Deferred-Acceptance Auctions for Multiple Levels of Service

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1 1 Deferred-Acceptance Auctons for Multple Levels of Servce VASILIS GKATZELIS, Drexel Unversty EVANGELOS MARKAKIS, Athens Unversty of Economcs & Busness TIM ROUGHGARDEN, Stanford Unversty Deferred-acceptance (DA) auctons are mechansms that are based on backward-greedy algorthms and possess a number of remarkable ncentve propertes, ncludng mplementaton as an obvously-strategyproof ascendng aucton. All exstng work on DA auctons consders only bnary sngle-parameter problems, where each bdder ether wns or loses. Ths paper generalzes the DA aucton framework to non-bnary settngs, and apples ths generalzed framework to obtan approxmately welfare-maxmzng DA auctons for a number of basc mechansm desgn problems: mult-unt auctons, problems wth polymatrod constrants or multple knapsack constrants, and the problem of schedulng jobs to mnmze ther total weghted completon tme. Our results requre the desgn of novel backward-greedy algorthms wth good approxmaton guarantees. CCS Concepts: Theory of computaton Approxmaton algorthms analyss; Algorthmc game theory and mechansm desgn; Addtonal Key Words and Phrases: deferred-acceptance auctons; mechansm desgn; socal welfare; approxmaton algorthms; schedulng 1 INTRODUCTION The work of Mlgrom and Segal [33] recently proposed the framework of Deferred-Acceptance (DA) Auctons, a famly of sngle-parameter mechansms wth a number of remarkable ncentve propertes. In fact, the reverse aucton part of the very recently concluded FCC Incentve Aucton for reallocatng spectrum was a DA aucton [14]. DA auctons can be thought of as runnng an adaptve backward greedy algorthm for decdng the set of accepted bdders. The mechansm mantans a score for each actve bdder (whch can change from round to round), and rejects n each round the bdder wth the lowest score, stoppng when the stll-actve bdders consttute a feasble soluton. Ths contrasts wth standard (forward) greedy algorthms, whch begn wth the empty set and teratvely add the bdder wth the hghest score (subject to feasblty constrants). DA auctons are appealng n practce for a number of reasons. Frst, DA auctons can be mplemented as ascendng clock auctons (or descendng auctons f appled to procurement auctons), whch has several desrable consequences. Truthful bddng s a domnant strategy, even n the sense of obvous strategyproofness formalzed by L [28]; see the dscusson n Secton 3. In contrast, a sealed-bd Vckrey aucton, for example, s not obvously strategyproof n ths sense. Ths s mportant for the partcpaton and behavor of non-expert users, who may not fully understand the computatons nvolved n settng the clock prces. Furthermore, every DA aucton satsfes Ths work s supported by the Natonal Scence Foundaton, under grants CCF , CCF , CCF , and CCF Part of ths work took place whle the authors were vstng the Smons Insttute for the Theory of Computng. Permsson to make dgtal or hard copes of all or part 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 bear ths notce and the full ctaton on the frst page. Copyrghts for components of ths work owned by others than ACM must be honored. Abstractng wth credt s permtted. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Request permssons from permssons@acm.org. EC 17, 2017June Assocaton 26 30, 2017, for Cambrdge, ComputngMassachusetts, Machnery. USA ACM. ISBN /17/06 $ XXXX-XXXX/2017/1-ART1 DOI: $ ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

2 1 Vasls Gkatzels, Evangelos Markaks, and Tm Roughgarden weak group-strategyproofness (WGSP). That s, no coalton of bdders can collectvely submt false bds n a way that makes every bdder of the coalton strctly better off. We refer the reader to [34] for further advantages and motvaton behnd DA auctons. Unfortunately, DA auctons do not always acheve a good approxmaton to the socal welfare. Backward greedy algorthms can be nferor to forward greedy algorthms, and the former need not even produce an ncluson-maxmal soluton. Drven by ths concern, Düttng et al. [11] subsequently explored the power and lmtatons of DA auctons from an approxmaton algorthms vewpont (see also related work below). Ther man results concern knapsack auctons and combnatoral auctons wth sngle-mnded bdders. For example, n the latter settng, novel DA auctons can nearly match the performance of arbtrary truthful and computatonally effcent mechansms. All work thus far on DA auctons has concerned bnary problems,.e., settngs where each bdder ether wns or loses. In many scenaros, however, a bdder receves some level of servce rather than a bnary decson. Thnk, for example, of an ar tcket purchase. Dependng on the buyer s wllngness to pay, the avalable choces may nclude basc economy class, economy class wth a more comfortable seat and more leg room, economy class wth extra bag allowance, or busness class. In a mult-unt aucton wth non-unt demand bdders, the level of servce corresponds to the number of tems awarded. In a cloud computng settng, levels of servce could correspond to possble completon tmes of a user s task. The goal of ths paper s to generalze the DA aucton framework to non-bnary settngs, and apply ths generalzed framework to a number of basc mechansm desgn problems. Our results: The frst contrbuton of ths paper s to extend the DA aucton framework to nonbnary settngs, where each bdder receves some level of servce, subject to feasblty constrants (Defntons 3.1 and 7.1). We consder bdders wth downward-slopng valuatons, meanng that a bdder s margnal value for upgradng to the next level of servce s non-ncreasng n the level. The key dea n the generalzaton s to complement the usual DA aucton scorng functon wth a clnchng functon that specfes the level of servce that each bdder has clnched as the aucton progresses. Just lke n a DA aucton, at each stage, a generalzed DA aucton decdes whch bdder to remove from the set of actve bdders. The dfference s that upon removal, the clnchng functon also determnes the level of servce receved by the bdder who leaves the aucton. We show that, provded the clnchng functons are monotone, these generalzed DA auctons possess the key ncentve guarantees of bnary DA auctons, ncludng an obvously strategyproof ascendng mplementaton and weak group-strategyproofness. We demonstrate the usefulness of ths framework by desgnng new DA auctons for four dfferent problems. () Allocaton problems where the servce levels of the bdders must obey a polymatrod constrant and where bdders have lnear valuatons. Here, we show that the VCG mechansm can be mplemented n our generalzed DA framework (Proposton 4.1). () Job schedulng for mnmzng the total weghted completon tme. Here, we gve a novel generalzed DA schedulng aucton that acheves a 2(1 + 2)-approxmaton of the optmal socal welfare (Secton 5). A pror, t s not obvous that a constant factor approxmaton can be acheved by any backward greedy algorthm. () Knapsack auctons wth multple knapsacks avalable, where the set of bdders assgned to the same level need to satsfy a knapsack constrant, and bdders have lnear valuatons. Even n the bnary settng wth one knapsack, no DA aucton can acheve an approxmaton rato better than O(logm), where m s the sze of the knapsack. We show that even wth multple knapsacks, there s a DA aucton achevng a O(log m)-approxmaton of the optmal socal welfare (Theorem 6.1). ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

3 Deferred-Acceptance Auctons for Multple Levels of Servce 1 (v) Mult-unt auctons where bdders have general downward-slopng valuatons. Even wth two bdders and two unts, the VCG mechansm cannot be mplemented as a generalzed DA aucton, and nether can any weakly group-strategyproof mechansm (Proposton 7.3). We gve a novel DA mult-unt aucton that acheves an O(log n)-approxmaton to the socal welfare, where n s the number of agents (Theorem 7.6). For settngs () (v), we are not aware of any exstng weakly group-strategyproof mechansms (DA or otherwse) wth non-trval approxmaton guarantees. Related Work. Mlgrom and Segal [33] ntated the study of procurement DA auctons, motvated by FCC auctons for reallocatng spectrum. In a procurement aucton t s the bdders who own the resources that the auctoneer wshes to buy. As dscussed n ther work, the propertes that they establshed, such as group-strategyproofness and mplementablty as a clock aucton, can be easly shown to hold also for the more common type of sellng auctons, whch s the focus of our work. In a recent verson of ther paper, Mlgrom and Segal [34] analyze the socal welfare that can be attaned by DA auctons n a procurement settng wth near matrod structure. They show that n ths settng a DA aucton attans near optmal performance, generalzng well-known results n combnatoral optmzaton whch show that forward and backward greedy algorthms are optmal on matrods [see, e.g., 26]. Followng up on the work of [33], Duttng et al. [11] explored the power and lmtatons of DA auctons from an approxmaton algorthms vewpont. They provded both postve and negatve results for knapsack auctons and auctons wth sngle-mnded bdders. On the postve sde, for sngle-mnded bdders, t was shown that we can have almost matchng approxmaton results by DA auctons as the exstng ones n the lterature, hence provdng stronger ncentve guarantees to the known approxmatons. In partcular, an O(d)-approxmaton was obtaned va a DA aucton, when the sze of each requested bundle s at most d, matchng the known d-approxmaton of [27], whch s obtaned by a forward greedy algorthm. A dfferent DA aucton was also shown to acheve a m logm-approxmaton, almost matchng the known m-approxmaton of [27], where m s the number of avalable tems. On the negatve sde, for knapsack auctons, t was shown that no DA aucton can acheve an approxmaton that s sublogarthmc n m, establshng a separaton wth the constant factor approxmatons that are known f we only want strategyproof mechansms. There have been several other works that have already appeared regardng the applcablty of the DA auctons framework. In the procurement settng, the performance of DA auctons has been further studed by Km [25], for spectrum and bandwdth reallocaton problems. Nguyen and Sandholm [37] provde an expermental evaluaton of the performance of varous DA heurstcs for the actual nterference constrants of the upcomng FCC aucton. The DA framework has also been extended to double auctons by Düttng et al. [12] and to mult-lateral markets by Blumrosen and Zohar [7]. The applcaton to double auctons s explored further by Marx and Loertscher [29]. These works also pont out the advantages of the DA framework n obtanng budget balance for double auctons. In a smlar sprt, Ensthaler and Gebe [13] and Jarman and Mesner [20] pont out the advantages of the DA auctons n the desgn of budget-constraned procurement auctons. Group-strategyproof mechansms have been studed extensvely pror to the DA auctons framework, among others, n the context of cost-sharng mechansms [10, 18, 19, 21, 22, 30, 35]. Ths lne of work has also analyzed the economc effcency that can be acheved wth these mechansms. As has been observed, often the stronger ncentve propertes come at a sgnfcant cost n terms of economc effcency [see, e.g., 39]. Fnally, the stronger noton of obvous strategyproofness has been recently ntroduced by [28]. ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

4 1 Vasls Gkatzels, Evangelos Markaks, and Tm Roughgarden There s an extensve lterature on ascendng auctons for both sngle-parameter and multparameter settngs pre-datng the DA framework. For sngle-parameter settngs, an ascendng aucton based on the backward greedy algorthm for matrods s descrbed n Bkhchandan et al. [4]. Babaoff et al. [3] provde a procedure convertng any algorthm to a domnant-strategy ascendng aucton (wth some loss n the approxmaton factor). For mult-parameter settngs, Ausubel [1] descrbes an ascendng aucton for dentcal tems and bdders wth decreasng margnal values. Ascendng auctons for non-dentcal tems and bdders wth gross substtute preferences appear n [2, 17, 24, 31]. For a comprehensve study of the computatonal and nformatonal aspects of teratve auctons see [5, 6]. 2 PRELIMINARIES We focus on problems where some type of servce needs to be assgned to each of n agents n a set N. Ths servce can be offered at dfferent qualty levels (ncludng no servce at all) but, due to scarcty or cost of ths servce, there are restrctons n the levels that can be allocated. We represent the levels of servce as ratonal numbers, hence, the feasble servce allocaton outcomes are defned by a set system I Q N. 1 We assume that I s non-empty and downward closed, meanng that f T I and T T, then T I. Note that the bnary settng, to whch prevous work on DA auctons was restrcted, corresponds to the specal case where I {0, 1} N. We assume that each bdder has a prvate valuaton functon such that v (l) s the value of bdder for obtanng the level of servce l. Although all of our mechansms can be mplemented as ascendng auctons, our presentaton focuses on drect-revelaton mechansms n the form M = (f,p), whch consst of an outcome rule f and a payment rule p. In our settng, gven a vector of bds b = (b ) N, where b denotes the bd reported by bdder (comng ether from a sngle-parameter or mult-parameter doman, dependng on the form of the functon v ( )), the outcome rule f computes a feasble soluton,.e., a set of I, so that every bdder s assgned a level of servce. On the same nput, the payment rule p computes payments p = (p (b)) N R n where p (b) denotes the payment of bdder. Gven a mechansm M = (f,p), and a bd vector b, let l = f (b) be the level of servce assgned to bdder by the mechansm. We assume that the bdders have quas-lnear utltes and hence bdder s utlty equals u M (b) = v (l ) p (b). Snce bdders can be strategc n reportng ther bds to the mechansm, towards maxmzng ther own utlty, we am for mechansms that provde ncentve guarantees. We say that a mechansm M s strategyproof or ncentve compatble f for any bdder, for any bd vector b = (b j ) j, and any bd b of bdder u M (v, b ) u M (b, b ). We are nterested n an even stronger form of resstance to manpulaton, namely resstance aganst coaltons of bdders. Gven a subset of bdders S N, and a bddng vector b, we denote by b S, the vector contanng only the bds of S, and by b S the bds of the remanng bdders. We say that a mechansm M s weakly group-strategyproof (WGSP) f for any coalton S N, for any vector b S = (b j ) j S, and for any vector b S = (b j ) j S of the bdders n S, t holds that u M (v S, b S ) u M (b S, b S ) for some S. Hence, there s no coalton that can make all ts members strctly better off by devatng from the truth. 2 1 In our exposton, we do not need to mpose a pror an upper bound on the maxmum possble level of servce. 2 There s an even stronger form of group-strategyproofness, requrng that there s no coaltonal devaton where some members are better off and the rest are not worse off, but DA auctons do not usually guarantee ths property. ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

5 Deferred-Acceptance Auctons for Multple Levels of Servce 1 Apart from the ncentve ssues, one of the man objectves we try to optmze n mechansm desgn s the socal welfare. If T = (l 1,..., l n ) I s a feasble soluton, the generated socal welfare s SW (T ) = v (l ). A mechansm M wth an allocaton rule f acheves an approxmaton rato of ρ f SW (OPT (v)) max ρ, v SW (f (v)) where OPT (v) = arg max T I {SW (T )} denotes a welfare-maxmzng outcome. 3 GENERALIZED SINGLE-PARAMETER DEFERRED-ACCEPTANCE AUCTIONS We begn by proposng a generalzaton of the DA auctons framework of [34] that captures nonbnary sngle-parameter settngs. We assume that each bdder can receve one of multple levels of servce, and for now, that her value for level l Q equals to v l. Ths lnearty assumpton s common n many applcatons, ncludng sponsored search advertsng and job schedulng. Secton 7 generalzes the framework to the mult-parameter settng of downward-slopng valuatons. We defne a generalzed DA aucton by combnng a scorng functon wth a provsonal allocaton functon, whch we wll refer to as the clnchng functon. Clnchng functons are the key nnovaton of our generalzed framework. The aucton begns wth all bdders beng actve, t operates n a sequence of stages, and after each stage t fnalzes the level of servce of some actve bdder. Ths process contnues, untl the level of servce for every bdder has been fnalzed. Prces are then set usng Myerson s Lemma, to enforce truthfulness (see below). Defnton 3.1. A DA aucton operates n dscrete stages t 1. We denote by A t N the set of currently actve bdders n the begnnng of each stage t; ntally, A 1 = N, and A t+1 A t, for every t 1. The DA aucton s fully defned by two collectons of functons: The scorng functons σ A t (b, b N \A ), that are non-decreasng n ther frst argument. t The clnchng functons д A t bdders,.e.: д A t +1 (b N \At +1 ) д A t (b N \A t ). (b N \A ), whch are non-ncreasng w.r.t. the set of actve t At each stage t, f A t, then the level of servce of some actve bdder arg mn At {σ A t (b, b )} N \A t s fnalzed, possbly wth the use of some te-breakng rule. That s, a bdder wth the lowest score stops beng actve, we set A t+1 = A t \ {}, and her level of servce s fnalzed at level д A t (b ). N \A t When we reach A t =, then the aucton termnates and the payment of each bdder s determned by Myerson s lemma (see Equaton (1)). The scorng and clnchng functons generally change throughout the stages. These functons can depend on the bds of non-actve bdders, b, and on the set of actve bdders, A N \A t t, though not the actual bds of actve bdders, for ncentve reasons. For a gven bd vector b = (b ) N R n +, the DA aucton mples an outcome rule f, whch defnes the level of servce f (b) of each bdder. If ths outcome rule always outputs a feasble soluton (a set of I), then we say that the DA aucton s feasble. The followng lemma shows that the allocaton rule f (b) mpled by a DA aucton s always monotone w.r.t. the bd b of bdder. The proof of the lemma s deferred to the full verson of the paper. Lemma 3.2. Any generalzed DA aucton yelds a monotone allocaton rule,.e., for any bd vector b, and any b < b, we have f (b, b ) f (b). Next we derve payments accordng to Myerson s lemma [36], gven the monotone allocaton functon f (b, b ), whch are approprate summatons of threshold values for each level of servce. Formally, suppose that under (b, b ), bdder receves level of servce l. Let L = { 1, 2,..., l } ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

6 1 Vasls Gkatzels, Evangelos Markaks, and Tm Roughgarden wth 1 < 2 < < l, be the levels of servce that are attanable, as we let the bd of ncrease from 0 to b, keepng b fxed. The monotoncty of the levels we encounter s ensured by the monotoncty of the allocaton rule f ( ). For each j L, let z j be the followng threshold value: z j (b ) = nf b 0{b : f (b, b ) j }. Then, the payment of bdder comes from the followng formula: p (b) = l z j (b ) ( j j 1 ), (1) j=1 where for j = 1, we set 0 = 0. We next show that the mechansm nduced by such an allocaton rule and correspondng payment rule s weakly group-strategyproof. Theorem 3.3. Every generalzed DA aucton s weakly group strategy-proof. Proof. Assume there exsts a coalton of bdders K N that can coordnate n msreportng ther values n a way that strctly mproves the utlty of every K. We show that ths assumpton leads to a contradcton. Let v denote the true value of each bdder and v be the value reported by K. Also, let α K be the bdder of the coalton who would be fnalzed frst (at stage t α ) accordng to the true values, and let β K be the bdder of the coalton who s fnalzed frst (at stage t β ) after the devaton. Frst, we show that t β > t α. Amng for a contradcton assume that, after the devaton, bdder β s fnalzed at stage t β t α, and ths devaton strctly ncreases her utlty. Accordng to the defnton of the generalzed DA auctons, the sequence of bdders fnalzed durng the frst t β 1 stages under the devaton s dentcal to the correspondng sequence of bdders fnalzed pror to the devaton. To verfy ths fact, note that the scores of these bdders are not affected by the devaton, snce all the bdders n K reman actve durng the frst t β 1 stages (recall that the scores at a stage t maybe affected by the bds of N \ A t and not by actve bdders). Hence, both n the orgnal profle as well as n the devaton, the level of servce that has been clnched by bdder β by stage t β, and the total payment for ths level s the same. Snce we have assumed t β t α, bdder β was recevng a weakly hgher level of servce pror to the devaton. Moreover, the cost for any addtonal levels of servce compared to the level she receves under the devaton, was at most v β. Hence, overall the utlty of β cannot have ncreased wth the devaton, leadng to a contradcton. Therefore, there exsts some bdder K who s rejected at stage t α after the devaton, and was fnalzed at some stage t > t α pror to the devaton. Snce bdder α was fnalzed before bdder pror to the devaton, and usng the fact that both of these bdders scores are the same before and after the devaton, we nfer that v α > v α. Also, f l α and l α s the level of servce that α receves before and after the devaton, ths means that l α > l α (otherwse the utlty of α would be the same after the devaton) and the threshold prce of any level hgher than l α s at least v α (snce α would be rejected at stage t α f she dd not strctly ncrease her bd). Then, accordng to Equaton 1, the addtonal payment of α, for the addtonal l α l α levels of servce s at least l α j=l α +1 v α ( j j 1 ) (l α l α )v α. Note that the payment of α for the frst l α levels s the same as before snce t β > t α. Hence, snce the value of α for these addtonal levels of servce s (l α l α )v α, ths means that bdder α could not have strctly ncreased her utlty upon devatng. ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

7 Deferred-Acceptance Auctons for Multple Levels of Servce 1 Apart from group-strategy-proofness, the DA framework also guarantees several other mportant propertes that are desrable for practcal purposes. In partcular, [34] showed that a (bnary) DA aucton can be mplemented by an ascendng clock aucton. Ths means that we can thnk of the aucton as runnng n a sequence of perods, and offerng an ascendng sequence of prces to every bdder. Our generalzaton can also be mplemented as such an ascendng aucton. As a result, t obvous for the bdders to verfy that truth-tellng s ther optmal strategy, even f they are not aware of how the clock prces are set. Ths s n contrast, e.g., to the VCG mechansm, where t s not at all clear a pror why a bdder should behave truthfully. Ths dstncton s formalzed under the noton of obvous strategy-proofness n [28]. The followng two propostons summarze the fact that such propertes carry over to the settng of generalzed DA auctons as well, whch further justfes ther appeal. The proof of the propostons as well as a dscusson regardng clock auctons can be found n the full verson of the paper. Proposton 3.4. Every generalzed DA aucton, can be mplemented as an ascendng clock aucton. We henceforth refer to the class of ascendng clock auctons derved from generalzed DA auctons accordng to Proposton 3.4 as generalzed DA ascendng auctons. Proposton 3.5. Every generalzed DA ascendng aucton s obvously strategyproof. 4 WARM-UP: POLYMATROID CONSTRAINTS We begn our study of generalzed DA auctons by consderng settngs where the set of feasble outcomes s defned by a polymatrod constrant (defned below). For nstance, consder a mult-unt aucton wth k dentcal copes of the same good, and addtve bdders, so that bdder s value for acqurng l unts of the good (a level of servce l ) s l v. The constrant n ths case s that l k,.e., the number of unts to be allocated s at most k. For another example, consder the keyword sponsored search auctons, where the sellers are competng for a sequence of q < n advertsng slots, and each slot j has a clck-through rate r j. If bdder s value for a clck s v, then ts value for slot j s r j v, and we say that receves a level of servce of l = r j. Ths tme, the constrant s that the outcome (l 1, l 2,..., l n ) needs to correspond to a matchng of bdders to slots, wth l = 0 for any bdder who s not matched to a slot. For more motvatng examples see also [4]. More generally, n ths secton we consder problems, where the set of feasble outcomes s defned va a gven submodular functon h : 2 n R +, as follows: { } P h = l N n l h(s) S N. S In the frst settng provded above,.e., the mult-unt aucton wth k avalable unts, the polymatrod constrant s defned by the constant (submodular) functon h, wth h(s) = k for every S N. In the second settng, f we assume that r 1 r 2 r q, the polymatrod constrant s defned by h(s) = S j=1 r j. Ths mples that any sngle bdder gets l r 1, any two bdders, get l + l r 1 + r 2, and so on. As a frst applcaton of our framework, we show that, n fact, there s a smple generalzed DA aucton that acheves the optmal socal welfare n any problem nstance nvolvng polymatrod constrants. Gven the submodular functon h of the polymatrod constrant, the scorng and clnchng functons of ths aucton are as follows: The polymatrod aucton scorng functon s σ A t (b, b ) = b N \A t. The polymatrod aucton clnchng functon s д A t (b ) = h(a N \A t t ) h(a t \ {}). ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

8 1 Vasls Gkatzels, Evangelos Markaks, and Tm Roughgarden Note that ths DA aucton has a very smple scorng functon, whch s not adaptve. In partcular, at every stage, the bdder who s fnalzed by ths DA aucton s the one wth the smallest bd, among the ones that are stll actve. What s more nterestng s the clnchng functon, accordng to whch, at each stage t each bdder A t has clnched a level of servce equal to her margnal contrbuton to the value of h(a t ). Snce h s submodular, ths margnal contrbuton weakly ncreases as A t shrnks, so ths s a vald clnchng functon. For a concrete example of how ths aucton works, we consder the two settngs descrbed above. In the mult-unt settng, ths aucton would keep fnalzng the lower valued bdders wthout lettng them clnch any tem, untl only the hghest bdder s actve, at whch pont ths bdder clnches all k unts. For each one of these unts, the prce s equal to the second hghest bd. In the keyword sponsored search settng, durng the frst n q stages, the n q lowest bdders are fnalzed wthout clnchng any slots. Then, at stage n q + 1 every actve bdder clnches a level of servce of r q, whch s the clck through rate of the worst slot, q, and the lowest value bdder A n q+1 s fnalzed at l = r q,.e., t s assgned to that slot. Note that, after stage n q + 1, every actve bdder s guaranteed a level of servce of at least r q, and the prce that the top q bdders pay for clnchng ths level of servce s b q+1 r q,.e., r q tmes the (q + 1)-th hghest bd. As the aucton moves on, more bdders get fnalzed, and the prce that they pay for the addtonal levels of servce that they clnch weakly ncreases over tme. To verfy that ths aucton always yelds the maxmum socal welfare, note that ts outcome s exactly the same as the one that would arse f we nstead used the followng forward greedy algorthm. Frst, gve the hghest bdder,, the hghest level of servce possble,.e., h({}). Then, gve the second hghest bdder the hghest level possble, gven the exstng assgnment to,.e., gets h({, }) h({}), and so on. Ths greedy algorthm s known to be optmal n polymatrod settngs [15]. In fact, the aucton presented above s exactly what the ascendng aucton of [4] reduces to f the valuatons of the bdders are addtve, and what the ascendng aucton of [16] reduces to f we remove the budget constrants. Furthermore, the allocaton and the payments of ths aucton s the same as that of the VCG aucton. Snce these auctons le wthn the generalzed DA framework, all of the ncentve guarantees of the prevous secton (Theorem 3.3 Proposton 3.5) apply. Proposton 4.1. When the set of feasble allocatons s defned by a polymatrod constrant, the VCG mechansm s a generalzed DA aucton. In contrast Proposton 4.1, we show n Secton 7 that the VCG mechansm s not generally a DA aucton when bdders have downward-slopng valuatons. 5 SCHEDULING CONSTRAINTS To exhbt the strength of our generalzaton and the dversty of settngs that t apples to, we now consder a well-studed non-bnary problem of job schedulng. Gven a set of m dentcal machnes and a set N of n jobs, each of whch needs processng tme p on any one of the machnes, a schedule s an assgnment of each job to a machne. Furthermore, for each machne j, the schedule defnes whch one of ts assgned jobs the machne wll process at any gven tme. As a result, f a schedule x assgns the jobs of bdders 1, 2, and 3 on the same machne, to be processed n that order wth no dle tme, the completon tmes of the jobs wll be c 1 (x) = p 1, c 2 (x) = p 1 +p 2, and c 3 (x) = p 1 +p 2 +p 3 respectvely. Dependng on ts urgency, or mportance, each job also has a weght v, and the objectve of ths schedulng problem s to compute a schedule x that mnmzes the weghted completon tme N v c (x). ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

9 Deferred-Acceptance Auctons for Multple Levels of Servce 1 Ths problem has receved consderable attenton (see [9] for a survey of relevant results) and t s known to be NP-hard [8]. But, the followng smple greedy algorthm acheves a (1 + 2)/ approxmaton for ths problem [23, 40]: order the jobs n non-ncreasng order of ther v /p rato and schedule them, n that order, on the frst avalable machne. In ths secton, rather than assumng that the scheduler knows the weght v of each job, we consder the harder problem where ths nformaton s prvate to agent, who owns ths job. The completon tme c of job then corresponds to the delay that user suffers, and the weght v corresponds to the value of agent for mprovng the completon tme of her job by a unt of tme. Every agent s guaranteed completon by some deadlne d (defned later on) but, dependng on ts value, v, the agent may be wllng to pay n order to mprove ts completon tme,.e., to get a hgher level of servce. Our goal s to desgn a DA aucton that elcts bds b correspondng to the prvate values v and outputs a schedule x, amng to mnmze the socal cost,.e., b c (x). As we have shown, every DA aucton can be mplemented as an ascendng aucton. But, what would an ascendng aucton for job schedulng look lke? Ascendng Job Schedulng Auctons. In an ascendng aucton for job schedulng, as the prces ncrease, the bdders who reman actve should be guaranteed outcomes of ncreasngly hgh qualty,.e., a lower completon tme. Ths means that rather than schedulng the jobs n a bottom-up fashon, startng from lower completon tmes and movng toward hgher ones, lke all known lst-processng algorthms do, an ascendng aucton schedules jobs top-to-bottom. In partcular, a DA ascendng aucton uses a clnchng functon to promse ncreasngly better completon tmes to the actve bdders and to decde ther actual completon tme when they are fnalzed. But, a clnchng decson s rreversble, and t cannot depend on the bds of jobs n A t,.e., the ones stll competng for lower completon tmes. Therefore, a generalzed DA aucton needs to ensure that, no matter the values of the actve jobs to be fnalzed later on, there exsts a feasble way of schedulng them that respects the clnchng promses and yelds a good approxmaton factor. Ths ntroduces a novel and non-trval trade-off for the desgner: on one hand, the desgner wants to allow the jobs to clnch low completon tmes, amng for a good approxmaton factor but, on the other hand, f the promsed completon tmes are too low, ths may lead to an nfeasble schedule. Amng for a good approxmaton, we desgn a backward greedy algorthm whose socal cost s wthn a constant factor of the greedy algorthm descrbed above. In fact, we ensure that our DA aucton assgns to every job a completon tme wthn a constant factor of ts completon tme n the forward greedy schedule. The most natural way to acheve that would be to let our DA aucton use the same scorng functon,.e., to fnalze bds n ncreasng order of ther b /p rato. But, how can we ensure that the completon tme of the job wth the worst rato, the one fnalzed frst, s wthn a constant of ts completon tme n the greedy schedule? Note that the clnchng functon needs to be oblvous to the values of the actve bds and, dependng on what these values are, the completon tme of job n the forward greedy outcome may vary sgnfcantly. For a concrete example, consder the case when, apart from the job wth the smallest value over sze rato, A t also comprses two sets of jobs X and Y, where X contans m jobs of sze 1 and Y contans m 1 jobs of sze m. If the jobs n Y have hgher ratos than those n X, the completon tme of job n the forward greedy outcome would be m + 1. On the other hand, f the jobs n X have hgher ratos than those n Y, the completon tme of job n the forward greedy outcome would be 2! But, we cannot fnalze wth a constant completon tme for any m, as ths would quckly lead to feasblty ssues, snce the average load per machne s not constant. ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

10 1 Vasls Gkatzels, Evangelos Markaks, and Tm Roughgarden 5.1 Deferred-Acceptance Schedulng Aucton In ths secton, we provde a DA aucton that determnes a feasble schedule of the tasks and acheves a 2(1 + 2) 4.83-approxmaton. At each stage t, our aucton wll be leveragng two parameters that depend on A t but not on the values of the bdders n t: a) M t = 1 m A t p,.e., the value of the fractonal makespan for the jobs n A t, and b) p t = max At {p },.e., the maxmum processng tme among the jobs n A t. Our aucton guarantees to every bdder a completon tme by the deadlne d = M 1 + 2p 1, and dependng on ther bds and processng tmes, some jobs get the opportunty to pay for mproved completon tmes. These mprovements of the completon tme, beyond the ntal deadlne, are captured by the clnchng functon, whch also decdes what the completon tme of a job wll be when t s fnalzed. Scorng functon. As long as p t M t /2, our DA schedulng aucton uses a scorng functon based on the greedy algorthm rato,.e., σ (b ) = b /p. But, when there exst jobs whose processng tme s hgher than M t /2, the aucton fnalzes them frst n a largest job frst fashon. Formally, σ A t (b ) = { b /p, f p t < M t /2 or p p t 0, f p t M t /2 and p = p t. Clnchng functon. For each machne j, we denote by E j (t) the mnmum startng tme among all the tasks scheduled on machne j pror to stage t, and let E max (t) = max j {E j (t)}. For t = 1, we set E j (1) = d for every j [m]. The clnchng functon s defned as: д A t (b N \A t ) = M 1 + 2p 1 mn { M t + 2p t, E max (t) }. In more detal, at each stage t, ths clnchng functon defnes how much earler than the deadlne (M 1 + 2p 1 ) job s guaranteed to termnate (recall that we want the clnchng functon to be nonncreasng w.r.t. A t, hence we take the dfference between the deadlne and the completon tme). In other words, at stage t, every job s guaranteed to complete no later than mn { M t + 2p t, E max (t) }. The job fnalzed at stage t s the one wth mnmum score σ A t (b ), and t s scheduled on some machne j arg max j {E j (t)}. If ths machne s not busy at tme M t + 2p t, then job s scheduled so that ts completon tme s c (x) = M t + 2p t. If, on the other hand, every machne s busy at that tme, then the job s scheduled so that ts completon tme s exactly E max (t) = E j(t) (t),.e., rght before the prevously scheduled tasks on the machne. Theorem 5.1. The DA schedulng aucton yelds a 2(1 + 2) 4.83 approxmaton. What s more demandng, s that we also need to verfy that the DA aucton always yelds a feasble schedule. That s, we need to ensure that there exsts a schedule where all the jobs can be processed by the completon tmes they have clnched. To do ths, we wll defne frst some approprate quanttes for every stage t, that wll gude us n the analyss. Let Goal(t) = M t + 2p t, and for every processor j [m], let δ j (t) = max{0,goal(t) E j (t)}. Hence, δ j s the dfference between the goal functon and the tme at whch j starts processng the jobs currently assgned to her. For techncal convenence, we set ths to 0 when E j (t) exceeds GOAL(t). The rest of the analyss focuses on argung about the behavor of the vector (δ j ( )) j [m] over tme, wth respect to the features: Max(t) = max j [m] δ j(t), Mn(t) = mn j [m] δ j(t), Avg(t) = 1 m δ j (t) For an llustraton of these quanttes, see Fgure 1. We can thnk of Goal(t) as the updated deadlne at stage t, whch keeps decreasng between successve stages. I.e., recall that Goal(1) = j [m] ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

11 Deferred-Acceptance Auctons for Multple Levels of Servce 1 d = M 1 + 2p 1, and t s easy to see that Goal(t + 1) Goal(t). Our proof examnes the area below the lne at tme Goal(t) to argue about the exstence of a feasble schedule. Fg. 1. An llustraton of the relevant quanttes at a stage t. To proceed, we prove the followng mportant lemma, whch wll be useful n verfyng that our DA aucton always yelds a feasble schedule. A rough ntuton behnd the proof s that, whenever, after stage t, Goal(t) drops to Goal(t + 1) t causes ether Max(t) Mn(t) or Avg(t) to drop (or t causes an aggregate drop) by enough tme so as to mantan that the nequalty remans true. Lemma 5.2. For every stage t of the aucton we have Max(t) Mn(t) + Avg(t) 2p t. (2) Proof. The proof s by nducton on t. For t = 1 ths s straghtforward to verfy, snce E j (1) = d for every j [m]. Hence, we now assume that Inequalty (2) holds for every stage up to t, and our goal s to show that t then remans true at stage t + 1. We dstngush three possble cases: a) Mn(t) > 0 and Mn(t + 1) > 0, b) Mn(t) > 0 and Mn(t + 1) = 0, and c) Mn(t) = 0. In the analyss that follows, we wll refer to a job as beng tght, f t s scheduled wthout leavng any dle tme, tll the next job. I.e., for a job scheduled on machne j at stage t, tghtness means that ts completon tme s E (t) (thus all machnes are busy at tme M t + 2p t ). a) If Mn(t) > 0 and Mn(t + 1) > 0, let t be the last stage before t when Mn(t ) = 0 (the exstence of such a stage s clearly guaranteed, e.g., t = 1). All the machnes are always busy n the tme nterval between Goal(t ) and Goal(t + 1) (by defnton of t and snce Mn(t + 1) > 0). Therefore Avg(t + 1) = Avg(t ) 2(p t p t+1 ). (3) If Max(t ) p t, then Max(t + 1) Mn(t + 1) Max(t ). Ths s true because all jobs scheduled between t and t + 1 are tght, and ther sze s no more than p t Max(t ). Hence, the gap between Max(t) and Mn(t) wll not ncrease when we go to stage t + 1. Usng Inequalty (3), ths mples that Max(t + 1) Mn(t + 1) + Avg(t + 1) Max(t ) + Avg(t ) 2(p t p t+1 ) = Max(t ) Mn(t ) + Avg(t ) 2(p t p t+1 ) 2p t 2(p t p t+1 ) = 2p t+1 The last nequalty above s true by the nducton hypothess appled to t. ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

12 1 Vasls Gkatzels, Evangelos Markaks, and Tm Roughgarden If Max(t ) < p t, then Max(t + 1) Mn(t + 1) p t. Ths s true because all jobs scheduled between t and t + 1 are tght, and ther sze s no more than p t Max(t ). Hence, the gap between Max(t + 1) and Mn(t + 1) may ncrease, but not beyond p t. Also, snce Max(t ) < p t we also have Avg(t ) < p t. Usng Inequalty (3), ths mples that Max(t + 1) Mn(t + 1) + Avg(t + 1) p t + Avg(t ) 2(p t p t+1 ) 2p t 2(p t p t+1 ) = 2p t+1 b) Mn(t) > 0 and Mn(t + 1) = 0. Ths case s more lengthy and complcated to analyze, yet t s smlar n sprt wth Case (a) and and we omt t from ths verson. c) If Mn(t) = 0, then Max(t) + Avg(t) 2p t, and we consder two possble cases: If the job scheduled at stage t s not the one wth the earlest startng tme at stage t + 1 (.e., f t does not affect Max(t + 1)), then Max(t + 1) Max(t) 2(p t p t+1 ) p t /m. Combned wth the fact that Avg(t + 1) Avg(t) + p t /m, we get that Max(t + 1) Mn(t + 1) + Avg(t + 1) Max(t + 1) + Avg(t + 1) Max(t) 2(p t p t+1 ) + Avg(t) 2p t 2(p t p t+1 ) = 2p t+1 If the job scheduled at stage t s the one wth the earlest startng tme at stage t + 1 (.e., f t defnes Max(t +1)), then Max(t +1) p t 2(p t p t+1 ) p t /m. Snce Avg(t +1) Max(t +1) we get Max(t + 1) Mn(t + 1) + Avg(t + 1) Max(t + 1) + Avg(t + 1) 2Max(t + 1) 2(p t 2(p t p t+1 ) p t /m) 2p t+1 We note that the guarantee provded n the prevous lemma s tght. Ths can be verfed by consderng for example an nstance wth two machnes n whch the jobs have processng tmes (lsted by order of beng scheduled): 1, 1/2, 1, 3/4, 1, 7/8,..., 1, (2 k 1)/2 k. Corollary 5.3. The DA schedulng aucton s always feasble. Proof. Assume that at some stage t there exsts a job that cannot ft n any machne,.e., max j {E j (t)} < p t. Let t be the last stage before t when Mn(t ) = 0. Gven the result of Lemma 5.2, at stage t we have Max(t ) + Avg(t ) 2p t, and hence Avg(t ) (Max(t ) + Avg(t ))/2 p t. Snce all subsequently scheduled jobs are tght, f P s the total processng tme of the jobs scheduled from t to t, the average busy tme below Goal(t ) ncreased by P/m, and the drop from Goal(t ) to Goal(t) s exactly P/m + 2(p t p t ). As a result, Avg(t) Avg(t ) 2(p t p t ) p t 2(p t p t ) 2p t p t p t. But, ths means that max j {E j (t)} Goal(t) p t p t, whch contradcts our assumpton that max j {E j (t)} < p t. ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

13 Deferred-Acceptance Auctons for Multple Levels of Servce 1 6 MULTIPLE KNAPSACK CONSTRAINTS As a thrd applcaton of the generalzed DA auctons framework, we also study settngs where the set of feasble solutons s defned va multple knapsack constrants though, due to space constrants, most of ths secton s deferred to the full verson of the paper. There are multple motvatng examples for knapsack constrants: n FCC auctons there are dfferent bands (VHF, UHF) each wth ts own capacty constrants, and wth a dfferent value. Smlarly, the economy, busness, and frst class seats n an arplane correspond to three dfferent knapsack constrants, each wth a dfferent value. As a fnal example, consder cross country movng trucks that are scheduled to move property or products from a source to a destnaton. Each truck has a capacty constrant and the earler the arrval date the hgher the value. Formally, n ths secton we assume that each possble level of servce l has a capacty of m slots, and we let s be the sze of each bdder,.e., the number of slots that the bdder needs to occupy, f assgned level l. The presence of multple knapsack constrants then mply that an outcome s feasble f the total sze of the set of bdders N l N fnalzed at a level of servce l s at most m,.e., N l s m for every possble level of servce l. Düttng et al. [11] showed that, even n the bnary settng, f the set of bdders that can be smultaneously accepted s defned by a sngle knapsack constrant of sze m, then no DA aucton can acheve an approxmaton better than O(logm). The followng theorem provdes a matchng postve result: a generalzed DA aucton that acheves the optmal logarthmc approxmaton. Theorem 6.1. There exsts a multple knapsack DA aucton achevng a O(logm) approxmaton for an arbtrary number of knapsacks. Just lke n the prevous secton, the DA aucton that we propose manages to strke a balance between allowng the bdders to clnch hgher levels of servce as quckly as possble, yet ensurng that these promses wll not eventually volate any knapsack constrants. 7 MULTI-PARAMETER DEFERRED-ACCEPTANCE AUCTIONS Ths secton further generalzes DA auctons to a mult-parameter settng: bdders wth downwardslopng (a.k.a. submodular) valuatons n the level of servce awarded. For notatonal smplcty, we assume that the possble levels of servce are represented by a set L = {1, 2,..., k} and that each bdder reports her margnal value for each addtonal level, as b (1) = v (1), b (2) = v (2) v (1), up to b (k) = v (k) v (k 1) wth b (1) b (2) b (k). Hence, b (j) s the added value for the bdder f she jumps from level of servce j 1 to j. Just lke the generalzed DA aucton framework that we defned n Secton 3, the mult-parameter framework that we propose here also uses a collecton of scorng functons and clnchng functons for each bdder. The mportant dfference s that the agents now report a sequence of bds that correspond to ther margnal valuatons for the dfferent levels of servce. Gven these margnal bds for each bdder, at each stage t, the mult-parameter DA aucton computes the score of each actve bdder usng her bd correspondng to the next level of servce that s consdered for clnchng. That s, f д A t (b N \A ) s the level of servce that has clnched by stage t, whch we henceforth t denote by just д t, her score s computed based on her margnal value for clnchng one more level of servce,.e, based on b (д t + 1). Defnton 7.1. A mult-parameter DA aucton operates n stages t 1. In each stage t a set of bdders A t N s actve; ntally, A 1 = N, and A t+1 A t, for every t 1. Just as n the sngle-parameter case, the DA aucton s fully defned by two collectons of functons: The scorng functons σ A t (, b N \A ) that are non-decreasng n ther frst argument. t ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

14 1 Vasls Gkatzels, Evangelos Markaks, and Tm Roughgarden The clnchng functons д A t (b N \A ), that are non-decreasng w.r.t. the set of actve bdders, t.e., f for every t 1, д A t +1 (b N \At +1 ) д A t (b ). N \A t At each stage t, f A t, the score of bdder A t s computed as: σ A t (b (д t + 1), b ), N \A t.e., the score s a functon of the bdder s margnal value for recevng a level ncrease, gven the level that she has already clnched. The bdder wth the smallest score s fnalzed at level д t = д A t (b ) and she s removed from the set of actve bdders (A N \A t t+1 = A t \ {}). All the ncentve guarantees for sngle-parameter DA auctons carry over to the present settng. The proof of these propertes follows essentally the same arguments as n Theorem 3.3 and Propostons 3.4 and 3.5. These propertes are summarzed below. Proposton 7.2. Every mult-parameter DA aucton s weakly group-strategyproof, and has an equvalent clock aucton mplementaton that s obvously strategyproof. 7.1 Multunt Auctons wth Decreasng Margnal Values To demonstrate an applcaton of mult-parameter DA auctons, we consder the settng of multunt auctons wth submodular bdders. Mult-unt auctons are beng deployed n practce n a wde range of applcatons, and more recently they have also been used by varous onlne brokers (for more, see e.g., [32, 38]). Such auctons are defned by a collecton of m dentcal unts of some good that need to be allocated among n agents. The level of servce n ths settng corresponds to the number of unts that each agent receves, and the value of an agent for a bundle of tems s a submodular functon of the bundle s sze. Hence the valuaton functon here s fully descrbed by a nonncreasng vector of margnal values, for bundles up to sze m. Ascendng Multunt Auctons. There s a well-known ascendng mult-unt aucton, the clnchng aucton of Ausubel [1]. Gven an ascendng prce per unt, the bdders n ths aucton respond by reportng the number of unts that they would be nterested n acqurng at the current prce,.e., the number of ther margnal valuatons that are at least as hgh as the prce. As the prce ncreases, the number of tems that the bdders request drops, and a bdder clnches a unt at the pont where, even f he were to leave the aucton, the total demand of the other bdders would leave that unt unallocated. Ths aucton s known to be strategy-proof and n fact mplements the VCG mechansm. However t s also known to be susceptble to demand reducton n group devatons, and hence s not weakly group-strategyproof. For a smple example, consder an nstance nvolvng two unts and two bdders: bdder a has margnal values (1, 1), whle bdder b has margnal values (0.6, 0.6). If we were to run the clnchng aucton of [1] n ths nstance, after the prce would exceed 0.6, bdder b would leave the aucton and bdder a would clnch both unts at that prce for a utlty of = 0.8. If, on the other hand, both agents reduced ther demand by clamng that they have no value for a second tem, they would get one unt each for a prce of 0! The equvalence of Ausubel s aucton wth the VCG mechansm mples the followng contrast to Proposton 4.1. Observaton 1. In the settng of mult-unt auctons wth downward-slopng valuatons, the VCG mechansm s not a DA aucton. We next defne a DA aucton for mult-unt auctons. But before that, we frst establsh that some loss n socal welfare wll be unavodable, even f we only nsst on a weakly group-strategyproof mechansm. In partcular, we have the followng mpossblty result, the proof of whch s deferred to the full verson. ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.

15 Deferred-Acceptance Auctons for Multple Levels of Servce 1 Proposton 7.3. Even for 2 players and 2 unts, no weakly group-strategyproof mechansm can guarantee an approxmaton factor to the socal welfare, that s better than 2. The DA aucton that we propose n the next subsecton provdes an O(log n)-approxmaton. We leave t as an open queston whether or not there s a DA (or at least a weakly group-strategyproof) mechansm wth a better approxmaton factor. 7.2 Deferred-Acceptance Multunt Aucton wth Decreasng Margnal Valuatons For notatonal smplcty, assume that n = 2 κ for some κ N +, and let λ = m/(n log n). Also, assume that m n log n, and hence λ 1. 3 In order to acheve the logarthmc approxmaton, we restrct the dstrbuton of unts to agents so that t obeys a very partcular structure: The number of unts that each agent may be allocated s exactly λ 2 r, for some r {0, 1,...,κ}. We therefore dstrbute the unts n blocks of exponentally ncreasng sze. The dstrbuton of these blocks of unts across the agents s such that the number of agents that receve at least λ 2 r unts s at most n/2 r. In ths restrcted settng, nstead of provdng m possble levels of servce to each bdder (the number of unts they can receve), the levels of servce are κ + 1 (the number of unt blocks they can receve). Hence, rather than requrng that the agents report all of ther m margnal valuatons, our aucton requres that the agents report only ther average margnal value for recevng exactly λ 2 r unts of the good, for r {0, 1,...,κ},.e., ther average margnal value for each one of the κ + 1 blocks. Therefore, the agents report a bd vector of sze κ + 1 contanng weakly decreasng margnal values. For nstance, f n = 4 and m = 8, then κ = 2 and λ = 1. If the margnal valuatons of an agent are [8, 7, 6, 4, 4, 3, 2, 1], then her average margnal bd vector would be [8, 7, avg(6, 4)] = [8, 7, 5]. Our aucton does not assgn more than a κ fracton of the unts to one agent, so the remanng margnal values are dsregarded. Defnton 7.4. The DA multunt aucton receves κ + 1 margnal bds from each bdder, correspondng to ther average margnal values for each block. At every stage the scorng functon of every bdder s the dentfy functon,.e., σ A t (b (д t + 1)) = b (д t + 1). The clnchng functon s д A t ( ) = n A t, whch means that ntally (when n = A t ), every bdder has clnched the frst block, whch contans λ 2 0 = λ unts. Then, after the n/2 agents wth the smallest score are fnalzed, bdders that reman actve clnch an addtonal block of λ 2 1 = 2λ unts. Every tme the set of actve bdders s halved, those that reman actve clnch another block, untl they are all fnalzed. For the analyss of the aucton, t s convenent to also descrbe t as an ascendng aucton. Smlarly to the Ausubel clnchng aucton, the blocks of unts are clnched when the demand of the compettors drops below a specfc threshold but, unlke the clnchng aucton, ours s a DA aucton, so t s weakly group-strategyproof. The ascendng aucton mplementaton of our DA aucton uses a prce per unt p, whch happens to be the same for every bdder, and s ntally set to p = 0. The aucton begns by assgnng λ unts to each bdder at a prce of p = 0. Then, the agents compete for an addtonal block of λ unts. The aucton offers an average prce p per unt for ths block, and p s gradually ncreased untl exactly n/2 agents are stll wllng to pay ths amount per unt of the block. The bdders who do not receve these extra λ unts are fnalzed, and they hence receve just the frst block of λ unts for a prce of 0. The n/2 bdders that reman actve clnch each unt of the second block for a threshold prce p and they then compete for the thrd block, whch 3 If n s not a power of 2, we may remove the bdders wth the lowest frst margnal value untl we reach a power of 2. Ths does not affect the order of magntude of the approxmaton guarantee that we establsh. The same goes for the assumpton that m n log n. ACM Transactons on Economcs and Computaton, Vol. 1, No. 1, Artcle 1. Publcaton date: January 2017.