SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 1 An Effcent Nash-Implementaton Mechansm for Dvsble Resource Allocaton Rahul Jan IBM T.J. Watson Research Center Hawthorne, NY 10532 rahul.jan@us.bm.com Jean Walrand EECS Department, Unversty of Calforna, Berkeley wlr@eecs.berkeley.edu Abstract We propose a mechansm for auctonng bundles of multple dvsble goods. Such a mechansm s very useful for allocaton of bandwdth n a network where the buyers want the same amount of bandwdth on each lnk n ther route. We allow for buyers to specfy multple routes (correspondng to a source-destnaton par). The total flow can then be splt among these multple routes. We frst propose a sngle-sded VCG-type mechansm. However, nstead of reportng ther valuaton functons, the players only reveal 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 weak 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. We show through an example that there are other Nash equlbra that are not effcent. Further, we provde a suffcent characterzaton of all effcent Nash equlbra. We then generalze ths to buyers gettng arbtrary amounts of varous goods. Ths requre each buyer to submt a bd separately for each good but ther utlty functon a general functon of allocatons of varous dvsble goods. Then, we present a doublesded aucton mechansm for multple dvsble goods wth buyers and sellers. We show that there exsts a Nash equlbrum of ths aucton game whch yelds the effcent allocaton. I. INTRODUCTION Many 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 objectve (such as maxmzaton of the sum of ndvdual objectve functons). There are nformaton asymmetres:
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 2 each entty knows only ts own objectve functon (henceforth, called a utlty functon) and the system admnstrator knows the class to whch the utlty functons belong but does not actually know 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 acheves that acheves a global objectve. Any such desgn s possble only f some nformaton ndcatve of the enttes utlty functons s 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 knowledge of realzed utlty functon by the system admnstrator, stll acheves an allocaton that maxmzes the global objectve functon. In some sense, ths s an nverse game theory problem. A theory that studes desgn of strategy-proof resource allocaton mechansms has been under development snce the 1960s ([5], [20] are good references). In ths paper, we are nterested n solvng network resource allocaton and exchange problems n a partcular envronment. Our problem s motvated by the problem of resource allocaton n communcaton networks where servce provders want bandwdth on a whole route, hence same bandwdth on all lnks n the route. The frst problem s allocatng multple dvsble 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 lnks that form a route. Let there be n agents and let R for agent denote a set of bundles, such as set of routes between a source-destnaton par. For each agent, hs allocaton mght be splt between r R (such as multple routes) but wthn each r, the share of allocaton on good l for route r has to same for all l r (such as requrng the same capacty on all lnks on a route). All the goods belong to an agency (the system admnstrator) whch 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 agency would lke to allocate the varous goods among the agents to maxmze the sum of utlty derved by all the agents. However, user utltes are unknown to the agency. Thus, t must elct some nformaton from the agents to determne the optmal allocaton. Ths can be done through an aucton mechansm wheren each agent s asked to reveal a bd sgnal
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 3 representatve of ts utlty functon. However, each agent s selfsh, acts strategcally and has 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 (multlateral tradng) envronment where there are 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 Resource allocaton mechansm desgn n a general settng has been extensvely studed by Economsts and Operatons 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 Vckrey- Clarke-Groves mechansm [23], [3], [6] s the most promnent postve result. Attenton was drawn to smlar resource allocaton problem n networks by the work of [11], [15], [16]. Ths work was largely motvated by the need to desgn and analyze dstrbuted prcng sgnal-based network congeston control algorthms. In partcular, [11], [12] showed that when agents n a network 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 work nspred a mechansm desgn (the Kelly mechansm) for allocaton of dvsble goods (such as bandwdth n a network) [18]. Ths mechansm was analyzed for the case when users are strategc n [9]. It was dscovered that wth a sngle dvsble good, the Kelly mechansm
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 4 can result n an effcency loss of upto 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 knowledge of all the players utlty functons. It was however shown n [7] that n the network verson of Kelly mechansm (a player submts a sngle bd for bandwdth on all lnks that consttute hs route), the effcency loss can be arbtrarly bad. Followng up on ths work, a generalzed class of proportonal allocaton (ESPA) mechansms was ntroduced n [18], and further analyzed n [17], [24]. It was shown that these are effcent for allocaton of a sngle dvsble good. Such ESPA mechansms requre one-dmensonal bdsgnals 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 [7], a smlar generalzaton of the proportonal allocaton mechansm was proposed for a sngle dvsble good. In [10], a general class of convex VCG-type mechansms were 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 [25]. 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 [14]. 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 work of Lazar and Semret [13], [14], [21]. They proposed a VCG-style aucton mechansm for a sngle dvsble good [14]. Attempts have been made to generalze ths mechansm to multple dvsble goods so that t can be useful for network resource allocaton problems [1], [13], [19]. The settng of [13] addresses the case where agents want bundles of lnks (goods), and a dfferent aucton s held for every lnk. However, each agent s utlty only depends on the mnmum allocaton t obtans on any lnk n ts route. A slghtly dfferent settng s provded n [21], chapter 3, wheren sellers place ask bds to sell
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 5 bandwdth on ndvdual lnks. Moreover, a buyer has to effectvely bd separately for bandwdth n each lnk n ts route. Thus, there s a separate double aucton for each lnk. 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, [1] consders the networked PSP mechansm of [13], [21] and proposed strateges for the agents that wll mprove the effcency of the outcome. But, n our vew, what s requred s a network aucton mechansm wheren agents can express ther bd for a whole path (such a settng s consdered n the combnatoral auctons lterature but for ndvsble goods). In [19], a varaton of the basc PSP mechansm s provded whch uses a hgher dmensonal bd-sgnal space to yeld the same effcency results. Ths n our opnon s not necessary as was also shown n the recent paper [10]. However, the results of [10] do not hold n the case where the routng matrx has full rank. The proposals n ths paper are nspred by [14]. We propose a VCG-style mechansm but nstead of reportng ther types (or complete utlty functons), agents only report a two-dmensonal 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 are 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, just as n the VCG mechansm. What s remarkable here s that for dvsble goods, when the utlty 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 need not have knowledge of the utlty functons of others, nor of the actons beng taken by them. II. 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. For example, a buyer mght desre
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 6 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 = I(r R ) and A lr = I(l r), where I s the ndcator functon. We wll call n S(x) = v (x ) =1 as 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 = H r z r,, 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 Objectve: 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.
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 7 Observe that t s a strct 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) ( C l ) A lr zr λ l = 0, l,r ν l r λ l = 0, r R, λ l, ν, x, zr 0, l, r R,. The above condtons are derved from the KKT necessary and suffcent condtons for optmalty n convex programs [2]. Note that t s possble for a system admnstrator to acheve ths objectve only f he knows 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 asks 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. Throughout the paper, we wll assume that Assumpton 1: For each lnk l, there are at least two buyers and j such that l r s, wth r R and s R j. Agent s Objectve: To pck 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 knowng the actual valuaton functons. III. THE NETWORK SECOND-PRICE MECHANISM WITH MULTIPLE 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 (wth r R a subset of L) 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 ).
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 8 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 ) = j β j ( x j x j ). (7) The above defnes the Network Second Prce (NSP) mechansm. Ths s a VCG-style mechansm [23] 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. 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. The strategy space of buyer s B = [0, ) [0, C ] where C = max r R mn l r C l. 1 A Nash equlbrum s a bd profle b = (b 1,, b n) such that u (b, b ) u (b, b ), b B. 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.
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 9 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. A. 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 capacty constrant 1 for good l, ν the Lagrange multpler correspondng to the flow balance constrant 2 and µ be the Lagrange multpler correspondng to the demand constrant 3 n the aucton optmzaton (6). (β ν µ x = 0, (8) ( C l ) A lr zr λ l = 0, l,r (d x ) µ = 0, ν l r λ l = 0, r R, λ l, ν, µ, x, zr 0, l, r R,. 1) Exstence of an Effcent Nash Equlbrum: We frst show 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 ε-nash equlbrum, a result obtaned n [14] for the specal case of a sngle good. 2) Ineffcent Nash Equlbra and Reserve Prces: However, not all Nash equlbra of the NSP mechansm are effcent. We show 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
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 10 s easy to check 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 a 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 networks. However, unless the auctoneer has some a prory 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. 3) A Suffcent Characterzaton of Effcent Nash Equlbra: We wll now provde a suffcent condton for a Nash equlbrum to be effcent. Note that the dual of the lnear program (6) s gven by mn{ l λ l C l + µ d : ν + µ β, l r λ l ν, r R,, λ l, ν, µ 0}. (9) Let λ l, ν, µ denote a soluton of the above wth Nash equlbrum b.
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 11 Theorem 2: Consder a Nash equlbrum b of the NSP game wth µ > 0 for all, then the correspondng allocaton s effcent. The above result mples that f at any Nash equlbrum, the allocaton s such that x = d for all, then t must be effcent. IV. THE NSP MECHANISM FOR ARBITRARY 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 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 (x ) as the socal welfare functon, whch s a strctly ncreasng concave functon. Our system objectve then s to determne an allocaton x that satsfes We wll call such an allocaton effcent. =1 max S(x 1,, x n ) (10) x l C l, l, x l 0,, l. As before, t s a strct 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 λ l x l = 0,, l (11) x l ( ) C l x l λ l = 0, l λ l, x l 0,, l.
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 12 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 s.t.,l β lx l (12) l x l C 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 ) = j Ths defnes the NSP mechansm for arbtrary bundles. As before, the payoff of buyer s u (b, b ) = v ( x (b)) P (b). β j ( x j x j ). (13) 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 3: There exsts a Nash Equlbrum b of the NSP mechansm whose correspondng allocaton x s effcent (.e., η(x ) = 1). The proof s n the appendx. V. THE NETWORK SECOND-PRICE DOUBLE-SIDED MECHANISM 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 j sells only one good L j 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
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 13 concave and dfferentable. And each seller has cost c j (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 (wth r R a subset of L) 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 j specfes the good L j, an ask-bd a j = (α j, s j ) where α j s the mnmum per unt prce that j s wllng to accept and can supply up to s j unts of the good L j. 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 j α j y j (14) Az y, Hz = x, x d, y s, x, y, z 0. Let ( x, ỹ ) denote the soluton to the above wth d = 0 and ( x j, ȳ j ) denote the soluton to the above wth s j = 0. Then, the money transfer (the payment) to be made by buyer s T (b, b, a) = k β k ( x k x k) j α j (ỹ j ỹ j ). (15) and the money transfer to be made by seller j (negatve would means transfer to the seller) T j (b, a j, a j ) = β ( x j x ) k j α k (ȳ j k ỹ k ). (16) 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),
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 14 and the payoff of seller j s ū j (b, a j, a j ) = T j (b, a) c j (ỹ j (b, a)). We wll say an allocaton (x, y ) s effcent f t s a soluton of the followng optmzaton problem max v (x ) j c j (y j ) (17) Az y, (18) Hz = x, (19) x, y, z 0. (20) Such an allocaton s necessarly Pareto-effcent snce no player can unlaterally mprove hs payoff wthout makng another player worse off. The strategy space of the buyer s B = [0, ) [0, ). The strategy space of seller j s A j = [0, ) [0, ). A Nash equlbrum for ths game s defned as before, and we say 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. Theorem 4: There exsts an effcent Nash Equlbrum (x, y ) n the NSP double-sded mechansm. VI. CONCLUSIONS AND FURTHER WORK We have proposed a mechansm for allocaton of multple dvsble goods such as bandwdth n a communcaton network. The mechansm s VCG-lke and the players are only asked 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 [14] to the network 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
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 15 we show through an example. A dstrbuted, computatonally effcent algorthm that yelds an ε-effcent ε-nash equlbrum can be obtaned as a generalzaton of the algorthm presented n [21]. We also present a double-sded mechansm whch has a Nash equlbrum wth effcent allocaton. Our work s also related to [10]. They present a lmted communcaton VCG-lke 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 s not a partcular case of ther mechansm. Further, expermental work n electrcty markets have shown that mechansms whch express both quantty and per-unt prce, such as n our mechansm, work better than one-dmensonal bd mechansms as proposed n [10] (see [4] for a dscusson). APPENDIX PROOFS OF THEOREMS Proof: (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 = l r:r R λ l, r R,. Consder the strategy profle d = x and β = v (d ). Note that ths mples x β = l r:r R λ l, r R,. Gven the bds b of the others as fxed, f buyer changes hs bd b to decrease hs allocaton by a small > 0, then the allocaton of all the other players does not change snce all of them already receve the maxmum quantty they ask for. From equaton (7), we get that the payment of player does not change. However, snce v s strctly ncreasng and concave, hs valuaton reduces by v (x ) v (x ). Thus, hs payoff actually reduces. Now, suppose player changes hs bd to b such that he ncreases hs allocaton x small > 0. Denote the change on route r by r so that = r R r. Let the resultng by a
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 16 allocaton be x. Then, the change n hs payment (later denoted P ) s gven by P (b, b ) P (b, b ) (21) = β j (x j x j ) j r R = r R (j,s):j,s R j,l r( l s l r λ l (j,s):j,s R j,l r λ l )(z js z js ) (z js z = ( ) r λ l r R l r = ν. (22) Frst note that due to buyer changng hs bd, hs allocaton ncreases by, but the allocaton of no other player can ncrease snce each player j, j already gets d j. The nequalty follows because there mght be other buyers affected who do not share a route r of. The followng equalty s just rearrangement of terms, whle the remanng are obvous. Now, snce v (x ) = l R λ l and v s strctly concave, v (x js ) + ) v (x ) < ν P, 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: (Theorem 2) It s suffcent to show that the ν 1,, ν n 0 are such that x (v (x ) ν ) = 0. Thus, we only need to consder agents wth x > 0. By assumpton µ > 0, whch mples that x = d. Suppose agent changes hs bd to d = d + to ncrease hs allocaton by (small enough) > 0. Let x denote the new allocaton. Then, by complementary slackness, x > x. Now, from senstvty analyss of lnear programs, we know that for small enough > 0, j β j(x j x j) = µ. Thus, the change n payment of agent s P = j β j (x j x j ) = (β µ ) = ν.
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 17 The last equalty follows from complementary slackness: x (ν + µ β ) = 0. Snce b s a Nash equlbrum strategy, t must be that v (x + ) v (x ) < ν. Now, suppose buyer wants to decrease hs allocaton by. Suppose he changes hs bd to d = d. Then, by complementary slackness x can see that the change n payment s ν and as 0, we get that < x. By a smlar argument as above, we and we establsh that v (x ) = ν. v (x ) v (x ) < ν Proof: (Theorem 3) Let x be an effcent allocaton. Then, there exst λ 1,, λ L 0 such that v (x ) x l mples = λ l,, l. Consder the strategy profle d l = x l and β l = v (d l ). Note that ths β l = λ l,, l. 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 x l by some small 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, (23) = x l = x l, for l > l. We consder two cases: () buyer changes hs bd to b to change hs allocaton from 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 to x. Then, buyer now gets l less for goods l l and 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 ask for. From equaton (13), we get then that the payment of player does not change. However, snce v reduces by v (x ) v ( x ). Thus, s strctly ncreasng and concave n each argument, hs valuaton strctly v ( x ) v (x ) < P ( b, b ) P (b, b ) = 0. (24)
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 18 Now, suppose buyer changes hs bd to 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 ) = j = l> l = l> l l> l β jl ( x jl x jl ) λ l ( x jl x j λ l l, 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> l From (24) and (25), we get that λ l l = P (b, b ) P ( b, b ). (25) 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. jl ) Proof: (Theorem 4) 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 = l r:r R λ l, and c j(yj ) = λ Lj, j. Consder the strategy profle d = x, β = v (d ), s j = yj and α j = c j(s j ). Note that ths mples β = λ l, and α j = λ Lj, j. (26) l r:r R Consder a buyer wth x changes hs bd b to decrease hs allocaton x > 0. Gven the bds (b, a) of the others as fxed, f buyer by a small > 0, then note that the allocaton of all the other buyers does not change but some sellers on lnk l r, r R sell less.
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 19 From equaton (15), we get the change n payment of buyer (later denoted T ) s T (b, b, a) T (b, a) = λ Lj (y j yj ) r R j:l j =l,l r = λ l r (27) r R l r = r R r ν = ν. The frst equalty s obtaned just by takng dfferences of the two payments, and the second equalty s obtaned notng that the allocatons of sellers change only for l r and fnal equalty by notng that the total change n allocaton over all r R s. Snce v s strctly ncreasng and concave, we get that.e., the net change n hs payoff s negatve. x Now, suppose buyer wth x v (x ) v (x ) < ν = T, (28) 0, changes hs bd to b such that t ncreases hs allocaton by a small > 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, T (b, b, a) T (b, a) (29) = β k (x k x k ) k r R = r R (k,s):k,s R k,l r( l s l r λ l (k,s):k,s R k,l r λ l )(z ks z ks ) (z ks z = ( ) r λ l r R l r = ν. (30) The reasonng s same as n the proof of Theorem 1. Further, snce v s strctly ncreasng and concave, we have v (x + ) v (x ) < ν T, (31) ks )
SUBMITTED TO IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS 20 From (28) and (31), 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. Now consder a seller j wth y j > 0. Suppose a seller j changes hs bd to ncrease y j a small > 0. Ths wll not affect the allocaton of the buyers but some sellers sellng good l mght get affected. Clearly, the net change n transfer of the seller s T j = λ l and snce c j s strctly ncreasng and convex, we get that c j (y j And f any seller j (sellng l) wth y j + ) c j (y j ) λ l = T j. > 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 T j = = And agan by strct convexty of :l r,r R r R :l r :l r,r R ( l r:r R λ l β (z r z r) ) ( r R :l r (x x λ l. (32) c j (y j ) c j (y j ) λ l T j, whch mples that a j s a best response of seller j to bds of other players (b, a j ). Thus, (b, a) s a Nash equlbrum. Moreover, the correspondng allocaton s effcent. ) ) by REFERENCES [1] M. BITSAKI, G. STAMOULIS AND C. COURCOUBETIS, A new strategy for bddng n the network-wde progressve second prce aucton for bandwdth, Proc. CoNEXT, 2005. [2] S. BOYD AND L. VANDENBERGHE, Convex Optmzaton, Cambrdge Unversty Press,2004 [3] E. CLARKE Multpart prcng of publc goods, Publc Choce, 2:19-33, 1971. [4] W. ELMAGHRABY AND S. OREN, The effcency of mult-unt electrcty auctons, The Energy Journal, 20(4):89-116, 1999. [5] D. FUDENBERG AND J. TIROLE, Chapter 8, Game Theory, MIT Press, 1991. [6] T. GROVES, Incentves n teams, Econometrca, 41:617-631, 1973.
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