A Duality Based Unified Approach to Bayesian Mechanism Design

Transcription

1 A Dualty Based Unfed Approach to Bayesan Mechansm Desgn Yang Ca McGll Unversty, Canada Nkhl R. Devanur Mcrosoft Resarch, USA S. Matthew Wenberg Prnceton Unversty, USA ABSTRACT We provde a unfed vew of many recent developments n Bayesan mechansm desgn, ncludng the black-box reductons of Ca et. al. [6, 7, 8, 9, 19], smple auctons for addtve buyers [25, 32, 1, 38], and posted-prce mechansms for untdemand buyers [11, 12, 13]. Addtonally, we show that vewng these three prevously dsont lnes of work through the same lens leads to new developments as well. Frst, we provde a dualty framework for Bayesan mechansm desgn, whch naturally accommodates multple agents and arbtrary obectves/feasblty constrants. Usng ths, we prove that ether a posted-prce mechansm or the VCG aucton wth per-bdder entry fees acheves a constant-factor of the optmal Bayesan IC revenue whenever buyers are unt-demand or addtve, unfyng prevous breakthroughs of Chawla et. al. and Yao, and mprovng both approxmaton ratos (from to 24 and 69 to 8). Fnally, we show that ths vew also leads to mproved structural characterzatons n the Ca et. al. framework. Categores and Subect Descrptors F.0 [Theory of Computaton]: General General Terms Theory, Economcs, Algorthms Keywords Revenue, Smple and Approxmately Optmal Auctons Supported by NSERC Dscovery RGPIN Work done n part whle the author was a Research Fellow at the Smons Insttute for the Theory of Computng. Work done n part whle the author was vstng the Smons Insttute for the Theory of Computng. Work done n part whle the author was a Mcrosoft Research Fellow at the Smons Insttute for the Theory of Computng. 1. INTRODUCTION The past several years have seen a tremendous advance n the feld of Bayesan Mechansm Desgn, based on deas and concepts rooted n Theoretcal Computer Scence. For nstance, due to a lne of work ntated by Chawla et. al., we now know that posted-prce mechansms are approxmately optmal wth respect to the optmal Bayesan Incentve Compatble 1 (BIC) mechansm whenever buyers are unt-demand, 2 and values are ndependent [11, 12, 13, 30]. Due to a lne of work ntated by Hart and Nsan [25], we now know that ether runnng Myerson s aucton separately for each tem or runnng the VCG mechansm wth a per-bdder entry fee 3 s approxmately optmal wth respect to the optmal BIC mechansm whenever buyers are addtve, and values are ndependent [32, 1, 38]. Due to a lne of work ntated by Ca et. al., we now know that optmal mechansms are dstrbutons over vrtual welfare maxmzers, and have computatonally effcent algorthms to fnd them n qute general settngs [6, 7, 8, 9, 4, 19, 17]. The man contrbuton of ths work s a unfed approach to all three of these prevously dsont research drectons. At a hgh level, we show how a new nterpretaton of the Ca- Daskalaks-Wenberg (CDW) framework provdes us a dualty theory, whch then allows us to strengthen the characterzaton results of Ca et. al., as well as nterpret the benchmarks used n [11, 12, 13, 30, 25, 10, 32, 1] as dual solutons. Surprsngly, we learn that essentally the same dual soluton yelds all the key benchmarks n these works. Ths nspres us to use t to desgn the frst non-trval benchmark wth respect to the optmal BIC revenue n settngs consdered n [12, 38], whch we then analyze to acheve better approxmaton factors n both cases. 1.1 Smple vs. Optmal Aucton Desgn 1 A mechansm s Bayesan Incentve Compatble (BIC) f t s n every bdder s nterest to tell the truth, assumng that all other bdders reported ther values. A mechansm s Domnant Strategy Incentve Compatble (DSIC) f t s n every bdder s nterest to tell the truth no matter what reports the other bdders make. 2 A valuaton s unt-demand f v(s) = max S{v({})}. A valuaton s addtve f v(s) = S v({}). 3 By ths, we mean that the mechansm offers each bdder the opton to partcpate for b, whch mght depend on the other bdders bds but not bdder s. If they choose to partcpate, then they play n the VCG aucton (and pay any addtonal prces that VCG charges them).

2 It s well-known by now that the optmal aucton suffers many propertes that are undesrable n practce, ncludng randomzaton, non-monotoncty, and others [26, 27, 5, 14, 15]. To cope wth ths, much recent work n mult-dmensonal mechansm desgn has turned to desgnng smple mechansms that are approxmately optmal. Some of the most exctng contrbutons from TCS to Bayesan mechansm desgn have come from ths drecton, and nclude a lne of work ntated by Chawla et. al. for unt-demand buyers, and Hart and Nsan for addtve buyers. In a settng wth m heterogeneous tems for sale and n untdemand buyers whose values for the tems are drawn ndependently, the state-of-the-art shows that a smple posted-prce mechansm (.e. a mechansm that vsts each buyer one at a tme and posts a prce for each tem) obtans a constant factor of the optmal BIC revenue [11, 12, 13, 30]. The man dea behnd these works s a mult- to sngle-dmensonal reducton. They consder a related settng where each bdder s splt nto m separate copes, one for each tem, wth bdder s copy nterested only n tem. The value dstrbutons are the same as the orgnal mult-dmensonal settng. One key ngredent drvng these works s that the optmal revenue n the orgnal settng s upper bounded by a small constant tmes the optmal revenue n the copes settng. In a settng wth m heterogeneous tems for sale and n addtve buyers whose values for the tems are drawn ndependently, the state-of-the-art result shows that the better of runnng Myerson s optmal aucton for each tem separately or runnng the VCG aucton wth a per-bdder entry fee obtans a constant factor of the optmal BIC revenue [25, 32, 1, 38]. One man dea behnd these works s a core-tal decomposton, that breaks the revenue down nto cases where the buyers have ether low (the core) or hgh (the tal) values. Although these two approaches appear dfferent at frst, we are able to show that they n fact arse from bascally the same dual n our dualty theory. Essentally, we show that a specfc dual soluton wthn our framework gves rse to an upper bound that decomposes nto the sum of two terms, one that looks lke the the copes benchmark, and one that looks lke the core-tal benchmark. In terms of concrete results, ths new understandng yelds mproved approxmaton ratos on both fronts. For addtve buyers, we mprove the rato provded by Yao [38] from 69 to 8. For unt-demand buyers, we mprove the approxmaton rato provded by Chawla et. al. [12] from to 24. In addton to these concrete results, we beleve our work makes the followng conceptual contrbutons as well. Frst, whle the sngle-buyer core-tal decomposton technques (frst ntroduced by L and Yao [32]) are now becomng standard [32, 1, 36, 2], they do not generalze naturally to multple buyers. Yao [38] ntroduced new technques n hs extenson to multbuyers termed β-adusted revenue and β-exclusve mechansms, whch are techncally qute nvolved. Our dualtybased proof can be vewed as a natural generalzaton of the core-tal decomposton to mult-buyer settngs. Indeed, the core-tal decomposton whch requred substantal work prevously s obtaned for free: t s as smple as breakng a summaton nto two parts. Second, we use bascally the same analyss for both addtve and unt-demand valuatons, meanng that our framework provdes a unfed approach to tackle both settngs. Fnally, we wsh to pont out that the key dfference between our proofs and those of [12, 1, 38] are our dualty-based benchmarks: we are able to mmedately get more mleage out of these benchmarks whle barely needng to develop new approxmaton technques. Indeed, the bulk of the work s n properly decomposng our benchmarks nto terms that can be approxmated usng deas smlar to pror work. All these suggest that our technques are lkely be useful n more general settngs. We vew our maor contrbuton as provdng a dualty based unfed framework for desgnng smple and approxmately optmal auctons. As an applcaton, we provde a smpler and tghter analyss for both addtve and unt-demand bdders. In partcular, the fact that we acheve both results from the same dual soluton provdes strong evdence that even ths partcular dual soluton (or at least the ntuton behnd t) s worthy of deeper study. 1.2 General Bayesan Mechansm Desgn Another recent contrbuton of the TCS communty s the CDW framework for generc Bayesan mechansm desgn problems. Here, t s shown that Bayesan mechansm desgn problems for essentally any obectve can be solved wth black-box access ust to an algorthm that optmzes a perturbed verson of that same obectve. One aspect of ths lne of work s computatonal: we now have computatonally effcent algorthms to fnd the optmal (or approxmately optmal) mechansm n numerous settngs of nterest. Another aspect s structural: we now know that n all settngs that ft nto ths framework, the optmal mechansm s a dstrbuton over vrtual obectve optmzers. A mechansm s a vrtual obectve optmzer f t pontwse maxmzes the sum of the orgnal obectve and the vrtual welfare. The vrtual welfare s gven by a vrtual valuaton/transformaton, whch s a mappng from valuatons to lnear combnatons of valuatons. Our contrbuton to ths lne of work s to mprove the exstng structural characterzaton. Prevously, these vrtual transformatons were thought to be randomzed and arbtrary, havng no clear connecton to the obectve at hand. Our dualty theory can say much more about what these vrtual transformatons mght look lke: every nstance has a strong dual n the form of n dsont flows, one for each agent. The nodes n agent s flow correspond to possble valuatons of ths agent, 4 and non-zero flow from type t ( ) to t ( ) captures that the ncentve constrant between t ( ) and t ( ) bnds. We show how a flow nduces a vrtual transformaton, and that the optmal dual gves a sngle, determnstc vrtual valuaton functon such that: 1. Ths vrtual valuaton functon can be found computatonally effcently. 2. In the specal case of revenue, the optmal mechansm has expected revenue = ts expected vrtual welfare, and every BIC mechansm has expected revenue ts expected vrtual welfare. 3. The optmal mechansm optmzes the orgnal obectve + vrtual welfare pontwse. 5 4 Both the CDW framework and our dualty theory only apply drectly f there are fntely many possble types for each agent. 5 Ths could be randomzed; there s always a determnstc

3 1.3 Other Related Work Recently, strong dualty frameworks for a sngle addtve buyer were developed n [14, 16, 21, 20, 22]. These frameworks show that the dual problem to revenue optmzaton for a sngle addtve buyer can be nterpreted as an optmal transport/bpartte matchng problem. More recent work of Hartlne and Haghpanah [24] can also be nterpreted as provdng an alternatve path-fndng dualty framework. When they exst, these paths provde a wtness that a certan Myerson-type mechansm s optmal, but the paths are not guaranteed to exst n all nstances. In addton to ther mathematcal beauty, these dualty frameworks also serve as tools to prove that mechansms are optmal. These tools have been successfully appled to provde condtons when prcng only the grand bundle [14], postng a unform tem prcng [24], or even employng a randomzed mechansm s optmal [22] when sellng to a sngle addtve or unt-demand buyer. However, none of these frameworks currently apply n mult-bdder settngs, and to date have been unable to yeld any approxmate optmalty results n the sngle bdder settngs where they do apply. We also wsh to argue that our dualty s perhaps more transparent than exstng theores. For nstance, t s easy to nterpret dual solutons n our framework as vrtual valuaton functons, and dual solutons for multple buyer nstances are ust tuples of duals for sngle buyers. In addton, we are able to re-derve and extend the breakthrough results of [11, 12, 13, 25, 32, 1, 38] usng essentally the same dual soluton. Stll, t s not our goal to subsume prevous dualty theores, and our new theory certanly doesn t. For nstance, prevous frameworks are capable of provng that a mechansm s exactly optmal when the nput dstrbutons are contnuous. Our theory as-s can only handle dstrbutons wth fnte support exactly. 6 However, we have demonstrated that there s at least one mportant doman (smple, approxmately optmal mechansms) where our theory seems to be more applcable. Organzaton. We provde prelmnares and notaton below. In Secton 3, we present our dualty theory for revenue maxmzaton, and n Secton 4 we present a canoncal dual soluton that proves useful n dfferent settngs. As a warm-up, we show n Secton 5 how to analyze ths dual soluton when there s ust a sngle buyer. In Secton 6, we provde the mult-bdder analyss, whch s more techncal. Due to space lmtatons, our extenson of the CDW framework n settngs beyond revenue can be found n the full verson. 2. PRELIMINARIES Optmal Aucton Desgn. For ths verson of the paper, we restrct our attenton to revenue maxmzaton n the followng settng (the full verson contans our extenson of the CDW framework n more general settngs): there s one copy of each of m heterogenous goods for sale to n buyers. The buyers are ether all addtve or all unt-demand, wth buyer havng value t for tem. We use t = (t 1,..., t m) to demaxmzer but n cases where the optmal mechansm s randomzed, the obectve plus vrtual welfare are such that there are numerous maxmzers, and the optmal mechansm randomly selects one. 6 Our theory can stll handle contnuous dstrbutons arbtrarly well. See Secton 2. note buyer s values for all the goods and t to denote every buyer except s values for all the goods. T s the set of all possble values of buyer for tem, T = T, T = T and T = T. All values for all tems are drawn ndependently. We denote by D the dstrbuton of t, D = D, D, = D, D = D, and D = D, and f (f, f,, f, etc.) the denstes of these fnte-support dstrbutons. The optmal aucton optmzes expected revenue over all BIC mechansms. For a gven value dstrbuton D, we denote by REV(D) the expected revenue acheved by ths aucton, and t wll be clear from context whether buyers are addtve or unt-demand. We defne F to be a set system over [n] [m] that descrbes all feasble allocatons. 7 Reduced Forms. The reduced form of an aucton stores for all bdders, tems, and types t, what s the probablty that agent wll receve tem when reportng t to the mechansm (over the randomness n the mechansm and randomness n other agents reported types, assumng they come from D ) as π (t ). It s easy to see that f a buyer s addtve, or unt-demand and receves only one tem at a tme, that ther expected value for reportng type t to the mechansm s ust t π (t ). We say that a reduced form s feasble f there exsts some feasble mechansm (that selects an outcome n F wth probablty 1) that matches the probabltes promsed by the reduced form. If P (F, D) s defned to be the set of all feasble reduced forms, t s easy to see (and shown n [6], for nstance) that P (F, D) s closed and convex. Smple Mechansms. Even though the benchmark we target s the optmal randomzed BIC mechansm, the smple mechansms we desgn wll all be determnstc and satsfy DSIC. For a sngle buyer, the two mechansms we consder are sellng separately and sellng together. Sellng separately posts a prce p on each tem and lets the buyer purchase whatever subset of tems she pleases. We denote by SREV(D) the revenue of the optmal such prcng. Sellng together posts a sngle prce p on the grand bundle, and lets the buyer purchase the entre bundle for p or nothng. We denote by BREV(D) the revenue of the optmal such prcng. For multple buyers the generalzaton of sellng together s the VCG mechansm wth an entry fee, whch offers to each bdder the opportunty to pay an entry fee e (t ) and partcpate n the VCG mechansm (payng any addtonal fees charged by the VCG mechansm). If they choose not to pay the entry fee, they pay nothng and receve no tems. We denote the revenue of the mechansm that charges the optmal entry fees to the buyers as BVCG(D), and VCG(D) the revenue of the VCG mechansm wth no entry fees. The generalzaton of sellng separately s a lttle dfferent, and descrbed below. Sngle-Dmensonal Copes. A benchmark that shows up n our decompostons relates the mult-dmensonal nstances we care about to a sngle-dmensonal settng, and orgnated n work of Chawla et. al. [11]. For any mult-dmensonal nstance D we can magne splttng bdder nto m dfferent 7 When bdders are addtve, F only allows allocatng each tem at most once. When bdders are unt-demand, F contans all matchngs between the bdders and the tems.

4 copes, wth bdder s copy nterested only n recevng tem and nothng else. So n ths new nstance there are nm sngle-dmensonal bdders, and copy (, ) s value for wnnng s t (whch s stll drawn from D ). The set system F from the orgnal settng now specfes whch copes can smultaneously wn. We denote by OPT COPIES (D) the revenue of Myerson s optmal aucton [34] n the copes settng nduced by D. 8 Contnuous versus Fnte-Support Dstrbutons. Our approach explctly assumes that the nput dstrbutons have fnte support. Ths s a standard assumpton when computaton s nvolved. However, most exstng works n the smple vs. optmal paradgm hold even for contnuous dstrbutons (ncludng [11, 12, 13, 25, 32, 1, 38, 36, 2]). Fortunately, t s known that every D can be dscretzed nto D + such that REV(D) [(1 ɛ)rev(d + ), (1 + ɛ)rev(d + )], and D + has fnte support. So all of our results can be made arbtrarly close to exact for contnuous dstrbutons. We conclude ths secton by provng ths formally. The followng theorem s shown n [36], drawng from pror works [29, 28, 3, 18]. THEOREM 1. [36, 18] For all, let D and D + be any two dstrbutons, wth coupled samples t ( ) and t + ( ) such that t + (x) t(x) for all x F. If δ( ) = t+ ( ) t( ), then for any ɛ > 0, REV(D + ) (1 ɛ)(rev(d) VAL(δ)), where VAL(δ) denotes the welfare of the VCG allocaton when buyer s type s drawn accordng to the random varable δ ( ). To see how ths mples that our dualty s arbtrarly close to exact for contnuous dstrbutons, let D ɛ be the dstrbuton that frst samples t ( ) from D, then outputs t ɛ ( ) such that t ɛ (x) = mn{t ɛ (x), 1/ɛ}. It s easy to see that as ɛ 0, REV(D ɛ ) REV(D). So we can get arbtrarly close whle only consderng dstrbutons that are bounded. Now for any bounded dstrbuton D, defne D +,ɛ to frst sample t ( ) from D, then output t +,ɛ ( ) such that t +,ɛ (x) = ɛ t (x)/ɛ. Smlarly defne D,ɛ wth the celng -1 nstead of the celng. Then t s clear that D +,ɛ, D, and D,ɛ can be coupled so that t +,ɛ (x) t (x) t,ɛ (x) for all x, and that takng ether of the two consecutve dfferences results n a δ( ) such that δ(x) ɛ for all x. So applyng Theorem 1, we see that for any desred ɛ, we have REV(D) [(1 ɛ)(rev(d,ɛ nɛ)), (1 + ɛ)(rev(d +,ɛ + nɛ))]. Fnally, we ust observe that REV(D +,ɛ ) = REV(D,ɛ ) + nɛ, as every buyer values every outcome at exactly ɛ more n D +,ɛ versus D,ɛ. So as ɛ 0, the revenues are the same, and both approach REV(D). Note that both D +,ɛ and D,ɛ have fnte support. 3. OUR DUALITY THEORY We begn by wrtng the LP for revenue maxmzaton (Fgure 1). For ease of notaton, assume that there s a specal type to represent the opton of not partcpatng n the aucton. That means π ( ) = 0 and p ( ) = 0. Now a Bayesan IR (BIR) constrant s smply another BIC constrant: for any type t, bdder wll not want to le to type. We let T + = 8 Note that when buyers are addtve that OPT COPIES s exactly the revenue of sellng tems separately usng Myerson s optmal aucton n the orgnal settng. Varables: p (t ), for all bdders and types t T, denotng the expected prce pad by bdder when reportng type t over the randomness of the mechansm and the other bdders types. π (t ), for all bdders, tems, and types t T, denotng the probablty that bdder receves tem when reportng type t over the randomness of the mechansm and the other bdders types. Constrants: π (t ) t p (t ) π (t ) t p (t ), for all bdders, and types t T, t T +, guaranteeng that the reduced form mechansm (π, p) s BIC and BIR. π P (F, D), guaranteeng π s feasble. t T f (t ) p (t ), the expected rev- Obectve: max n enue. =1 Fgure 1: A Lnear Program (LP) for Revenue Optmzaton. T { }. To proceed, we ll ntroduce a varable λ (t, t ) for each of the BIC constrants, and take the partal Lagrangan of LP 1 by Lagrangfyng all BIC constrants. The theory of Lagrangan multplers tells us that the soluton to LP 1 s equvalent to the prmal varables solvng the partally Lagrangfed dual (Fgure 2). Lagrangan multplers have been used for mechansm desgn before [33, 31, 37, 4], however, our results are the frst to obtan useful approxmaton benchmarks from ths approach. DEFINITION 1. Let L(λ, π, p) be a the partal Lagrangan defned as follows: L(λ, π, p) n ( = f (t ) p (t ) + λ (t, t ) =1 t T t T t T + ( t (π(t ) π(t ) ) ( p (t ) p (t ) ))) (1) n = p ( (t ) f (t ) + λ (t, t ) λ (t, t ) ) =1 t T + n =1 t T π (t ) ( t T + t T t T + t λ (t, t ) t λ (t, t ) ) t T (π ( ) = 0, p ( ) = 0) (2) 3.1 Useful Propertes of the Dual Problem DEFINITION 2 (USEFUL DUAL VARIABLES). A feasble dual soluton λ s useful f max π P (F,D),p L(λ, π, p) <. LEMMA 1 (USEFUL DUAL VARIABLES). A dual soluton λ s useful ff for each bdder, λ forms a vald flow,.e., ff the followng satsfes flow conservaton at all nodes except the source and the snk:

5 Varables: λ (t, t ) for all, t T, t T +, the Lagrangan multplers for Bayesan IC constrants. Constrants: λ (t, t ) 0 for all, t T, t T +, guaranteeng that the Lagrangan multplers are non-negatve. Obectve: mn λ max π P (F,D),p L(λ, π, p). Fgure 2: Partal Lagrangan of the Revenue Maxmzaton LP. Nodes: A super source s and a super snk, along wth a node t for every type t T. Flow from s to t of weght f (t ), for all t T. Flow from t to t of weght λ (t, t ) for all t T, and t T + (ncludng the snk). PROOF. Let us thnk of L(λ, π, p) usng expresson (2). Clearly, f there exsts any and t T such that f (t ) + λ(t, t ) λ(t, t ) 0, t T t T + then snce p (t ) s unconstraned and has a non-zero multpler n the obectve, max π P (F,D),p L(λ, π, p) = +. Therefore, n order for λ to be useful, we must have f (t ) + λ(t, t ) λ(t, t ) = 0 t T t T + for all and t T. Ths s exactly sayng what we descrbed n the Lemma statement s a flow. The other drecton s smple, whenever λ forms a flow, L(λ, π, p) only depends on π. Snce π s bounded, the maxmzaton problem has a fnte value. DEFINITION 3 (VIRTUAL VALUE FUNCTION). For each λ, we defne a correspondng vrtual value functon Φ( ), such that for every bdder, every type t T, Φ (t ) = t t T λ(t, t )(t t ). 1 f (t ) THEOREM 2 (VIRTUAL WELFARE REVENUE). For any set of useful duals λ and any BIC mechansm M = (π, p), the revenue of M s the vrtual welfare of π w.r.t. the vrtual value functon Φ( ) correspondng to λ. That s: n =1 t T f (t ) p (t ) n =1 t T f (t ) π (t ) Φ (t ). Furthermore, let λ be the optmal dual varables and M = (π, p ) be the revenue optmal BIC mechansm, then the expected vrtual welfare wth respect to Φ (nduced by λ ) under π equals the expected revenue of M, and π argmax π P (F,D) n =1 f (t )π (t )Φ (t ). t T PROOF. When λ s useful, we can smplfy L(λ, π, p) by removng all terms assocated wth p (because all such terms have a multpler of zero, by Lemma 1), and replace the terms t T + λ(t, t ) wth f (t ) + t T λ(t, t ). After the smplfcaton, we have L(λ, π, p) = n =1 t T f (t ) ( ) π (t ) equal to n =1 t 1 t f (t ) T λ(t, t )(t t ), whch s t T f (t ) π (t ) Φ (t ), exactly the vrtual welfare of π. Now, we only need to prove that L(λ, π, p) s greater than the revenue of M. Let us thnk of L(λ, π, p) usng Expresson (1). Snce M s a BIC mechansm, ( π(t ) π(t ) ) ( p (t ) p (t ) ) 0 for any and t T, t T +. Also, all the dual varables λ are nonnegatve. Therefore, t s clear that L(λ, π, p) s at least as large as the revenue of M. When λ s the optmal dual soluaton, by strong dualty we know max π P (F,D),p L(λ, π, p) equals the revenue of M. But we also know that L(λ, π, p ) s at least as large as the revenue of M, so π necessarly maxmzes the vrtual welfare over all π P (F, D), wth respect to the vrtual transformaton Φ correspondng to λ. 4. CANONICAL FLOW AND VIRTUAL VALUATION FUNCTION In ths secton, we present a canoncal way to set the Lagrangan multplers/flow that nduces our benchmarks. We use P (t ) to denote the prce that bdder could pay to receve exactly tem n the VCG mechansm aganst bdders wth types t. 9 We wll partton the type space T nto m + 1 regons: () R (t ) 0 contans all types t such that t < P (t ), ; () R (t ) contans all types t such that t P (t ) 0 and s the smallest ndex n argmax k {t k P k (t )}. Ths parttons the types nto subsets based on whch tem has the largest surplus (value mnus prce), and we break tes lexcographcally. For any bdder and any type profle t of everyone else, we defne λ (t ) to be the followng flow. 1. For every type t n regon R (t ) 0, the flow goes drectly to (the super snk). 2. > 0, any flow enterng R (t ) s from s (the super source) and any flow leavng R (t ) s to. 3. t and t n R (t ) ( > 0), λ (t, t ) > 0 only f t and t only dffers on the -th coordnate. We wll now spend the maorty of ths secton buldng our canoncal flow and provng that t acheves certan desrable propertes. We begn by establshng some nce propertes of Φ (t ) ( ) of any flow λ (t ) constructed accordng to the above partal descrpton. CLAIM 1. For any type t R (t ), ts correspondng vrtual value Φ (t ) k (t ) for tem k s exactly ts value t k for all k. 9 Note that when buyers are addtve, ths s exactly the hghest bd for tem from buyers besdes. When buyers are untdemand, buyer only ever buys one tem, and ths s the prce she would pay for recevng.

6 PROOF. By the defnton of Φ (t ) ( ), Φ (t ) (t ) = t 1 f (t ) t λ (t ) (t, t )(t k t k ). Snce t R, by the defnton of the flow λ (t ), for any t such that λ (t ) (t, t ) > 0, t k t k = 0 for all k, therefore Φ (t ) k (t ) = t. Next, we study Φ (t ) (t ) for coordnate when t s n. Ths turns out the to be closely related to the ( roned ) R (t ) vrtual value functon defned by Myerson [34] for a sngle dmensonal dstrbutons. For each,, we use ϕ ( ) and ϕ ( ) to denote the Myerson vrtual value and roned vrtual value functon for dstrbuton D respectvely. k CLAIM 2. For any type t R (t ), f we only allow flow from type t to t, where t k = t k for all k and t s the successor of t (the largest value smaller than t n the support of D ), then Φ (t ) (t ) = ϕ (t ) = t (t t ) Pr t D [t>t ] f (t ). PROOF. Let us fx t,, and prove ths s true for all choces of t,. If t s the largest value n T, then there s no flow comng nto t except the one from the source, so Φ (t ) (t ) = t. For every other value of t, the flow comng from ts predecessor (t, t, ) s exactly f k (t k ) f (v) v T :v>t k = Π k f k (t k ) Pr t D [t > t ]. Ths s because each type of the form (v, t, ) wth v > t s also n R (t ). So all flow that enters these types wll be passed down to t (and possbly further, before gong to the snk), and the total amount of flow enterng all of these types from the source s exactly Π k f k (t k ) v T :v>t f (v). Therefore, Φ (t ) (t ) = ϕ (t ). When D s regular, ths s the canoncal flow we use. When the dstrbuton s not regular, we also need to ron the vrtual values lke n Myerson s work. Indeed, we use the same procedure: frst convexfy the revenue curve, then take the dervates of the convexfed revenue curve as the roned vrtual values. To convexfy the revenue curve, we only need to add loops to the flow we descrbed n Clam 2. The next Lemma states that there exsts a flow that performs ths procedure and the resultng vrtual value functon Φ (t ) (t ) s upper bounded by the Myerson s roned vrtual value functon ϕ (t ) f t R (t ). LEMMA 2 λ (t ) (IRONING). For any, t, there exsts a flow such that the correspondng Φ (t ) (t ) satsfes: for, Φ (t ) (t ) ϕ (t ). any > 0 any t R (t ) PROOF. Frst, we show how to modfy a flow to fx nonmonotonctes n Φ (t ) ( ). Then we show how to use ths procedure to ron. If we have two types, t and t such that Φ (t ) (t ) > Φ (t ) (t ), but t < t (and t, = t, ), let s consder Fgure 3: An example of λ (t ) for two tems. addng a cycle between t and t wth weght w. Specfcally, ncrease both λ (t ) (t, t ) and λ (t ) (t, t ) by w. What affect does ths have on Φ (t ) ( )? Frst, t s clear that ths s stll a vald flow, as we ve only added a cycle. Second, t s clear that we don t change Φ (t ) (t ) at all, for any t / {t, t }. Next, we see that we don t change Φ (t ) k (t ) or Φ (t ) k (t ) for any k. Fnally, we see that Φ (t ) (t ) decreases by exactly w(t t )/f (t ) and Φ (t ) (t ) ncreases by exactly w(t t )/f (t ). So by settng w approprately, we see that we can update λ (t ) so that Φ (t ) (t ) = Φ (t ) (t ), but wthout changng the average vrtual value for tem among these two types, nor ther vrtual value for any other tem, nor any other type s vrtual values for any tem. Now, observe that Myerson ronng can always be mplemented n the followng way: pck a dsont set of ntervals I 1,..., I k that we wsh to ron. Ths s decded by the convex hull of the revenue curve for the correspondng dstrbuton. In partcular, nsde each nterval I l, the average vrtual values of the hghest N (for any N) types s no larger than the average vrtual values n the whole nterval. Iteratvely, fnd two adacent types t, t I l (for any l) such that Φ (t ) (t ) > Φ (t ) (t ), but t < t (and t, = t, ). Then update each type s roned vrtual value to the average of ther prevous (roned) vrtual values. The end result wll be that all types n I wll have the same roned vrtual value, whch s equal to the average vrtual value on that nterval. We have shown that we can certanly mplement ths procedure va the adustments above. The only catch between exact Myerson ronng and what we wsh to do n our flow s that we are not ronng the entre support of D, but only the porton above some cutoff, C. The only effect ths has s that t possbly truncates some nterval I l at C nstead of ts true (lower) lower bound. By the nature of ronng, we know that ths necessarly mples that the average vrtual value on I k [0, C) s larger than the average vrtual value on I k [C, ) (recall: the ronng procedure s only

7 other tems ther values. Ths transformaton s feasble only f we know exactly t and could use a dfferent dual soluton for each t. Snce we can t, a natural dea s to defne a flow by takng an expectaton over t. Ths s ndeed our flow. We conclude ths secton wth one fnal lemma and our man theorem regardng the canoncal flow. Both proofs are mmedate corollares of the flow defnton and Theorem 2. Fgure 4: An example of λ (wth ronng) for a sngle bdder. to fx non-monotonctes. If the average vrtual value on the lower nterval were to be less than the average vrtual value on the hgher nterval, we wouldn t ron them to the same roned nterval). So the vrtual values we are left wth after our procedure are certanly smaller than the true roned vrtual values, completng the proof. LEMMA 3. There exsts a flow λ (t ) satsfes the followng propertes: such that Φ (t ) (t ) For any > 0, t R (t ), Φ (t ) (t ) ϕ (t ), where ϕ ( ) s Myerson s roned vrtual value for D. For any, t R (t ), Φ (t ) k (t ) = t k for all k. In partcular, Φ (t ) (t ) = t, t R (t ) 0. PROOF. Combne Lemma 2 and Clam 1. Lemma 3 sn t exactly the flow we want to use: note that we ve defned several flows that depend on t, but we only get to select one flow for bdder, and t doesn t get to change dependng on t. Below we defne a sngle flow essentally by averagng across all t accordng to the dstrbutons. DEFINITION 4 (FLOW). The flow for bdder s λ = t T f (t )λ (t ). Accordngly, the vrtual value functon Φ of λ s Φ ( ) = t T f (t )Φ (t ) ( ). Intuton behnd Our Flow: The socal welfare s a trval upper bound for revenue, whch can be arbtrarly bad n the worst case. To desgn a good benchmark, we want to replace some of the terms that contrbute the most to the socal welfare wth more manageable ones. The flow λ (t ) ams to acheve exactly ths. For each bdder, we fnd the tem that contrbutes the most to the socal welfare when awarded to. Then we turn the vrtual value of tem nto ts Myerson s sngledmensonal vrtual value, and keep the vrtual value of all the LEMMA 4. For all,, t, Φ (t ) t Pr v D [t / R (v ) ] + ϕ (t ) Pr v D [t R (v ) ]. THEOREM 3. Let M be any BIC mechansm wth ( π, p ) as ts reduced form. The expected revenue of M s upper bounded by the expected vrtual welfare of the same allocaton rule wth respect to the canoncal vrtual value functon Φ ( ). In partcular, f (t ) p (t ) t T f (t ) π (t ) Φ (t ) t T f (t ) π (t ) t T ( t Pr [t / R (v ) ] v D ) + ϕ (t ) Pr [t R (v ) ] (3) v D 5. WARM UP: SINGLE BIDDER As a warm up, we start wth the sngle bdder case. Throughout ths secton, we keep the same notatons but drop the subscrpt and superscrpt (t ) whenever s approprate. Canoncal Flow for a Sngle Bdder. Snce the canoncal flow and the correspondng vrtual valuaton functons are defned based on other bdders types t, let us see how t s smplfed when there s only a sngle bdder. Frst, the VCG prces are all 0, therefore λ s smply one flow nstead of a dstrbuton of dfferent flows. Second, for the same reason, the regon R 0 s empty and regon R contans all types t wth t t k for all k (see Fgure 4 for an example). Ths smplfes Expresson (3) to f(t) π (t) t T = t T f(t) π (t) t I[t / R ] + t T ( t I[t / R ] + ϕ (t ) I[t R ] f(t) π (t) ϕ (t ) I[t R ] (NON-FAVORITE) (SINGLE) We bound SINGLE below, and NON-FAVORITE dfferently for unt-demand and addtve valuatons. LEMMA 5. For any feasble π( ), SINGLE OPT COPIES. PROOF. Assume M s the mechansm that nduces π( ). Consder another mechansm M for the Copes settng, such that for every type profle t, M serves agent ff M allocates )

8 tem n the orgnal settng and t R. As M s feasble n the orgnal settng, M s clearly feasble n the Copes settng. When agent s type s t, ts probablty of beng served n M s t f (t ) π (t, t ) I[t R ] for all and t. Therefore, SINGLE s the roned vrtual welfare acheved by M wth respect to ϕ( ). Snce the copes settng s a sngle dmensonal settng, the optmal revenue OPT COPIES equals the maxmum roned vrtual welfare, thus no smaller than SIN- GLE. Upper Bound for a Unt-demand Bdder. As mentoned prevously, the bulk of our work s n obtanng a benchmark and properly decomposng t. Now that we have a decomposton, we can use technques smlar to those of Chawla et. al. [11, 12, 13] to approxmate each term. LEMMA 6. When the types are unt-demand, for any feasble π( ), NON-FAVORITE OPT COPIES. PROOF. Indeed, we wll prove that NON-FAVORITE s upper bounded by the revenue of the VCG mechansm n the Copes settng. Defne S(t) to be the second largest number n {t 1,, t m}. When the types are unt-demand, the Copes settng s a sngle tem aucton wth m bdders. Therefore, f we run the Vckrey aucton n the Copes settng, the revenue s t T f(t) S(t). If t R, then there exsts some k such that t k t, so t I[t R ] S(t) for all. Therefore, t T f(t) π(t) t I[t / R] t T f(t) π(t) S(t) t T f(t) S(t). The last nequalty s because the bdder s unt demand, so π(t) 1. Combnng Lemma 5 and Lemma 6, we recover the result of Chawla et al. [13]: THEOREM 4. For a sngle unt-demand bdder, the optmal revenue s upper bounded by 2OPT COPIES. Upper Bound for an Addtve Bdder. When the bdder s addtve, we need to further decompose NON-FAVORITE nto two terms we call CORE and TAIL. Let r = SREV. Agan, we remnd the reader that most of our work s already done n obtanng our decomposton. The remanng porton of the proof s ndeed nspred by pror work of Babaoff et. al. [1]. However, t s worth notng that the coretal decomposton presented here s perhaps more transparent: we are smply splttng a sum nto two parts dependng on whether the buyer s value for tem s larger than some threshold. f(t) π (t) t I[t / R ] t T f(t) t I[t / R ] t T = f (t ) t f (t ) I[t / R ] t >r t + f (t ) t f (t ) I[t / R ] t t r f (t ) t Pr [t / R ] t D t >r + f (t ) t t r LEMMA 7. TAIL r. (CORE) PROOF. By the defnton of R, for any gven t, (TAIL) Pr [t / R ] = Pr [ k, t k t ]. t D t D It s clear that by settng prce t on each tem separately, we can make revenue at least t Pr t D [ k, t k t ]. The buyer wll certanly choose to purchase somethng at prce t whenever there s an tem she values above t. So we see that ths term s upper bounded by r. Thus, TAIL r t >r f(t) = r Prt D [t > r] = the revenue of sellng each tem separately at prce r, whch s also r. LEMMA 8. If we sell the grand bundle at prce CORE 2r, the bdder wll purchase t wth probablty at least 1/2. In other words, BREV CORE r, or CORE 2BREV + 2 2SREV. PROOF. We wll frst need a techncal lemma (also used n [1], but proved here for completeness). LEMMA 9. Let x be a postve sngle dmensonal random varable drawn from F of fnte support, 10 such that for any number a, a Pr x F [x a] B where B s an absolute constant. Then for any postve number s, the second moment of the random varable x s = x I[x s] s upper bounded by 2B s. PROOF. Let {a 1,..., a l } be the ntersecton of the support of F and [0, s], and a 0 = 0. E[x 2 s] = = l Pr (x = a k) a 2 k x F k=0 l (a 2 k a 2 k 1) k=1 l Pr (x = a d) x F d=k l (a 2 k a 2 k 1) Pr [x a k] x F k=1 l k=1 2B 2B s 2(a k a k 1 ) a k Pr x F [x a k] l (a k a k 1 ) k=1 The penultmate nequalty s because a k Pr x F [x a k ] B. Now wth Lemma 9, for each defne a new random varable c based on the followng procedure: draw a sample r 10 The same statement holds for contnuous dstrbuton as well, and can be proved usng ntegraton by parts.

9 from D, f r les n [0, r], then c = r, otherwse c = 0. Let c = c. It s not hard to see that we have E[c] = t r f(t) t. Now we are gong to show that c concentrates because t has small varance. Snce the c s are ndependent, Var[c] = Var[c] E[c2 ]. We wll bound each E[c 2 ] separately. Let r = max x{x Pr t D [t x]}. By Lemma 9, we can upper bound E[c 2 ] by 2r r. On the other hand, t s easy to see that r = r, so Var[c] 2r2. By the Chebyshev nequalty, Therefore, Pr t D [ Pr[c < E[c] 2r] Var[c] 4r t E[c] 2r] Pr[c E[c] 2r] 1 2. So BREV E[c] 2r 2, as we can sell the grand bundle at prce E[c] 2r, and t wll be purchased wth probablty at least 1/2. THEOREM 5. For a sngle addtve bdder, the optmal revenue s 2BREV + 4SREV. PROOF. Combnng Lemma 5, 7 and 8, the optmal revenue s upper bounded by OPT COPIES +SREV + 2BREV + 2SREV. It s not hard to see that OPT COPIES = SREV, because the optmal aucton n the copes settng ust sells everythng separately. So the optmal revenue s upper bounded by 2BREV + 4SREV. 6. MULTIPLE BIDDERS In ths secton, we show how to use the upper bound n Theorem 3 to show that determnstc DSIC mechansms can acheve a constant fracton of the (randomzed) optmal BIC revenue n mult-bdder settngs when the bdders valuatons are all unt-demand or addtve. Smlar to the sngle bdder case, we frst decompose the upper bound (Expresson 3) nto three components and bound them separately. In the last expresson n what follows, we call the frst term NON-FAVORITE, the second term UNDER and the thrd term SINGLE. We further break NON-FAVORITE nto two parts, OVER and SUR- PLUS and bound them separately. The followng are the approxmaton factors we acheve: THEOREM 6. For multple unt-demand bdders, the optmal revenue s upper bounded by 4OPT COPIES. THEOREM 7. For multple addtve bdders, the optmal revenue s upper bounded by 6OPT COPIES +2BVCG. Note that a smple posted-prce mechansm acheves revenue OPT COPIES /6 when buyers are unt-demand [12, 30], and sellng each tem separately usng Myerson s aucton acheves revenue OPT COPIES when buyers are addtve. Therefore, the CHMS/KW [12, 30] posted-prce mechansm acheves a 24- approxmaton to the optmal BIC mechansm (prevously, t was known to be a approxmaton), and Yao s approxmaton ratos [38] are mproved from 69 to 8. Some parts of the followng analyss draw nspraton from pror works of Chawla et. al. [12] and Yao [38], however, much of the analyss also represents new technques. In partcular, t s worth pontng out that our proof of Theorem 7 looks smlar to our sngle-bdder case, whereas Yao s orgnal proof requred the entrely new machnery of β-adusted revenue and βexclusve mechansms. Below s our decomposton, frst nto NON-FAVORITE, UNDER, and SINGLE, then further decomposng NON-FAVORITE nto OVER and SURPLUS. ( f (t ) π (t ) t Pr [t / R (v ) ] v D t T ) + ϕ (t ) Pr [t R (v ) ] v D f (t ) π (t ) t f (v ) t T v T [ ( k I, tk P k (v ) t P ) (v ) ( t < P )] (v ) + f (t )π (t ) ϕ (t ) Pr [t R (v ) ] v D t T = f (t ) π (t ) t f (v ) t T v T [ ( k I, tk P k (v ) t P ) (v ) ( t P )] (v ) (NON-FAVORITE) + f (t ) π (t ) t T v T t f (v ) I[t < P (v )] (UNDER) + f (t ) π (t ) ϕ (t ) t T Pr v D [t R (v ) ] (SINGLE) NON-FAVORITE f (t ) π (t ) t T P (v )f (v )I[t P (v )] (OVER) v T + f (t ) π (t ) (t P (v )) t T v T [ ( k f (v ) I, tk P k (v ) t P ) (v ) ( t P )] (v ) (SURPLUS) Analyzng SURPLUS for Unt-demand Bdders: The proof of ths lemma s smlar n sprt to Lemma 6. LEMMA 10. When the types are unt-demand, for any feasble π( ), SURPLUS OPT COPIES. PROOF. Indeed, we wll prove that SURPLUS s bounded above by the revenue of the VCG mechansm n the Copes settng. For any defne S (t, v ) to be the second largest

10 number n {t 1 P 1(v ),, t m P m(v )}. Now consder runnng the VCG mechansm on type profle (t, v ). An agent (, ) s served n the VCG mechansm n the Copes settng, ff tem s allocated to n the VCG mechansm n the orgnal settng, whch s equvalent to sayng t P (v ) 0 and t P (v ) t k P k (v ) for all k. The Copes settng s sngle-dmensonal, therefore any agent s payment s her threshold bd. For agent (, ), her threshold bd s P (v ) + max{0, max k t k P k (v )} whch s at least S (t, v ). On the other hand, for any, whenever, t P (v ) 0, there exsts some such that (, ) s served n the VCG mechansm. Combnng the two conclusons above, we show that on any profle (t, v ), the payment n the VCG mechansm collected from agents n {(, )} [m] s at least S (t, v ) I[, t P (v ) 0]. So the total revenue of the VCG Copes mechansm s at least: (t,v ) T f(t, v ) S (t, v ) I[, t P (v ) 0]. Next we argue for any and (t, v ), the followng nequalty holds. (t P (v )) ( k ] I[, tk P k (v ) t P (v ) 0 S (t, v ) I[, t P (v ) 0] (4) We only need to consder the case when the LHS s non-zero. In that case, the RHS has value S (t, v ), and also there exsts some k such that t k P k (v ) t P (v ), so t P (v ) S (t, v ). So now we can rewrte SURPLUS and upper bound t wth the revenue of the VCG mechansm n the Copes settng. f (t ) π (t ) (t P (v )) t T v T [ ] f (v ) I k, t k P k (v ) t P (v ) 0 = f(t, v ) π (t ) (t P (v )) (t,v ) T [ ] I k, t k P k (v ) t P (v ) 0 f(t, v ) π (t ) S (t, v ) (t,v ) T I[, t P (v ) 0] (Inequalty (4)) f(t, v ) S (t, v ) (t,v ) T I[, t P (v ) 0] ( π (t ) 1, t ) The last lne s upper bounded by the revenue of the VCG mechansm n the Copes settng by our work above, whch s clearly upper bounded by OPT COPIES. Analyzng SURPLUS for Addtve Bdders:. Smlar to the sngle bdder case, we wll agan break the term SURPLUS nto the CORE and the TAIL, and analyze them separately. Before we proceed, we frst defne the cutoffs. Let r (v ) = max x P (v ){x Pr t D [t x]}. The observant reader wll notce that ths s bdder s ex-ante payment for tem n Ronen s sngle-tem mechansm [35] condtoned on other bdders types beng v, but ths connecton s not necessary to understand the proof. Further let r (v ) = r(v ), r = Ev D [r (v )] and r = r, the expected revenue of runnng Ronen s mechansm separately for each tem (agan, the connecton to Ronen s mechansm s not necessary to understand the proof). We frst bound TAIL and CORE, usng arguments smlar to the sngle tem case (Lemmas 7 and 8), SURPLUS f (v ) v T (t P (v )) t P (v ) t, T, f, (t, ) f (t ) I[ k, t k P k (v ) t P (v )] = f (v ) f (t ) v T + t P (v ) (t P (v )) Pr t, D, [ k, t k P k (v ) t P (v )] f (v ) f (t ) v T t >P (v )+r (v ) (t P (v )) Pr t, D, [ k, t k P k (v ) t P (v )] f (v ) v T LEMMA 11. TAIL r. PROOF. Frst, by unon bound (TAIL) t [P (v ),P (v )+r (v )] f (t ) (t P (v )) (CORE) Pr [ k, t k P k (v ) t P (v )] t, D, Pr [ t k P k (v ) t P (v )]. t k D k k By the defnton of r k (v ), we certanly have r k (v ) (P k (v ) + t P (v )) Pr tk D k [ t k P k (v ) t P (v )], so we can also derve: Pr [ t k P k (v ) t P (v )] t k D k r k (v ) P k (v ) + t P r k (v ) (v ) t P. (v ) Usng these two nequaltes, we can upper bound TAIL:

11 f (v ) v T r k (v ) =r k f (v ) r (v ) v t >P (v )+r (v ) f (v ) r (v ) v t >P (v )+r (v ) f (t ) f (t ) (Defnton of r (v )) LEMMA 12. BVCG CORE 2 r. In other words, 2r + 2BVCG CORE. PROOF. Fx any v T, let t D, defne two new random varables and b (v ) = (t P (v ))I[t P (v )] c (v ) = b (v )I[b (v ) r (v )]. Clearly, c (v ) s supported on [0, r (v )]. Also, we have = E[c (v )] t [P (v ),P (v )+r (v )] f (t ) (t P (v )). So we can rewrte CORE as f (v ) E[c (v )]. v T Now we wll descrbe a VCG mechansm wth per bdder entry fee. Defne an entry fee functon for bdder dependng on v as e (v ) = E[c(v )] 2r(v ). We wll show that for any and other bdders types v T, bdder accepts the entry fee e (v ) wth probablty at least 1/2. Snce bdders are addtve, the VCG mechansm s exactly m separate Vckrey auctons, one for each tem. So P (v ) = max l {v l }, and for any set of S, ts Clarke Pvot prce for to receve set S s S P(v ). That also means b(v ) s the random varable that represents bdder s utlty n the VCG mechansm when other bdders bds are v. If we can prove Pr[ b(v ) e (v )] 1/2 for all v, then we know bdder accepts the entry fee wth probablty at least 1/2. It s not hard to see for any nonnegatve number a, a Pr[b (v ) a] (a + P (v )) Pr[t a + P (v )] r (v ). Therefore, because each c (v ) [0, r (v )], by Lemma 9 we can agan bound the second moment as: E[c (v ) 2 ] 2r (v )r (v ). Snce c s are ndependent, Var[ c (v )] = E[c (v ) 2 ] r (v ) 2. By Chebyshev nequalty, we know Pr[ c (v ) Var[ c(v )] 4r (v ) 2 1/2. Var[c (v )] E[c (v )] 2r (v )] Therefore, as b (v ) c (v ), we can conclude: Pr[ b (v ) e (v )] 1/2 So the entry fee s accepted wth probablty at least 1/2 for all and v. So: BVCG 1 2 = CORE 2 r. v T f (v ) ( E[c (v )] 2r (v ) ) Analyzng SINGLE, OVER and UNDER: Frst we consder SINGLE, whch s smlar to Lemma 5. LEMMA 13. For any feasble π( ), SINGLE OPT COPIES. PROOF. Assume M s the ex-post allocaton rule that nduces π( ). Consder another ex-post allocaton rule M for the copes settng, such that for every type profle t, f M allocates tem to bdder n the orgnal settng then M serves agent (, ) wth probablty Pr v D [t R (v ) ]. As M s feasble n the orgnal settng, M s clearly feasble n the Copes settng. When agent (, ) has type t, her probablty of beng served n M s f, (t, ) π (t, t, ) t, Pr [(t, t, ) R (v ) ] v D for all and t. Therefore, SINGLE s the roned vrtual welfare acheved by M wth respect to ϕ( ). Snce the copes settng s a sngle dmensonal settng, the optmal revenue OPT COPIES equals the maxmum roned vrtual welfare, thus no smaller than SINGLE. Next, we move onto OVER. We begn wth the followng techncal propostons: PROPOSITION 1. Let π( ) be any reduced form of a BIC mechansm n the orgnal settng. Defne Π (t ) = E t, D, [π (t )]. Then Π (t ) s monotone n t.

12 PROOF. In fact, for all t,, we must have π (, t, ) monotone ncreasng n t. Assume for contradcton that ths were not the case, and let t < t wth π (t, t, ) > π (t, t, ). Then (t, t, ), (t, t, ) form a 2-cycle that volates cyclc monotoncty. Ths s because both types value all tems except for exactly the same. PROPOSITION 2. For any v T, any π( ) that s a reduced form of some BIC mechansm, OPT COPIES f (t ) π (t ) P (v ) I[t P (v )]. t T PROOF. Recall from Proposton 1 that every BIC nterm form π( ) n the orgnal settng corresponds to a monotone nterm form n the copes settng, Π( ). Let M be any (possbly randomzed) allocaton rule that nduces Π( ), and p( ) a correspondng prce rule (wlog we can let (M, p) be expost IR). Consder the followng mechansm nstead: on nput t, frst run (M, p) to (possbly randomly) determne a set of potental wnners. Then, f (, ) s a potental wnner, offer (, ) servce at prce max{p (t), P (v )). Whenever (, ) s a potental wnner, t p (t). It s clear that n the event that (, ) s a potental wnner, and t P (t ), (, ) wll accept the prce and pay at least P (v ). Therefore, for any t as long as (, ) s served n M, then the payment from (, ) n the new proposed mechansm s at least P (v )I[t P (v )]. That means the total revenue of the new mechansm s at least t T f(t) π(t) P (v ) I[t P (v )], whch s upper bounded by OPT COPIES. LEMMA 14. OVER OPT COPIES. PROOF. Ths can be proved by rewrtng OVER and then applyng Proposton 2. OVER = f (t ) π (t ) t T P (v )f(v)i[t P (v )] = v T v T v T f(v) f (t ) π (t ) P (v ) t T I[t P (v )] f(v) OPT COPIES = OPT COPIES When there s only one bdder, UNDER s always 0. Here, UNDER OPT COPIES. We apply Proposton 3 (below) once for each type profle t, usng the allocaton of ths mechansm on type profle t to specfy (, ) and let x = t. Then takng the convex combnaton of the RHS of Proposton 3 for all profles t wth multplers f(t) gves UNDER OPT COPIES. PROPOSITION 3. Let {(, )} S [m] be a feasble allocaton n the copes settng. For all choces x 1,..., x m 0, OPT COPIES v T f(v) S x I[P (v ) > x ]. PROOF. Before begnnng the proof of Proposton 3, we wll need the followng defnton and theorem due to Gul and Stacchett [23]. DEFINITION 5. Let W T (S) be the maxmum attanable welfare usng only bdders n T and tems n S. THEOREM 8. ([23]) If all bdders n T have gross substtute valuatons, then W T (S) s submodular. Now wth Theorem 8, consder n the Copes settng the VCG mechansm wth lazy reserve x for each copy (, ). Specfcally, we wll frst solct bds, then fnd the max-welfare allocaton and call all (, ) who get allocated temporary wnners. Then, f (, ) s a temporary wnner, (, ) s gven the opton to receve servce for the maxmum of ther Clarke pvot prce and x. It s clear that n ths mechansm, whenever any agent (, ) receves servce, the prce she pays s at least x. Also, t s not hard to see the allocaton rule s monotone, thus ths s a truthful mechansm. Next, we argue for any v T and S, whenever P (v ) > x, there exsts some such that (, ) s served n the mechansm above. By the defnton of Clarke pvot prce, we know P (v ) = W [n] { }([m]) W [n] { }([m] {}). Frst, we show that f tem s allocated to some bdder n the max-welfare allocaton n the orgnal settng then v P (v ). Assume S to be the set of tems allocated to bdder. Snce the VCG mechansm s truthful, the utlty for wnnng set S s better than wnnng set S {}: v k (W [n] {} ([m]) W [n] {} ([m] S )) k S v k (W [n] {} ([m]) k S {} Rearrangng the terms, we get v W [n] {} ([m] S + {})). W [n] {} ([m] S + {}) W [n] {} ([m] S ) W [n] {} ([m]) W [n] {} ([m] {})) (T heorem 8) =P (v ). Now we stll need to argue that whenever P (v ) x, tem s always allocated n the max-welfare allocaton to some bdder wth v x. 1. If agent (, ) s a temporary wnner, v P (v ) > x. Therefore, agent (, ) wll accept the prce. 2. If agent (, ) s not a temporary wnner, let S be the set of tems that are allocated to bdder n the welfare maxmzng allocaton n the orgnal settng. Snce W [n] { }([m] S ) W [n] { }([m] S {}) W [n] { }([m]) W [n] { }([m] {}) = P (v ), and P (v ) > x, that means () tem s awarded to some bdder n the welfare maxmzng allocaton, () v > x because otherwse W [n] { }([m] S ) W [n] { }([m] S {})+x,