Automatica. An efficient Nash-implementation mechanism for network resource allocation

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1 Automatca 46 ( Contents lsts avalable at ScenceDrect Automatca ournal homepage: An effcent Nash-mplementaton mechansm for networ resource allocaton Rahul Jan a,b,, Jean Walrand b,c a Electrcal Engneerng Department, Unversty of Southern Calforna, Los Angeles, CA 90089, Unted States b ISE Department, Unversty of Southern Calforna, Los Angeles, CA 90089, Unted States c EECS Department, Unversty of Calforna, Bereley, CA 94720, Unted States a r t c l e n f o a b s t r a c t Artcle hstory: Receved 22 January 2009 Receved n revsed form 15 January 2010 Accepted 21 Aprl 2010 Avalable onlne 9 June 2010 Keywords: Game theory Nash equlbrum Mechansm desgn Networ auctons Bandwdth allocaton We propose a mechansm for auctonng bundles of multple dvsble goods n a networ where buyers want the same amount of bandwdth on each ln n ther route. Buyers can specfy multple routes (correspondng to a source destnaton par. The total flow can then be splt among these multple routes. We frst propose a one-sded VCG-type mechansm. Players do not report a full valuaton functon but only a two-dmensonal bd sgnal: the maxmum quantty that they want and the per-unt prce they are wllng to pay. The proposed mechansm s a wea Nash mplementaton,.e., t has a non-unque Nash equlbrum that mplements the socal-welfare maxmzng allocaton. We show the exstence of an effcent Nash equlbrum n the correspondng aucton game, though there may exst other Nash equlbra that are not effcent. We then generalze ths to arbtrary bundles of varous goods. Each buyer submts a bd separately for each good but ther utlty functon s a general functon of allocatons of bundles of varous dvsble goods. We then present a double-sded aucton mechansm for multple dvsble goods. We show that there exsts a Nash equlbrum of ths aucton game whch yelds the effcent allocaton wth strong budget balance Elsever Ltd. All rghts reserved. 1. Introducton Many networ resource allocaton problems nvolve multple dvsble resources (.e., those that can be dvded nfntely, e.g., bandwdth when t s avalable n any real fracton of Mbps whch are to be shared among many enttes. The allocaton of resources s to be done to acheve a global obectve (such as maxmzaton of the sum of ndvdual obectve functons. There are nformaton asymmetres: each entty nows only ts own obectve functon (henceforth, called a utlty functon and the system admnstrator nows the class to whch the utlty functons belong but does not actually now the ndvdual realzed utlty functons. The system admnstrator, gven ths lmted nformaton, s to desgn a system that determnes an allocaton to the varous enttes that acheves a global obectve. Any such desgn s possble only f some nformaton ndcatve of the enttes utlty functons s The materal n ths paper was partally presented at the 45th IEEE Conference on Decson and Control, San Dego, CA, USA, December 13 15, 2006, and the 2nd IEEE Internatonal Worshop on Bandwdth on Demand, Salvador da Baha, Brazl, Aprl 11, Ths paper was recommended for publcaton n revsed form by Assocate Edtor Mchèle Breton under the drecton of Edtor Berç Rüstem. Correspondng author at: Electrcal Engneerng Department, Unversty of Southern Calforna, Los Angeles, CA 90089, Unted States. Tel.: ; fax: E-mal addresses: rahul.an@usc.edu (R. Jan, wlr@eecs.bereley.edu (J. Walrand. elcted from them, and used to determne the allocaton. However, each of the enttes s an ndependent, self-nterested and strategc player, and thus may attempt to manpulate the system to ts advantage by msreportng nformaton about ts utlty functon. A basc queston then s how can we desgn rules of nteracton or a game that, despte strategc behavor on the part of players, and wthout a pror nowledge of the realzed utlty functon by the system admnstrator, stll acheves an allocaton that maxmzes the global obectve functon. We can see ths as an nverse game theory problem,.e., how to desgn games that acheve certan obectves. A theory that studes the desgn of strategy-proof resource allocaton mechansms has been under development snce the 1960s (Fudenberg & Trole, 1991(chap 8 and Mas-Colell, Whnston, & Green, 1995(chap 23 are good references. In ths paper, we are nterested n solvng networ resource allocaton and exchange problems n a partcular envronment. Our formulaton s motvated by the problem of resource allocaton n communcaton networs where servce provders want bandwdth on a whole route, and hence the same bandwdth on all lns n the route. The frst problem s allocatng multple dvsble networ resources among strategc agents. Let there be L dvsble goods avalable n quanttes C 1,..., C L. Let r [1 : L] denote a bundle of goods, such as those lns that form a route. Let there be n agents and let R for agent denote a set of bundles, such as the set of routes between a source destnaton par. For each agent, hs allocaton mght be splt between r R (such as multple routes, /$ see front matter 2010 Elsever Ltd. All rghts reserved. do: /.automatca

2 R. Jan, J. Walrand / Automatca 46 ( but wthn each r, the share of allocaton on good l for route r has to be the same for all l r (such as requrng the same capacty on all lns on a route. All the goods belong to the system admnstrator who must determne how the goods should be allocated among the agents. Each agent derves dfferent satsfacton from ownng a certan quantty of the varous goods,.e., they have dfferent utlty functons. The system admnstrator would le to allocate the varous goods among the agents to maxmze the sum of utlty derved by all the agents. However, user utltes are unnown to the system admnstrator. Thus, he must elct some nformaton from the agents to determne the optmal allocaton. Ths can be done through an aucton mechansm wheren each agent s ased to reveal a bd sgnal representatve of ts utlty functon. However, each agent s selfsh, acts strategcally and may have an ncentve to msrepresent ts bd sgnal. Thus, we must desgn an aucton mechansm that s robust to such strategc manpulaton by the agents. We then generalze ths to the case where the allocated bundle to a buyer can be arbtrary. The second problem addresses a more general envronment for multlateral tradng among many buyers and many sellers. We wll assume that each buyer wants capacty between a source destnaton par (over multple routes. Each seller sells goods ndvdually (.e., wthout formng bundles, and for smplcty we wll assume that each seller sells only one type of good, though there may be multple sellers sellng the same good. We wll requre each buyer and each seller to reveal a bd sgnal representatve of hs utlty or cost functon. And our goal s to determne an allocaton of resources that maxmzes the socal welfare (sum of utlty derved by all buyers mnus sum of cost ncurred by all the sellers. Each of the agents has hs own utlty and cost functon, and acts strategcally. Thus, t mght be dffcult to obtan an optmal allocaton. Our goal s to desgn an exchange mechansm whch despte strategc behavor by the partcpants yelds an allocaton that maxmzes the socal welfare. Lterature revew. Mechansm desgn for resource allocaton s a classcal problem studed by Economsts and Operaton Researchers. Unfortunately, most of the fundamental results are negatve (such as the varous mpossblty theorems that specfy economc envronments for whch t s mpossble to desgn mechansms wth certan specfed propertes. The Vcrey Clare Groves (VCG mechansm (Vcrey, 1961 s the most promnent postve result. Attenton was drawn to smlar resource allocaton problem n networs by the wor of Kelly (1997, Low and Varaya (1993 and Mace-Mason and Varan (1995. Ths wor was largely motvated by the need to desgn and analyze dstrbuted prcng sgnal-based networ congeston control algorthms. In partcular, Kelly (1997 and Kelly, Maullo, and Tan (1998 showed that when agents n a networ do not act strategcally, the resource allocaton problem can be solved effcently n a dstrbuted manner. In fact, t was suggested that the nternet transport control protocol (TCP can be understood to be dong exactly such a dstrbuted optmzaton. Ths wor nspred a mechansm desgn (the Kelly mechansm for allocaton of dvsble goods (such as bandwdth n a networ (Maheswaran & Basar, Ths mechansm was analyzed for the case when users are strategc n Johar and Tstsls (2004. It was dscovered that, wth a sngle dvsble good, the Kelly mechansm can result n an effcency loss of up to 25%,.e., the value of the socal welfare functon at the equlbrum outcome allocaton s 25% less than the one determned by a centralzed mechansm wth complete nowledge of all the players utlty functons. It was however shown n Hae and Yang (unpublshed that, n the networ verson of Kelly mechansm (a player submts a sngle bd for bandwdth on all lns that consttute hs route, the effcency loss can be arbtrarly bad. Followng up on ths wor, a generalzed class of proportonal allocaton (ESPA mechansms was ntroduced and analyzed n Maheswaran and Basar (2004. It was shown that these are effcent for allocaton of a sngle dvsble good. Such ESPA mechansms requre one-dmensonal bd sgnals and have a unque Nash equlbrum at whch the allocaton s effcent. However, the mechansms trade off domnant-strategy mplementaton, a very desrable property, for ease n mplementaton as compared to the VCG class of mechansms. In Hae and Yang (unpublshed, a smlar generalzaton of the proportonal allocaton mechansm was proposed for a sngle dvsble good. In Johar and Tstsls (2009, a general convex VCG-type mechansm was ntroduced that requred one-dmensonal bd sgnals. It was establshed that there exsts one Nash equlbrum at whch the correspondng allocaton s effcent. Condtons were provded under whch the Nash equlbrum s unque and the outcome s guaranteed to be effcent. A proposal, very smlar n sprt, and really a sub-case of the above, was presented n Yang and Hae (2007. Both these mechansm requre that the pseudo-utlty functons that the players report be twce contnuously dfferentable. A mechansm for a sngle good, n the same sprt but wth non-dfferentable pseudo-utlty functons, was frst reported n Semret (1999. Note that all the mechansms mentoned above are sngle-sded,.e., they only nvolve the auctoneer and multple buyers. Double-sded mechansms wth both buyers and sellers are of nterest for actual bandwdth exchanges. Our contrbuton. Ths paper s drectly related to the wor of Lazar and Semret (Lazar & Semret, 1997; Semret, They proposed a VCG-style aucton mechansm for a sngle dvsble good (Semret, Attempts have been made to generalze ths mechansm to multple dvsble goods so that t can be useful for networ resource allocaton problems (Btsa, Stamouls, & Courcoubets, 2005; Lazar & Semret, 1997; Malle & Tuffn, The settng of Lazar and Semret (1997 addresses the case where agents want bundles of lns (goods, and a dfferent aucton s held for every ln. However, each agent s utlty only depends on the mnmum allocaton t obtans on any ln n ts route. A slghtly dfferent settng s provded n Semret (1999, chap 3, wheren sellers place as bds to sell bandwdth on ndvdual lns. Moreover, a buyer has to effectvely bd separately for bandwdth n each ln n ts route. Thus, there s a separate double aucton for each ln. Such auctons when agents have complementartes across goods can lead to outcomes where an agent does not get all goods n ts bundle, and thus mght end up wth zero valuaton for hs allocaton. In fact, Btsa et al. (2005 consders the PSP mechansm of Lazar and Semret (1997 and Semret (1999 as t would be mplemented n a networ context wheren agents mae separate bds for each ln n a route. Ths however maes the bddng strateges of agents very complex. It s more desrable to have a mechansm wheren agents can mae a sngle bd on a whole path (or an end-to-end route. In Malle and Tuffn (2004, a varaton of the basc PSP mechansm s provded for a sngle good whch uses a hgher-dmensonal bd-sgnal space. Ths allows for a one-shot mechansm that acheves effcency at an equlbrum wthout the tme-consumng bddng convergence phase of the PSP mechansm. However, the proposed mechansm wored only for a sngle good and thus had lmtatons n terms of topology. The proposals n ths paper are nspred by Semret (1999. We propose a VCG-style mechansm, but nstead of reportng ther types (or complete utlty functons, agents only report a twodmensonal bd: a per-unt prce β and the maxmum quantty d that the agent s wllng to buy at that prce. Note that ths corresponds to a valuaton functon ˆv(x = β mn{x, d} whch s contnuous, concave, non-decreasng but non-dfferentable. The mechansm determnes an allocaton whch maxmzes the socal welfare correspondng to the reported utlty functons. The payment of each agent s exactly the externalty t mposes on the others through ts partcpaton, ust as n the VCG mechansm. What s remarable here s that for dvsble goods, when the utlty

3 1278 R. Jan, J. Walrand / Automatca 46 ( functons are strctly ncreasng, strctly concave and dfferentable, t suffces for agents to report only a quantty and ther margnal valuaton at that quantty (nstead of the full valuaton functon for the mechansm to yeld the effcent outcome at a Nash equlbrum. What s lost s the domnant-strategy mplementaton of VCG mechansms,.e., truthful reportng of utlty functons s not a domnant strategy equlbrum: each agent may not have nowledge of the utlty functons of others, nor of the actons beng taen by them. The man reason for ths s that domnant-strategy mplementaton requres the bd-message space to be rch enough so that each agent can report hs exact utlty functon, whch for dvsble goods he must be able to specfy hs utlty u at each real value x. 2. Problem statement Consder L dvsble goods, L = {1,..., L}, wth C l unts of good l beng avalable. Let Γ be the power set of L. Let there be n buyers. Buyer wants a bundle of goods r R Γ and wants the same quantty x of all goods n ths bundle. We call such bundles routes. For example, a buyer mght desre any route between a source destnaton par. R would then denote all the routes r between ths source destnaton par. Moreover, we would allow that the buyer s total flow s splt between varous routes n R, e.g., buyer mght receve z 1 = x /2 on route r 1 R and z 2 = x /2 on route r 2 R for a total of x. We wll assume that each buyer has a quas-lnear utlty functon u (x, ω = v (x ω, where ω s the payment made by buyer and v (x s a strctly ncreasng, strctly concave and twce dfferentable valuaton functon. Denote x = (x 1,..., x n and C = (C 1,... C L. We wll denote by z r the flow of carred on route r R. We wll use the notaton H r = 1 (r R and A lr = 1 (l r, where 1 s the ndcator functon. When there are multple sellers as well, seller sellng capacty on ln l, we denote B l = 1(l = l. We wll call S(x = n v =1 (x the socal welfare functon, whch s a strctly ncreasng concave functon. We wll requre capacty constrants A lr z r C l, l L, (1,r x = r R H r z r,, (2 x, z r 0,, r. (3 The frst constrant smply says that t s not possble to allocate more than the avalable quantty of any good, the second constrant says that total flow allocaton to a buyer equals the sum of flow allocatons to hm on varous routes r R, and the thrd constrant says that only non-negatve allocatons are allowed. The three constrants together determne a convex doman. Let λ l and ν be the Lagrange multplers correspondng to constrants (1 and (2. System obectve: To determne an allocaton x that satsfes max S(x (4 Az C, Hz = x, x, z 0. We wll call such an allocaton effcent. Observe that ths s a convex optmzaton problem. Thus, a soluton exsts and moreover t s unque. It s characterzed by the followng set of condtons: (v (x ν x = 0, (5 v (x ν 0, ( C l,r A lr z r = 0, l C l,r ν ( A lr z r 0, l 0, r R, ν z = 0, r r R, λ, l ν, x, z 0, l, r r R,. The above condtons are derved from the KKT necessary and suffcent condtons for optmalty n convex programs. Note that t s possble for a system admnstrator to acheve ths obectve only f he nows the valuaton functons of all the agents exactly. Ths however may not be true n dstrbuted systems wth selfsh agents who may not reveal ther actual valuaton functons. In that case, we need an ncentvzed mechansm (( x 1, P 1,..., ( x n, P n whch ass agent to report a sgnal b ndcatve of ts valuaton functon v, and determnes an allocaton x and a payment P to be made by t. Agent s obectve: To pc a b to maxmze ts net utlty u (b ; b = v (x (b, b P (b, b, where b are the bd sgnals of all the other agents. Ths gves rse to a strategc game between the agents. The allocaton and payment rule s to be desgned n such a way that each agent reports a sgnal that enables the system admnstrator to determne the allocaton x even wthout nowng the actual valuaton functons. 3. The networ second-prce (NSP mechansm wth multple routes We now propose a mechansm to be used by the system admnstrator (also called the auctoneer to allocate multple dvsble goods avalable n certan quanttes among many buyers. The buyers specfy R 1,..., R n Γ and correspondng bds b 1,..., b n. The bd b = (β, d specfes the maxmum per-unt prce β that s wllng to pay and demands up to d unts of R. Denote d = (d 1,..., d n. Note that any buyer derves zero utlty f he gets flow on a route r R. Thus, he has no ncentve to not truthfully report R. The auctoneer then determnes an allocaton x = ( x 1,..., x n as a soluton of the followng optmzaton problem: max β x (6 s.t. Az C, Hz = x, x d, x, z 0. Let x denote the soluton of the above wth d = 0. Then, the payment to be made by buyer s P (b, b = β ( x x. (7 The above defnes the networ second-prce (NSP mechansm. Ths s a VCG-style mechansm (Vcrey, 1961 where the players, nstead of reportng ther type or a full valuaton functon, only report the parameters (β, d of the revealed valuaton functon ˆv (x = β mn(x, d. The payment of s the externalty or the decrease n socal welfare that the buyer mposes on all the other players by hs partcpaton based on ths revealed valuaton functon. Note that the soluton to the aucton optmzaton (6 need not be unque. Ths problem occurs n most auctons that determne the allocaton by solvng an optmzaton problem that may not have a unque soluton ncludng the VCG mechansm. One way to get around ths problem s to use any determnstc te-breang rule to pc one among the many optmal solutons. Snce the

4 R. Jan, J. Walrand / Automatca 46 ( players compute a strategy nowng full well the propertes the allocaton wll have (namely, that t wll solve the aucton optmzaton problem, and the payment that they wll have to mae based on the bds and the allocaton, ths does not affect the analyss. The payoff of buyer s u (b, b = v ( x (b P (b. Recall that an allocaton x s effcent f t s a soluton of the optmzaton (3. Note that such an allocaton cannot be changed to mprove any player s payoff wthout decreasng some other player s payoff and hence s Pareto-effcent. 1 The strategy space of buyer s B = [0, [0, C ], where C = r R mn C l. A Nash equlbrum s a bd profle b = (b,..., 1 b n such that u (b, b u (b, b, b B. Nash equlbra whch yeld effcent allocaton wll be sad to be effcent. For any Nash equlbrum allocaton x, we wll say that ts relatve effcency s η := v (x / v (x. Note that ths wll le n [0,1], where η = 1 wll mean that full effcency s acheved. The worst (least relatve effcency over all Nash equlbra of game s ts called the prce of anarchy. The best (hghest relatve effcency over all Nash equlbra of a game s called ts prce of stablty (Anshelevch et al., Propertes of the NSP mechansm We frst note the KKT condtons for the aucton optmzaton problem. Let λ l be the Lagrange multpler correspondng to the frst (capacty constrant for good l, ν be the Lagrange multpler correspondng to the second (flow balance constrant and µ be the Lagrange multpler correspondng to the thrd (demand constrant n the aucton optmzaton (6. (β ν µ x = 0, (8 β ν µ 0, ( C l,r C l,r A lr z r A lr z r 0, d x 0, (d x µ = 0, ν ( = 0, l l 0, r R, ν z = 0, r r R, λ, ν l, µ, x, z 0, l, r r R, Exstence of an effcent Nash equlbrum We frst show the exstence of a Nash equlbrum n the correspondng resource allocaton game by constructon. Theorem 1. There exsts a Nash equlbrum b of the NSP mechansm whose correspondng allocaton x s effcent (.e., η(x = 1. The proof s by constructon. We relegate t to the Appendx. Note that the above result mples the exstence of an ε-effcent 1 Note that we are after allocatve effcency (socal welfare maxmzaton here, whch also happens to be a Pareto-effcent allocaton. However, there would be other Pareto-effcent allocatons as well. ε-nash equlbrum, a result obtaned n Semret (1999 for the specal case of a sngle good. Remars. (1 It s worth notng here that a unque Nash equlbrum such as acheved n domnant-strategy mplementaton mechansms (e.g., the VCG mechansm s not possble here snce such mechansms requre reportng of the whole utlty functon. Ths s practcally mpossble wth dvsble goods wth general utlty functons (unless they are fntely parameterzable. (2 There s a multplcty of equlbra but, as we dscuss next, some of the Nash equlbra can be elmnated by ntroducng reserve prces. If the mechansm s repeated, then there s the possblty of elmnatng all but the effcent Nash equlbrum. Ths would requre a learnng scheme whch s outsde the scope of ths paper, and wll be addressed n future wor Ineffcent Nash equlbra and reserve prces However, not all Nash equlbra of the NSP mechansm are effcent. We show the exstence of an neffcent one through an example. Example 1. Consder two players wth lnear valuaton functons, v (x = θ x for one good wth C = 1, and wth θ 1 > θ 2. Thus, the effcent allocaton s (1, 0. Let player 2 bd β 2 = (θ 1, 1 ɛ and player 1 bd β 1 = (θ 2, ɛ. The allocaton s (ɛ, 1 ɛ and the payments are (0, 0. It s easy to chec that t s a Nash equlbrum. Further, the relatve effcency s (θ 2 (1 ɛ + θ 1 ɛ/θ 1. For ɛ and θ 2 arbtrarly small, ths can be made arbtrarly close to zero. Note that, n the example above, we assumed that the valuaton functons are lnear. Theorem 1 assumes that the utlty functons are strctly concave. However, one can magne strctly concave valuaton functons arbtrarly close to beng lnear. Thus, for any 0 < ɛ < 1, there exst valuaton functons and Nash equlbra n the two-player aucton game above whch have relatve effcency smaller than ɛ. But note that ths arbtrarly large effcency loss can be mtgated by ntroducng reserve prces and elmnatng some of the neffcent Nash equlbra. Example 2. Let p be a reserve prce, the prce that any partcpant has to pay. Then, n the example above, the players bd β 1 = (θ 1, d 1 and β 2 = (θ 2, d 2 wth θ 1 > θ 2 f there s a d 2 such that v 2 (d 2 θ 2 (d 2 (1 d 1 p v 2 (1 d 1 p 0. The nequalty follows because, wth such bds, player 2 prefers to be the wnner and get d 2 and pay p + θ 2 (d 2 + d 1 1. Smlarly, player 1 bds β 1 < β 2 and a d 1 such that v 1 (1 d 2 p v 1 (d 1 θ 1 (d 1 + d 2 1 p 0. And agan ths nequalty follows because player 1 prefers to lose and get 1 d 2 and pay only the reserve prce. The two above yeld that d 1 1 p/θ 2 and d 2 1 p/θ 1. Thus, d 2 cannot be arbtrarly close to 1, and clearly, the worst relatve effcency of any Nash equlbra has now mproved. Ths dea extends to general networs. However, unless the auctoneer has some a pror nformaton about user valuaton functons (such as a dstrbuton on user types, t cannot be guaranteed that reserve prcng wll not elmnate the effcent Nash equlbrum as well. 4. The NSP mechansm for arbtrary bundles We now consder a slghtly dfferent settng. There are stll L dvsble goods, L = {1,..., L}, wth C l unts of good l beng avalable. And there are n buyers. But now each buyer wants an arbtrary bundle,.e., buyer wants x = (x 1,..., x L. Its valuaton functon now s v (x 1,..., x L, whch depends on amounts of varous goods obtaned. We stll assume that these functons are

5 1280 R. Jan, J. Walrand / Automatca 46 ( nce, n the sense that they are strctly ncreasng, strctly concave and twce dfferentable n each argument. We wll call S(x 1,..., x n = n v =1 (x the socal welfare functon, whch s a strctly ncreasng concave functon. Our system obectve then s to determne an allocaton x that satsfes max S(x 1,..., x n (9 x l C l, l, x l 0,, l. We wll call such an allocaton effcent. As before, ths s a convex optmzaton problem and thus, a soluton exsts and s unque. Let λ l be the Lagrange multplers correspondng to the capacty constrant. Then, the optmal soluton s characterzed by the followng set of condtons: ( v x l = 0,, l (10 x l v 0,, l x ( l C l x l = 0, l C l x l 0, l λ, l x l 0,, l. The buyers specfy bds b 1,..., b n, where b = (β, d, where β = (β 1,..., β L, d = (d 1,..., d L, whch specfes the maxmum per-unt prce β l that s wllng to pay for good l and demands up to d l unts of t. The auctoneer then determnes an allocaton x = ( x 1,..., x n as a soluton of the followng optmzaton problem: max β l x l (11 s.t.,l x l C l, l, 0 x l d l,, l. Let x denote the soluton of the above wth d = 0. Then, the payment to be made by buyer s P (b, b = β ( x x. (12 Ths defnes the NSP mechansm for arbtrary bundles. As before, the payoff of buyer s u (b, b = v ( x (b P (b. The strategy space of buyer s B = [0, L l [0, C l ]. The Nash equlbrum s then defned as before. We can now show exstence of a Nash equlbrum n the correspondng resource allocaton game by constructon. Theorem 2. There exsts a Nash equlbrum b of the NSP mechansm whose correspondng allocaton x s effcent (.e., η(x = 1. The proof s gven n the Appendx. Remars. As for the mechansm n the prevous secton, the NSP mechansm for arbtrary bundles can also have multple Nash equlbra. From the same examples as n the prevous secton, one can conclude that some of the neffcent Nash equlbra can be elmnated by ntroducng reserve prces. 5. The NSP double-sded mechansm Consder L dvsble goods, L = {1,..., L}, wth C l unts of good l beng avalable. Let Γ be the power set of L. Let there be n buyers. Buyer wants a bundle of goods r R Γ and wants the same quantty (or flow x of all goods n ths bundle r. Let there be m L sellers; seller sells only one good l and there can be more than one seller sellng the same good. We wll assume that each buyer has valuaton functon v (x, whch s strctly ncreasng, strctly concave and dfferentable. And each seller has cost c (y, whch s strctly ncreasng, convex and dfferentable. Note that ths also ncludes the case where the costs are lnear. The buyers specfy R 1,..., R n Γ and correspondng bds b 1,..., b n. The bd b = (β, d specfes the maxmum per-unt prce β that s wllng to pay and demands up to d unts of the bundle R. Denote d = (d 1,..., d n. Seller specfes the good l and an as-bd a = (α, s, where α s the mnmum per-unt prce that s wllng to accept and can supply up to s unts of the good l. Denote s = (s 1,..., s m. The auctoneer then determnes an allocaton ( x, ỹ as a soluton of the followng optmzaton problem: max β x α y (13 Az By, Hz = x, x d, y s, x, y, z 0. Let ( x, ỹ denote the soluton to the above wth d = 0 and ( x, ȳ denote the soluton to the above wth s = 0. Then, the money transfer (the payment to be made by buyer s T (b, b, a = β ( x x α (ỹ ỹ. (14 and the money transfer to be made by seller (negatve would mean transfer to the seller T (b, a, a = β ( x x α (ȳ ỹ. (15 Recall that these transfer are the externalty that the agents mpose on the others through ther partcpaton. The payoff of buyer s ũ (b, b, a = v ( x (b, a T (b, a, and the payoff of seller s ū (b, a, a = T (b, a c (ỹ (b, a. We wll say that an allocaton (x, y s effcent f t s a soluton of the followng optmzaton problem: max v (x c (y (16 Az By, (17 Hz = x, (18 x, y, z 0. (19 Such an allocaton s necessarly Pareto-effcent snce no player can unlaterally mprove hs payoff wthout mang another player worse off. The strategy space of the buyer s B = [0, [0,. The strategy space of seller s A = [0, [0,. A Nash equlbrum for ths game s defned as before, and we say that t s effcent f the correspondng allocaton s effcent. We can show the exstence of a Nash equlbrum n the double-sded mechansm by constructon. Moreover, an mportant property a double-sded mechansm must have s strong budget balance,.e., T = T. In words, the total payments made by the buyer equal the total payments made to the sellers.

6 R. Jan, J. Walrand / Automatca 46 ( Theorem 3. There exsts an effcent Nash equlbrum (x, y wth strong budget balance n the NSP double-sded mechansm. The proof can be found n the Appendx. Remars. (1 Whle the above theorem establshed the exstence of an effcent, strongly budget-balanced Nash equlbrum, there mght exst other Nash equlbra whch are ether not effcent, or not budget balanced, or nether. (2 We also note that the double-sded NSP mechansm can be extended n a natural way (as n Secton 4 to the case where buyers have arbtrary bundles whle sellers sell on ndvdual lns. We leave t to the reader to confrm that the analogue of Theorem 3 wll stll hold. 6. Conclusons and further wor We have proposed a mechansm for the allocaton of multple dvsble goods such as bandwdth n a communcaton networ. The mechansm s VCG-le and the players are only ased to report two numbers: a prce per unt, and the maxmum quantty demanded, as opposed to the VCG mechansm, whch requres the full valuaton functon. Our mechansm s a generalzaton of that presented n Semret (1999 to the networ case. We show the exstence of a Nash equlbrum where the allocaton s effcent. Ths mmedately mples the exstence of an ε-nash equlbrum (where each player, gven strateges of all other players, chooses a response whch s wthn ε of the best response whch s ε- effcent (.e., an allocaton whch s wthn κε of the socal welfare maxmzng allocaton, where κ s a constant. However, not all Nash equlbra are effcent, as we show through an example. A dstrbuted, computatonally effcent algorthm that yelds an ε- effcent ε-nash equlbrum for the sngle good case was presented n Semret (1999 whle we presented ts generalzaton to a multple goods settng wth two players n Dmas, Jan, and Walrand (2006. The generalzaton to multple players s not avalable yet, and t s not clear f usng such an algorthm s a Nash equlbrum at all. We also present a double-sded mechansm whch has a Nash equlbrum wth effcent allocaton and strong budget balance. We note that snce, for each of the mechansms presented, we demonstrated the exstence of an effcent Nash equlbra, the prce of stablty (PoS (see Anshelevch et al. (2004 of the mechansms s 1, and the prce of anarchy s 0 as shown by Example 1. Our wor s also related to Johar and Tstsls (2009. They present a lmted communcaton VCG-le mechansm that yelds an effcent Nash equlbrum and gves condtons under whch all equlbra are effcent, some of whch are restrctve. Further, whle they requre the revealed utlty functons to be dfferentable for every parameter, our revealed utlty functons are not dfferentable and hence ths s not a partcular case of ther mechansm. Further, expermental wor n electrcty marets has shown that mechansms whch express both quantty and per-unt prce, such as n our mechansm, wor better than one-dmensonal bd mechansms (see Elmaghraby and Oren (1999 for a dscusson. Appendx. Proofs of theorems Proof of Theorem 1. Let x be an effcent allocaton. Then, there exst ν 1,..., ν n 0, λ 1,..., λ L 0 such that v (x ν = :r R λ l, r R, wth equalty f x > 0 (a strct nequalty s possble wth x = 0. Consder the strategy profle d = x and β = v (d. Note that x s an aucton outcome wth λ s and ν s as above, and µ = 0,,.e., these solve (8. Ths mples that such that x > 0 β = λ l, r R. (20 :r R Consder a buyer wth x > 0. Gven the bds b of the others as fxed, f buyer changes hs bd b to decrease hs allocaton x by a δ > 0 (a buyer wth x = 0 cannot decrease hs allocaton, then the allocaton of all the other players does not change snce all of them already receve the maxmum quantty they as for. From Eq. (7, we get that the payment of player does not change. However, snce v s strctly ncreasng and concave, hs utlty reduces by v (x v (x δ. Thus, hs net payoff actually reduces. Now, consder a buyer (wth x 0 changes hs bd to b such that he ncreases hs allocaton x by a δ > 0. Denote the change on route r by δ r so that δ = r R δ r. Let the resultng allocaton be x. Denote L = {l r : r R }, the set of lns on whch buyer s traffc can flow for some route r R. Let L = {l s : s R, }, the set of lns on whch any other buyer s traffc can flow for some route s R. We frst note that, for l L L, (,s:,s R,l s s s r R : r r. (21 Ths s because, f λ l > 0, then the capacty constrant on l s tght, and any ncrease n s share of bandwdth on l s at the expense of other buyers, who also want l. (Note that we have an nequalty because buyer s flow need not completely obtan the capacty vacated by the other buyers. If the capacty constrant on l s not tght, then λ l = 0, and the above stll holds. We further note that, for l L \ L,.e., the set of lns on whch no other buyer s traffc can flow, f the capacty constrant s tght, then an ncrease n s flow cannot come from ncreases for r R such that l r (though there may be a decrease, hence the nequalty. If the capacty constrant s not tght, then λ l = 0. In other words, r R : r r 0, l L \ L. (22 The change n buyer s payment (later denoted P as he changes hs bd to b to ncrease hs allocaton by δ s gven by P (b, b P (b, b = (x = s s λ l s R l s = s λ s l (,s:,s R (,s:,s R = l L L l L L l L = r R = r R r = r R = δ ν. r l s s l s L x β s λ l (23 s (,s:,s R,l s r R : r R : r r r r λ l r ( r ν s (24 r (25 r (26 λ l

7 1282 R. Jan, J. Walrand / Automatca 46 ( The frst equalty follows by defnton. The second equalty follows by defnton, by Eq. (20 and by notng that snce buyer ncreases hs bd, for any buyer, x x and z s z s. The thrd follows merely by compactfyng notaton and wrtng a double sum over and s as a sngle sum over (, s. Inequalty (23 follows because the sum over l has fewer terms than before. Equalty (24 s arrved at by rearrangement of terms. Inequalty (25 follows from nequalty (21. Inequalty (26 follows from nequalty (22. The remanng are obvous. Now, snce v s strctly concave, v (x + δ v (x < ν δ P. Ths holds even f x = 0. Thus, gven the bds b of all the other players, the best response of player s to bd b so that he obtans x. Ths mples that b = (b 1,..., b n s a Nash equlbrum and the correspondng allocaton s effcent. Proof of Theorem 2. Let x be an effcent allocaton. Then, there exst λ 1,..., λ L 0 such that v (x λ l,, l (wth equalty f x l > 0. Consder the strategy profle d l = x l and β l = v (d l. x l Note that ths mples that β l = λ l,, l : x l > 0. Gven the bds b of the others as fxed, suppose that buyer changes hs bd b to change hs allocaton x to some other x by some l = x l x l. Wthout loss of generalty, assume that there s some l such that l < 0 for l l and l 0 for l > l. Defne x l = x l l, for l l, (27 = x l, for l > l. We consder two cases: ( buyer changes hs bd to b to change hs allocaton from x to x, and ( buyer changes hs bd to b to change hs allocaton from x to x. So, frst consder that the buyer changes hs bd to change hs allocaton from x to x. Then, buyer now gets l less for goods l l and the same as before for the other goods. The allocaton of all the other players does not change snce all of them already receve the maxmum quantty they as for. From Eq. (12, we get then that the payment of player does not change. However, snce v s strctly ncreasng and concave n each argument, hs valuaton strctly reduces by v (x v ( x. Thus, v ( x v (x < P ( b, b P (b, b = 0. (28 Now, suppose that buyer changes hs bd from b to b such that he changes hs allocaton from x to x. Note that now hs allocaton changes by l 0 for l > l. Then, the change n hs payment s gven by P (b, b P ( b, b = = l> l = l> l x l l> l: xl >0 λ l : x l >0 l, β l ( x l x l ( x l x where the last equalty follows snce the total change n allocaton of all other players on an tem l > l s l, whch s how much more buyer gets of l. Now, v s strctly concave n each argument. Thus, v (x v ( x < l = P (b, b P ( b, b. (29 l> l l From (28 and (29, we get that v (x v (x < P (b, b P (b, b, whch mples that, gven the bds b of all the other players, the best response of player s to bd b so that he obtans x. Thus, b = (b 1,..., b n s a Nash equlbrum and the correspondng allocaton s effcent. Proof of Theorem 3. Let (x, y be an effcent allocaton. Then, there exst ν 1,..., ν n 0 and λ 1,..., λ L 0 such that v (x ν = :r R λ l, and c (y λ l,. Consder the strategy profle d = x, β = v (d, s = y and α = c (s. Note that ths mples that β = λ l, : x > 0 and α = λ l, : y > 0. (30 :r R Consder a buyer wth x > 0. Gven the bds (b, a of the others as fxed, f buyer changes hs bd b to decrease hs allocaton x by a > 0, then note that the allocaton of all the other buyers does not change but some sellers on lns L = {l r, r R } sell less. From Eq. (14, we get the change n payment of buyer (later denoted T s T (b, b, a T (b, a = α (y y l L :l =l,y >0 = λ l (y y l L :l =l,y >0 = λ l (x r x r r R :x r >0 = r R :x r >0 = λ l r r R :x r >0 r ν = ν. The frst equalty s obtaned ust by tang dfferences of the two payments, the second equalty by (30, and the thrd follows by an argument smlar to (21. The last two are obvous. Snce v s strctly ncreasng and concave, we get that v (x v (x < ν = T, (31.e., hs net payoff decreases. Now, suppose the buyer, wth x 0, changes hs bd to b such that t ncreases hs allocaton x by a > 0, then note that, whle the allocaton of all the sellers remans unchanged, that of some buyers decreases. Let the resultng allocaton of buyers be x. Then, as n the proof of Theorem 1, T (b, b, a T (b, a (32 = s λ s l (,s:,s R,z s >0 l s s (,s:,s R,z s >0 l s L = l L L = l L = r r R r R : r R : r r s λ l r r r λ l = ν. The reasonng s same as before. Further, snce v s strctly ncreasng and concave, we have

8 R. Jan, J. Walrand / Automatca 46 ( v (x + v (x < ν T. (33 From (31 and (33, we get that, gven the bds (b, a of all the other players, the best response of a buyer s to bd b so that he obtans x. 0. Suppose a seller changes hs bd to ncrease y by a > 0. Ths wll not affect the allocaton of the buyers but some sellers sellng good l mght get affected. Clearly, the net change n payment to the seller s T = λ l (follows easly from (30 and, snce c s strctly ncreasng and convex, we get that c (y Now consder a seller wth y + c (y > = T. (34 And f any seller (sellng l, wth y > 0, were to change hs bd to decrease hs allocaton by > 0, then the allocaton to other sellers does not change but some buyers get less. Thus, the net change n seller s transfer s gven by T = β (zr :l L r R :,z r >0 λ l (zr :l L r R :,z r >0 =. r (35 r And agan, by strct convexty of c, c (y c (y > λ l T. (36 From (34 and (36, we get that a s a best response of seller to bds of other players (b, a. Thus, (b, a s a Nash equlbrum. Moreover, the correspondng allocaton s effcent. To prove strong budget balance at ths Nash equlbrum, we note that x = ỹ, l, :l R :l=l x :l R ỹ :l=l = :l=l ỹ, : l R = :l R x, : l = L. We can now wrte the payments for all and as T = λ l x (ỹ ỹ, l :l=l and T = l :l R ( x ( λ l ( x x (ȳ ỹ, :l R :l=l whch usng the facts noted above yeld T = l R λ l x, and T = λ L ỹ, from whch we easly get T = T,.e., strong budget balance at the Nash equlbrum (b, a. References Anshelevch, E., Dasgupta, A., Klenberg, J., Tardos, E., Wexler, T., & Roughgarden, T. (2004. The prce of stablty for networ desgn wth far cost allocaton. In Proc annual symp. on found. of computer scence, FOCS. Btsa, M., Stamouls, G., & Courcoubets, C. (2005. A new strategy for bddng n the networ-wde progressve second prce aucton for bandwdth. In Proc. CoNEXT conference. Dmas, A., Jan, R., & Walrand, J. (2006. Mechansms for effcent allocaton n dvsble capacty networs. In Proc. 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IEEE Journal of Selected Areas of Communcatons, 25, Rahul Jan s an Assstant Professor of Electrcal Engneerng at the Unversty of Southern Calforna, Los Angeles, CA. Pror to onng USC, he was a postdoctoral member of the research staff at the Mathematcal Scences Dvson of the IBM T.J. Watson Research Center, Yortown Heghts, NY. He receved hs Ph.D. n EECS n 2004, and an M.A. n Statstcs n 2002, both from the Unversty of Calforna, Bereley. He also receved an M.S. n ECE from Rce Unversty n 1999 and completed hs undergraduate wor wth a B.Tech n EE n 1997 from the Indan Insttute of Technology, Kanpur. He won a Best Paper Award at The ValueTools Conference 2009 and was a recpent of the CAREER Award from the Natonal Scence Foundaton n He has dverse research nterests, wth current focus on Game Theory and Economcs for Networs, and Stochastc Control and Learnng. Jean Walrand receved hs Ph.D. n EECS from UC Bereley and has been on the faculty of that department snce He s the author of An Introducton to Queueng Networs (Prentce Hall, 1988 and of Communcaton Networs: A Frst Course (2nd ed, McGraw-Hll, 1998 and co-author of Hgh-Performance Communcaton Networs (2nd ed, Morgan Kaufman, 2000 and of Communcaton Networs: A Concse Introducton (Morgan & Claypool, Hs research nterests nclude stochastc processes, queung theory, communcaton networs, game theory and the economcs of the Internet. Prof. Walrand s a Fellow of the Belgan Amercan Educaton Foundaton and of the IEEE and a recpent of the Lanchester Prze and of the Stephen O. Rce Prze.