure prce of anarchy s c, whch cannot be acheved va smoothness snce ths bound does not hold for the Bayesan prce of anarchy. 4. FIRST-PRICE MECHANISMS In ths secton we analyze greedy algorthms pared wth a frst prce payment scheme. More precsely (wth the excepton of results relatng to correlated Bayesan equlbrum where we wll consder more specfc greedy allocatons), gven a strongly loserndependent algorthm A, we wll be studyng the performance of the frst-prce mechansm M 1 (A) at equlbrum. Our frst step wll be to show that a utlty-maxmzng declaraton of an agent never nvolves overbddng on a set that he may possbly be allocated. Ths wll mply that agents do not employ overbddng strateges at equlbrum. It may appear at frst glance that any strategy that recommends overbddng on sets s obvously domnated for any allocaton algorthm, snce wnnng any bd larger than one s true value leads to negatve utlty. However, we must also show that an agent cannot fnd t advantageous to overbd on some set S n order to affect hs probablty of wnnng some other set T. We wll demonstrate that such stuatons cannot occur when allocatons are chosen by a strongly loser ndependent algorthm. For a type v and a declaraton d, we wll wrte d for the declaraton defned as d (S) = mn{v (S), d (S)}. That s, d agrees wth d, except that the declared value of each set can be at most the true value for that set. Note that d = d precsely f d does not overbd on any set. We now show that any declaraton d that overbds on a set that could potentally be won s weakly domnated by strategy d. LEMMA 4.1. For any monotone strongly loser ndependent allocaton rule A, valuaton v, and declaraton profle d, we have u (d) u (d, d ). Moreover, the nequalty s strct when d (A(d)) > v (A(d)). PROOF. Let S = A (d). Suppose frst that d (S) > v (S). Then u (d) = v (S) d (S) < 0. Snce v (T ) d (T ) 0 for every set T, ths mples that u (d, d ) > u (d), as requred. Next suppose that d (S) v (S), so that d (S) = d (S). We clam that A (d, d ) = S. Suppose not, for contradcton. Then we can construct a sequence of declaratons (d 1, d 2,..., d k ), wth d 1 = d and d k = d, such that adjacent declaratons dffer only on a sngle set and declared values only decrease. Suppose j s mnmal such that Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

14 A:14 A. Borodn and B. Lucer A (d j, d ) S; such a j > 1 must exst snce, by assumpton, A (d, d ) S. Then (a) d j 1 and d j dffer only on the value assgned to some set T, (b) d j 1 (T ) > d j (T ), (c) A (d j 1, d ) = S, and (d) A (d j, d ) S. Strong loser-ndependence then mples that A (d j, d ) = T. However, the fact that d j 1 (T ) > d j (T ) then contradcts the component-wse monotoncty of A. We conclude by contradcton that A (d, d ) = S. Snce S s also A(d), we have u (d) = v (S) d (S) = u (d, d ) as requred. An mmedate corollary s that f b s a bddng strategy, and there exsts a type v and set S such that (b (v ))(S) > v (S), then b s weakly domnated by the strategy b. Moreover, b s strctly better, n terms of utlty, under any dstrbuton of declaratons n whch agent wns set S wth postve probablty. We conclude that at any BNE of mechansm M 1 (A), no player wll overbd on a set that he wns wth postve probablty. COROLLARY 4.2. For any monotone strongly loser ndependent allocaton rule A, BNE b, type v, and set S, f Pr v F [A (b(v)) = S] > 0 then (b (v ))(S) v (S) Pure Nash Equlbra We are now ready to bound the prce of anarchy of M 1 (A). We begn wth a result for pure Nash equlbra, rather than the fully general BNE case. THEOREM 4.3. Suppose A s a c-approxmate monotone strongly loser ndependent allocaton rule for a combnatoral allocaton problem. Then the prce of anarchy of M 1 (A) s at most c. PROOF. Fx type profle v and suppose that b forms a pure Nash equlbrum. Snce the Nash equlbrum s pure, we wll wrte d = b(v) for notatonal convenence. Let y be an optmal allocaton for v, and let x( ) denote the allocaton rule for A. Lemma 3.3 mples d (x (d)) 1 θ (y, d ). (2) c Choose arbtrarly small ɛ > 0 and let d be the sngle-mnded declaraton for set y at value θ (y, d ) + ɛ. Then x (d, d ) = y (from the defnton of crtcal values) and hence u (d, d ) = v (y ) θ (y, d ) ɛ. Snce d s a Nash equlbrum, t must be that v (y ) θ (y, d ) ɛ = u (d, d ) u (d, d ) = v (x (d)) d (x (d)). Summng over all and applyng (2) and Corollary 4.2 we have v (y ) θ (y, d ) d (x (d)) + v (x (d)) + nɛ (c 1) d (x (d)) + v (x (d)) + nɛ c v (x (d)) + nɛ Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

15 Equlbra of Greedy Combnatoral Auctons A:15 whch, takng ɛ 0, mples SW (x(d), v) = v (x (d)) as requred. 1 v (y ) c = 1 c SW OP T (v) The power of Theorem 4.3 s marred by the fact that, for some problem nstances, the mechansm M 1 (A) s not guaranteed to have a pure Nash equlbrum. An example s gven n Appendx A Bayes-Nash Equlbra We are now ready to bound the mxed Bayesan prce of anarchy for mechansm M 1 (A). THEOREM 4.4. Suppose A s a monotone strongly loser ndependent allocaton rule for a combnatoral allocaton problem. Then the Bayesan prce of anarchy of M 1 (A) s at most 8 c 1 e for every ndependent type dstrbuton F. c c 1 e c We note that c ( ) e = c + O(c/e c ). The remander of ths subsecton s c dedcated to the proof of Theorem 4.4. Fx a product dstrbuton F over type profles and let b( ) be a (possbly mxed) Bayes-Nash equlbrum wth respect to F. Choose some type declaraton v and let y v denote an optmal allocaton for v. Followng the proof of Theorem 4.3, we would lke to bound the expected value of θ (y v, d ) wth respect to v (y v) and u (b(v)) for each. We encapsulate ths bound n Lemma 4.6 and Corollary 4.7, below. Ths wll allow us to use Lemma 3.3 to obtan a relaton between the expected welfare of A and the expected optmal welfare; ths relatonshp s gven n Lemma 4.5. LEMMA 4.5. Suppose that A s a c-approxmate monotone strongly loser ndependent allocaton rule and that there exst constants γ 0 and σ [0, c] for [n] such that, whenever b s a Bayes-Nash equlbrum for M 1 (A), t s the case that for all, all v, and all S M, E v [θ (S, b (v ))] γv (S) σ E v [u (b(v))]. Then E v [SW (A(b(v)), v)] γ c E v[sw OP T (v)]. LEMMA 4.6. Suppose that b s a Bayes-Nash equlbrum for mechansm M 1 (A) and dstrbuton F. Then for all, all v, and all S M, ( ) v (S) E v [θ (S, b (v ))] v (S) 1 + ln E v [u (b(v))]. E v [u (b(v))] Before provng Lemmas 4.5 and 4.6, let us show how they mply Theorem 4.4. We frst note the followng smple corollary of Lemma In the ntal conference verson of ths work, we presented a bound of c + O(log c) on the BPOA. Subsequently, ths bound was ndependently mproved by Lucer [Lucer 2011] to c + O(c 2 /e c ) and by Syrgkans and Tardos [Syrgkans and Tardos 2013] to c + O(c/e c ). We present here a slghtly modfed verson of the argument from Lucer [Lucer 2011], whch yelds the mproved Syrgkans and Tardos [Syrgkans and Tardos 2013] bound of c + O(c/e c ). Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.

16 A:16 A. Borodn and B. Lucer COROLLARY 4.7. Suppose that b s a Bayes-Nash equlbrum for mechansm M 1 (A) and dstrbuton F. Then for all, all v, and all S M, E v [θ (S, b (v ))] (1 e c ) v (S) c E v [u (b(v))]. PROOF. Fx agent. By Lemma 4.6, we know ( E v [θ (S, b (v ))] v (S) 1 + ln v (S) E v [u (b(v))] ) E v [u (b(v))]. (3) ( ) v Note that f 1 + ln (S) E v [u (b(v))] c then (3) mmedately mples the desred result. We can therefore assume otherwse, and choose α > 0 such that ( ) v (S) 1 + ln = c + α. E v [u (b(v))] Rearrangng, we get that v (S) = e α e c 1 E v [u (b(v))]. Applyng these two equaltes to (3), we have Snce α e α E v [θ (S, b (v ))] v (S) (c + α) E v [u (b(v))] = v (S) α e α v(s) e c 1 c E v [u (b(v))]. acheves ts maxmum value of 1/e at α = 1, we can conclude that E v [θ (S, b (v ))] v (S) 1 e c v (S) c E v [u (b(v))] as requred. Theorem 4.4 follows drectly from Corollary 4.7 and Lemma 4.5. We next complete the proof of Theorem 4.4 by provng Lemmas 4.5 and 4.6. Proof of Lemma 4.5 : Fx dstrbuton F over type profles and let b( ) be a (possbly mxed) Bayes-Nash equlbrum wth respect to F. Choose some type declaraton v and let y v denote an optmal allocaton for v. We know that for all [n] and v, E v [θ (y v, b (v ))] γv (y v ) σ E v [u (b (v ), b (v )))]. Note the dstncton between v, over whch we are takng expectatons, and v, whch s the type profle fxed to defne y v. Now, summng over and takng expectaton over all choces of v, we have ] ] ] E v [ E v [θ (y v, b (v ))] γe v [ v (y v ) E v [ We now consder each of the three terms n (4). Frst, note that ] E v [ v (y v ) σ E v [u (b (v ), b (v ))] (4) = E v [SW OP T (v)]. (5). Journal of the ACM, Vol. V, No. N, Artcle A, Publcaton date: January YYYY.