Altruism, Selfishness, and Spite in Traffic Routing

Transcription

1 Altruism, Slfishnss, and Spit in Traffic Routing Po-An Chn Univrsity of Southrn California David Kmp Univrsity of Southrn California ABSTRACT In this papr, w study th pric of anarchy of traffic routing, undr th assumption that usrs ar partially altruistic or spitful. W modl such bhavior by positing that th cost prcivd by a usr is a linar combination of th actual latncy of th rout chosn slfish componnt), and th incras in latncy th usr causs for othrs altruistic componnt). W show that if all usrs hav a cofficint of at last β > 0 for th altruistic componnt, thn th pric of anarchy is boundd by /β, for all ntwork topologis, arbitrary commoditis, and arbitrary smi-convx latncy functions. W xtnd this rsult to giv mor prcis bounds on th pric of anarchy for spcific classs of latncy functions, vn for β < 0 modling spitful bhavior. In particular, w show that if all latncy functions ar linar, th pric of anarchy is boundd by 4/3 + 2β β 2 ). W nxt study non-uniform altruism distributions, whr diffrnt usrs may hav diffrnt cofficints β. W prov that all such gams, vn with infinitly many typs of playrs, hav a Nash Equilibrium. W show that if th avrag of th cofficints for th altruistic componnts of all usrs is β, thn th pric of anarchy is boundd by / β, for singl commodity paralll link ntworks, and arbitrary convx latncy functions. In particular, this rsult gnralizs, albit non-constructivly, th Stacklbrg routing rsults of Roughgardn and of Swamy. Mor gnrally, w bound th pric of anarchy basd on th class of allowabl latncy functions, and as a corollary obtain tightr bounds for Stacklbrg routing than a rcnt rsult of Swamy. Catgoris and Subjct Dscriptors: G.2.3 [Mathmatics of Computing]: Discrt Mathmatics Applications Gnral Trms: Economics, Thory Kywords: altruism, spit, slfishnss, routing, anarchy Supportd in part by NSF CAREER award , and NSF grant DDDAS-TMRP Prmission to mak digital or hard copis of all or part of this work for prsonal or classroom us is grantd without f providd that copis ar not mad or distributd for profit or commrcial advantag and that copis bar this notic and th full citation on th first pag. To copy othrwis, to rpublish, to post on srvrs or to rdistribut to lists, rquirs prior spcific prmission and/or a f. EC 08, July 8 2, 2008, Chicago, Illinois, USA. Copyright 2008 ACM /08/07...$ INTRODUCTION On of th most basic and important problms in managing ntworks is to rout traffic so as to mak th latncy xprincd by th avrag usr small. This problm can b solvd ffctivly whn all th traffic submits to th control of a cntral authority. Howvr, nithr in road ntworks nor in larg-scal dcntralizd computr ntworks such as th Intrnt) is it fasibl to stablish such a cntral authority. Rathr, individual usrs of th ntwork hav control ovr th paths thy choos from thir origin to thir dstination. Th prvailing assumption is that usrs will xrt this powr to choos th rout minimizing thir individual latncy, rgardlss of th ffcts that such a choic may hav on othr usrs. A natural qustion is thn how much th avrag latncy incrass as a rsult of such slfish bhavior, compard to a cntral authority balancing th latncis of diffrnt usrs. Th ratio btwn th socially optimal outcom and th outcom of slfish choics has bn trmd Pric of Anarchy PoA) by Koutsoupias and Papadimitriou [23]. Roughgardn and Tardos [36] pionrd th study of th PoA for traffic routing ntworks. Thy analyz a modl proposd by Wardrop and Bckmann t al. [42, 2], in which dgs possss traffic-dpndnt latncy functions. Whn usrs choos a crtain path, thy incras th traffic on all dgs of th path, and thus also th latncy xprincd by all othr usrs sharing th path. This modl of slfishnss assums, in accordanc with much of th gam thory litratur, that usrs choos thir routs compltly without rgard to th dlay that thir choic may caus for othr usrs in th systm. Th assumption of slfishnss as wll as that of rationality) has bn rpatdly qustiond by conomists and psychologists. Tim and again, xprimnts hav shown that vn for simpl gams in controlld nvironmnts, participants do not act slfishly [24, 25]; thir bhavior can b ithr altruistic or malicious. Many xplanations hav bn considrd for this phnomnon, including an innat sns of fairnss [2], rciprocity among agnts [7], or spit and altruism [25]. In this work, w invstigat th qustion whthr and how th Pric of Anarchy in traffic routing will chang if usrs ar assumd to b not ntirly slfish. To this nd, w considr a natural modl of altruism and spit, xtnding on proposd by Ldyard [24, p. 54]. Intuitivly, w want to modl that usrs will trad off th bnfit to thmslvs against th bnfit to othrs. This can b modld by assuming that th utility of ach playr is a linar combination of his own a priori payoff and th payoffs of othr playrs. In

2 th contxt of traffic routing, th prcivd cost of a playr is a linar combination of his own latncy and th incras in latncy th playr causs othrs prcis dfinitions ar givn in Sction 2). By varying th altruism cofficint β, w can smoothly tun th altruism from spitful β = ), through slfish β = 0), to ntirly altruistic β = ). Our first rsult is that if all usrs ar at last) β-altruistic, and β > 0, thn th Pric of Anarchy is always boundd by /β, for all ntworks, arbitrarily many commoditis, and arbitrary smi-convx latncy functions on th dgs. Thus, if a constant amount of altruism is introducd into th systm, thn th PoA is boundd by a constant. A mor gnral vrsion of our rsult charactrizs prcisly th worst-cas PoA for any class of latncy functions; from this gnral rsult, bttr bounds can b obtaind for mor rstrictd classs of functions. Among othrs, our rsult implis a bound of 4 3+2β β 2 on th PoA if all latncy functions ar linar. Th gnral bound also lts us analyz th spit rsistanc of a class of latncy functions: th most spit undr which th PoA would still b finit. W nxt xtnd our rsults byond uniform altruism, and considr arbitrary distributions of altruism among th playrs. In that scnario, vn th xistnc of Nash Equilibria is not obvious; w us a thorm of Mas-Colll [28] to prov that such gams with infinitly many agnts indd hav Nash Equilibria. Evn for singl-commodity flows in arbitrary graphs, prohibitiv lowr bounds on th PoA ar known [3], so w focus hr on paralll link ntworks, studid for instanc by Roughgardn [32]. For paralll link ntworks, w show that for any nonngativ distribution of altruism ovr th usrs in th ntwork with avrag altruism lvl β, th pric of anarchy with convx dg latncy functions is always boundd by / β. In th spcific cas whr th distribution of altruism has only compltly altruistic or compltly slfish usrs, this matchs a bound obtaind with a polynomial-tim algorithm) by Roughgardn [32]. Th bound of / β follows from a mor gnral rsult charactrizing th PoA for arbitrary classs of convx functions. In fact, that mor gnral rsult, whn applid to th cas of a distribution ovr ntirly slfish and ntirly altruistic usrs, implis tightr bounds for Stacklbrg routing compard with a rcnt rsult of Swamy [40]. Finally, w show that for th bound w driv, th worst cas is in fact attaind by th {0, } altruism distribution, whil th bst cas is whn all usrs ar β-altruistic.. Rlatd Work Th study of th inffctivnss of Slfish Routing was pionrd within th thory community by th groundbraking work of Roughgardn and Tardos [36]. It was prcdd by work in th conomics and traffic nginring communitis on congstion modls, traffic routing, and th impact of tolls [29, 42, 2]. Sinc th original papr by Roughgardn and Tardos, a lot of progrss has bn mad on diffrnt aspcts of th problm, including diffrnt objctivs [3], Stacklbrg stratgis in which an altruistic cntral authority controls a fraction of all traffic [22, 32, 40], th impact of tolls or taxs on th infficincy [8, 9, 4, 5, 2], atomic gams whrin usrs control non-infinitsimal amounts of Roughgardn s bound for Stacklbrg routing on paralll link ntworks applis to arbitrary functions, whras ours rquirs convxity. traffic [0, 8, 34], and th ffcts of ntwork structur on th infficincy [27, 30, 35]. For an xcllnt ovrviw of many of ths rsults, s th book by Roughgardn [33]. Among othr things, our rsults draw a connction btwn Stacklbrg stratgis and tolls on usrs, in that th altruistic componnt of a usr s utility can b considrd as a traffic-dpndnt) toll, and ntirly altruistic usrs act as though thy submittd to th control of a bnvolnt authority. A mor dtaild discussions of th connctions and rlatd rsults is givn in Sction 2.2, aftr a formal dfinition of our modl of altruism and spit. Qustions about th accuracy of th assumption that usrs ar slfish and rational hav bn as old as th fild of gam thory s,.g., [24]). Diffrnt modls hav bn proposd to modl usr prfrncs mor accuratly [24, 25]. A modl somwhat similar to ours was rcntly studid in th contxt of contributions to P2P systms by Fldman t al. [3], who positd an intrinsic gnrosity paramtr of usrs, thir willingnss to contribut to th systm. Thy thn study contribution dynamics and thir quilibria, akin to many collctiv bhavior scnarios studid by Schlling [37]. Tradoffs btwn individual optimization and social optimum in th contxt of traffic routing ar also considrd by Jahn t al. [9]. Thy posit that usrs will b willing to incur latncy somwhat xcding a lowst possibl baslin if advisd by a traffic routing systm. Thy xprimntally valuat how cntralizd routing of usrs undr this rstriction compars with unrstrictd cntralizd routing which may plac vry havy burdns on som usrs). Mor rcntly, Babaioff t al. [] studid th impact of spitful bhavior on th outcom of routing gams. In thir modl, thr ar two typs of playrs: slfish rational playrs, and malicious playrs, who sk to maximiz th avrag dlay xprincd by th rational playrs whil not caring about thir own dlay). Thy quantify th impact of malicious playrs on th quilibrium, and show that th pric of anarchy can somtims b incrasd, and in fact dcrasd at othr tims. Th notion of spitful bhavior by individuals, and modls similar to th ons w ar proposing hr, hav rcntly bn studid in th contxt of auctions. For singl-itm auctions, Brandt and Wiss [5] study th bhavior of antisocial agnts, whos utility dcrass in thir comptitors profit. Thy analyz optimum bidding stratgis in a fullinformation stting in this modl. Following up on this work, Brandt t al. [4] study th Baysian stting, and driv symmtric Baysian-Nash quilibria for spitful agnts in firstpric and scond-pric sald bid auctions. Thy show that th xpctd rvnu in scondnd-pric auctions is highr than th xpctd rvnu in first-pric auctions whn all agnts ar nithr compltly slfish nor compltly spitful. Thy also prov that in th prsnc of spit, complt information rducs th rvnu in scond-pric auctions, whil it incrass th rvnu in first-pric auctions. Vtsikas and Jnnings [4] gnraliz som of ths rsults for multi-unit auctions, driving symmtric Bays-Nash quilibria for spitful agnts in both m th and m + ) th pric sald bid auctions. Similarly, Liang and Qi [26] study th ffcts of cooprativ or vindictiv bidding stratgis on th rvnu of sponsord sarch auctions and th xistnc of truthful stratgis and quilibria. Finally, mchanism dsign for spitful agnts in schduling is considrd by Garg t al. [6].

3 2. PRELIMINARIES Our modl is basd on th modl of Wardrop [42], as dscribd by Roughgardn and Tardos [33, 36]. W ar givn a dirctd) graph G = V, E), in which ach dg is quippd with a flow-dpndnt latncy function c x). Th maning is that if th total flow on th dg is x, thn ach usr xprincs a dlay c x) on that dg. W assum that ach c is a continuously diffrntiabl and monoton nondcrasing function. In addition, for som of our rsults, w will assum that ach c is convx, and for othrs that ach c is smi-convx, i.., that x c x) is convx. W assum that usrs/agnts ar non-atomic, i.., infinitsimally small. Thus, w can think of th total traffic as a multi-commodity flow with rats r i btwn sourc-sink pairs s i, t i ), whr th total flow from s i to t i is r i. If f dnots th total flow on dg, thn th total latncy xprincd by a usr on a path P is c P f) := P c f ). Th total latncy xprincd by all usrs is thus Cf) := f c f ). An instanc of th routing problm is thus a tripl G, r, c) whr r and c ar th vctors of flow rats and dg cost functions). Th socially optimum solution for G, r, c) is th flow f minimizing Cf), and thus th solution to th convx program Minimiz subjct to f cf) f is a fasibl multi-commodity flow for G, r, c). Th constraints ar th standard linar multi-commodity flow constraints; th objctiv function is convx so long as ach c is smi-convx. Thus, th optimum can b computd in polynomial tim using convx programming. Slfish usrs do not car about th cost Cf). Thir sol goal is to slct a path P minimizing thir own latncy c P f). As th goals of diffrnt slfish usrs in minimizing thir latncy ar conflicting with ach othr, th traffic routing problm can b considrd a gam, and th outcom of this gam will b a Nash Equilibrium: a multi-commodity flow f such that, givn f, no usr has an incntiv to choos a diffrnt path. Thus, a flow f is at Nash Equilibrium if and only if for ach commodity i, all s i -t i paths P with f P > 0 hav th sam latncy c P f), and all othr s i-t i paths hav at last th sam latncy. Nash Equilibria, too, can b computd as solutions to a convx program: Proposition 2. [33], Proposition 2.6.). Th Nash flows of an instanc G, r, c) ar xactly th solutions to th following convx program, and can thus b computd in polynomial tim. Minimiz subjct to f c 0 t)dt f is a fasibl multi-commodity flow for G, r, c). If f is a flow at Nash Equilibrium, and f th socially optimum flow, thn an intrsting qustion, first invstigatd in dtail by Roughgardn and Tardos [36], is how much largr Cf) can b than Cf ). Th ratio ρg, r, c) := Cf)/Cf ) is calld th pric of anarchy of th instanc G, r, c). Roughgardn and Tardos [36] gav a gnralization of Pigou s xampl [29], showing that if th cost functions can b arbitrary, thn th pric of anarchy is unboundd, vn for ntworks consisting of two nods and two paralll links. On th othr hand, thy provd that if all functions ar linar c x) = a x + b, thn th pric of anarchy is at most 4/3. 2. Altruism and Spit Th assumption that usrs ar ntirly slfish is simplistic, and not warrantd in many scnarios. Indd, xprimnts in conomics hav found tim and again that usrs bhav nithr rationally nor slfishly, vn in th absnc of prsonal intraction or rpatd xprimnts [24, 25]. Diffrnt modls of such bhavior hav bn proposd, including basd on rciprocity [7], an innat notion of fairnss [2], or altruism and spit [25]. W bas our tratmnt on a simpl and lgant suggstion of Ldyard [24]. In a gam with n playrs, th utility of a playr i givn an action vctor a is p i a) + β i n j p ja), whr th p i ar th individuals payoff functions. Th paramtr β i capturs how important th avrag social wlfar is to playr i. W modify this approach slightly, and posit that usr i s utility is th combination β i )p i a) + β i n j p ja), whr β i [, ] is th usr s altruism lvl. This has th advantag of making all utilitis comparabl on th sam scal, and allowing us to modl ntirly altruistic bhavior by stting β i =. 2 W call p i a) th slfish part of playr i s utility, and n j p ja) th altruistic part. If β i < 0, thn playr i drivs utility from a dcras in social utility; w call such playrs spitful. In ordr to apply this modl to our scnario of traffic routing, w dfin th payoff of usr i on path P as p i = c P f), whr f is th total flow, dtrmind by th actions of all othr playrs. Thn, maximizing utility is quivalnt to minimizing latncy. Th traffic routing modl assums that thr ar infinitly many usrs, ach of whom is infinitsimally small. W can still dfin th utility function analogously, using th wll-dfind) avrag latncy of all usrs as th altruistic part. Howvr, bcaus usrs ar infinitsimally small and latncy functions continuous, th avrag latncy of othr usrs will not dpnd on an individual usr s action. Thus, as long as β, ach partially altruistic usr will act xactly lik a slfish usr. A natural modl considring th ffct th usr has on othrs should instad b basd on th rat at which th usr s action will affct othr usrs. W thus us th following dfinition of a β-altruistic usr 3 : Dfinition 2.2. Each β-altruistic usr for β [, ]) chooss a path P so as to minimiz th cost function c β) P f) := β) P c f ) + β P f c f )). Th trm P c f ) is th slfish part of th cost, whil P fcf)) is th altruistic part. f c f )) dnots th drivativ with rspct to f. Notic that w can rwrit c β) P f) = P cf) + β P fc f ). Dfinition 2.2 is similar to th dfinition of th valuation of a usr with a tim/mony tradoff of β in th cas of ntwork routing with tolls [8]. Howvr, notic that unlik th standard modl for tolls, th dg toll τ a usr incurs in our modl is traffic-dpndnt, namly τ := f c f ). W say that th usrs ar uniformly β-altruistic if all usrs ar β-altruistic. Mor gnrally, w allow for th cas of arbitrary distributions of altruism among th usrs. In th 2 Th rstriction to valus β i is justifid in Sction 3. 3 Whil our dfinition is motivatd mathmatically, thr is a psychological intrprtation of th undrlying choic: In ordr to bhav partially) altruistically, infinitsimally small usrs must giv infinitsimally small wight to thir own payoff, which is achivd implicitly by making th altruistic componnt th drivativ of th social wlfar.

4 gnral cas, for ach commodity i, w ar givn an arbitrary altruism dnsity function ψ i on th intrval [, ]. W only rquir that all ths functions ψ i b indd distributions, i.., forming a Borl masur of total masur. If th rat for commodity i is r i, thn th ovrall altruism dnsity function is ψ = i r i i r iψ i. Th avrag altruism of a distribution ψ is thn tψt)dt. An instanc of th partially altruistic traffic routing problm is thus th quadrupl G, r, c, ψ i)). If thr is a singl commodity with distribution ψ, w writ G, r, c, ψ), and if th altruism is uniform, w simplify furthr to G, r, c, β). Proposition 2.3. Lt G, r, c, β) b an instanc with uniform altruism β 0. Thn, th Nash flows ar th optima of th convx program f c β) 0 Minimiz t)dt subjct to f is a fasibl multi-commodity flow for G, r) In particular, th instanc G, r, c, β) always posssss a Nash Equilibrium for β 0. Th proof of this proposition is virtually idntical to that of Proposition 2.6. from [33]. Th proof thr only usd th fact that ach agnt was minimizing a sum of monoton incrasing functions gf) to conclud that th Nash Equilibrium was th flow minimizing th convx) objctiv f g 0 t)dt. Thus, it applis qually to g t) := c β) t). Th situation is not quit as straightforward for th cas of non-uniform altruism distributions ψ, or for ngativ β. Evn for two diffrnt valus of altruism, thr appars to b no natural convx programming formulation for Nash Equilibria. Howvr, using a thorm of Mas-Colll [28], w can still prov th xistnc of Nash Equilibria. Thorm 2.4. Each instanc G, r, c, ψ i )) has a Nash Equilibrium. Proof. Thorm of Mas-Colll [28] provs that ach gam of infinitly many playrs has a Nash Equilibrium. A gam is charactrizd by a distribution Borl masur) ovr utility functions which ar continuous in th action of th playr, and th distribution of actions by th rmaining playrs. It is asy to s that ach playr in th routing gam has a utility function c β) P f) continuous in th choic of path P trivially, sinc th spac of all simpl s i-t i paths is finit) and in th distribution of othr playrs stratgis f by continuity of ach c ). Th utility for paths not conncting s i to t i is or an appropriatly ngativ constant). Th distribution of altruism valus β implis a corrsponding distribution ovr utility functions. Thus, th thorm of Mas-Colll implis th xistnc of Nash Equilibria for routing gams. Th proof by Mas-Colll is inhrntly non-constructiv; accordingly, Thorm 2.4 dos not imply any algorithm for finding such quilibria. Sinc thr always xists a Nash Equilibrium of instancs G, r, c, ψ i)), w can again dfin th Pric of Anarchy PoA), as ρg, r, c, ψ i)) = Cf)/Cf ), whr f is a Nash flow for G, r, c, ψ i )), and f a socially optimal flow for G, r, c). 2.2 Taxs and Stacklbrg Stratgis Our dfinition of partial altruism naturally rlats to two stratgis that hav bn proposd in th litratur for daling with th slfishnss of usrs: Pigou taxs and Stacklbrg stratgis. Th ida of taxs or tolls on dgs is to charg usrs a f for using an dg. Th assumption is that mony and latncy can b masurd on th sam scal, and usrs will minimiz th wightd) sum of th two. It is wll-known [29] that if th toll chargd on ach dg quals th marginal cost to othrs f c f )) at th optimum solution, thn th Nash Equilibrium will minimiz Cf), i.., b optimal. Our modl of partial altruism can thus b intrprtd as charging usrs a traffic-dpndnt constant fraction of th marginal tax, i.., with rspct to th currnt flow. Whn th altruism is not uniform, diffrnt usrs will b chargd diffrnt taxs β i f c f ) on dgs. Our modl can thus b considrd as invstigating th pric of anarchy whn diffrnt usrs hav diffrnt tradoffs btwn taxs and latncy, but thir tradoff stays constant across diffrnt dgs. Similar modls wr considrd,.g., in [, 39]. Col t al. [9, 8] also study optimization problms arising from non-uniform taxation in ntworks. Howvr, thir goal is to minimiz th total tolls, subjct to forcing th flow to optimal, whras w study th pric of anarchy givn th taxation schm of charging a usr-dpndnt) fraction of th marginal tax on ach dg. A diffrnt stratgy for lowring th pric of anarchy is availabl whn a bnvolnt cntral authority controls a λ fraction of th total traffic. Th cntral authority s goal is to rout this fraction so as to minimiz th total cost Cf), subjct to th fact that th rmaining usrs will subsquntly rout thir traffic slfishly. Algorithms for routing flows with this objctiv ar calld Stacklbrg stratgis, and th corrsponding asymmtric gams Stacklbrg gams s,.g., [32]). Whn th altruism distribution has support {0, }, and th cumulativ distribution function of ψ is th stp function whos valu at 0 is λ, and whos valu at is, th altruistic usrs can b intrprtd as a cntral authority, and thir flow as a Stacklbrg stratgy with th corrsponding pric of anarchy. Whn th cntral authority controls a λ fraction of th traffic, thn th avrag altruism is xactly λ, and thus, any bound on th pric of anarchy for avrag altruism λ givs ris to th sam bound for Stacklbrg routing. Notic that th convrs is not ncssarily tru: at th momnt, it is not known if vry optimal Stacklbrg stratgy givs ris to a Nash Equilibrium of th routing gam with altruism support {0, }. Such Stacklbrg routing stratgis hav bn studid xtnsivly. In gnral, th pric of anarchy can still b unboundd, vn for singl-commodity flows whr a cntral authority controls a larg constant fraction of th traffic [3]. For linar latncy functions, Karakostas and Kolliopoulos [22] rcntly showd an uppr bound of 4 X)/3 on th Pric of Anarchy whr X = λ)3 λ+) 2 λ+ ) for arbitrary ntworks and commoditis in which a cntral authority controls a λ fraction of traffic. For arbitrary latncy functions in sris-paralll ntworks, Swamy [40] bounds th pric of anarchy by + /λ. For paralll link ntworks with latncy functions from a class C with an uppr bound ρc) on th pric of anarchy in Pigou xampls, h shows an up-

5 pr bound of λ + λ)ρc). In th contxt of Stacklbrg routing, a convrs dirction has bn studid by Sharma and Williamson [38] and Kaporis and Spirakis [20]. Thy ask how much traffic nds to b controlld by a cntral authority to guarant any improvmnt in avrag latncy [38] calld Stacklbrg thrshold) or to guarant optimality of th rsulting Nash Equilibrium [20] calld Pric of Optimum). 3. UNIFORM ALTRUISM In this sction, w focus on th modl of uniformly altruistic usrs: ach usr is β-altruistic for β. Thus, th prcivd cost of an dg to th usr is c β) x) = β)c x) + β d xcx)) = cx) + dx βxc x). Notic that for β = 0, this coincids with slfishnss; β = corrsponds to complt altruism, and c ) x) is xactly th marginal cost of. For β =, th usrs ar compltly spitful.) Our first rsult follows dirctly from th dfinitions of flows at Nash quilibrium and optimum, and givs a tight) uppr bound on th Pric of Anarchy for arbitrary ntworks, commoditis, and arbitrary smi-convx cost functions. 4 Proposition 3.. If all cost functions c ar nondcrasing and smi-convx, thn for all ntworks G and flow rats r, and any altruism lvl β 0, ], ρg, r, c, β) /β. Proof. Lt ˆf b a Nash Equilibrium flow, minimizing th potntial function Φf) = f c β) 0 t)dt, th objctiv function of th convx program in Proposition 2.3. Also, lt f th optimum flow, minimizing th total cost Cf) = f tc 0 t)) dt. Simply from th dfinition of c β) t), it follows that for any flow f, w hav Φf) Cf) Φf). β Applying th first inquality to f and th scond to ˆf, and using th optimality of ˆf for Φ, w obtain C ˆf) β Φ ˆf) β Φf ) β Cf ). Mor gnrally, w driv a rsult bounding th pric of anarchy whn all cost functions c ar drawn from a givn class of cost functions. Our charactrization will b in trms of th anarchy valu α β) C) of a st C of functions for β- altruistic usrs, which is dfind as a gnralization of th anarchy valu of functions in [33]. Dfinition For any cost function c, th anarchy valu α β) c) of c for β-altruistic usrs is dfind as α β) c) = sup r,x 0 r x cx)+r x) c β) r), whr 0/0 is dfind to. 2. For any class C of cost functions, th anarchy valu for β-altruistic usrs α β) C) is sup c C,c 0 α β) c). Th motivation for this dfinition of α β) c) is that it capturs th pric of anarchy for uniformly β-altruistic usrs in a two-nod two-link ntwork, whr on link has latncy function c and th othr has a worst-cas constant. Indd, w will prov this to b th cas in Lmma 3.7 blow. Notic that Lmma 3.7 immdiatly implis that αc) is a lowr 4 W thank an anonymous rviwr for th simplifid proof. bound on th pric of anarchy in th worst cas whn all dg latncy functions ar chosn from C. Our main thorm in this sction shows that it is also an uppr bound for all ntworks and arbitrary commoditis. W ar mostly intrstd in α β) C) whn it is finit. In particular, this suggsts dfining th spit rsistanc of C as th last altruistic bhavior that C could support. Formally, b c = inf{β α β) c) < }, and b C = inf c C b c. It is not difficult to show that b c = inf r, and that rc r) αβ) c) = for β b c. Using L Hôpital s rul, on ss that th monotonicity and convxity of c imply that b c lim r rc r) for all c, which also motivats our arlir rstriction to altruism valus β. Thorm 3.3. Lt C b a st of cost functions, and G, r, c) an instanc with cost functions c C. Thn, ρg, r, c, β) α β) C). Proof. Fix an instanc G, r, c) with cost functions c C. Lt f b an optimal flow and f a Nash flow for β-altruistic usrs. By rarranging Dfinition 3.2, w obtain th bound x c x) r c r) α β) C) +x r) cβ) r) for any x, r 0. Applying this bound to ach dg, with x = f and r = f, w bound Cf ) = E f c f ) α β) C) E fcf) + E f f ) c β) f ) Cf) = + α β) C) E f f ) c β) f ). It rmains to show that f c β) f ) f c β) f ). To this nd, rcall that f is a Nash flow for β-altruistic usrs if and only if it minimizs c β) P f) f P ovr all fasibl flows f. In particular, applying this variational inquality to f and f provs th dsird inquality. As a corollary of Thorm 3.3, w can obtain a tight bound in th cas whr th cost functions ar polynomials of dgr at most p with non-ngativ cofficints. W dnot this class by C p. Thorm 3.4. If G, r, c) has cost functions in C p, thn for any altruism valu β /p, ], ) ρg, r, c, β) +βp +p )/p +βp βp) + + βp +p. Proof. First, notic that b Cp = /p. It can b asily vrifid that all subsqunt calculations stay valid for β > /p, whil for β /p, th pric of anarchy is unboundd. As obsrvd in [33], it suffics to focus only on polynomials cx) = ax i with x p. For any instanc G, r, c) with arbitrary polynomials can b quivalntly transformd into on with only such monomials, by rplacing ach dg with cost function c x) = p i=0 a ix i by a dirctd path of p + dgs, th i th dg of which has cost function c,ix) = a ix i. In ordr to comput th anarchy valu αc) of a nonzro polynomial function cx) = ax i, w us th quivalnt charactrization that α β) c) = sup r 0 λcλr) + λ) + βrc r) )), whr λ [0, ] solvs c ) λr) = c β) r), and 0/0 is dfind to. To prov this quivalnt charactrization, w first obsrv that d dλ cλr)λr + cβ) r)r λr)) = c ) λr)r r = 0,

6 so thr is indd a valu of λ [0, ] solving c ) λr) = c β) r). By Lmma 3.7 blow, α β) c) is th pric of anarchy in a two-nod two-link ntwork, on of whos links has th cost function cx), th othr link having constant cost c β) r). Routing λr units of flow on th link with cost cx), and th rst on th link with cost c β) r), provids an optimal flow, whil th Nash Equilibrium has all of its flow on th link with cost cx). Thus, th ratio of th cost of a Nash flow to that of an optimal flow is = r cλr)λr++βrc r))r λr) λcλr) + λ) + βrc r) )). Solving for λ in th spcial cas cx) = ax i, w obtain λ = +βi +i )/i, and thus cλr) = +βi and c r) = i. Thn, +i r α β) c) = +βi +i )/i +βi βi) + + βi) +i, which is indpndnt of a and incrasing in i by a drivativ tst). Hnc, th largst αc) is attaind for c = x p, giving ) α β) C p ) = +βp +p )/p +βp βp) + + βp +p, as claimd. It is not difficult to vrify that th prvious bound convrgs to as p ; th worst cas bhavior is in fact attaind with polynomials of high dgr. Howvr, for p =, β Thorm 3.4 also allows us to obtain a tightr bound in th spcial cas that all latncy functions ar linar. Corollary 3.5. If G, r, c) has linar cost functions, thn for any β, ], ρg, r, c, β) 4 3+2β β 2. Notic that for any β > 0, this bound improvs on th bound by Roughgardn and Tardos [36] of 4/3 whn all usrs ar compltly slfish. As th bound can also b shown to b tight, it thus charactrizs xactly th gain by partial positiv altruism with linar cost functions, and th spit rsistanc of linar cost functions. In particular, it shows that linar costs hav th highst spit rsistanc among all classs of cost functions. Rmark 3.6. Our rsults in this sction xtnd straightforwardly to gnral non-atomic congstion gams not ncssarily ntwork congstion gams), so long as all cost functions ar nondcrasing. S, for instanc, [7].) In a gnral congstion gam, ach playr s stratgy consists of a st of rsourcs, and th cost of th stratgy dpnds simply on th numbr of playrs using ach rsourc. Thus, th prcivd cost of a playr s stratgy S with altruism β is c β) S) = S cβ) x ) = S cx) + βxc x ), whr x is th total masur of playrs using rsourc, and c is a nondcrasing function. With th sam dfinitions of α β) C), th proofs of th abov proposition and thorms naturally carry ovr to this mor gnral stting. Finally, w show that th bounds drivd in Thorm 3.3 ar indd tight, vn for two-nod two-link ntworks: Lmma 3.7. Considr a two-nod two-link ntwork with flow rat r =, and cost functions c x) = cx) on th first link, and constant cost function c 2 x) = c β) r) = + βrc r) for th scond link. Thn, th pric of anarchy of this instanc is α β) c). Proof. It is asy to obsrv from th dfinition of c 2 that all β-altruistic usrs will nd up using link, so that th total cost of th Nash Equilibrium is c) = r, whil th socially optimum solution has total cost inf x x cx) + r x) + βr x)c r)). Hnc, th pric of anarchy is xactly αc). By applying this charactrization togthr with Thorm 3.4 and ltting th dgr of th polynomial go to, w obtain instancs G, r, c) whos pric of anarchy approachs /β arbitrarily closly. Similarly, by choosing p =, w obtain that th bound in Corollary 3.5 is tight. 4. NON-UNIFORM ALTRUISM In this sction, w xtnd our rsults to th mor gnral and ralistic cas whr diffrnt usrs can hav diffrnt altruism lvls. In th most gnral cas, w ar givn a distribution ψ of altruism. Th xistnc of Nash Equilibria in this modl was shown non-constructivly as Thorm 2.4. Evn for a singl commodity and an altruism distribution with support {0, }, and arbitrarily larg constant β, a rcnt rsult on Stacklbrg routing du to Bonifaci t al. [3] shows that th pric of anarchy can bcom unboundd. Thus, w focus hr on th cas of singl-commodity traffic in paralll link ntworks. Paralll link ntworks hav bn studid by Roughgardn [32]; among othrs, thy naturally modl th assignmnt of infinitsimally small jobs to machins with load-dpndnt latncis. Formally, a paralll link ntwork has two nods s, t, and m paralll dgs,..., m from s to t. Our main thorm in this sction givs a tight) uppr bound on th Pric of Anarchy in th prsnc of partial altruism for singl commodity paralll link ntworks and arbitrary convx) cost functions. Thorm 4.. If all cost functions c ar convx and nondcrasing, thn for paralll link ntworks G and flow rats r, and any ovrall altruism dnsity function ψ with nonngativ support and avrag altruism β, ρg, r, c, ψ) / β. W will prov Thorm 4. as a corollary of th following mor gnral rsult, bounding th pric of anarchy in trms of th st of functions prmissibl as dg latncis. Thorm 4.2. If all cost functions c ar convx and nondcrasing, thn for paralll link ntworks G and flow rats r, and any ovrall altruism dnsity function ψ with nonngativ support, ) ρg, r, c, ψ) ψt) dt 0 α t) C). Proof. Lt f dnot th flow at Nash Equilibrium. W first show that without loss of gnrality, w can assum that ach link contains only on typ of usrs i.., if usrs hav diffrnt altruism valus β, β, thn thy do not shar a link) and that th support of ψ is finit. To s this, assum that f has usrs of altruism valus β < β sharing an dg. Now rplac all usrs on with altruism β by usrs with altruism β. f must still b a flow at Nash Equilibrium for

7 th nw instanc bcaus β -altruistic usrs ar on link in Nash Equilibrium). By rpating this procss, w vntually obtain an instanc with altruism dnsity ψ which stochastically dominats ψ and has finit support. For this nw ψ, th bound on th pric of anarchy for f providd by th right-hand sid of Thorm 4.2 can only b smallr, giving us an vn bttr bound than rquird. Thus, w can from now on focus on th cas dscribd abov. Lt 0 β < β 2 <... < β k b th finit) support of ψ, whr th rat of β i -altruistic usrs is r i so k i= r i = r). W nd to show that for all flows g of rat r in particular th optimum flow), w hav Cg) k i= r i r ) Cf), ) α β i) C) which w will do by induction on k. Th bas cas k = 0 is of cours trivial. For th inductiv stp, lt f b a Nash Equilibrium flow, and g any flow of rat r. For ach i, lt E i b th st of dgs with positiv flow of β i-altruistic usrs undr f. Notic that by our assumption, th sts E i ar disjoint. For any st E of dgs, lt fe ) = E f similarly, ge )) dnot th total flow on E. Lt E := E \E dnot th st of all dgs not usd by β -altruistic usrs. Intuitivly, bcaus th mor altruistic usrs prfr th dg st E ovr E, w would xpct a good flow g to do th sam. Indd, w first show that th latncy undr f on all dgs in E is no largr than in E, whil th drivativ is no largr in E than in E. Lt E, E j, j > b arbitrary links with positiv flow f. Thus, all usrs on hav altruism β, whil all usrs on hav altruism β j > β. Bcaus f is at Nash Equilibrium, c f ) + β f c f ) c f ) + β f c f ), 2) c f ) + β jf c f ) c f ) + β jf c f ). 3) Combining appropriatly scald vrsions of Inquality 2) and Inquality 3) givs us that c f ) c f ), 4) f c f ) f c f ). 5) Our high-lvl stratgy will b to bound th Nash Equilibrium flow on E against a rstriction g of g of rat r r on E by induction, and us a comparison argumnt for th flow on E. W will construct a flow h of rat r whos cost is chapr than a componnt of g of th sam rat, and which is optimal for modifid rsidual dg costs. W can thus compar it against th flow f on E using Thorm 3.3. Dfin f to b th rstriction of f to th dg st E, i.., f = f for E, and f = 0 for E. Thus, f is a flow of rat r := r r. Dfin th modifid cost function c x) := c f + x) + β f c f ) for all dgs. Thus, c x) is th cost incurrd by flow on if f is unaltrabl, but not considrd part of th actual flow, plus a suitabl constant trm to mimic th altruistic componnt. This dfinition of c x) implis that th prcivd cost of dg to β -altruistic usrs is c β ) x) = c f + x) + β xc f + x) + β f c f ). Thus, for E, w hav that c β ) x) c β) f ) for all x 0, whil for E, bcaus f = 0, c β ) x) = c β) x + f ). In particular, this implis that th β -altruistic usrs ar at Nash Equilibrium with rspct to th modifid cost functions c x). Hnc, by Thorm 3.3, and bcaus c x) = c x) for all E, w gt Cf f ) = Cf f ) α β) C) C f) whr f is an optimum flow of rat r with rspct to th modifid dg cost functions c. In ordr to compar f against th part of g on th dg st E, it will b usful to assum that ge ) fe ). W will show nxt that w can mak this assumption w.l.o.g. For assum that it did not hold. Thn, lt E, E b dgs with g) > f) > 0 and g ) < f ). Th xistnc of, follows from th assumption ge ) < fe )). By th bound on th drivativs in Inquality 5), and using th convxity of th dg latncy functions, w obtain that g c g ) f c f ) f c f ) g c g ). Thus, g can b mad chapr by moving som of its flow from to. By rpating this procss, w can thus assum that ge ) fe ). Lt γ b such that Cf f ) = γcf). Bcaus f and f f us disjoint dg sts, w gt Cf ) = γ)cf). Notic that th assumption of disjoint dg sts is indd crucial hr. Du to th non-constant cost of dgs, in gnral, it dos not hold that Cf) + Cf ) = Cf + f ).) By Lmma 4.3 blow, w can dcompos g = h+g, whr g is a flow of rat r ntirly on E, and h is a flow of rat r satisfying th proprty 7), namly C f) hcg) + g c g ) c g )). W can thus apply induction on th flows f and g of rat r on th modifid graph with dg st E. Notic that whil f may not b an Equilibrium flow on E, it is indd an Equilibrium flow on E. Thus, w obtain that Cg) = Cg ) + = k i=2 h)c g ) + g c g ) c g )) r i r α β i) C) ) Cf ) + α β ) C) Cf f ) 6) k i=2 r i r α β i) C) ) γ) + α β ) C) γ ) Cf). r i r α β i ) C) α β ) C) W nxt show that γ r r. By Inquality 4), vry usr on E incurs lowr dlay than vry usr on E j, and consquntly on E. Thus, th avrag dlay r Cf f ) of usrs on E is at most th avrag dlay Cf) of all usrs, r so Cf f ) r r Cf). Th lowr bound 6) is a convx combination of th nonngativ trms k i=2 and, with cofficints γ) and γ. Th anarchy valu α β) C) is a monoton non-incrasing function of β, so th wightd avrag rciprocal anarchy valu for altruism lvls β 2,..., β k is at last th rciprocal for β. Thus, th convx combination is minimizd whn th cofficint γ of th smallr trm α β ) C) is as larg as possibl, i.., whn γ = r /r. Substituting this bound, Cg) k = k i= r i i=2 r r + ) r r α β i ) C) α β ) C) r ) Cf) r i ) Cf), r α β i ) C) complting th inductiv stp, and thus th proof. Lmma 4.3. Lt f b a flow of rat r using only dgs from E, and dfin c x) := c f + x) + β f c f ). Lt g b any flow of rat r = r + r, with ge ) r. Lt f b th optimum flow of rat r with rspct to dg costs c. Thn, g can b dcomposd as g = h + g, whr g is a flow of rat

8 r on E, satisfying C f) h c g ) + g c g ) c g )). 7) Proof. Lt := ge ) r 0 b th amount of xcss flow that g snds on E, compard to f. W bgin by stting h = g for all dgs E, giving us a flow of rat r. So w nd to add mor units of flow to h. Lt E := { E g f } b th st of dgs in E on which g snds mor flow than f. Thus, w hav that ge ) f E ) ge ) f E ) =. In particular, w can dfin a flow h of total rat on E, such that h g f for all E. For all othr dgs, w st h = 0, and thus obtain a flow h of rat r, such that h g for all dgs. W thn hav that hcg) = E h c h ) + E h c g ) E h c h ) + E h c f + h ), whr th inquality follows from th monotonicity of th latncis c. Nxt, bcaus g f for all E, and th c latncy functions ar convx, g ) c g ) h c f ) for all E with h > 0. Combining this bound with th fact that β, w obtain that g c g ) c g )) E g c g ) c g )) E f β h c f ). Summing th prvious two inqualitis now givs us h c g ) + g c g ) c g )) E h c h ) + E h c f + h ) + E h β f c f ) = h c h ) C f) whr th final inquality follows from th optimality of f with rspct to th cost functions c. Proof of Thorm 4.. If C is spcifically th st of all incrasing smi-convx functions, Proposition 3. implis that α t) C) t. Substituting this bound into th intgral givs us that ρg, r, c, ψ) ) 0 = / β. It would of cours b dsirabl to xtnd Thorms 4. and 4.2 to distributions including ngativ support. Howvr, such an xtnsion is in gnral not possibl. On can construct scnarios in which almost all of th latncy is incurrd by a small fraction of spitful usrs who togthr congst a link with vry stp incras. At th sam tim, all altruistic usrs us links with vry small constant latncy. Thn, th PoA is much largr than, whil th bounds of both thorms would rquir it to b clos to. An immdiat corollary of Thorm 4. can b obtaind by choosing th distribution with a rat of λ usrs bing compltly altruistic, and λ usrs bing compltly slfish. Sinc β = λ for this distribution, Thorm 4. immdiatly implis Corollary 4.4. In paralll link ntworks, th pric of anarchy undr Stacklbrg routing with a λ-fraction of traffic bing controlld by a cntral authority is at most /λ. This rsult was of cours alrady provd constructivly and giving fficint algorithms) by Roughgardn [32]; nvrthlss, it is intrsting that it follows dirctly from our gnral rsult. Mor gnrally, by using th sam distribution with support {0, } in Thorm 4.2, w obtain th following corollary: Corollary 4.5. In paralll link ntworks, th pric of anarchy undr Stacklbrg routing with a λ-fraction of traffic controlld by a cntral authority is at most λ αc) + λ). Notic that Corollary 4.5 improvs albit in a non-constructiv way) a rcnt rsult of Swamy [40] for Stacklbrg routing: W bound th PoA undr Stacklbrg routing by th wightd harmonic man of th PoA for slfish and altruistic usrs, whras Swamy s bounds giv th arithmtic man. It is known that th harmonic man is always boundd abov by th arithmtic man. W can also show that th cas of Stacklbrg routing is in fact th worst cas for th bound of Thorm 4.2, in th sns that th right-hand sid is maximizd. Whil th bound of Thorm 4.2 will in gnral not b tight, this nvrthlss givs ris to th philosophical intrprtation that, conditiond on a givn avrag altruism lvl β, th scnario in which compltly altruistic usrs or a cntral authority compnsat for compltly slfish usrs is th worst cas, whil uniform altruism through th population is th bst cas. Proposition 4.6. Conditiond on th man of ψ bing any givn β, ) th quantity ψt) dt 0 α t) C) is maximizd whn ψ has point mass of β on and β on 0. It is minimizd whn ψ has a point mass of on β. Proof. W will show that is concav as a function α β) C) of β. Both rsults thn follow radily from Jnsn s Inquality. To prov concavity, lt p, p 2 0 satisfy p + p 2 =. For any cost function c C, Dfinition 3.2 thus givs us α p β +p 2 β 2 ) c) λcλr)+ λ)+ λ)p = inf β +p 2 β 2 )c r) λ p = inf λcλr)+ λ)+ λ)β c r)) λ + p 2 λcλr)+ λ)+ λ)β 2 c r)) p inf λcλr)+ λ)+ λ)β c r)) λ + inf λ p λcλr)+ λ)+ λ)β 2 c r)) = p α β ) c) + p2 α β 2 ) c). Finally, w tak an infimum ovr all c C on both sids to complt th proof of concavity. 5. CONCLUSIONS In this papr, w proposd a simpl modl of altruism and spit in traffic routing, whr usrs utilitis or prcivd costs ar linar combinations of thir own latncy and th incras thy caus in othr usrs latncis. W provd a /β bound on th pric of anarchy vn for worst-cas ntworks, latncy functions, and commoditis, undr th assumption that all usrs ar at last) β-altruistic, and β > 0. W xtndd this rsult to non-uniform altruism distributions for singl-commodity flows in paralll link ntworks. Among othrs, this rsult rcovrs and improvs rcnt bounds on Stacklbrg routing by Roughgardn and by Swamy.

9 Our work suggsts many intrsting dirctions for furthr rsarch. First, th rsults should b gnralizd to mor gnral) ntwork topologis instad of paralll links. Notic that any such rsult would immdiatly imply corrsponding bounds on Stacklbrg routing, so th lowr bound of Bonifaci t al. [3] prcluds an xtnsion to arbitrary singlcommodity flows. Howvr, an xtnsion to sris-paralll graphs sms plausibl at this point. Whil w provd th xistnc of Nash Equilibria for all routing gams with non-atomic usrs, rgardlss of th distributions of altruism, th proof is non-constructiv. Th work of Roughgardn [32] implis that finding th bst Stacklbrg stratgy is NP-complt. Howvr, it would b intrsting whthr Stacklbrg stratgis mting our bound can always b found fficintly. Altrnativly, in light of rcnt rsults proving that finding Nash Equilibria is PPADcomplt [6], it may b possibl that finding Nash Equilibria for traffic routing gams with two or mor) altruism valus is also PPAD-complt. Finally, th study of partially altruistic and spitful bhavior can and should b xtndd byond th ralm of traffic routing. Svral natural gams studid in th CS community ar known to hav unboundd or larg PoA undr th assumption of slfish agnts. It would b intrsting to obsrv if th introduction of finit amounts of altruism uniformly or not) into such gams will also lad to a constant or othrwis rducd PoA. This holds in particular bcaus partial altruism appars to b a mor natural modl of actual usr bhavior. Acknowldgmnts W would lik to thank Tim Roughgardn and Pirr-Olivir Will for usful discussions and rfrncs, and svral anonymous rfrs for dtaild and usful commnts. 6. REFERENCES [] M. Babaioff, R. Klinbrg, and C.Papadimitriou. Congstion gams with malicious playrs. In Proc. 9th ACM Conf. on Elctronic Commrc, [2] M. Bckmann, C. McGuir, and C. Winstn. Studis in th Economics of Transportation. Yal Univrsity Prss, 956. [3] V. Bonifaci, T. Harks, and G. Schäfr. Th impact of stacklbrg routing in gnral ntworks. Tchnical Rport COGA Prprint , TU Brlin, [4] F. Brandt, T. Sandholm, and Y. Shoham. Spitful bidding in sald-bid auctions. In Proc. 20th Intl. Joint Conf. on Artificial Intllignc, pags , [5] F. Brandt and G. Wiss. Antisocial agnts and vickry auctions. In Proc. 8th Workshop on Agnt Thoris, Architcturs and Languags, pags , 200. [6] X. Chn and X. Dng. Sttling th complxity of two-playr nash-quilibrium. In Proc. 47th IEEE Symp. on Foundations of Computr Scinc, pags , [7] S. Chin and A. Sinclair. Convrgnc to approximat nash quilibria in congstion gams. In Proc. 8th ACM Symp. on Discrt Algorithms, [8] R. Col, Y. Dodis, and T. Roughgardn. Pricing ntwork dgs for htrognous slfish usrs. In Proc. 35th ACM Symp. on Thory of Computing, pags , [9] R. Col, Y. Dodis, and T. Roughgardn. How much can taxs hlp slfish routing? Journal of Computr and Systm Scincs, 72: , [0] R. Comintti, J. Corra, and N. Stir-Moss. Ntwork gams with atomic playrs. In Proc. 33rd Intl. Colloq. on Automata, Languags and Programming, pags , [] S. Dafrmos. Th traffic assignmnt problm for multiclass-usr transportation ntworks. Transportation Scinc, 6:73 87, 972. [2] E. Fhr and K. Schmidt. A thory of fairnss, comptition, and coopration. Th Quartrly Journal of Economics, 4:87 868, 999. [3] M. Fldman, C. Papadimitriou, J. Chuang, and I. Stoica. Fr-riding and whitwashing in pr-to-pr systms. In Proc. ACM SIGCOMM 04 Workshop on Practic and Thory of Incntivs in Ntworkd Systms PINS), [4] L. Flischr. Linar tolls suffic: Nw bounds and algorithms for tolls in singl sourc ntworks. Thortical Computr Scinc, 348:27 225, [5] L. Flischr, K. Jain, and M. Mahdian. Tolls for htrognous slfish usrs in multicommodity ntworks and gnralizd congstion gams. In Proc. 45th IEEE Symp. on Foundations of Computr Scinc, pags , [6] N. Garg, D. Grosu, and V. Chaudhary. An antisocial stratgy for schduling mchanisms. In Proc. 9th IEEE Intl. Paralll and Distributd Procssing Symp., [7] H. Gintis, S. Bowls, R. Boyd, and E. Fhr. Moral Sntimnts and Matrial Intrsts : Th Foundations of Coopration in Economic Lif. MIT Prss, [8] A. Hayraptyan, E. Tardos, and T. Wxlr. Th ffct of collusion in congstion gams. In Proc. 37th ACM Symp. on Thory of Computing, pags 89 98, [9] O. Jahn, R. Möhring, A. Schulz, and N. Stir-Moss. Systm-optimal routing of traffic flows with usr constraints in ntworks with congstion. Oprations Rsarch, 53:600 66, [20] A. Kaporis and P. Spirakis. Th pric of optimum in stacklbrg gams on arbitrary singl commodity ntworks and latncy functions. In Proc. 9th ACM Symp. on Paralll Algorithms and Architcturs, pags 9 28, [2] G. Karakostas and S. Kolliopoulos. Edg pricing of multicommodity ntworks for htrognous usrs. In Proc. 45th IEEE Symp. on Foundations of Computr Scinc, pags , [22] G. Karakostas and S. Kolliopoulos. Stacklbrg stratgis for slfish routing in gnral multicommodity ntworks. Tchnical Rport 2006/08, McMastr Univrsity, [23] E. Koutsoupias and C. Papadimitriou. Worst-cas quilibria. In Proc. 7th Annual Symp. on Thortical Aspcts of Computr Scinc. Springr, 999. [24] J. Ldyard. Public goods: A survy of xprimntal rssarch. In J. Kagl and A. Roth, ditors, Handbook of Exprimntal Economics, pags 94. Princton Univrsity Prss, 997.

10 [25] D. Lvin. Modling altruism and spitfulnss in xprimnts. Rviw of Economic Dynamics, : , 998. [26] L. Liang and Q. Qi. Cooprativ or vindictiv: Bidding stratgis in sponsord sarch auctions. In Proc. 3rd Workshop on Intrnt and Ntwork Economics WINE), pags 67 78, [27] H. Lin, T. Roughgardn, E. Tardos, and A. Walkovr. Brass s paradox, fibonacci numbrs, and xponntial inapproximability. In Proc. 32nd Intl. Colloq. on Automata, Languags and Programming, pags , [28] A. Mas-Colll. On a thorm of Schmidlr. J. of Mathmatical Economics, 3:20 206, 984. [29] A. Pigou. Th Economics of Wlfar. Macmillan, 920. [30] T. Roughgardn. Th pric of anarchy is indpndnt of th ntwork topology. Journal of Computr and Systm Scincs, 67:34 364, [3] T. Roughgardn. Th maximum latncy of slfish routing. In Proc. 5th ACM Symp. on Discrt Algorithms, pags , [32] T. Roughgardn. Stacklbrg schduling stratgis. SIAM J. on Computing, 33: , [33] T. Roughgardn. Slfish Routing and th Pric of Anarchy. MIT Prss, [34] T. Roughgardn. Slfish routing with atomic playrs. In Proc. 6th ACM Symp. on Discrt Algorithms, pags 84 85, [35] T. Roughgardn. On th svrity of brass s paradox: Dsigning ntworks for slfish usrs is hard. Journal of Computr and Systm Scincs, 72: , [36] T. Roughgardn and E. Tardos. How bad is slfish routing? In Proc. 4st IEEE Symp. on Foundations of Computr Scinc, [37] T. Schlling. Micromotivs and Macrobhavior. Norton, 978. [38] Y. Sharma and D. Williamson. Stacklbrg thrsholds in ntwork routing gams or th valu of altruism. In Proc. 9th ACM Conf. on Elctronic Commrc, [39] M. Smith. Th marginal cost taxation of a transportation ntwork. Transportation Rssarch, Part B, 3: , 979. [40] C. Swamy. Th ffctivnss of stacklbrg stratgis and tolls for ntwork congstion gams. In Proc. 8th ACM Symp. on Discrt Algorithms, [4] I. Vtsikas and N. Jnnings. Outprforming th comptition in multi-unit sald bid auctions. In Proc. 6th Intl. Joint Conf. on Autonomous Agnts and Multiagnt Systms, pags , [42] J. Wardrop. Som thortical aspcts of road traffic rsarch. In Proc. of th Institut of Civil Enginrs, Pt. II, volum, pags , 952.