Nash Social Welfare Approximation for Strategic Agents

Transcription

1 Nash Socal Welfare Approxmaton for Strategc Agents Smna Brânze Hebrew U. of Jerusalem Vasls Gkatzels Drexel Unversty Ruta Mehta U.I. Urbana-Champagn Abstract The far dvson of resources among strategc agents s an mportant age-old problem that has led to a rch body of lterature. At the center of ths lterature les the queston of whether there exst mechansms that can mplement far outcomes, despte the agents strategc behavor. A fundamental objectve functon used for measurng far outcomes s the Nash socal welfare (NSW), mathematcally defned as the geometrc mean of the agents values n a gven allocaton. Ths objectve functon s maxmzed by wdely known soluton concepts such as Nash barganng and the compettve equlbrum wth equal ncomes. In ths work we focus on the queston of (approxmately) mplementng ths objectve. We analyze two classcal mechansms, the Fsher market and the Tradng Post mechansm, and provde bounds on the qualty of ther allocatons as measured by the NSW objectve. We consder two extreme classes of valuatons functons, namely perfect substtutes and perfect complements. For perfect substtutes we show that both mechansms have a prce of anarchy (PoA) of at most 2 and at least e 1 e ( 1.44). For perfect complements we fnd that the Fsher market mechansm has arbtrarly bad PoA, but Tradng Post performs surprsngly well, achevng a PoA of (1+ǫ) for every ǫ > 0. In fact, we fnd that all the equlbra of the Tradng Post mechansm are pure, so ths bound extends beyond the pure PoA. Ths work was done n part whle the authors were vstng the Smons Insttute for the Theory of Computng. Smna was also supported by ISF grant 1435/14 admnstered by the Israel Academy of Scences and Israel-USA B-natonal Scence Foundaton (BSF) grant , as well as the I-CORE Program of the Plannng and Budgetng Commttee and The Israel Scence Foundaton. Vasls was addtonally supported by grants NSF , NSF , and NSF

2 1 Introducton The problem of allocatng resources among multple partcpants n a far manner s as old as human socety tself, wth some of the earlest recorded nstances of the problem datng back to more than 2500 years ago 1. The mathematcal study of far dvson began wth the work of Stenhaus durng the second world war, whch led to an extensve and growng body of work on far dvson protocols wthn economcs and poltcal scence, e.g., [4, 7, 52, 61, 65]. Recent years have seen an ncreased amount of work on far dvson comng from computer scence (see, e.g., [8, Part II]), partly motvated by problems related to allocatng computatonal resources such as CPU, memory, and bandwdth among the users of a computng system. Ths work has focused on both settngs wth dvsble goods (e.g., [9, 11, 18, 22, 57, 59]) and ndvsble ones (e.g., [2, 14, 20, 60]). One of the most basc questons underlyng the far dvson problem s that of defnng farness to begn wth. A large body of work n economcs, especally socal choce theory, s concerned wth ths very queston, and numerous soluton concepts have been proposed n response. Our farness concept of choce heren s the Nash socal welfare (NSW), whch dates back to the fftes [45, 54] and has been proposed by Nash as a soluton for barganng problems, usng an axomatc approach. Ths objectve ams to choose an outcome x maxmzng the geometrc mean of the utltes (u (x)) of the n partcpatng agents and, lke other standard welfare objectves, t s captured by the followng famly of functons known as generalzed (power) means: M p (x) = ( 1 n ) 1/p [u (x)] p. In partcular, the NSW corresponds to M 0 (x),.e., the lmt of M p (x) as p goes to zero,.e., (Π u (x)) 1 n. Whle an extended treatment of the NSW can be found, for example, n Mouln [52], we hghlght a fundamental property of the NSW objectve, namely that t acheves a natural compromse between ndvdual farness and effcency. Twootherwell-studedfunctonscapturedbyM p (x)arethe()egaltaran(max-mn) objectve attaned as p, and () utltaran (average) objectve attaned at p = 1, whch correspond to extreme farness and extreme effcency, respectvely. However, the former may cause vast neffcences, whle the latter can completely neglect how unhappy some agents mght be. The NSW objectve les between these two extremes and strkes a natural balance between them, snce maxmzng the geometrc mean leads to more balanced valuatons, but wthout neglectng effcency. The hghly desred farness and effcency trade-off that the NSW objectve provdes can be verfed va ts close connecton wth market equlbrum outcomes n the Fsher market model one of the fundamental resource allocaton models n mathematcal economcs. Ths model was developed by Fsher [6] and studed n an extensve body of lterature [15, 19, 26, 31, 32, 35 37, 44, 56]. The basc settng nvolves a seller who brngs multple dvsble goods to the market and a set of buyers equpped wth monetary endowments, or budgets. The seller s goal s to set prces that extract all the money from the buyers, each of whch ams to spend all ts budget n acqurng the best possble bundle at the gven prces. A market equlbrum s an outcome where supply meets demand, and t has been shown to exst under very general condtons. When every buyer has the same budget, ths outcome s also known as the compettve equlbrum from equal ncomes (CEEI) [64]. For a broad famly of valuatons, ncludng the ones consdered n ths paper, the market equlbrum allocaton s known to maxmze the NSW objectve when the budgets are equal. In other words, the seller s goal can be acheved by computng the allocaton of the goods that maxmzes the NSW whch, n turn, mples the desred prce for each good. However, n order to be able to maxmze the NSW, the seller needs to know the valuatons of the partcpants. When these valuatons are prvate nformaton of the buyers, a natural soluton would be to use the Fsher market mechansm: ask the buyers to report ther valuatons, and then compute the NSW maxmzng allocaton based on the reports. Unfortunately, t s well known that buyers can fegn dfferent nterests and eventually get better allocatons [1, 3, 10, 16, 17]. Ths strategc behavor can result n the mechansm computng a market equlbrum wth respect to utlty functons whch may have lttle to do wth realty, leadng to unfar allocatons. In ths work we address the followng basc queston: 1 See, e.g., Hesod s Theogony, where a protocol known as Cut and Choose s mentoned. 2

3 How well does the Fsher market mechansm optmze ts own objectve the Nash socal welfare when the partcpants are strategc? Are there wdely used mechansms wth better guarantees? Ths queston falls under the general umbrella of mplementaton theory [24], and partcularly, of mplementng markets [5, 30, 40, 53, 58]. In ths lterature, the goal s to dentfy mechansms (game forms) for whch the set of Nash equlbra concdes wth the set of market equlbrum allocatons for every possble state of the world [24]. In general ths can be acheved only n the lmt, as the number of players goes to nfnty and every player s nfntesmal compared to the entre economy [29]. In ths work we show that, even for small markets, there exst classcal mechansms achevng outcomes that closely approxmate the optmal NSW on every nstance. We measure the qualty of a mechansm usng the prce of anarchy [55], defned as the rato between the optmal NSW and the NSW of the worst Nash equlbrum outcome obtaned by the mechansm. 1.1 Our Results We study the queston of approxmately mplementng the NSW objectve for two extensvely studed classes of valuatons, namely lnear (or addtve) and Leontef [19, 35, 55]; two extremes. Lnear valuatons capture goods that are perfect substtutes,.e., that can replace each other n consumpton, such as Peps and Coca- Cola. Leontef valuatons capture perfect complements,.e., goods that have no value wthout each other, such as left and rght shoes. Our frst set of results concerns the Fsher market mechansm, whch collects the bdders valuatons n the form of bds and then computes the market equlbrum based on those bds: The Nash equlbra of the Fsher market mechansm approxmate the NSW objectve wthn a factor of 2 for lnear valuatons. For Leontef valuatons, the approxmaton degrades lnearly wth the number of players. These bounds reveal sgnfcant dfferences between the qualty of the Nash equlbra of the Fsher market mechansm for complements and substtutes. Much more strkngly, we fnd that a classc mechansm known as Tradng Post, orgnally ntroduced by Shapley and Shubk [62] and studed n a long lne of work n dfferent scenaros [29, 43, 46, 49], offers yet stronger guarantees. At a hgh level, rather than collectng the players preferences and computng the market equlbrum, the Tradng Post mechansm gves each partcpant drect control over how to spend ts budget. Once the agents choose how to dstrbute ther budget over the avalable goods, they receve a fracton from each good that s proportonal to the amount they spent on t. Our man results for the Tradng Post mechansm are: The Nash equlbra of the Tradng Post mechansm approxmate the NSW objectve wthn a factor of 2 for lnear valuatons. But, for Leontef valuatons, the Tradng Post mechansm acheves an approxmaton of 1+ǫ for any ǫ > 0. In other words, not only does the Tradng Post mechansm acheve the same approxmaton as the Fsher market mechansm for lnear valuatons, but t also almost perfectly mplements the market equlbrum outcome for Leontef valuatons! We vew ths as an mportant result that testfes to the usefulness and robustness of the Tradng Post mechansm. An nterpretaton of the result s that the Tradng Post mechansm lmts the extent to whch an agent can affect the outcome, thus also lmtng the extent to whch thngs can go awry. Specfcally, when an agent devates n the Tradng Post mechansm, ths devaton has no effect on the way that the other agents are spendng ther money. On the other hand, an agent s devaton n the Fsher market mechansm can lead to a market equlbrum where the other agents spendng and allocaton has changed sgnfcantly. In addton to ths, n the Fsher market mechansm an agent can affect the prce of an tem even f the agent does not end up spendng on that tem n the fnal outcome. Ths s n contrast to the Tradng Post mechansm where an agent can affect only the prces of the tems that ths agent s spendng on, so the agents are forced to put ther money where ther mouth s. Fnally, we prove that the set of mxed Nash equlbra of the Tradng Post mechansm concdes wth the set of pure Nash equlbra, whch extends our PoA bounds for ths mechansm to mxed PoA as well. 3

4 1.2 Related Work The paper most closely related to our work s that of Cole et al. [22] whch proposes truthful mechansms for approxmately maxmzng the Nash socal welfare objectve. One of the truthful mechansms that they propose, the Partal Allocaton mechansm, guarantees a approxmaton of the optmal NSW for both lnear and Leontef valuatons. In fact, the Partal Allocaton mechansm guarantees that every agent receves a approxmaton of the value that t would receve n the market equlbrum. But, n order to ensure truthfulness, ths mechansm s forced to keep some of the goods unallocated, whch makes t napplcable for many real world settngs. Complementng ths mechansm, our work analyzes smple and well-studed mechansms that allocate all the goods fully. Most of the lterature on far dvson startng from the 1940 s deals wth the cake-cuttng problem, whch models the allocaton of a dvsble heterogeneous resource such as land, tme, and mneral deposts, among agents wth dfferent preferences [4, 7, 52, 61, 65]. Some recent work has studed the agents ncentves n cake cuttng. In partcular, Chen et al. [18] study truthful cake-cuttng wth agents havng pecewse unform valuatons and provde a polynomal-tme mechansm that s truthful, proportonal, and envy-free, whle Mossel and Tamuz [51] shows that for general valuatons there exsts a protocol that s truthful n expectaton, envy-free, and proportonal for any number of players. The work of Maya and Nsan [48] shows that truthfulness comes at a sgnfcant cost n terms of effcency for drect revelaton mechansms, whle Brânze and Mltersen [9] show that the only strategyproof mechansms n the standard query model for cake cuttng are dctatorshps (even for two players; a smlar mpossblty holds for n > 2). The standard cake cuttng model assumes addtve valuatons, and so t does not capture resources wth Leontef valuatons, whch we also analyze n ths paper. The resource allocaton lterature has seen a resurgence of work studyng far and effcent allocaton for Leontef valuatons [28, 39, 42, 57]. 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. [39] 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. Parkes et al. [57] assessed DRF n terms of the resultng effcency, showng that t performs poorly. Dolev et al. [28] proposed an alternate farness crteron called Bottleneck Based Farness, whch Gutman and Nsan [42] subsequently showed s satsfed by the proportonally far allocaton. Gutman and Nsan [42] also posed the study of ncentves related to ths latter noton as an nterestng open problem. It s worth notng that Ghods et al. [39] acknowledge that the CEEI,.e., the NSW maxmzng allocaton would actually be the preferred far dvson mechansm n ther settng, and that the man drawback of ths soluton s the fact that t cannot be mplemented truthfully. Our results show that the Tradng Post mechansm can, n fact, approxmate the CEEI outcome arbtrarly well, thus sheddng new lght on ths settng. The Tradng Post mechansm, also known as the Shapley-Shubk game [62], has been studed n an extensve body of lterature over the years, sometmes under very dfferent names, such as Chnese aucton [47], proportonal sharng mechansm [34], and the Tullock contest n rent seekng [33, 50, 63], the latter beng a varant of the game wth a dfferent success probablty for tems that nobody bd on. Tradng Post can also be nterpreted as a congeston game (see, e.g., [38]), or an all-pay aucton when the budgets are ntrnscally valuable to the players. The fact that, facng the Fsher mechansm, the agents may gan by bddng strategcally s well known. Adsul et. al. [1] studed the agents ncentves and proved exstence and structural propertes of Nash equlbra for ths mechansm. Extendng ths work, Chen et al. [16, 17] proved bounds on the extent to whch an agent can gan by msreportng for varous classes of valuaton functons, ncludng addtve and Leontef. Fnally, Brânze et al. [10] showed bounds for the prce of anarchy of ths mechansm wth respect to the socal welfare objectve, and Cole and Tao [21] studed large markets under mld randomness and showed that ths prce of anarchy converges to one. Fnally, recent work on the NSW has revealed addtonal appealng propertes of ths objectve. For ndvsble tems and addtve valuatons, the NSW can be approxmated n polynomal tme [20] and ts optmal allocaton s approxmately envy-free [13]. On the other hand, for dvsble tems, t can be used as an ntermedate step toward approxmatng the normalzed socal welfare objectve [23]. 4

5 2 Prelmnares Let N = {1,...,n} be a set of players (agents) and M = {1,...,m} a set of dvsble goods. Player s utlty for a bundle of goods s represented by a non-decreasng valuaton functon u : [0,1] m R +. An allocaton x s a partton of the goods to the players such that x,j represents the amount of good j receved by player. Our goal wll be to allocate all the resources fully; t s wthout loss of generalty to assume that a sngle unt of each good s avalable, thus the set of feasble allocatons s F = {x x,j 0 and n =1 x,j = 1}. Our measure for assessng the qualty of an allocaton s ts Nash socal welfare. At a gven allocaton x t s defned as follows ( n NSW(x) = u (x ) =1 In order to also capture stuatons where the agents may have dfferent mportance or prorty, such as clout n barganng scenaros, we also consder the weghted verson of the Nash socal welfare objectve. Note ths s the objectve maxmzed by the Fsher market equlbrum soluton when the buyers have dfferent budgets. We slghtly abuse notaton and refer to the weghted objectve as the Nash socal welfare (NSW) as well. If B 1 s the budget of agent and B = n =1 B s the total budget, the nduced market equlbrum n Fsher s model maxmzes the followng objectve: NSW(x) = ( n =1 ) 1 n u (x ) B ) 1 B. Note that we get back the orgnal defnton when all players have the same budget. We would lke to fnd mechansms that maxmze the NSW objectve n the presence of strategc agents whose goal s to maxmze ther own utlty. We measure the qualty of the mechansms usng the prce of anarchy [55] wth respect to the NSW objectve. Gven a problem nstance I and some mechansm M that yelds a set of pure Nash equlbra E, the prce of anarchy (PoA) of M for I s the maxmum rato between the optmal NSW obtaned at some allocaton x and the NSW at an allocaton x E: PoA(M,I) = max x E. { NSW(x ) NSW(x) The prce of anarchy of M s the maxmum of ths value over all possble nstances,.e., max I {PoA(M,I)}. Valuaton Functons. We focus on two very common and extensvely studed valuaton functons that le at the two extremes of perfect substtutes and perfect complements. For both, let v = (v,1,...,v,m ) R m + be a vector of valuatons for agent, where v,j captures the lkng of agent for good j. Perfect substtutes, defned mathematcally through addtve valuatons, represent goods that can replace each other n consumpton, such as Peps and Coca-Cola. In the addtve model, the utlty of a player for bundle x s u (x ) = m v,j x,j. On the other hand, perfect complements, represented by Leontef utltes, capture scenaros where one good may have no value wthout the other, such as a left and a rght shoe, or the CPU tme and computer memory requred for the completon of a computng task. In the Leontef model, the utlty of a player for a bundle x s u (x ) = mn m {x,j /v,j }; that s, player desres the tems n the rato v,1 : v,2 :... : v,m. In the Leontef model the coeffcents can be rescaled freely, and so we assume w.l.o.g. that v,j 1. 3 The Fsher Market Mechansm In the Fsher market model, gven prces p = (p 1,...,p m ), each buyer demands a bundle x that maxmzes her utlty subject to her budget constrants; we call ths an optmal bundle of buyer at prces p. Prces p nduce a market equlbrum f buyers get ther optmal bundle and market clears. Formally, prces p and allocaton x consttute a market equlbrum f 1. Optmal bundle: N and y : y p B, u (x ) u (y) } 5

6 2. Market clearng: Each good s fully sold or has prce zero,.e., j M, m x,j 1, and equalty holds f p j > 0. Each buyer exhausts all ts budget,.e., N, m x,jp j = B. For lnear and Leontef valuatons, the market equlbra can be computed usng the Esenberg-Gale (EG) convex program formulatons that follow. max s.t. Lnear n =1 B logu u = m v,jx,j, N n =1 x,j 1, j M x,j 0, N,j M max s.t. Leontef n =1 B logu u x,j v,j, N,j M n =1 x,j 1, j M x,j 0, N,j M (1) Let p j be the dual varable of the second nequalty (for good j) n both cases, whch corresponds to prce of good j. Snce due to strong dualty Karush-Kuhn-Tucker (KKT) condtons capture solutons of the formulatons, we get the followng characterzaton for market equlbra. For lnear valuatons, an outcome (x,p) s a market equlbrum f and only f, Ln 1 N and j M, x,j > 0 v,j p j = max k M v,k p k. Ln 2 N, j M x,jp j = B. j M, ether N x,j = 1 or p j = 0. For Leontef valuatons, an outcome (x,p) s a market equlbrum f and only f. Leo 1 N and j M, v,j > 0 u = x,j has to spend to get unt utlty. v,j = Leo 2 N, j M x,jp j = B. j M, ether N x,j = 1 or p j = 0. B j M v,jpj. Note that j M v,jp j s the amount buyer The Fsher market mechansm asks that the agents report ther valuatons and then t computes the market equlbrum allocaton wth respect to the reported valuatons usng the EG formulatons. Defnton 1 (Fsher Market Mechansm). The Fsher Market Mechansm s such that: The strategy space of each agent conssts of all possble valuatons the agent may pose: S = {s s R m 0 }. We refer to an agent s strategy as a report. Gven a strategy profle s = (s ) n =1, the outcome of the game s a market equlbrum of the Fsher market gven by B,s, after removng the tems j for whch N s (j) = 0. If there exsts a market equlbrum E preferred by all the agents (wth respect to ther true valuatons), then E s the outcome of the game on B,s. Otherwse, the outcome s any fxed market equlbrum. ItswellknownthatthebuyershavencentvestohdethertruevaluatonsnaFshermarketmechansm. We llustrate ths phenomenon through an example. Example 1. Consder a Fsher market wth players N = {1, 2}, tems M = {1, 2}, addtve valuatons v 1,1 = 1, v 1,2 = 0, v 2,1 = v 2,2 = 0.5, and budgets equal to 1. If the players are truthful, the market equlbrum allocaton s x 1 = (1,0), x 2 = (0,1). However, f player 2 pretended that ts value for tem 2 s a very small v 2,2 = ǫ > 0, then player 2 would not only get tem 2, but also a fracton of tem 1. In the rest of ths secton we study the performance of the Fsher mechansm when the tems are substtutes and when they are complements. For both of these cases we provde upper and lower bounds for the prce of anarchy. 6

7 3.1 Fsher Market: Perfect Substtutes In ths secton we study effcency loss n the Fsher market due to strategc agents wth addtve valuatons. We note that pure Nash equlbra n the nduced game are known to always exst due to Adsul et al. [1]. 2 Our frst man result states that the Fsher market approxmates the Nash Socal Welfare wthn a small constant factor, even when the players are strategc. Theorem 1. The Fsher Market Mechansm wth addtve valuatons has prce of anarchy at most 2. We frst prove a useful lemma, whch bounds the change n prces due to a unlateral devaton n the Fsher market. Lemma 1. Let p be the prces n a Fsher market equlbrum. Suppose that buyer unlaterally changes ts reported values to v, leadng to new market equlbrum prces p. Then, p j B + p j. j: p j >pj j: p j >pj Proof. Let M +, M, and M = be the sets of goods whose prces have strctly ncreased, strctly decreased, and remaned unchanged, respectvely, when transtonng from p to p. If some player k s buyng any fracton of a good from M or M = at p, then the player wll cannot be buyng anythng from M + at p. To verfy ths fact, assume that player k was spendng on some tem α M M = at prces p and s now spendng on some tem β M + at prces p. Snce both these allocatons are market equlbra, Ln 1 mples that v k,α p α v k,β p β at prces p and v k,α p α v k,β p β at prces p. But, snce p α p α and p β > p β, ths leads to the contradcton that v k,α > v k,α. Thus, any player, other than, who s buyng goods from M + at prces p had to be spendng all ts budget on these tems at prces p. Ths mples that the only reason why the sum of the prces of the goods n M + could ncrease s because player s contrbutng more money on these tems. But, snce the budget of s B, the total ncrease n these prces s at most B, whch concludes the proof. In addton to ths, we wll be usng the weghted arthmetc and geometrc mean nequalty, whch upper bounds the weghted geometrc mean usng the weghted arthmetc mean. Lemma 2. For any nonnegatve numbers ρ 1,ρ 2,...,ρ n and w 1,w 2,...,w n such that W = n =1 w, ( n We can now prove the man theorem. =1 ρ w ) 1/W n =1 ρ w. W Proof. (of Theorem 1) Gven a problem nstance wth addtve valuatons, let x be the allocaton that maxmzes the Nash socal welfare.e., the market equlbrum allocaton wth respect to the true valuatons and x and p the allocaton and prces respectvely obtaned under some Nash equlbrum of the market, where the players report fake valuatons ṽ. Addtonally, for each player, let x be the allocaton that would arse f every player k reported ṽ k whle reported ts true value v j for every tem j wth x j > 0, and zero elsewhere. Snce ṽ s an equlbrum, ths unlateral devaton of cannot ncrease ts utlty: u ( x ) u (x ) = x jv j. (2) Snce x s a market equlbrum wth respect to the reported values then, accordng to the KKT condton (Ln 1 ), x j > 0 mples v j/p j v k/p k for any other tem k, where p j s the prce of tem j n x. Also, the KKT condton (Ln 2 ) mply that j x j p j = B for every bdder. Therefore, x provdes at least as 2 The exstence s shown under a conflct-free te breakng rule, whch tres to allocate best bundle to as many agents as possble when there s a choce. 7

8 much value as any other bundle that can afford facng prces p,.e., any bundle that costs at most B. In partcular, consder the allocaton x such that x j = B x j m. k=1 x k p k For ths allocaton we have: x jp j = ( B x ) j m k=1 x k p k p j = B m x j p j m k=1 x k p k = B. Therefore, player can afford ths bundle of tems usng the budget B, whch mples that x provdes at least as much value,.e., u (x ) u (x ) = v,j x,j = B u (x ) m. x,j p j Let ρ denote the rato u(x ) u ( x ). Usng the Nash equlbrum nequalty (2), we get: Usng Lemma 1, we get: u ( x ) B u (x ) m x,j p j u (x m ) u ( x ) x,j p j B ρ B x,jp j. ρ B x,jp j x,j p j + j: p j pj x,jp j B + x,j p j, j: p j > pj and summng over all players: n ρ B =1 n B + =1 x,j p j 2B. (3) Substtutng B for w n Lemma 2 and usng Inequalty (3) yelds: ( n =1 ρ B ) 1/B n =1 ρ B B 2. Next we show a lower bound for the prce of anarchy of the Fsher mechansm. We construct a collecton of problem nstances whose PoA goes to e 1/e as number of players grows. Theorem 2. The Fsher mechansm has a prce of anarchy no better than e 1/e Gven some value of n, we construct a market wth n+2 agents and n+1 goods. Fx an nteger k n; we wll set ts value later. Each player k lkes only good,.e., v, = 1 and all other v,j = 0. On the other hand, every agent [k +1,n], apart from havng v, = 1, also has some small, but postve, value v,j = ǫ for all tems j k. The rest of that agent s v,j values are zero. Agent n+1 has a small but postve value ǫ for goods j [k +1,n] and value 2 for good n+1. Fnally, agent n+2 values only good n+1 at value 2. Here ǫ << ǫ, and we wll set ther values later. In the allocaton where every agent [1,n] gets all of good, whle agents n+1 and n+2 share good n+1 equally, the NSW s equal to 1. Next, we construct a Nash equlbrum strategy profle s of the above market where the NSW approaches (1/e) 1/e as n. We defne a strategy profle s where the frst k agents and the last agent,.e., agent n+2, bd truthfully, whle every bdder [k+1,n] msreports n a way such that t ends up spendng some small amount δ = 2ǫ on tem and the rest of ts budget, namely (1 δ), s equally dvded on tems n [1,k]. We later set a value for δ such that agent n+1 would want to buy only good n+1. 8

9 We now show that the above profle s a Nash equlbrum for carefully chosen values of ǫ, and ǫ. Note that, snce bdders [1,k] and bdder n + 2 bds truthfully and values just one tem, each one of these bdders wll spend all of ts budget on the correspondng tems, no matter what the remanng bdders report. Therefore, the prce of tems j [1,k] and tem n+1 wll be exactly 1 f we exclude the spendng of bdders [k+1,n+1], no matter what these bdders report. Also, f the prce of tems j [k+1,n] s equal to δ n profle s, then ths prce wll not drop below δ, rrespectve of what bdder n+1 reports. In the next two lemmas, we show that a devaton from s does not help the bdders, even when the prces are held constant for the goods where an agent starts spendng more money. Lemma 3. If δ = 2ǫ then, even f the prces of goods j [k +1,n] are fxed at δ, the (n+1) th bdder has no ncentve to spend money on any good other than good n+1. Proof. If agent n+1 spends all of ts budget on tem n+1, then ts prce becomes 2 and the agent receves half of that tem,.e., a utlty of 1. On the other hand, f the agent spends at total of γ > 0 on goods j [k+1,n] then, even f the prce of these tems remans δ, her utlty would be ǫ γ δ +21 γ 2 γ. For ths to be strctly greater than 1 we have to have δ < 2ǫ γǫ, whch contradcts our assumpton that δ = 2ǫ. Furthermore, usng a smlar analyss as that of Lemma 3, t follows that f agent [k+1,n] devates n a way that decreases ts spendng on good to 2ǫ τ for τ > 0, then the Fsher market mechansm outcome wll have the (n+1) th τ agent spendng at least 1+ǫ on that tem. Thus, such a devaton would cause agent to lose ts monopoly on good. Next, we show that ths s not advantageous for agent f we set ǫ = 1 n 4 and ǫ = 1 n, whch mples δ = 2 n. Lemma 4. For any τ 0, f agent s spendng δ τ on good and agent (n+1) s spendng τ 1+ǫ, when others are bddng accordng to s, then agent s utlty s maxmzed at τ = 0. Proof. Note that at τ = 0, agent [k + 1,n] s spendng δ = 2ǫ on good and, accordng to Lemma 3, agent n+1 would not be nterested n spendng on good. Thus, agent s buyng good exclusvely and spends 1 δ k on each of the frst k goods. Thus the prce each one of the frst k goods s 1+(n k) 1 δ k. Agent s utlty at ths allocaton s 1+ 1 δ k (1 δ)kǫ 1+(n k) 1 δ kǫ = 1+ k +(n k)(1 δ) k As mentoned above, f agent reduces spendng on good by τ, then the (n + 1) th agent wll end up τ spendng at least 1+ǫ on ths tem. Ths may lead to ncreased prces for the frst k goods, but we show that ths devaton would not beneft even f these prces remaned the same. The utlty of agent after such a devaton would be δ τ (1 δ +τ)kǫ δ τ + τ + 1+ǫ k +(n k)(1 δ) The latter utlty s greater than the former only when τkǫ k +(n k)(1 δ) > τ 1+ǫ δ τ + τ ǫ > k +(n k)(1 δ) 1+ǫ k(δ +δǫ τǫ ) > 1, whch contradcts the fact that ǫ = 1 n 4. Lemmas3and4mplythatssaNashequlbrum. Theprceofthefrstk goodsatthsnashequlbrum s 1+ (n k)(1 δ) k = k+(n k)(1 δ) k. Thus, the utlty of buyer [1,k], who spends all of ts $1 on good, s u =. On the other hand, the utlty of buyer [k+1,n], who gets all of good and some of k k+(n k)(1 δ) the frst k goods, s u = 1+ (1 δ)kǫ k+(n k)(1 δ). Agents (n+1) and (n+2) get half of good n+1 and thereby get utlty of 1 each. Snce ǫ = 1 n 4 and ǫ = 1 n, then [1,k], lm n u = k n and [k+1,n]lm n u = 1. Thus, NSW at ths bd profle as n s ( k n ) k n, and thereby PoA s at least ( n k )k/n. Lettng k = n e, ths becomes e 1/e, whch concludes the proof of Theorem 2. 9

10 3.2 Fsher Market: Perfect Complements On the other hand, the Fsher market wth perfect complements has a prce of anarchy that grows lnearly wth the number of players. The exstence of pure Nash equlbra n Fsher markets wth Leontef valuatons was establshed by Brânze et al. [10]. Theorem 3. The Fsher Market Mechansm wth Leontef valuatons has a prce of anarchy of n and the bound s tght. Proof. Our tool s the followng theorem, whch states that Fsher markets wth Leontef utltes have Nash equlbra where players copy each others strateges. Lemma 5 ([10]). The Fsher Market mechansm wth Leontef preferences always has a Nash equlbrum where every buyer reports the unform valuaton (1/m,...,1/m). For completeness, we nclude the worst case example. Consder an nstance wth n players of equal budgets (B = 1) and n tems, where each player lkes tem and nothng else; that s, v, = 1, for all N and v,j = 0, N, j. Then the optmal Nash Socal Welfare s obtaned n the Fsher market equlbrum, where the prce of each tem j s p j = 1 and the allocaton s x, = 1, N and x,j = 0, N, j. However, the strategy profle y = (y 1,...,y n ), where y = (1/n,...,1/n) s a Nash equlbrum n whch each player gets a fracton of 1/n from every tem, yeldng utlty 1/n for every player. It follows that the prce of anarchy s n. For a general upper bound, we note that any Nash equlbrum must be proportonal,.e. each player gets a fracton of at least B /B of ts best possble utlty, OPT. Let (p,x) be a Nash equlbrum of the market, acheved under some reports v. Suppose for a contradcton that there exsts player wth u (x ) < B /B OPT. Then f reported nstead ts true valuaton v, the new market equlbrum, (p,x ), acheved under valuatons (v,v ), should satsfy the nequalty u (x ) B /B OPT. If ths were not the case, the outcome would not be a market (v,v ), snce can afford the bundle y = (B 1/B,...,B m /B): p(y) = p j B B = B B p j = B, (4) Moreover, u (y) = B /B OPT, whch together wth Identty 4 contradcts the market equlbrum property of (p,x ). Thus n any Nash equlbrum (p,x) we have that u (x ) B /B OPT, and so the Nash Socal Welfare s NSW(x) n ( B =1 B OPT ) B B. Then the prce of anarchy can be bounded as follows: n OPT B/B n ( PoA ( B =1 B OPT ) ) ( ) n B B /B =1 B B B B/B = n, B =1 B where for the last nequalty we used the fact that the weghted geometrc mean s bounded by the weghted arthmetc mean (Lemma 2).Ths completes the proof of the theorem. 4 The Tradng Post Mechansm The mportant dfference between the Tradng Post mechansm and the Fsher Market mechansm s the strategy space of the agents. More precsely, unlke the Fsher market mechansm, where the agents are asked to report ther valuatons, the Tradng Post mechansm nstead asks the agents to drectly choose how to dstrbute ther budgets. Once the agents have chosen how much of ther budget to spend on each of the goods, the total spendng on each good j s treated as ts prce, and each agent s allocated a fracton of good { j proportonal to the amount that s spendng on j. Therefore, the strategy set of each player s b [0,1] m m b,j = B }. A bd profle b = (b 1,...,b n ) yelds the allocaton: x,j = b,j n k=1 b k,j 10

11 For the Tradng Post mechansm nstances we assume that every good s desred by at least two players,.e., for each good j, there exst buyers,k such that v,j 0,v k,j 0. If ths were not the case, then we could ether dscard that good or gve t away for free. Ths s a commonly assumed property called perfect competton. Our man results n ths secton are that the Nash equlbra of Tradng Post approxmate the NSW objectve wthn a factor of (2e) 1/e for lnear valuatons (the bound becomes 2 for arbtrary budgets), and wthn a factor of 1+ǫ for every ǫ > 0 for Leontef valuatons. Moreover, n both cases, all the Nash equlbra of the game are pure. 4.1 Tradng Post: Perfect Substtutes The exstence of pure Nash equlbra n the Tradng Post mechansm for addtve valuatons was establshed by Feldman, La, and Zhang [34] under perfect competton. Wthout competton, even very smple games may not have pure Nash equlbra. To see ths, consder for nstance a game wth two players, two tems, and addtve valuatons v 1,1 = 1, v 1,2 = 0, v 2,1 = v 2,2 = 0.5. Through a case analyss t can be seen that both players wll compete for tem 1, whle player 2 s the only one that wants tem 2, reason for whch ths player wll successvely reduce ts bd for 2 to get a hgher fracton from tem 1. However, n the lmt of ts bd for the second tem gong to zero, player 2 loses the tem. Our frst result for Tradng Post quantfes the degradaton n the Nash Socal welfare value for addtve valuatons. Theorem 4. The Tradng Post Mechansm wth addtve valuatons has prce of anarchy at most 2. Proof. Gven a problem nstance wth addtve valuatons, let x be the allocaton that maxmzes the Nash socal welfare and p be the correspondng market equlbrum prces. Also, let x be the allocaton obtaned under a Nash equlbrum where each player bds b and the prce of each tem j s p j = b,j. For some player, let x be the allocaton that arses f every player k bds b k whle agent unlaterally devates to b,j = b,j +δ,j for each tem j and some δ,j R such that j δ,j = 0. Let δ,j be such that for some β > 0 and every tem j b,j +δ,j = x,j. (5) p j +δ,j β Ths bd b s mpled by the soluton of the followng program wth varables δ,j for each j M: mnmze: β subject to: β = x,j ( p j +δ,j ), j M b,j +δ,j δ,j = 0, j M The allocaton nduced by ths unlateral devaton of s x,j = b,j p j +δ,j = b,j +δ,j p j +δ,j = x,j β. Therefore, the utlty of player after ths devaton s u (x )/β. But the outcome x s a Nash equlbrum, so ths devaton cannot yeld a hgher utlty for, whch mples that u ( x) u (x )/β. By defnton of b, we get j b,j = j x,j ( p j +δ,j ) = B. Therefore, replacng for x,j = x,j /β, we get B β = x,j( p j +δ,j ) (6) Snce x,j 0, Equaton (5) mples that δ,j b,j for every tem j. Therefore, the sum of the negatve δ,j values s no less than B. Ths mples that δ,j B, snce j δ,j = 0, so for every agent we have x,j( p j +δ,j ) B + x,j p j. (7) 11

12 Usng Inequaltes (6) and (7), and summng over all players gves n u (x B ) n n B β B + u ( x ) =1 =1 =1 x,j p j 2B. (8) Note that Inequalty (8) s the same as nequalty (3) n the proof of Theorem 1. Thus, followng the same arguments as those n that proof proves the theorem. The next theorem complements the upper bound on the prce of anarchy wth a lower bound of approxmately Theorem 5. The Tradng Post mechansm has a prce of anarchy no better than e 1/e Proof. In order to prove ths theorem we can use the same famly of problem nstances used n the proof of Theorem 2. In fact, verfyng that the market equlbrum outcome nduced by the strategy profle s n that constructon s a Nash equlbrum for the Tradng Post mechansm as well s much more straghtforward snce an agent s devaton may affect only the way that partcular agent ends up spendng ts budget. 4.2 Tradng Post: Perfect Complements We begn by characterzng the precse condtons under whch the Tradng Post mechansm has exact pure Nash equlbra for Leontef utltes; the proof s ncluded n the appendx. Theorem 6. The Tradng Post mechansm wth Leontef utltes has pure Nash equlbra f and only f the correspondng market equlbrum prces are all strctly postve. When ths happens, the Nash equlbrum utltes n Tradng Post are unque and the prce of anarchy s 1. Ths theorem establshes a correspondence between the pure Nash equlbra of Tradng Post and the correspondng market equlbra wth respect to the agents (true) valuatons. We observe however that exstence of pure Nash equlbra n Tradng Post s not guaranteed for Leontef utltes. Example 2. Consder a game wth two players and two tems, where player 1 has values v 1,1 = v 1,2 = 0.5 and player 2 has v 2,1 = 0.9, v 2,2 = 0.1. Assume there s a pure Nash equlbrum profle b. Snce both players requre a non-zero amount from every tem for ther utlty to be postve, we have that b,j > 0 for all,j {1,2}. Denote b 1 = b 1,1 and b 2 = b 2,1 ; then b 1,2 = 1 b 1 and b 2,1 = 1 b 2. Note that each player must receve the two tems n the same rato relatve to ts valuaton; that s: u (b) = ( b b 1 +b 2 ) 1 v,1 = ( 1 b b 1 +b 2 ) 1 v,2 (9) Otherwse, f the two ratos were not equal, then a player could transfer weght among the tems to mprove the smaller fracton. Then the requrement n 9 are equvalent to the followng equatons: ( ) ( ) b1 1 b 1 +b = 1 b1 1 b 1 +b b 1 = b 2 (10) and ( b2 b 1 +b 2 ) ( ) = 1 b2 1 b 1 +b b2 2 +8b 1 b 2 = 9b 1 +7b 2 (11) Combnng equatons 10 and 11, we get that b 1 = 1 and b 2 = 1, whch contradcts the requrement that b 1,b 2 (0,1). Thus the equlbrum profle b cannot exst. The ssue llustrated by ths example s that the Tradng Post cannot mplement market outcomes when there exst tems prced at zero n the correspondng market equlbrum. Ths motvates us to ntroduce an entrance fee n the Tradng Post mechansm, denoted by a parameter > 0, whch s the mnmum amount that an agent needs to spend on an tem n order to receve any of t. We denote the correspondng mechansm by T P( ). The value of can be arbtrarly small, so ts mpact on the outcome of the game 12

13 s nsgnfcant. Formally, gven a bd profle b = (b 1,...,b n ), let b = (b 1,...,b n ) be the effectve bd profle whch, for every N and j M, satsfes b,j = b,j f b,j, and b,j = 0 otherwse. Then, the bd profle b yelds the allocaton: b,j x,j = n k=1 b k,j Clearly, n any Nash equlbrum, we have that for every player and tem j we have that b,j or b,j = 0(the latter dentty holds for those goods j that are outsde of player s demand). The man result of ths secton s that, for every ǫ (0,1/m), the Tradng Post mechansm wth ǫ/m 2 has a prce of anarchy of at most 1+ǫ. We frst show that Tradng Post has a pure Nash equlbrum for Leontef utltes for every strctly postve entrance fee. The proof uses an applcaton of Glcksberg s theorem for contnuous games and s ncluded n the appendx. Theorem 7. The parameterzed Tradng Post mechansm T P( ) s guaranteed to have a pure Nash equlbrum for every > 0. We start by defnng a noton of approxmate market equlbrum that wll be useful Defnton 2 (ǫ-market equlbrum). Gven a problem nstance and some ǫ > 0, an outcome (p,x) s an ǫ-market equlbrum f and only f: All the goods wth a postve prce are completely sold. All the buyers exhaust ther budget. Each buyer gets an ǫ-optmal bundle at prces p; that s, for every bundle y [0,1] m that could afford at these prces (p y B ), we have u (y) u (x )(1+ǫ). The followng theorem states that for every small enough ǫ > 0, all the PNE of the Tradng Post game wth a small enough entrance fee correspond to ǫ-market equlbrum outcomes. Theorem 8. Let ǫ > 0. Then for every 0 < < mn { ǫ m 2, 1 m}, every pure Nash equlbrum of the mechansm T P( ) wth Leontef valuatons corresponds to an ǫ-market equlbrum. Proof. Let b be a pure Nash equlbrum of T P( ) and x the nduced allocaton. For each player, let D = {j M v,j > 0} be the set of tems that requres, and let m = D. We also overrde notaton and refer to u (b) as the utlty of player when the strategy profles are b. Frst note that b,j > 0 for each player and tem j D. If ths were not the case, then player would get zero utlty at strategy profle b; ths s worse than playng the unform strategy z = (B /m,...,b /m), whch guarantees a postve value regardless of the strateges of the other players b, namely: u (z, b ) = mn j D { z,j z,j + k b k,j 1 v,j } mn j D { B /m B /m+ k B k For each player and tem j D, denote the fracton of utlty that derves from j by: φ,j = b,j n k=1 b k,j Then u ( b) = mn j D φ,j. Sort the tems n D ncreasngly by ther contrbuton to s utlty: φ,1 φ,2... φ,m ; t follows that u ( b) = φ,1. Let S = {j D φ,j = φ,1 } be the tems receved n the smallest fracton (equal to s utlty). If S = M, then the analyss s smlar to the exact equlbrum case, where the prces are strctly postve. The dffcult case s when S M. Then player s gettng a hgher than necessary fracton from some resource j M \S. Thus would mprove by shftng some of the mass from tem j to the tems n S. Snce b s an equlbrum, no such devaton s possble. Then t must be the case that b,j = for all j D \S. 1 v,j 1 v,j } > 0 13

14 Now nterpret the bds and allocaton as a market equlbrum wth Leontef utltes v and budgets B, by settng the prces to p = (p 1,...,p m ), where p j = n =1 b,j for all j M, and the allocaton to x, the same as the one nduced by the bds b n the tradng post game. We argue that (p,x) s an ǫ-market equlbrum. Clearly at the outcome (p,x) all the goods are sold and each buyer exhausts ther budget. Moreover observe that all the prces are strctly postve. We must addtonally show that each player gets an ǫ-optmal bundle at (p,x). Fx an arbtrary buyer. Let y be an optmal bundle for gven prces p, and let q,j be the amount of money spent by to purchase y,j unts of good j at these prces. An upper bound on the optmal value u (y ) s attaned when buyer shfts all the money spent on purchasng tems outsde S to purchase nstead hgher fractons from the tems n S. Snce the strategy profle b s an exact equlbrum n the game T P( ), the amount of money spent by player on tems outsde S s at most (m 1) ; thus spends at most B (m 1) on the remanng tems n S. By an averagng argument, there exsts a good j S on whch spends the greatest amount of ts money,.e. b,j B (m 1). S Ths wll be the tem for whch the gan brought by the devaton n spendng s modest. Formally, the maxmum fracton of utlty that can get from tem j wthout decreasng the ratos at whch the other tems n S are receved s: φ,j = q,j b,j +(m 1) ( p j v n,j = φ,j + ) k=1 b k,j v,j (m 1) ( n ) φ,j + ( n ) k=1 b k,j v,j k=1 b k,j v,j = φ,j (1+ǫ) = u (x )(1+ǫ) b,j ǫ where n the nequaltes we addtonally used that < ǫ 2 /m, B 1 N, and S m 1. The denttes hold snce tem j s n the tght set S. Then u (y ) φ,j u (x )(1+ǫ). Thus each player gets an ǫ-optmal bundle, and so (p, x) s an ǫ-market equlbrum. The followng theorem, whch we beleve s of ndependent nterest, states that n Fsher markets wth Leontef utltes, approxmate market equlbra are close to exact equlbra n terms of ther Nash Socal Welfare. Theorem 9. The Nash socal welfare at an ǫ-market equlbrum for Leontef utltes s at least a 1 (1+ǫ) factor of the optmal Nash socal welfare. Proof. For any gven problem nstance, let (p,x ) be an ǫ-market equlbrum and let (p,x ) be exact market equlbrum prces and allocaton. By abuse of notaton let u (p ) denote the optmal utlty player can obtan at prces p,.e., u (p ) = max{u (y) y 0; p y B }. For Leontef utlty functons, convex formulaton of (1) captures the market equlbrum allocaton. Note that, n order to get a utlty of 1 at prces p, agent would need to spend a total amount of money equal to φ (p) = j v jp j. Devanur [27] derved the followng dual of ths convex program: mn : j p j B log(φ(p))+ B log(b ) B s.t. j : p j 0 Note that the term ( B log(b ) B ) s a constant for a gven market snce B s are constants, and hence s omtted n [27]. Snce (p,x ) s a market equlbrum, usng strong dualty and the fact that agents spend all ther money at equlbrum,.e., j p j = B : B log(u (x )) = B log(φ(p ))+ B log(b ) (12) 14

15 Furthermore, at the ǫ-market equlbrum (x,p ) all the agents spend all ther money, mplyng j p j = B. Snce p s a feasble dual soluton, B log(φ(p ))+ B log(b ) B log(φ(p ))+ B log(b ). Substtutng the left hand sde usng Equaton (12), and takng an antlogarthm on both sdes yelds u (x ) B ( ) B B φ(p. (13) ) Snce the optmal utlty that agent gets at prces p s u (p ), whch she derves usng B money, and whle for unt utlty she needs φ(p ) money, we get : u (p ) = B φ(p ) Snce (x,p ) s an ǫ-market equlbrum, each agent gets an ǫ-optmal bundle, so u (p ) u (x )(1+ǫ). B Accordng to (14), ths mples that φ(p ) u (x )(1+ǫ), whch we combne wth (13) to get: u (x ) B ( ) m B φ(p (1+ǫ) B u (x ) ) B Snce the Nash socal welfare at x s (Π u (x ) B ) 1 B, the result follows. Fnally, we can state the man result of ths secton. { } ǫ Theorem 10. For every ǫ > 0, the Tradng Post game T P( ) wth entrance fee 0 < < mn 2 m, 1 m has a prce of anarchy of 1+ǫ, even for arbtrary budgets. Proof. By Theorem 8, every PNE of T P( ) corresponds to an ǫ-market equlbrum. By Theorem 9, every ǫ-market equlbrum attans at least a fracton 1 1+ǫ of the optmal Nash socal welfare. Thus, the prce of anarchy of T P( ) s 1+ǫ, whch completes the proof. 4.3 Tradng Post: Beyond Pure Nash Equlbra So far we have bound Prce of Anarchy wth respect to pure Nash equlbra n case of both the mechansms. A natural queston s what f NSW at mxed Nash equlbra are bad. In case of Tradng Post mechansm we rule out ths case by showng that all ts Nash equlbra are pure, be t for lnear or Leontef valuatons. Detals of ths result may be found n Appendx B, where we show the followng theorem. Theorem 11. For market wth lnear utltes, every Nash equlbrum of the correspondng tradng post game s pure (Theorem 12). For market wth Leontef utltes, every Nash equlbrum of the correspondng tradng post game TP( ), for > 0, s pure (Theorem 13). Fnally, gven such an effcency acheved by Tradng Post mechansm one may wonder f these mechansms n-fact have unque equlbrum. We rule ths out through Examples 3 and 4 for lnear and Leontef valuatons respectvely gven n Appendx B. 5 Dscusson In ths paper we analyzed two well-known mechansms, namely the Fsher market mechansm and the Tradng Post mechansm, n terms of ther prce of anarchy wth respect to the Nash socal welfare objectve. We showed that both mechansms manage to obtan a small prce of anarchy n the lnear valuatons case but, when t comes to Leontef valuatons, the two mechansms perform very dfferently: the Fsher market (14) 15

16 mechansm has a very hgh prce of anarchy, whle the Tradng Post mechansm s essentally optmal. As an nterpretaton of ths result we observe that, although both of these mechansms are closely connected to the market equlbrum n the Fsher model, the Fsher market mechansm allows a unlateral devaton of one agent to affect the way the other agents end up spendng ther budgets, whle the Tradng Post mechansm does not provde the agent wth the same power. It therefore appears that ths lmtaton s helpful n avodng undesred Nash equlbra. Furthermore, we show that the bounds for the Tradng Post mechansm go beyond pure Nash equlbra, as the set of mxed Nash equlbra s equal to the set of pure Nash equlbra for ths mechansm. A natural extenson s to see f these nce propertes extend to general CES valuaton functons. Note that CES valuatons are parameterzed usng some ρ (, 1], and lnear and Leontef correspond to ρ = 1 and ρ = respectvely. Another mportant queston s to acheve good approxmaton of the NSW usng truthful non-wasteful mechansms. There has been some work along ths lne, but wth other objectves as ther focus. References [1] B. Adsul, Ch. Sobhan Babu, J. Garg, R. Mehta, and M. Sohon. Nash equlbra n Fsher market. In SAGT, pages 30 41, [2] H. Azz and S. Mackenze. A dscrete and bounded envy-free cake cuttng protocol for four agents. In Proceedngs of STOC, [3] M. Babaoff, B. Lucer, N. Nsan, and R. Paes Leme. On the effcency of the Walrasan mechansm. In EC, pages , [4] J.B. Barbanel. The Geometry of Effcent Far Dvson. Cambrdge Unv. Press, [5] C. Beva, L. Corchón, and S. Wlke. Implementaton of the Walrasan correspondence by market games. Revew of Economc Desgn, 7: , [6] W. C. Branard and H. E. Scarf. How to compute equlbrum prces n Cowles Foundaton Dscusson Paper, 1270, [7] S. Brams and A. Taylor. Far Dvson: from cake cuttng to dspute resoluton. Cambrdge Unversty Press, Cambrdge, [8] F. Brandt, V. Contzer, U. Endrss, J. Lang, and A. D. Procacca. Handbook of Computatonal Socal Choce. Cambrdge Unversty Press, [9] S. Brânze and P. B. Mltersen. A dctatorshp theorem for cake cuttng. In Proceedngs of IJCAI, pages , [10] S. Brânze, Y. Chen, X. Deng, A. Flos-Ratskas, S. Frederksen, and J. Zhang. The Fsher market game: Equlbrum and welfare. In AAAI, pages , [11] S. Brânze, I. Caraganns, D. Kurokawa, and A.D. Procacca. An Algorthmc Framework for Strategc Far Dvson. In Proceedngs of AAAI, [12] A. Cambn and L. Marten. Generalzed Convexty and Optmzaton: Theory and Applcatons. Sprnger, [13] I. Caraganns, D. Kurokawa, H. Mouln, A.D. Procacca, N. Shah, and J. Wang. The unreasonable farness of maxmum Nash welfare. In EC, 2016 (to appear). [14] D. Chakrabarty, J. Chuzhoy, and S. Khanna. On allocatng goods to maxmze farness. In FOCS, pages , [15] N. Chen, Xate Deng, Xaomng Sun, and Andrew Yao. Fsher Equlbrum Prce wth a class of Concave Utlty Functons. In ESA, pages ,