Mechanisms for Efficient Allocation in Divisible Capacity Networks

Transcription

1 Mechansms for Effcent Allocaton n Dvsble Capacty Networks Antons Dmaks, Rahul Jan and Jean Walrand EECS Department Unversty of Calforna, Berkeley {dmaks,ran,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 frst propose a sngle-sded VCG-type mechansm. However, nstead of reportng types, 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. We show the exstence of an effcent Nash equlbrum n the correspondng aucton game of the mechansm. We show through an example that not all Nash equlbra are effcent but provde a dstrbuted algorthm that yelds the effcent one. Further, we provde a suffcent characterzaton of all effcent Nash equlbra. We then present a double-sded aucton mechansm for multple dvsble goods, and show that there exsts a Nash equlbrum of the aucton game whch yelds the effcent allocaton. I. INTRODUCTION We address two related problems of network resource allocaton and exchange n ths paper. 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 there be n agents, each of whom wants a bundle of goods, say R for agent. The 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. All the goods belong to an agency 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. Thus, the agency would lke to allocate the varous goods among the agents to maxmze the sum of utlty derved by all the agents (there could be other obectves as well). However, user utltes are unknown to the agency. Thus, t must elct some nformaton from the agents so that t can determne the optmal allocaton. Ths can be done through an aucton mechansm wheren each agent s asked to reveal a bd sgnal 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. The second problem s of desgnng a multlateral tradng envronment. Here, there are many buyers and many sellers. Each buyer wants the same capacty on a bundle of goods but each seller sells only one type of good. We 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. The frst problem has receved a lot of attenton n the lterature, [5], [9], [10] beng some key papers. Kelly [5], [6] showed that when agents n a network act as prce-takers (.e., do not act strategcally), the resource allocaton problem can be solved effcently n a dstrbuted manner. Inspred by ths work, a proportonal allocaton (PA) mechansm for dvsble goods was proposed [12]. The case when users are strategc and prce-antcpatng was consdered n [3]. In the case of a sngle dvsble good, they showed that ths can result n an effcency loss of 25%. A smlar result s obtaned for the network case, when players desre multple goods (such as capacty on lnks along a route) when an temzedbd PA mechansm s consdered. In the temzed-bd PA mechansm, the player specfes a bd on each lnk n ts route. Ths however s not realstc for large networks. Thus a sum-bd PA mechansm was proposed n [2]. It was however shown that n ths case, the effcency loss of the proportonal allocaton mechansm can be 100%. A generalzed class of proportonal allocaton (ESPA) mechansms was frst ntroduced n [11], and further analyzed n [12], [15]. These have been shown to be 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 for ease n mplementaton as compared to the VCG class of mechansms. In [2], a smlar generalzaton of the proportonal allocaton mechansm was proposed for a sngle dvsble good. But ts extenson to multple dvsble goods has not been provded. Ths paper s more drectly related to the work of Lazar and Semret [7], [8], [14]. They proposed a VCG-style aucton mechansm for a sngle dvsble good [8]. Attempts have been made to generalze ths mechansm to multple dvsble goods so that t can be useful for network resource allocaton problems [1], [7], [13]. The settng of [7] addresses the case where agents want bundles of lnks (goods), and a dfferent

2 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 [14], chapter 3, wheren sellers place ask bds to sell 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 [7], [14] 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 [13], 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 [4]. However, the results of [4] do not hold n the case where the routng matrx has full rank. The proposals n ths paper are nspred by [8]. 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 utlty functon ˆv(x) = β max{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 other through ts partcpaton, ust 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 domnantstrategy mplementaton of VCG mechansms,.e., truthful reportng of utlty functons s not a domnant strategy equlbrum. Thus, each agent need not have knowledge of the utlty functons of others, nor of the actons beng taken by them. Each agent needs to know only hs own utlty functon but the agents must n some way coordnate ther actons so that they can reach a Nash equlbrum. Thus, they must have some nformaton about the actons beng taken by others. Wthout such nformaton, t s dffcult to envsage how a game wth multple Nash equlbra can be mplemented n a real settng. Thus, as n [8], an teratve algorthm for computng a Nash equlbrum (preferably effcent) s hghly desrable. We present such an algorthm and prove ts convergence n a smple case. Ths can be generalzed to the network case wth multple players. II. THE NETWORK SECOND-PRICE MECHANISM Consder L dvsble goods, L = {1,, L}, wth C l unts of good l beng avalable. Let there be n buyers. Buyer wants a bundle of goods R L and wants the same quantty x of all goods n hs bundle. We wll assume that each buyer has quas-lnear utlty functon u (x) = v (x) ω wth a strctly ncreasng, strctly concave and dfferentable valuaton functon v (x), and ω s the payment made by buyer. The buyers specfy bundles of goods R 1,, R n 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. The auctoneer then determnes an allocaton x = ( x 1,, x n ) as a soluton of the followng optmzaton problem: max s.t. β x (1) : x C l, l L, x [0, d ], = 1,, n. 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 ). (2) Recall that ths s the externalty that the buyer mposes on the other player based on hs revealed utlty functon ˆv (x) = β max(x, d ). Now, the payoff of buyer s u (b, b ) = v ( x (b)) P (b). We wll say an allocaton x s effcent f t s a soluton of the followng max{ v (x ) : : x C l, l L}. (3) Note that such an allocaton wll be Pareto-effcent. The strategy space of the buyer s B = [0, ) [0, C ] where C = mn l R 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 v (x )/ v (x ). Note that ths wll le n [0,1], where 1 wll mean that full effcency s acheved.

3 III. PROPERTIES OF THE NSP MECHANISM A. Exstence of an Effcent Nash Equlbrum We frst show exstence of a Nash equlbrum n the correspondng aucton game by constructon. Theorem 1: There exsts an effcent Nash Equlbrum n the NSP aucton game. Proof: Let x be an effcent allocaton. Then, there exst λ 1,, λ L 0 such that v (x the strategy profle d = x mples β = ) = λ l,. Consder and β = v (d ). Note that ths : λ l,. Gven the bds b of the others as fxed, f buyer changes hs bd b to decrease hs allocaton x 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 (2), 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 by a small > 0. Let the resultng allocaton be x. Then, the change n hs payment (later denoted P ) s gven by P (b, b ) P (b, b ) = β (x = λ l (x l R = l R : ) λ l (x ) ) λ l, (4) where the last nequalty follows for the followng reason: Frst note that due to player changng hs bd, hs allocaton ncreases by, but the allocaton of no other player can ncrease snce each player, already gets d. Further, as player s allocaton ncreases by, there must be at least one player on each lnk l R whose allocaton decreases by. However, ths can happen on other lnks l / R as well, hence the nequalty. But v (x ) = λ l and v s strctly concave. Thus, v (x + ) v (x ) < λ l 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. Note that the above result mples the exstence of an ε- effcent ε-nash equlbrum, a result obtaned n [8] for the specal case of a sngle good. B. Ineffcent Nash Equlbra and Reserve Prces 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 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, player 2 bds β 2 = θ 1 > β 1 = θ 2 f there s a d 2 such that v 2 (d 2 ) θ 2 (d 1 + d 2 1) p v 2 (1 d 1 ) p 0. 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. 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. C. A Suffcent Characterzaton of Effcent Nash Equlbra We wll now provde a suffcent condton for a Nash equlbrum to be effcent. For a Nash equlbrum b, let λ l be the Lagrange multpler correspondng to the capacty constrant for good l and µ be the Lagrange multpler correspondng to the x d demand constrant n the aucton optmzaton (1). Note that the dual of the lnear program (1) s gven by mn{ λ l C l + µ d : λ l +µ θ,, λ l, µ 0}. l (5) Let λ l, µ denote a soluton of the above wth Nash equlbrum b. Theorem 2: Consder a Nash equlbrum b of the NSP game wth µ > 0 for all, then the correspondng allocaton s effcent. Proof: It s suffcent to show that the λ 1,, λ L 0 are such that x (v (x ) λ l ) = 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.

4 Now, from senstvty analyss of lnear programs, we know that for small enough > 0, θ (x x ) = µ. Thus, the change n payment of agent s P = θ (x x ) = (θ µ ) = λ l. The last nequalty follows from complementary slackness: x ( λ l + µ θ ) = 0. Snce b s a Nash equlbrum strategy, t must be that v (x + ) v (x ) < λ l. Now, suppose buyer wants to decrease hs allocaton by. Suppose he changes hs bd to d = d. Then, by complementary slackness x < x. By a smlar argument as above, we can see that the change n payment s λ l and as 0, we get that v (x ) v (x ) < λ l and we establsh that v (x ) = λ l. The above result mples that f at any Nash equlbrum, the allocaton s such that x = d for all, then t must be effcent. D. Dstrbuted Computaton of an Effcent Nash Equlbrum We now provde a dstrbuted algorthm that computes an effcent Nash equlbrum. We wll provde the algorthm for the smple settng of one good and two agents. Ths can be generalzed to multple users and multple goods Let (θ1, 0 x 0 1) = (θ2, 0 x 0 2) = (0, 0) be the ntal bds. The algorthm works as follows: At each step only one player makes a bd. If at the nth step, for n 1, t s player s turn, he checks f v (xn 1 ) > θī n 1. (Here ī denotes the other player). If yes, then he computes the new bd (θ n, xn ) and the resultng allocaton x n ī, of the other user, accordng to: v (x n ) = θ n 1 ī (6) xī = 1 x n (7) θ n = θ n 1 ī + ɛ, (8) for some fxed ɛ > 0, common to all agents. In the case where v (x n 1 ) θī n 1, (9) player stops. Snce player ī cannot further ncrease hs surplus, both players do not have any ncentve to move and the algorthm termnates. If at step n, t s player s turn, and (9) does not hold, snce (8) mples θ n > θ n 2, we have v (xn ) > v (xn 2 ). Strct convexty of valuatons mples x n < x n 2. Thus, x n, xn 2, x n 4,... s ncreasng and x n 1, x n 3,... s decreasng, for = 1, 2. In partcular, the monotoncty of the latter mples that the algorthm wll termnate snce the left hand sde of (9) decreases, whle t s rght hand sde ncreases by a constant amount ɛ > 0, each tme. We now show that f at step n player decdes to stop, then the fnal allocaton (x1 n 1, x n 1 2 ) s arbtrarly close to the effcent allocaton (x 1, x 2 ). The dea s to show that the margnal valuatons, v (xn 1 ), v ī(x n 1 ī ) are close to each other, n whch case the allocatons must be close to the effcent ones as well, as we wll see below. By the mean value theorem, v ī(x ī n 1 ) v ī(x n 3 ī ) = v ī (ψ)(xī n 1 xī n 3 ) (10) for some ψ (xī n 3, xī n 1 ). Usng (6) for player ī, and steps n 1, n 2, (10) yelds x n 3 x n 1 = x n 1 ī x n 3 ī = 2ɛ v ī (ψ). (11) Now, the dual problem of the socal welfare optmzaton problem gves the upper bound v 1 (x 1) + v 2 (x 2) v 1 (x 1 (λ)) + v 2 (x 2 (λ)) λ(x 1 (λ) + x 2 (λ) 1), λ 0, where (x 1 (λ), x 2 (λ)) are such that v 1(x 1 (λ)) = v 2(x 2 (λ)) = λ. (If ths cannot hold for any λ, then x 1 (λ) = x 2 (λ) =.) Now, set yī = x n 1 ī and defne y such that v (y ) = θī n 1. By the stoppng condton (9), v (y ) v (xn 1 ), so y x n 2. Ths and (7), yelds y + yī 1. Hence, v 1 (x 1) + v 2 (x 2) vī(x n 1 ī ) v (x n 1 ) v 1 (x 1) + v 2 (x 2) v 1 (y 1 ) v 2 (y 2 ) θī n 1 (1 y 1 y 2 ) = θī n 1 (x n 1 y ) θī n 1 (x n 1 x n 3 ) θ n 1 ī 2ɛ u ī (ψ), Where the last nequalty follows from (11). ths shows that provded max x [0,1] v (x) < 0, the neffcency can be made arbtrarly small, provded one pcks small enough ɛ. IV. THE NETWORK SECOND-PRICE DOUBLE AUCTION MECHANISM Consder L dvsble goods, L = {1,, L}. Let there be n buyers, buyer wants a bundle of goods R and wants the same quantty x of all goods n hs bundle. Let there be m sellers, seller sells only one good L. 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. Buyers specfes a bundle of goods R and correspondng bd b = (β, d ) whch specfes the maxmum per unt prce β that s wllng to pay, and demands up to d unts of the bundle R. Seller specfes the good L, an ask-bd a = (α, s ) where α s the mnmum per unt prce that

5 s wllng to accept and can supply up to s unts of the good L. The auctoneer then determnes an allocaton ( x, ỹ) as a soluton of the followng optmzaton problem: max s.t. β x α y (12) : x :l=l y, l, x [0, d ],, y [0, s ],. 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 k ( x k x k) k α (ỹ ỹ ). (13) and the money transfer to be made by seller (negatve would means transfer to the seller) T (b, a, a ) = β ( x x ) α k (ȳ k ỹ k). (14) k 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 an allocaton (x, y ) s effcent f t s a soluton of the followng optmzaton problem max{ v (x ) c (y ) : x y, l}. : :l=l (15) 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 s A = [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 now show exstence of a Nash equlbrum n the double aucton game by constructon. Theorem 3: There exsts an effcent Nash Equlbrum n the NSP double aucton game. Proof: Let (x, y ) be an effcent allocaton. Then, there exst λ 1,, λ L 0 such that v (x ) = 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 β = λ l, and α = λ L,. (16) : 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 small > 0, then note that the allocaton of all the other buyers does not change but some sellers on lnk l R sell less. From equaton (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 = λ l. The frst equalty s obtaned ust 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 the total change n allocaton of all sellers of a good l s. Snce v s strctly ncreasng and concave, we get that v (x ) v (x ) < λ l = T, (17).e., the net change n hs payoff s negatve. Now, suppose player wth x 0, changes hs bd to b such that t ncreases hs allocaton x 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) = k β k (x k = k ) λ l (x l k R k k ) λ l, (18) where the last nequalty follows because of the ncrease n the allocaton to buyer on goods R and no change n allocaton to sellers (ther supply constrants are already actve) whch must result n a decrease n allocaton to at least one buyer on each l R. Further, v s strctly ncreasng and concave. Thus, v (x + ) v (x ) < λ l T, (19) From (17) and (19), 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 wth y > 0. Suppose a seller changes hs bd to ncrease y by 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 = λ l and snce c s strctly ncreasng and convex, we get that c (y + ) c (y ) λ l = T. And f any seller wth y > 0, were to change hs bd to decrease hs allocaton by > 0 then the allocaton to other

6 sellers does not change but some buyers get less. Thus, the net change n seller s transfer s T = λ l (x x ) = λ l (x x ) l R : λ l. (20) And agan by strct convexty of c (y ) c (y ) λ l T, whch mples 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. V. CONCLUSIONS AND FURTHER WORK We have proposed an aucton mechansm for allocatng 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 [8] 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 whch s ε-effcent. However, not all Nash equlbra are effcent as we show through an example. But we present a dstrbuted algorthm that yelds an ε-effcent ε-nash equlbrum. Ths algorthm s dfferent from that presented n [14] whch s dffcult to generalze to the network case. We also present a doublesded mechansm whch has a Nash equlbrum wth effcent allocaton. Our work s also related to [4]. 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. As part of further work, we would lke to obtan a dstrbuted algorthm for computng the effcent Nash equlbrum n the double-sded case. [6] F. KELLY, A. MAULLO AND D. TAN, Rate control n communcaton networks: Shadow prces, proportonal farness and stablty, J. Operatonal Research Soc., 49: , [7] A. LAZAR AND N. SEMRET, The progressve second prce aucton mechansm for network resource sharng, Proc. Int. Symp. on Dynamc Games and Applcatons, [8] A. LAZAR AND N. SEMRET, Desgn and analyss of the progressve second prce aucton for network bandwdth sharng, Telecommuncaton Systems - Specal ssue on Network Economcs, [9] S. LOW AND P. VARAIYA, A new approach to servce provsonng n ATM networks, IEEE/ACM Trans. on Networkng, 1(5): , [10] J. MACKIE-MASON AND H. VARIAN, Prcng congestble network resources, IEEE J. Selected Areas n Comm., 13(7): , [11] R. MAHESWARAN AND T.BASAR, Socal welfare of selfsh agents: Motvatng effcency for dvsble resources, Proc. CDC, [12] R. MAHESWARAN AND T.BASAR, Nash equlbrum and decentralzed negotaton n auctonng dvsble resources, J. Group Decson and Negotaton, 13(2), [13] P. MAILLE AND B. TUFFIN, Mult-bd auctons for bandwdth allocaton n communcaton networks, Proc. IEEE INFOCOM [14] N. SEMRET, Market Mechansms for Network Resource Sharng, PhD Dssertaton, Columba Unversty, [15] S.YANG AND B.HAJEK, Revenue and stablty of a mechansm for effcent allocaton of a dvsble good, manuscrpt, 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, [2] B.HAJEK AND S.YANG, Strategc buyers n a sum-bd game for flat networks, manuscrpt, [3] R.JOHARI AND J.TSITSIKLIS, Effcency loss n a network resource allocaton game, Mathematcs of Operatons Research, [4] R.JOHARI AND J.TSITSIKLIS, Communcaton requrements of VCGlke mechansms n convex envronments, manuscrpt, [5] F. KELLY, Chargng and rate control for elastc traffc, Euro. Trans. on Telecommuncatons, 8(1):33-37, 1997.