Mechanism Design for Fair Division

Transcription

1 Mechansm Desgn for Far Dvson Rchard Cole New York Unversty Vasls Gkatzels New York Unversty Gagan Goel Google Research Abstract We revst the classc problem of far dvson from a mechansm desgn perspectve, usng proportonal farness as a benchmark. In partcular, we am to allocate a collecton of dvsble tems to a set of agents whle ncentvzng the agents to be truthful n reportng ther valuatons. For the very large class of homogeneous valuatons, we desgn a truthful mechansm that provdes every agent wth at least a 1/e fracton of her proportonally far valuaton. To complement ths result, we show that no truthful mechansm can guarantee more than a 0.5 fracton, even for the restrcted class of addtve lnear valuatons. We also propose another mechansm for addtve lnear valuatons that works really well when every tem s hghly demanded. To guarantee truthfulness, our mechansms dscard a carefully chosen fracton of the allocated resources; we conclude by uncoverng nterestng connectons between our mechansms and celebrated solutons from the mechansm desgn lterature that use money nstead. 1 Introducton Ths paper studes the problem of allocatng a collecton of scarce resources among a set of selfnterested agents. We approach ths problem from the perspectve of a central polcy maker who desgns the mechansms dctatng the resource allocaton outcome. For nstance, the polcy maker could be the Federal Avaton Admnstraton FAA), whch s responsble for desgnng the schedules regulatng access to the U.S. arspace and arports, or the Federal Communcatons Commsson FCC), whch desgns the auctons of electromagnetc spectrum lcenses n the U.S. The agents n these examples,.e., the arlnes competng for landng slots, or the telecommuncatons companes competng for spectrum, may have dverse preferences regardng the outcome, so the polcy maker strves to desgn mechansms that optmze some trade-off between effcency and farness [1]. In many nstantatons of ths general framework, ncludng the examples mentoned above, the polcy maker faces a crucal obstacle: the agents preferences may not be known n advance, and wthout ths nformaton no mechansm can ensure that the resources are allocated effectvely. One soluton would be to ask the agents to report ther preferences to the mechansm but, unless the mechansm s desgned very carefully, the agents may be better off msreportng. The man goal of ths paper s to study the extent to whch the polcy maker can guarantee the farness of the resultng allocaton usng mechansms that ncentvze the agents to report ther true preferences. The need to address the tenson between the goals of the agents, who am to optmze ther own allocaton, and the objectve of the polcy maker, has been the foundaton of mechansm desgn. In Ths work was supported n part by NSF grants CCF and CCF The second author was an ntern at Google when part of ths work took place. 1

2 ths lterature, a mechansm comprses an allocaton rule and a payment rule; the former decdes how the resources are allocated, and the latter defnes the monetary payment that each agent needs to contrbute n exchange for the resources she s allocated. A mechansm s truthful f ts allocaton and payment rule guarantee that reportng the truth s always to the agent s best nterest. A classc result n mechansm desgn s the Vckrey-Clarke-Groves VCG) aucton, a truthful mechansm that maxmzes effcency. In partcular, f the valuaton of each agent for a resource allocaton outcome x s represented usng some functon v x), then the VCG mechansm outputs an allocaton that maxmzes the aggregate value,.e., v x). An mpressve property of VCG s that t combnes truthfulness and effcency for a very large famly of nstances. On the negatve sde, the effcency objectve that VCG optmzes, known as utltaran socal welfare, can be extremely unfar to ndvdual agents, whch makes t napproprate for many applcatons. Even when farness s not the prmary concern, e.g., n the FAA ar traffc schedulng example, unfar mechansms mght not be mplementable n practce [2]. Hence, n desgnng practcal mechansms, the polcy maker often has to satsfy equtablty constrants. Avodng nequtable allocatons s even more crtcal when farness s among the prmary objectves; the followng settngs are just a few llustratons of very dfferent szes where ths s true: Dvorce settlement negotatons. When a marred couple decdes to get a dvorce, the martal property needs to be dvded between them, and the laws governng ths process am to reach an outcome that s far to both of the ndvduals. Allocatng resources wthn a company. Companes need to dstrbute ther resources among dfferent groups of employees n an equtable fashon. For example, engneers wthn Google need to share access to the servers processng tme and memory) and the network. Prvatzaton auctons. Several mass prvatzaton auctons took place n formerly socalst countrese.g., Czechoslovaka) n the early 90s. These governments sought to prvatze, n a far manner, the state owned frms datng from the then recently ended communst era [3]. Despte the mportance of equtablty consderatons, the progress on far mechansms that guarantee truthfulness has been very lmted. In partcular, there s no known far counterpart of the VCG mechansm combnng truthfulness and farness for a large class of nstances. Ths lack of progress s n contrast to the very rch lterature on far dvson, whch has proposed varous notons of farness [4, 5, 6, 7, 8]. One noton of farness that has receved a lot of attenton s the maxmn crteron, also known as egaltaran socal welfare, whch was suggested by John Rawls [9]. Unlke VCG, whch maxmzes the aggregate value, a mechansm usng the maxmn objectve outputs an allocaton x that maxmzes the mnmum value over all agents,.e., mn {v x)}. Example 1.1 Utltaran versus egaltaran). Two arlne companes are competng for landng slots: Arlne 1 has a value v 1 for each slot, and Arlne 2 has a slghtly hgher value v 2 > v 1 because t owns planes that carry more passengers. Accordng to the utltaran objectve, all the avalable landng slots are allocated to Arlne 2 snce ts value s hgher. On the other hand, accordng to the egaltaran objectve, both of the arlnes receve some slots, but Arlne 1 receves more! In partcular, for every v 1 slots that Arlne 2 receves, the egaltaran soluton allocates v 2 slots to Arlne 1 so that the total value of the two arlnes s equalzed. Ths toy example exhbts the fact that both the utltaran and the egaltaran objectve lead to extreme outcomes: the former sacrfces farness n favor of effcency, and the latter sacrfces effcency n favor of satsfyng the least happy agent. Ths effect can become even more pronounced 2

3 as the number of resources and agents ncreases, whch calls for a more reasonable trade-off between farness and effcency. A well studed soluton that strkes a very appealng compromse between these two extreme objectves s the Compettve Equlbrum from Equal Incomes CEEI), whch s wdely regarded as the deal soluton for farness n mcroeconomcs [10, 11, 12, 13]. The preferred allocaton accordng to ths soluton corresponds to the compettve equlbrum outcome of the market that would arse f each agent were to be allotted the same amount of an artfcal currency, whch she could use to buy resources. In ths outcome, CEEI prces are computed for the resources, and every agent spends her budget optmally,.e., on the resources that yeld the best value for scrp) money. The followng smple example provdes some ntuton regardng what ths outcome looks lke. Example 1.2 CEEI). Three farmers produced 54 sacks of corn and 54 sacks of wheat, and they need to share t. The frst producer values each sack of corn 3 tmes more than a sack of wheat, the second values wheat 3 tmes more than corn, and the last one values wheat 1.5 tmes more than corn. If each of the producers were to be allocated a sngle unt of scrp money, then the CEEI prce for buyng all the corn n ths nstance would be 6/5, and the prce of wheat would be 9/5. At these prces, the frst producer would prefer to spend all of her 1 unt of scrp money on corn, whch would buy her a 5/6 fracton of t 45 sacks), and the second producer would spend all of her budget on wheat, whch would buy her a 5/9 fracton of t 30 sacks). Fnally, for the thrd producer, both of the crops have the same value for money, so she would spend 1/5 of her budget on corn and 4/5 on wheat, whch would buy her all the remanng sacks, thus clearng the market. The CEEI soluton s far n a very straghtforward way: t s the outcome that would arse f every agent had exactly the same buyng power. The fact that each agent spends her budget on the tems she prefers mples that ths allocaton s envy free; that s, no agent would prefer to swap the resources that she s allocated wth those of someone else. Envy-freeness s a hghly desred and very natural crteron that a far allocaton should meet. In addton to ts farness propertes, the CEEI s also Pareto effcent. Another remarkable property of the CEEI s that, for a very large famly of nstances, t s equvalent to the Nash barganng soluton [14, 15], whch s the result of an axomatc characterzaton of the propertes that a far soluton should satsfy [16]. Ths mples that the CEEI s scale-free, whch means that, the scale n whch an agent reports her values for the resources does not affect the outcome. In other words, f a producer n the CEEI example above has a value of v c = 1 for a sack of corn and v w = 2 for a sack of wheat, reportng v c = 2 and v w = 4 nstead would not affect the outcome. To verfy ths fact, note that the Nash barganng soluton chooses the allocaton x that maxmzes the product of the agents values,.e., v x). Furthermore, the outcome prescrbed by these solutons s the de facto bandwdth sharng method used n the networkng communty. Ths soluton whch, n the TCP congeston control context s known as proportonal farness PF), was ntroduced n the semnal work of Kelly [17], and t s currently the most wdely mplemented soluton n practce for nstance see [18]) 1. In ths paper we study nstances where these three solutons concde, so, for notatonal smplcty, we henceforth refer to the CEEI outcome as PF. The man drawback of the PF allocaton s that t cannot be mplemented usng truthful mechansms; even for smple nstances nvolvng just two agents and two tems, t s not dffcult to show 1 We note that some of the earler work on proportonal farness such as [17] and [19] have and ctatons respectvely n google scholar, ndcatng the mportance and usage of ths soluton. 3

4 that no payments rule can be combned wth the PF allocaton rule to make t truthful. In fact, the scale ndependence of the PF allocaton renders the use of actual monetary payments useless: t s easy to verfy that, when the allocaton rule of a truthful mechansm s scale-free then ts payment rule needs to be scale-free as well. But a payment rule that does not depend on the scale of the agents values cannot guarantee ndvdual ratonalty,.e., that no agent ever pays more than the value of the resources that she was allocated. Therefore, usng monetary payments to ensure the truthfulness of the partcpants s not an opton. In the absence of monetary payments, the only tool for algnng the ncentves of the agents wth the objectves of the polcy maker s what Hartlne and Roughgarden referred to as money burnng [20]. That s, the polcy maker can choose to ntentonally keep some of the resources unallocated n order to approprately nfluence the ncentves of the agents. Ths wthholdng of resources can often be nterpreted as an mplct form of payment, but snce these payments do not correspond to actual trades, they are essentally burned. 1.1 Our results In ths work we ntroduce a method for applyng money burnng n order to desgn truthful mechansms that closely approxmate the PF allocaton. We focus on nstances nvolvng multple dvsble resources ot tems) and we provde some surprsng postve results for the nduced problem n mult-dmensonal mechansm desgn wthout payments. To measure the farness of our mechansms we use the PF soluton as a benchmark, and our goal s to guarantee that every agent receves a good approxmaton of the value that she should be recevng accordng to the PF allocaton. Usng ths measure, we follow a worst-case analyss approach, accordng to whch the qualty of the mechansm s measured n the worst possble nstances that may arse, and our goal s to mnmze the nequtablty n these cases. The man contrbuton of ths paper s the Partal Allocaton mechansm. In Secton 3 we analyze ths mechansm and we prove that t s truthful and t guarantees that every player wll receve at least a 1/e fracton of her PF valuaton. These results hold for the general class of nstances where players have arbtrary homogeneous valuaton functons. Ths ncludes a wde range of well studed valuaton functons, from addtve lnear and Leontef, to Constant Elastcty of Substtuton and Cobb-Douglas [21]. We later extend these results to homogeneous valuatons of any degree. To complement ths postve result, we provde a negatve result showng that no truthful mechansm can guarantee to every player an allocaton wth value greater than 0.5 of the value of the PF allocaton, even f the mechansm s restrcted to the class of addtve lnear valuatons. In provng the truthfulness of the Partal Allocaton mechansm we reveal a connecton between the amount of resources that the mechansm dscards and the payments n VCG mechansms. In a nutshell, multplcatve reductons n allocatons are analogous to payments. As a result, we antcpate that ths approach may have a sgnfcant mpact on other problems n mechansm desgn wthout money. Indeed, we have already appled ths approach to the problem of maxmzng socal welfare wthout payments for whch a specal two-agent verson of the Partal Allocaton mechansm allowed us to mprove upon a settng for whch mostly negatve results were known [22]. In Secton 4 we show that, restrctng the set of possble nstances to ones nvolvng players wth addtve lnear valuatons 2 and tems wth hgh prces n the compettve equlbrum from 2 Note that our negatve results mply that the restrcton to addtve lnear valuatons alone would not be enough to allow for sgnfcantly better approxmaton factors. 4

5 equal ncomes 3, wll actually allow for the desgn of even more effcent and useful mechansms. We present the Strong Demand Matchng SDM) mechansm, a truthful mechansm that performs ncreasngly well as the compettve equlbrum prces ncrease. More specfcally, f p j s the prce of tem j, then the approxmaton factor guaranteed by ths mechansm s equal to mn j p j / p j ). It s nterestng to note that scenaros such as the prvatzaton aucton mentoned above nvolve a number of bdders much larger than the number of tems; as a rule, we expect ths to lead to hgh prces and a very good approxmaton of the partcpants PF valuatons. 1.2 Related Work Our settng s closely related to the large topc of far dvson or cake-cuttng [4, 5, 6, 7, 8], whch has been studed snce the 1940 s, usng the [0,1] nterval as the standard representaton of a cake. Each agent s preferences take the form of a valuaton functon over ths nterval, and then the valuatons of unons of subntervals are addtve. Note that the class of homogeneous valuaton functons of degree one takes us beyond ths standard cake-cuttng model. Leontef valuatons for example, allow for complementartes n the valuatons, and then the valuatons of unons of subntervals need not be addtve. On the other hand, the addtve lnear valuatons settng that we focus on n Secton 4 s very closely related to cake-cuttng wth pecewse constant valuaton functons over the [0, 1] nterval. Other common notons of farness that have been studed n ths lterature are, proportonalty 4, envy-freeness, and equtablty [4, 5, 6, 7, 8]. Despte the extensve work on far resource allocaton, truthfulness consderatons have not played a major role n ths lterature. Most results related to truthfulness were weakened by the assumpton that each agent would be truthful n reportng her valuatons unless ths strategy was domnated. Very recent work [23, 24, 25, 26] studes truthful cake cuttng varatons usng the standard noton of truthfulness accordng to whch an agent need not be truthful unless dong so s a domnant strategy. Chen et al. [23] study truthful cake-cuttng wth agents havng pecewse unform valuatons and they provde a polynomal-tme mechansm that s truthful, proportonal, and envy-free. They also desgn randomzed mechansms for more general famles of valuaton functons, whle Mossel and Tamuz [24] prove the exstence of truthful n expectaton) mechansms satsfyng proportonalty n expectaton for general valuatons. Zvan et al.[25] am to acheve envyfree Pareto optmal allocatons of multple dvsble goods whle reducng, but not elmnatng, the agents ncentves to le. The extent to whch untruthfulness s reduced by ther proposed mechansm s only evaluated emprcally and depends crtcally on ther assumpton that the resource lmtatons are soft constrants. Very recent work by Maya and Nsan [26] provdes evdence that truthfulness comes at a sgnfcant cost n terms of effcency. The recent papers of Guo and Contzer [27] and of Han et al. [28] also consder the truthful allocaton of multple dvsble goods; they focus on addtve lnear valuatons and ther goal s to maxmze the socal welfare or effcency) after scalng every player s reported valuatons so that her total valuaton for all tems s 1. Guo and Contzer [27] study two-agent nstances, provdng both upper and lower bounds for the achevable approxmaton; Han et al. [28] extend these results and also study the multple agents settng. For problem nstances that may nvolve an arbtrary number of tems both papers provde negatve results: no non-trval approxmaton factor can 3 The prces nduced by the market equlbrum when all bdders have a unt of scrp money; also referred to as PF prces. 4 It s worth dstngushng the noton of PF from that of proportonalty by notng that the latter s a much weaker noton, drectly mpled by the former. 5

6 be acheved by any truthful mechansm when the number of players s also unbounded. For the two-player case, after Guo and Contzer [27] studed some classes of dctatoral mechansms, Han et al. [28] showed that no dctatoral mechansm can guarantee more than the trval 0.5 factor. Interestngly, we recently showed [22] that combnng a specal two-player verson of the Partal Allocaton mechansm wth a dctatoral mechansm can actually beat ths bound, achevng a 2/3 approxmaton. The resource allocaton lterature has seen a resurgence of work studyng far and effcent allocaton for Leontef valuatons [29, 30, 31, 32]. These valuatons exhbt perfect complements and they are consdered to be natural valuaton abstractons for computng settngs where jobs need resources n fxed ratos. Ghods et al. [29] defned the noton of Domnant Resource Farness DRF), whch s a generalzaton of the egaltaran socal welfare to multple types of resources. Ths soluton has the advantage that t can be mplemented truthfully for ths specfc class of valuatons; as the authors acknowledge, the CEEI soluton would be the preferred far dvson mechansm n that settng as well, and ts man drawback s the fact that t cannot be mplemented truthfully. Parkes et al. [31] assessed DRF n terms of the resultng effcency, showng that t performs poorly. Dolev et al. [30] proposed an alternate farness crteron called Bottleneck Based Farness, whch Gutman and Nsan [32] subsequently showed s satsfed by the proportonally far allocaton. Gutman and Nsan [32] also posed the study of ncentves related to ths latter noton as an nterestng open problem. Our results could potentally have sgnfcant mpact on ths lne of work as we are provdng a truthful way to approxmate a soluton whch s recognzed as a good benchmark. It would also be nterestng to study the extent to whch the Partal Allocaton mechansm can outperform the exstng ones n terms of effcency. Most of the papers mentoned above contrbute to our understandng of the trade-offs between ether truthfulness and farness, or truthfulness and socal welfare. Another drecton that has been actvely pursued s to understand and quantfy the trade-off between farness and socal welfare. Bertsmas et al. [2, 1], and Caraganns et al. [33] measured the deteroraton of the socal welfare caused by dfferent farness restrctons, the prce of farness. More recently, Cohler et al. [34] desgned algorthms for computng allocatons that approxmately) maxmze socal welfare whle satsfyng envy-freeness. Also, the work of Cohen et al. [35], and of Feldman and La [36] consders the queston of fndng mechansms that satsfy both truthfulness and envy-freeness wthout sacrfcng any socal welfare. Our results ft nto the general agenda of approxmate mechansm desgn wthout money, explctly ntated by Procacca and Tennenholtz [37]. The underlyng connecton of our man mechansm wth VCG proposes a framework for desgnng truthful mechansms wthout money and we antcpate that ths mght have a sgnfcant mpact on ths lterature. 2 Prelmnares Let M denote the set of m tems and N the set of n bdders. Each tem s dvsble, meanng that t can be dvded nto arbtrarly small peces, whch are then allocated to dfferent bdders. An allocaton x of these tems to the bdders defnes the fracton x j of each tem j that each bdder wll be recevng; let F = {x x j 0 and x j 1} denote the set of feasble allocatons. Each bdder s assgned a weght b 1 whch allows for nterpersonal comparson of valuatons, and can serve as prorty n computng applcatons, as clout n barganng applcatons, or as a budget for the market equlbrum nterpretaton of our results. We assume that b s defned by the mechansm 6

7 as t cannot be truthfully elcted from the bdders. The preferences of each bdder N take the form of a valuaton functon v ), that assgns nonnegatve values to every allocaton n F. We assume that every player s valuaton for a gven allocaton x only depends on the bundle of tems that she wll be recevng. We wll present our results assumng that the valuaton functons are homogeneous of degree one,.e. player s valuaton for an allocaton x = f x satsfes v x ) = f v x), for any scalar f > 0. We later dscuss how to extend these results to general homogeneous valuatons of degree d for whch v x ) = f d v x). A couple of nterestng examples of homogeneous valuatons functons of degree one are addtve lnear valuatons and Leontef valuatons; accordng to the former, every player has a valuaton v j for each tem j and v x) = j x jv j, and accordng to the latter, each player s type corresponds to a set of values a j, one for each tem, and v x) = mn j {x j /a j }..e. player desres the tems n the rato a 1 : a 2 :... : a m.) An allocaton x F s Proportonally Far PF) f, for any other allocaton x F the weghted) aggregate proportonal change to the valuatons after replacngx wth x s not postve,.e.: b [v x ) v x )] v x 0. 1) ) N Ths allocaton rule s a strong refnement of Pareto effcency, snce Pareto effcency only guarantees that f some player s proportonal change s strctly postve, then there must be some player whose proportonal change s negatve. The Proportonally Far soluton can also be defned as an allocaton x F that maxmzes [v x)] b, or equvalently after takng a logarthm), that maxmzes b logv x); we wll refer to these two equvalent objectves as the PF objectves. Note that, although the PF allocaton need not be unque for a gven nstance, t does provde unque bdder valuatons [38]. We also note that the PF soluton s equvalent to the Nash barganng soluton. John Nash n hs semnal paper [16] consdered an axomatc approach to barganng and gave four axoms that any barganng soluton must satsfy. He showed that these four axoms yeld a unque soluton whch s captured by a convex program; ths convex program s equvalent to the one defned above for the PF soluton. Another well-studed allocaton rule whch s equvalent to the PF allocaton s the Compettve Equlbrum. Esenberg [14] showed that f all agents have valuaton functons that are quas-concave and homogeneous of degree 1, then the compettve equlbrum s also captured by the same convex program as the one for the PF soluton. Gven a valuaton functon reported from each bdder, we want to desgn mechansms that output an allocaton of tems to bdders. We restrct ourselves to truthful mechansms,.e. mechansms such that any false report from a bdder wll never return her a more valuable allocaton. Snce proportonal farness cannot be mplemented va truthful mechansms, we wll measure the performance of our mechansms based on the extent to whch they approxmate ths benchmark. More specfcally, the approxmaton factor, or compettve factor of a mechansm wll correspond to the mnmum value of ρi) across all relevant nstances I, where ρi) = mn N { v x) v x ) and x,x are the allocaton generated by the mechansm for nstance I and the PF allocaton of I respectvely. }, 7

8 3 The Partal Allocaton Mechansm In ths secton, we defne the Partal Allocaton PA) mechansm as a novel way to allocate dvsble tems to bdders wth homogeneous valuaton functons of degree one. We subsequently prove that ths non-dctatoral mechansm not only acheves truthfulness, but also guarantees that every bdder wll receve at least a 1/e fracton of the valuaton that she deserves, accordng to the PF soluton. Ths mechansm depends on a subroutne that computes the PF allocaton for the problem nstance at hand; we therefore later study the runnng tme of ths subroutne, as well as the robustness of our results n case ths subroutne returns only approxmate solutons. The PA mechansm elcts the valuaton functon v ) from each player and t computes the PF allocaton x consderng all the players valuatons. The fnal allocaton x output by the mechansm gves each player only a fracton f of her PF bundle x,.e. for every tem j of whch the PF allocaton assgned to her a porton of sze x j, the PA mechansm nstead assgns to her a porton of sze f x j, where f [0,1] depends on the extent to whch the presence of player nconvenences the other players; the value of f may therefore vary across dfferent players. The followng steps gve a more precse descrpton of the mechansm. Algorthm 1: The Partal Allocaton mechansm. 1 Compute the PF allocaton x based on the reported bds. 2 For each player, compute the PF allocaton x that would arse n her absence. 3 Allocate to each player a fracton f of her PF bundle x, where f = [v ) x )] b 1/b [v. 2) x )]b Lemma 3.1. The allocaton x produced by the PA mechansm s feasble. Proof. Snce the PF allocaton x s feasble, to verfy that the allocaton produced by the PA mechansm s also feasble, t suffces to show that f [0,1] for every bdder. The fact that f 0 s clear snce both the numerator and the denomnator are non-negatve. To show that f 1, note that x = argmax x F v x ). Snce x remans a feasble allocaton x F) after removng bdder we can just dscard bdder s share), ths mples v x ). v x ) 3.1 Truthfulness Wenowshowthat, desptethefactthatthsmechansmsnotdctatoralanddoesnotusemonetary payments, t s stll n the best nterest of every player to report her true valuaton functon, rrespectve of what the other players do. 8

9 Theorem 3.2. The PA mechansm s truthful. Proof. In order to prove ths theorem, we approach the PA mechansm from the perspectve of some arbtrary player. Let v ) denote the valuaton functon that each player reports to the PA mechansm. We assume that the valuaton functons reported by these players may dffer from ther true ones, v ). Player s faced wth the optons of, ether reportng her true valuaton functon v ), or reportng some false valuaton functon v ). After every player has reported some valuaton functon, the PA mechansm computes the PF allocaton wth respect to these valuaton functons; let x T denote the PF allocaton that arses f player reports the truth and x L otherwse. Fnally, player receves a fracton of what the computed PF allocaton assgned to her, and how bg or small ths fracton wll be depends on the computed PF allocaton. Let f T denote the fracton of her allocaton that player wll receve f x T s the computed PF allocaton and f L otherwse. Snce the players have homogeneous valuaton functons of degree one, what we need to show s that f T v x T ) f L v x L ), or equvalently that [f T v x T )] b [f L v x L )] b. Note that the denomnators of both fractons f T and f L, as gven by Equaton 2), wll be the same snce they are ndependent of the valuaton functon reported by player. Our problem therefore reduces to provng that [v x T )] b [ v x T )] b [v x L )] b [ v x L )] b. 3) To verfy that ths nequalty holds we use the fact that the PF allocaton s the one that maxmzes the product of the correspondng reported valuatons. Ths means that x T = argmax x F [v x)] b [ v x)] b, and snce x L F, ths mples that Inequalty 3) holds, and hence reportng her true valuaton functon s a domnant strategy for every player. The arguments used n the proof of Theorem 3.2 mply that, gven the valuaton functons reported by all the other players, player can effectvely choose any bundle that she wshes, but for each bundle the mechansm defnes what fracton player can keep. One can therefore thnk of the fracton of the bundle thrown away as a form of non-monetary payment that the player must suffer n exchange for that bundle, wth dfferent bundles matched to dfferent payments. The fact that the PA mechansm s truthful mples that these payments, n the form of fractons, make the bundle allocated to her by allocaton x the most desrable one. We revst ths nterpretaton n Secton Approxmaton Before studyng the approxmaton factor of the PA mechansm, we frst state a lemma whch wll be useful for provng Theorem 3.4 ts proof s deferred to the Appendx). 9

10 Lemma 3.3. For any set of pars δ,β ) wth β 1 and β δ b the followng holds where B = β ) 1+δ ) β 1+ B) b B. Usng ths lemma we can now prove tght bounds for the approxmaton factor of the Partal Allocaton mechansm. As we show n ths proof, the approxmaton factor depends drectly on the relatve weghts of the players. For smplcty n expressng the approxmaton factor, let b mn denote the smallest value of b across all bdders of an nstance and let B = N b ) bmn be the sum of the b values of all the other bdders. Fnally, let ψ = B/b mn denote the rato of these two values. Theorem 3.4. The approxmaton factor of the Partal Allocaton mechansm for the class of problem nstances of some gven ψ value s exactly 1+ 1 ψ) ψ. Proof. The PA mechansm allocates to each player a fracton f of her PF allocaton, and for the class of homogeneous valuaton functons of degree one ths means that the fnal valuaton of player wll be v x) = f v x ). The approxmaton factor guaranteed by the mechansm s therefore equal to mn {f }. Wthout loss of generalty, let player be the one wth the mnmum value of f. In the PF allocaton x that the PA mechansm computes after removng player, every other player experences a value of v x ). Let d denote the proportonal change between the valuaton of player for allocaton x and allocaton x,.e. Substtutng for v x ) n Equaton 2) yelds: v x ) = 1+d )v x ). Snce x s a PF allocaton, Inequalty 1) mples that ) 1/b 1 f = 1+d. 4) )b N b [v x ) v x )] v x ) 0 b d + b [v x ) v x )] v x 0 ) b d b. 5) The last equvalence holds due to the fact that v x ) = 0, snce allocaton x nothng to player. clearly assgns 10

11 Let B = b ; usng Inequalty 5) and Lemma 3.3 on substtutng b for b, d for δ, b for β, and B for B), t follows from Equaton 4) that f 1+ b ) B b. 6) B Toverfythatthsboundstght, consderanynstancewthjustonetemandthegvenψ value. The PF soluton dctates that each player should be recevng a fracton of the tem proportonal to the player s b value. The removal of a player therefore leads to a proportonal ncrease of exactly b /B for each of the other players PF valuaton. The PA mechansm therefore assgns to every player a fracton of her PF allocaton whch s equal to the rght hand sde of Inequalty 6). The player wth the smallest b value receves the smallest fracton. The approxmaton factor of Theorem 3.4 mples that f 1/2 for nstances wth two players havng equal b values, and f 1/e even when ψ goes to nfnty; we therefore get the followng corollary. Corollary 3.5. The Partal Allocaton mechansm always yelds an allocaton x such that for every partcpatng player v x) 1 e v x ). To complement ths approxmaton factor, we now provde a negatve result showng that, even for the specal case of addtve lnear valuatons, no truthful mechansm can guarantee an approxmaton factor better than n+1 2n for problem nstances wth n players. Theorem 3.6. There s no truthful mechansm that can guarantee an approxmaton factor greater than n+1 2n +ǫ for any constant ǫ > 0 for all n-player problem nstances, even f the valuatons are restrcted to beng addtve lnear. Proof. For an arbtrary real value of n > 1, let ρ = n+1 2n, and assume that Q s a truthful resource allocaton mechansm that guarantees a ρ + ǫ) approxmaton for all n-player problem nstances, where ǫ s a postve constant. Ths mechansm receves as nput the bdders valuatons and t returns a vald fractonal) allocaton of the tems. We wll defne n + 1 dfferent nput nstances for ths mechansm, each of whch wll consst of n bdders and m = k +1)n tems, where k > 2 ǫ wll take very large values. In order to prove the theorem, we wll then show that Q cannot smultaneously acheve ths approxmaton guarantee for all these nstances, leadng to a contradcton. For smplcty we wll refer to each bdder wth a number from 1 to n, to each tem wth a number from 1 to k +1)n, and to each problem nstance wth a number from 1 to n+1. We start by defnng the frst n problem nstances. For n, let problem nstance be as follows: Every bdder has a valuaton of kn+1 for tem and a valuaton of 1 for every other tem; bdder has a valuaton of 1 for all tems. In other words, all bdders except bdder have a strong preference for just one tem, whch s dfferent for each one of them. The PF allocaton for such addtve lnear valuatons dctates that every bdder s allocated only tem, whle bdder s allocated all the remanng kn + 1 tems. Snce Q acheves a ρ + ǫ approxmaton for ths nstance, t needs to provde bdder wth an allocaton whch the bdder values at least at ρ+ǫ)kn+1). In order to acheve ths, mechansm Q can assgn to ths bdder fractons of the set M of the n 1 tems that the PF soluton allocates to the other bdders as well as fractons 11

12 of the set M of the kn + 1 tems that the PF allocaton allocates to bdder. Even f all of the n 1 tems of M were fully allocated to bdder, the mechansm would stll need to assgn to ths bdder an allocaton of value at least ρ+ǫ)kn + 1) n 1) usng tems from M. Snce k > 2 ǫ, n 1 < ǫ 2 kn+1), and therefore mechansm Q wll need to allocate to bdder a fractonal assgnment of tems n M that the bdder values at least at ρ+ 2) ǫ kn + 1). Ths mples that there must exst at least one tem n M of whch bdder s allocated a fracton of sze at least ρ+ ǫ 2). Snce all the tems n M are dentcal and the numberng of the tems s arbtrary, we can, wthout loss of generalty, assume that ths tem s tem. We have therefore shown that, for every nstance n mechansm Q wll have to assgn to bdder at least ρ+ 2) ǫ of tem, and an allocaton of tems n M that guarantees her a valuaton of at least ρ+ 2) ǫ kn+1). We now defne problem nstance n + 1, n whch every bdder has a valuaton of kn + 1 for tem and a valuaton of 1 for all other tems. The PF soluton for ths nstance would allocate to each bdder all of tem, as well as k tems from the set {n+1,...,k +1)n} or more generally, fractons of these tems that add up to k). Clearly, every bdder can unlaterally msreport her valuaton leadng to problem nstance nstead of ths nstance; so, n order to mantan truthfulness, mechansm Q wll have to provde every bdder of problem nstance n+1 wth at least the value that such a devaton would provde her wth. One can quckly verfy that, even f mechansm Q when faced wth problem nstance provded bdder wth no more than a ρ+ 2) ǫ fracton of tem, stll such a devaton would provde bdder wth a valuaton of at least ρ+ ǫ ) kn+1) + 2 ρ+ ǫ ) kn 2 ρ+ ǫ ) 2kn. 2 The frst term of the left hand sde comes from the fracton of tem that the bdder receves and the second term comes from the average fracton of the remanng tems. If we substtute ρ = n+1 2n, we get that the truthfulness of Q mples that every bdder of problem nstance n+1 wll have to receve an allocaton of value at least n+1 2n + ǫ ) 2kn = kn+k +ǫkn. 2 For any gven constant value of ǫ though, snce k > 2 ǫ and n > 1, every bdder wll need to be assgned an allocaton that she values at more than kn+k+2, whch s greater than the valuaton of kn+k +1 that the player receves n the PF soluton. Ths s obvously a contradcton snce the PF soluton s Pareto effcent and there cannot exst any other allocaton for whch all bdders receve a strctly greater valuaton. Theorem 3.6 mples that, even f all the players have equal b values, no truthful mechansm can guarantee a greater than 3/4 approxmaton even for nstances wth just two bdders, and ths bound drops further as the number of bdders ncreases, fnally convergng to 1/2. To complement the statement of Corollary 3.5, we therefore get the followng corollary. Corollary 3.7. No truthful mechansm can guarantee that t wll always yeld an allocaton x such that for any ǫ > 0 and for every partcpatng player ) 1 v x) 2 +ǫ v x ). 12

13 3.3 Envy-Freeness We now consder the queston of whether the outcomes that the Partal Allocaton mechansm yelds are envy-free; we show that, for two well studed types of valuaton functons ths s ndeed the case, thus provdng further evdence of the farness propertes of ths mechansm. We start by showng that, f the bdders have addtve lnear valuatons, then the outcome that the PA mechansm outputs s also envy-free. Theorem 3.8. The PA mechansm s envy-free for addtve lnear bdder valuatons. Proof. Let x denote the PF allocaton ncludng all the bdders, wth each bdder s valuatons scaled so that v x ) = 1. Let v x j ) denote the value of bdder for x j, the PF share of bdder j n x, and let x denote the PF allocaton that arses after removng some bdder. The PA mechansm allocates each unweghted) bdder a fracton f of her PF share, where k f = [v kx )] 1 k [v kx )] = k [v kx )]. In order to prove that the PA mechansm s envy-free, we need to show that for every bdder, and for all j, f v x ) f j v x j ), or equvalently 1 k [v kx )] v x j ) k j [v kx j )] [v k x j)] v x j) [v k x )]. 7) k j k To prove the above nequalty, we wll modfy allocaton x so as to create an allocaton x j such that [v k x j )] v x j) [v k x )]. 8) k j k Clearly, for any feasble allocaton x j t must be the case that [v k x j)] [v k x j )], 9) k j k j sncex j s, bydefnton, thefeasbleallocatonthatmaxmzesthsproduct. Therefore, combnng Inequaltes 8) and 9) mples Inequalty 7). To construct allocaton x j, we use allocaton x and we defne the followng weghted drected graph G based on x : the set of vertces corresponds to the set of bdders, and a drected edge from the vertex for bdder j to that for bdder k exsts f and only f x allocates to bdder j portons of tems that were nstead allocated to bdder k n x. The weght of such an edge s equal to the total value that bdder j sees n all these portons. Snce the valuatons of all bdders are scaled so that v j x ) = 1 for all j, ths mples that, f the weght of some edge j,k) s v w.r.t. these scaled valuatons), then the total value of bdder k for those same portons that bdder j values at v, s at least v. If that were not the case, then x would not have allocated those portons to bdder k; allocatng them to bdder j nstead would lead to a postve aggregate proportonal change to the valuatons. Ths means that we can assume, wthout loss of generalty, that the graph s a drected acyclc one; f not, we can rearrange the allocaton so as to remove any drected cycles from ths graph wthout decreasng any bdder s valuaton. 13

14 Also note that for every bdder k t must be the case that v k x ) v kx ). To verfy ths fact, assume that t s not true, and let k be the bdder wth the mnmum value v k x ). Snce v k x ) < v kx ) = 1, t must be the case that x does not allocate to bdder k all of her PF share accordng to x, thus the vertex for bdder k has ncomng edges of postve weght n the drected acyclc graph G, and t therefore belongs to some drected path. The very frst vertex of ths path s a source of G that corresponds to some bdder s; the fact that ths vertex has no ncomng edges mples that v s x ) v sx ) = 1. Snce v k x ) < 1 we can deduce that there exsts some drected edge α,β) along the path from s to k such that v α x ) > v βx ). Returnng some of the portons contrbutng to ths edge from bdder α to bdder β wll lead to a postve aggregate proportonal change to the valuatons, contradctng that x s the PF allocaton excludng bdder. Havng shown that v k x ) v kx ) for every bdder k other than, we can now deduce that the total weght of ncomng edges for the vertex n G correspondng to any bdder k s at most as much as the total weght of the outgong edges. Fnally, ths also mples that the only snk of G wll have to be the vertex for bdder. The frst step of our constructon starts from allocaton x and t reallocates some of the x allocaton, leadng to a new allocaton x. Usng the drected subtree of G rooted at the vertex of bdder j, we reduce to zero the weghts of the edges leavng j by reducng the allocaton at j, ncreasng the allocaton at, and sutably changng the allocaton of other bdders. More specfcally, we start by returnng all the portons that bdder j was allocated n x but not n x, back to the bdders who were allocated these portons n x. These bdders to whom some portons were returned then return portons of equal value that they too were allocated n x but not n x ; ths s possble snce, for each such bdder, the total ncomng edge weght of ts vertex s outweghed by the total outgong edge weght. We repeat ths process untl the snk, the vertex for bdder, s reached. One can quckly verfy that v x) v j x ) v j x); 10) n words, the value that bdder ganed n ths transton from x to x s at least as large as the value that bdder j lost n that same transton. Fnally, n allocaton x, whatever value v j x) bdder j s left wth comes only from portons that were part of her PF share n x. Bdder j s total valuaton for any portons of her PF share n x that are allocated to other bdders n x s equal to 1 v j x). Thus, bdder s valuaton for those same portons wll be at most 1 v j x); otherwse modfyng x by allocatng these portons to would lead to a postve aggregate change to the valuatons. Ths means that for bdder the portons remanng wth bdder j n allocaton x have value at least v x j ) 1 v j x)). We conclude the constructon of allocaton x j by allocatng all the remanng portons allocated to bdder j n x to bdder, leadng to v x j ) v x)+v x j) 1 v j x)) v j x ) v j x)+v x j) 1 v j x)) v j x ) 1+v x j) [v j x ) 1]v x j)+v x j) = v j x )v x j). The second nequalty s deduced by substtutng from Inequalty 10); the last nequalty can be verfed by usng the fact that v x j ) 1, and multplyng both sdes of ths nequalty wth the 14

15 non-negatve value v j x ) 1, leadng to [v jx ) 1]v x j ) v jx ) 1. Also note that for all k / {,j}, v k x j ) = v k x ). We therefore conclude that the second nequalty of 8) s true. The frst nequalty s of course also true snce both x j and x j are feasble, but the former s, by defnton, the one that maxmzes that product. Followng the same proof structure we can now also show that the PA mechansm s envy-free when the bdders have Leontef valuatons. Theorem 3.9. The PA mechansm s envy-free for Leontef bdder valuatons. Proof. Just as n the proof of Theorem 3.8, let x denote the PF allocaton ncludng all the bdders, wth each bdder s valuatons scaled so that v x ) = 1. Also, let v x j ) denote the value of bdder for x j, the PF share of bdder j n x, and let x denote the PF allocaton that arses after removng some bdder. Followng the steps of the proof of Theorem 3.8 we can reduce the problem of showng that the PA mechansm s envy-free to constructng an allocaton x j that satsfes Inequalty 8),.e. such that [v k x j)] [v k x j )] v x j) [v k x )]. k j k j k To construct allocaton x j, we start from allocaton x and we reallocate the bundle of tem fractons allocated to bdder j n x to bdder nstead, whle mantanng the same allocatons for all other bdders. Therefore, after smplfyng the latter nequalty usng the fact that v k x j ) = v k x ) for all k,j, what we need to show s that v x j ) v x j)v j x ). 11) Note that, gven the structure of Leontef valuatons, every bdder s nterested n bundles of tem fractons that satsfy specfc proportons. Ths means that the bundle of tem fractons allocated to bdder j n x and the one allocated to her n x both satsfy the same proportons; that s, there exsts some constant c such that, for every one of the tems, bdder j receves n x exactly c tmes the amount that she receves n x. As a result, gven the fact that Leontef valuatons are homogeneous of degree one, v j x ) = c v jx ) = c usng the fact that v j x ) = 1). Smlarly, snce x j allocates to bdder the bundle of bdder j n x, and usng the homogeneous structure of Leontef valuatons, ths mples that v x j ) = c v x j ). Substtutng these two equaltes n Inequalty 11) verfes that the nequalty s true, thus concludng the proof. 3.4 Runnng Tme and Robustness The PA mechansm has reduced the problem of truthfully mplementng a constant factor approxmaton of the PF allocaton to computng exact PF allocatons for several dfferent problem nstances, as ths s the only subroutne that the mechansm calls. If the valuaton functons of the players are affne, then there s a polynomal tme algorthm to compute the exact PF allocaton [39, 40]. We now show that, even f the PF soluton can be only approxmately computed n polynomal tme, our truthfulness and approxmaton related statements are robust wth respect to such approxmatons all the proofs of ths subsecton are deferred to the Appendx). More specfcally, 15

16 we assume that the PA mechansm uses a polynomal tme algorthm that computes a feasble allocaton x nstead of x such that [ 1/B [ [v x)] ] b 1 ǫ) 1/B n [v x )] ] b, where B = b. =1 Usng ths algorthm, the PA mechansm can be adapted as follows: Algorthm 2: The Approxmate Partal Allocaton mechansm. 1 Compute the approxmate PF allocaton x based on the reported bds. 2 For each player, remove her and compute the approxmate PF allocaton x that would arse n her absence. 3 Allocate to each player a fracton f of everythng that she receves accordng to x, where f = mn 1, [v ) 1/b x)]b [v x )] b. 12) For ths adapted verson of the PA mechansm to reman feasble, we need to make sure that f remans less than or equal to 1. Even f, for some reason, the allocaton x computed by the approxmaton algorthm does not satsfy ths property, the adapted mechansm wll then choose f = 1 nstead. We start by showng two lemmas verfyng that ths adapted verson of the PA mechansm s robust both wth respect to the approxmaton factor t guarantees and wth respect to the truthfulness guarantee. Lemma The approxmaton factor of the adapted PA mechansm for the class of problem nstances of some gven ψ value s at least 1 ǫ) 1+ ψ) 1 ψ. Lemma If a player msreports her preferences to the adapted PA mechansm, she may ncrease her valuaton by at most a factor 1 ǫ) 2. Fnally, we show that f the valuaton functons are, for example, concave and homogeneous of degree one, then a feasble approxmate PF allocaton can ndeed be computed n polynomal tme. Lemma For concave homogeneous valuaton functons of degree one, there exsts an algorthm that computes a feasble allocaton x n tme polynomal n log1/ǫ and the problem sze, such that [v x)] b 1 ǫ) [v x )] b. 16

17 3.5 Extenson to General Homogeneous Valuatons We can actually extend most of the results that we have shown for homogeneous valuaton functons of degree one to any valuaton functon that can be expressed as v f x) = g f) v x), where g ) s some ncreasng nvertble functon; for homogeneous valuaton functons of degree d, ths functon s g f) = f d. If ths functon s known for each bdder, we can then adapt the PA mechansm as follows: nstead of allocatng to bdder a fracton f of her allocaton accordng to x as defned n Equaton 2), we nstead allocate to ths bdder a fracton g 1 f ), where g 1 ) s the nverse functon of g ). If, for example, some bdder has a homogeneous valuaton functon of degree d, then allocatng her a fracton f 1/d of her PF allocaton has the desred effect and both truthfulness and the same approxmaton factor guarantees stll hold. The dea behnd ths transformaton s that all that we need n order to acheve truthfulness and the approxmaton factor s to be able to dscard some fracton of a bdder s allocaton knowng exactly what fracton of her valuaton ths wll correspond to. 4 The Strong Demand Matchng Mechansm The man result of the prevous secton shows that one can guarantee a good constant factor approxmaton for any problem nstance wthn a very large class of bdder valuatons. The subsequent mpossblty result shows that, even f we restrct ourselves to problem nstances wth addtve lnear bdder valuatons, no truthful mechansm can guarantee more than a 1/2 approxmaton. In ths secton we study the queston of whether one can acheve even better factors when restrcted to some well-motvated class of nstances. We focus on addtve lnear valuatons, and we provde a postve answer to ths queston for problem nstances where every tem s hghly demanded. More formally, we consder problem nstances for whch the PF prce or equvalently the compettve equlbrum prce) of every tem s large when the budget of every player s fxed to one unt of scrp money 5. The motvaton behnd ths class of nstances comes from problems such as the one that arose wth the Czech prvatzaton auctons [3]. For such nstances, where the number of players s much hgher than the number of tems, one naturally antcpates that all tem prces wll be hgh n equlbrum. For the rest of the chapter we assume that the weghts of all players are equal and that ther valuatons are addtve lnear. Let p j denote the PF prce of tem j when every bdder s budget b s equal to 1. Our man result n ths secton s the followng: Theorem 4.1. For addtve lnear { valuatons there exsts a truthful mechansm that acheves an approxmaton factor of mn j p j / p j }. Note that f k = mn j p j, then ths approxmaton factor s at least k/k +1). We now descrbe our soluton whch we call the Strong Demand Matchng mechansm SDM). Informally speakng, SDM starts by gvng every bdder a unt amount of scrp money. It then ams to dscover mnmal tem prces such that the demand of each bdder at these prces can be satsfed usng a fracton of) just one tem. In essence, our mechansm s restrcted to computng allocatons that assgn each bdder to just one tem, and ths restrcton of the output space renders 5 Remark: Our mechansm does not make ths assumpton, but the approxmaton guarantees are much better wth ths assumpton. 17