VCG-Kelly Mechanisms for Allocation of Divisible Goods: Adapting VCG Mechanisms to One-Dimensional Signals

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST VCG-Kelly Mechanisms fo Allocation of Divisible Goods: Adapting VCG Mechanisms to One-Dimensional Signals Sichao Yang and Buce Hajek Abstact The VCG-Kelly mechanism is poposed, which is obtained by composing the communication efficient, onedimensional signaling idea of Kelly with the VCG mechanism, poviding efficient allocation fo stategic buyes at Nash equilibium points. It is shown that the evenue to the selle can be maximized o minimized using a paticula one-dimensional family of suogate valuation functions. Index Tems I. INTRODUCTION In this pape, we addess the mechanism design poblem, which specifies the ules fo allocation and payment, fo esouce allocation in a netwok with a single netwok opeato and multiple stategic buyes. The netwok opeato, without knowing the valuation functions of the buyes, aims to select an outcome that is efficient, i.e. maximizes the sum of buyes valuations. Kelly and his colleagues [1, 2] showed that, unde the fundamental assumption that all the buyes ae pice-takes, the optimization poblem can be solved in a decentalized way. We call thei method the Kelly mechanism. Johai and Tsitsiklis [3] showed that, fo stategic buyes using the Kelly mechanism on each of the links, the sum of valuations at the equilibium point is no less than 75% of the efficient value. We give two examples which show the undesiable pefomance of the Kelly mechanism fo stategic buyes when each buye pesents a single bid fo a path, athe than bids fo individual links. The celebated Vickey-Clak-Goves VCG mechanism [4 6] is such that tue epoting of valuation functions is a dominant stategy equilibium, but in the context of divisible goods, such as communication ate, the valuation functions ae infinite dimensional, and so have pohibitive communication equiements. In [7, 8], efficient mechanisms with onedimensional bids ae poposed fo a single link. We intoduce a mechanism with one dimensional bids fo esouce allocation in a netwok in which efficiency is achieved at Nash equilibium points NEPs. We call the mechanism the VCG-Kelly mechanism. Although buyes do not have dominant stategies unde the VCG-Kelly mechanism Manuscipt eceived July 1, 2006; evised Febuay 15, The authos ae with the Depatment of Electical and Compute Engineeing, and the Coodinated Science Laboatoy, Univesity of Illinois at Ubana-Champaign {syang8,b-hajek}@uiuc.edu. Digital Object Identifie /JSAC xx /07/$25.00 c 2007 IEEE in contast to the VCG mechanism, thei bids ae onedimensional. Simultaneously and independently of ou wok, Johai and Tsitsiklis [9] developed a family of VCG-like mechanisms using simple scale bids pe playe. Thei wok is moe geneal than ous hee, in that thei famewok involves a convex constaint set and is not limited to netwok esouce allocation settings. Howeve, in the moe limited setting of ou pape, unde a egulaity assumption, we find that the payoff function of a buye is a quasiconcave function of the buye s bid, that all NEPs ae efficient, that the NEP is unique if the efficient allocation is unique, and we chaacteize the ange of possible payments. Semet [10] poposed an efficient allocation mechanism in which each buye submits a two dimensional bid, one dimension fo unit pice, and the othe fo maximum quantity. Recently, othe wok using two dimensional bids appeaed: [11, 12]. The mechanism of [12] can be viewed as an instance of the VCG-Kelly mechanism, but fo two-paamete suogate functions. The ecent pape of Ausubel [13] descibes an allocation mechanism which is a genealization of ascending pice auctions, which can povide efficient allocation using one dimensional bids fo the allocation poblem we addess, but the payments ae computed by integating ove pice tajectoies. The pape is oganized as follows. Section II descibes the netwok model, the Kelly mechanism, with eithe pice-taking o stategic buyes including ou two examples, and the VCG mechanism. Section III intoduces the VCG-Kelly mechanism and pesents the esults about it mentioned above. Section IV discusses futue wok, and poofs ae given in the appendix. II. SYSTEM MODELS Conside a netwok with a finite set of buyes, R and a finite set of links, J. The buyes ae in one-to-one coespondence with paths, whee a path has an associated set of links. We wite j if link j is in the path of buye. The pathlink incidence matix A is defined by A j = 1 if j, and A j = 0 othewise. We efe to buyes in gende neutal tems, because in the context of communication netwoks, they ae typically compute pocesses. The buyes have elastic demands and the valuation function of a buye is epesented by U x, which is an inceasing, concave, and continuously diffeentiable function of its ate allocation x, ove R +. In this pape we assume that the valuation functions of the buyes ae in monetay units, so

2 2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 that if buye makes a payment of m fo ate x, the buye s payoff has the quasilinea fom Ux m. The links have fixed capacities C j fo all j J. A. System Poblem and Efficient Allocations A ate allocation vecto x = x : R is called efficient 1 if it is the solution to the system poblem: Buye 1 U 1 Buye 2 U 2... Buye U w 1 w 2 w log log log Netwok Poblem Solve Allocations x Pices p System Poblem U, A, C max U x 1 R whee X C = {x : x 0 and Ax C}. The system poblem is a convex optimization poblem with linea constaints, so fo any solution x thee is an associated vecto µ of Lagange multiplies satisfying the following optimality conditions: U x µ j, with equality if x > 0, R 2 j µ j 0, with equality if x < C j, j J 3 :j Thoughout the pape, X denotes the set of all efficient allocation vectos. B. Kelly Mechanism Kelly and his colleagues [1, 2] poposed a decentalized method fo solving the system optimization poblem. In this section, we biefly descibe thei method fom the mechanism design viewpoint. In the Kelly mechanism, the communication between any buye and the netwok consists of a one-dimensional bid w given by buye to the netwok, and a pice 2 p given by the netwok to buye. Fom the bid w of buye, the netwok constucts a suogate valuation function w log x. The netwok detemines the allocation and payment by maximizing the sum of the suogate valuations: Kelly mechanism allocation ule A, C: x K w = ag max w log x, 4 R The optimization poblem associated with this allocation ule is called the netwok poblem in [2]. To ensue that the mapping is well defined, we equie that x K w = 0 if w = 0. The solution x K w can be chaacteized as follows. It is the vecto x X C which, togethe with an associated vecto µ of Lagange multiplies, satisfies the following conditions. w x x = 0, R such that w = 0 = j µ j, R such that w > 0 µ j 0, with equality if x < C j, j J :j 1 An altenative definition of efficiency, which accounts fo the payments of the buyes, is commented on in Section III-B. 2 In the pape, pice always epesents pice-pe-unit. Fig. 1. The Kelly mechanism Kelly mechanism payment ule A, C: m K = p x whee p = j µ j. Figue 1 depicts the Kelly mechanism. The feedback aow is not a pat of the payment and allocation ules of the mechanism, but is shown to indicate that the equilibium bid of each buye depends on the bids of the othe buyes. Next, we discuss two scenaios: the Kelly mechanism applied to pice-taking buyes as in [1, 2] and the Kelly mechanism applied to stategic buyes. 1 Pice-taking buyes: A pice-taking buye sees p as a fixed pice set by the netwok and it ties to solve the poblem of maximizing the payoff function Π w = U w p w, which is called the use poblem in [2]. It is shown in [2] that, if x and µ satisfy 2 and 3, then fo bids w = x j µ j and pices p = j µ j, the following is tue: x solves the system poblem, and fo each, w solves the use poblem. Thus, the solution to the system poblem yields solutions to the use and netwok poblems. The system poblem is decomposed into a netwok poblem, which does not involve the valuation functions, and the use poblems, which do not involve the netwok topology. 2 Stategic buyes: A stategic buye takes into account how its pice is influenced by its bid, w. This induces a game among the buyes, with the payoff fo buye given by w Π w ; w = U w. p w Next, in two examples, we show that if buyes using the Kelly mechanism ae stategic, NEPs may not exist, and if an NEP does exist, the atio between the sum of valuations at the NEP to the maximum possible sum the efficiency atio can be abitaily small 3. Conside a netwok with L links, numbeed 1 though L, with capacities C l. Thee ae L+1 buyes, numbeed 0 though L. The path of buye zeo contains all the links. The path of buye l fo 1 l L contains just the single link l. We define the ate allocation in the seial netwok as follows: 4 If w = 0, then x = 0. Othewise, the netwok allocates ate to buye 3 These two examples wee fist pesented in [14]. 4 This is a slight modification of the Kelly mechanism defined ealie, because a single-link use bidding zeo can sometimes get a nonzeo ate, but only if the pice on the associated link is zeo.

3 YANG and HAJEK: VCG-KELLY MECHANISMS FOR ALLOCATION OF DIVISIBLE GOODS 3 U 0 U U 2 8 8! 2 w 0, w 1,w w Fig. 3. Nonconcavity of payoff function of buye 2 Fig. 2. The uniqueness of x 0 and ate x l = C l to each single link buye, whee is selected to minimize the sum of the suogate valuations: w 0 log + w l log C l. 5 1 l L Example 1: Suppose L = 2, and W = {w : x i w > 0, i = 0, 1, 2}, o equivalently W = {w 0 > 0, w 1 > 0, w 2 > 0} w 0 { < C 2, w 0 > 0, w 1 > 0, w 2 = 0} w 0 + w 1 C 1 w 0 { < C 1, w 0 > 0, w 1 = 0, w 2 > 0}. w 0 + w 2 C 2 It can be shown that all NEPs ae in W, and thee exists a bid vecto w in W satisfying the fist ode necessay condition fo an NEP. Let x =, C 1, C 2 be the coesponding ate allocation. The value of fo all such w s is uniquely defined by U 0 x { 0 U = max 1 C 1, U 2 C } 2 6 C 1 C 2 If the thee cuves U 0 x0, U 1 C1x0 C 1 and U 2 C2x0 C 2 do not all meet at a single point, then the value of w w w is also unique. Conside the case C 1 = 5, C 2 = 8, U 0 x = U 2 x = x, and U 1 x = 1 8 x. Thee is no tiple intesection, w = 2, 0, 2 is the unique bid vecto satisfying the fist ode necessay condition, and its associated ate vecto is x = 4, 1, 4. Howeve, Π 2 w 0, w 1, 0 = 3 > Π 2 w 0, w 1, w 2 = 2 so that w is not an NEP. Also because the point satisfying the fist ode necessay condition is unique, we conclude that thee is no NEP in this example. Intuitively, in this specific case, C 1 is close to, so that buye 2 does not lose much capacity by deceasing its bid fom 2 to 0. Figue 3 shows the payoff function Π 2 w 0, w 1, w 2 as a function of w 2. Note that it is not concave, which helps explain the lack of an NEP. Example 2: Let C l = 1 and U l x = x fo 1 l L, and U 0 x = γx, whee 0 < γ < L. In this case, the efficient allocation is = 0 and x l = 1 fo 1 l L. A bid vecto 2 w is an NEP if and only if w 0 = γ γ+1 and 1 l L w l = γ γ+1. While the NEP is not unique, all NEPs have the same 2 ate allocation vecto = γ γ + 1, x l = 1 γ l L Thus, the sum of valuations fo the NEP is γ 2 +L/γ+1 and the esulting efficiency atio fo L γ is γ 2 +L/Lγ+1. As L the efficiency atio conveges to 1/γ + 1. Also, taking γ to be an intege and L = γ 2 yields the efficiency atio 2/1 + γ. The atio conveges to zeo as γ. Thus, the loss of efficiency at the NEP can be made abitaily close to 100% by selecting γ to be a lage enough intege, and L = γ 2. C. VCG mechanism The VCG mechanism, defined as follows, takes epoted valuation functions, W, R, as inputs. VCG mechanism allocation ule A, C: x VCG W = ag max W x 7 R VCG mechanism payment ule A, C: m VCG W = max,x =0 W s x s W s x VCG s 8 The payment of buye in the VCG mechanism is othe buyes maximum suplus with buye not pesent minus the othe buyes suplus with buye pesent. The payoff of buye is Π x VCG = U x VCG + max W s x VCG,x =0 s W s x s 9 The last tem in 9 does not depend on x. If buye epots tuthfully, i.e. W U, then, by 7, the selle seeks a choice of x that maximizes the sum of the fist two tems of the buye s payoff. Hence, tuth epoting is an optimal stategy fo buye, no matte what the othe buyes epot. All the buyes epoting thei tue valuations is an NEP fo the VCG mechanism, which implies the allocation deived at the NEP of the VCG mechanism is efficient.

4 4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 Fig. 4. Buye 1 U 1 Buye 2 U 2... Buye U w 1 w 2 w VCGKelly Mechanism f 1 f 2 f The VCG-Kelly mechanism VCG Mechanism III. THE VCG-KELLY MECHANISM Allocations x Chages m We popose a new mechanism defined as follows: the buyes send one-dimensional bids and the mechanism foms suogate valuation functions as in the Kelly mechanism, and then the VCG mechanism is applied to the suogate valuation functions. We name the mechanism the VCG-Kelly mechanism because it combines the ideas of the VCG mechanism and the Kelly mechanism. See Figue 4. We intoduce a genealization beyond log functions, so that the suogate functions ae V w, x = w f x fo R, whee the f s ae stictly inceasing, stictly concave, and twice diffeentiable with f 0+ = + and f x < 0 ove R +. We allow f 0 =, with the convention that V 0, 0 = 0. One family of functions fom which to select the f s is { 1 α 1 φ α x 1α, if α > 0 and α 1, x = 10 log x, if α = 1, which is intoduced in [15]. The VCG-Kelly mechanism is then defined as follows. VCG-Kelly mechanism allocation ule V, A, C: 7.5 x VCGK w = ag max V w, x. 11 R VCG-Kelly mechanism payment ule V, A, C: w = max,x =0 V s w s, x s V s w s, x VCGK s 12 The ules 11 and 12 ae the same as 7 and 8, but with bids W x eplaced by the suogate functions V w, x, based on one-dimensional bids. The ules x VCGK and, togethe with the tue valuation functions, U, detemine a game among the buyes. The payoff of buye is Π w ; w = U x VCGK w w. 13 A bid vecto w NE is an NEP of the game if and only if Π w NE ; wne Π w ; w NE, w 0, R. A. Efficiency of NEPs fo the VCG-Kelly mechanism The following assumption ensues that each buye is faced with competition on each link of its path. Special Buyes Assumption Fo each link j thee ae at least two single-link buyes using the link, with valuation functions satisfying U 0+ = +. Such buyes ae called special. Special buyes paticipating in the VCG-Kelly mechanism always submit stictly positive bids. If the special buyes assumption does not hold, thee can be many NEPs induced by the VCG-Kelly mechanism that ae not efficient. Howeve, instead of using the VCG-Kelly mechanism, one can use an ɛ-elaxation of the VCG-Kelly mechanism, obtained by intoducing two fictitious buyes pe link, with valuation functions of the fom ɛ logx. As ɛ, the payments of the fictitious buyes convege to zeo. These issues ae exploed futhe in [8] in the case of a single link. Next we conside the chaacteizations of efficient allocations and the VCG-Kelly allocation ule unde the special buyes assumption. The special buyes assumption insues that all links ae satuated fo the efficient allocation, so that the optimality conditions 2 and 3 can be simplified. Unde the special buyes assumption, x X C is a solution to the system poblem efficient if and only if thee is an associated vecto µ with nonnegative coodinates, so that U x j µ j, R with x = 0 14 U x = j µ j, R with x > 0 15 Ax = C 16 Conside a fixed link j. Let denote one of the special onelink buyes using that link. Then U j x = µ j. Thus, although thee can be multiple distinct efficient allocation vectos x, fo each choice of x, the coesponding vecto µ is uniquely detemined. Moeove, unde the special buyes assumption and the assumption that the valuation functions of the buyes ae continuously diffeentiable, the dual poblem is stictly convex, so that µ does not depend on the choice of x. Let λ = j µ j fo each R, o in vecto fom, λ λ = Aµ. Then λ is the pice paid by buye fo the Kelly mechanism with pice-taking buyes denoted by p in Section II-B. We denote the set of special buyes by R s, and the space of admissible bid vectos by W = {w R R + : w > 0 R s }. Unde the special buyes assumption, all links ae satuated by x VCGK w fo any w W. Theefoe, x VCGK w is chaacteized by the following conditions. Fo some vecto µ, with nonnegative coodinates, x = 0, R with w = 0 17 w f x = j µ j, R with w > 0 18 Ax = C 19 The same obsevations made fo the efficient allocation vectos imply that the choice of µ in 18 is uniquely detemined by w, and we can hence denote it as µ VCGK w. Let λ VCGK j µvcgk = j fo each R, o in vecto fom, λ VCGK = Aµ VCGK.

5 YANG and HAJEK: VCG-KELLY MECHANISMS FOR ALLOCATION OF DIVISIBLE GOODS 5 The following poposition states some egulaity popeties of the VCG-Kelly mechanism unde the special buyes assumption. It also identifies λ VCGK as the maginal pice fo buye. Poposition 3.1: Regulaity of the VCG-Kelly mechanism Suppose the special buyes assumption holds. The mapping x VCGK : W X C is continuous. Fo any buye, and w fixed, x VCGK / > 0, λ VCGK / > 0, / / x VCGK / = λ VCGK w, and the payoff function Π w, w, is stictly quasiconcave in w. Let W NE denote the set of all NEPs fo the game, with stategy space W and payoff functions given by 13, induced by the VCG-Kelly mechanism. Ou main theoem on the existence and efficiency of NEPs is the following. Poposition 3.2: Suppose the special buyes assumption holds. The estiction of the mapping x VCGK to W NE is a bijection fom W NE to X. In paticula, NEPs exist and coespond to efficient allocations, and if the efficient allocation is unique fo example, if the tue valuation functions U ae stictly concave, then the NEP is unique. B. Pices and evenue unde the VCG-Kelly mechanism Thee is consideable flexibility in choosing a VCG-Kelly mechanism, because of the wide ange of possible choices of the functions f s, s R, used to define the suogate valuation functions. Fo a given NEP, λ epesents the maginal pice fo buye unde the VCG-Kelly mechanism at an NEP, fo any choice of the suogate valuation functions. In contast, the pices not the maginal pices, depend heavily on the choice of the suogate valuation functions. Fix an efficient allocation vecto x, and conside a buye with x > 0. Thee is a with allocation x, and the pice paid by = mvcgk w NE /x. The next poposition identifies the ange of possible pices as the suogate functions vay. Poposition 3.3: Suppose the special buyes assumption holds. Fix a choice of the valuation functions U, fix an efficient unique NEP w NE buye at that NEP is defined by p NE allocation x, and conside a buye such that x > 0, Then, unde the NEP of the VCG-Kelly mechanisim, p NE 0, λ. Futhemoe, if the function f s used to define the suogate valuation function fo each buye s R is taken to be φ α defined in 10, then lim α + p NE λ = 0 and lim α 0 p NE = Ċoollay 3.4: The total evenue of the selle is in the inteval 0, j J µ j C j. If the function f fo each buye is taken to be φ α, then the total evenue conveges to zeo as α +, and to j J µ j C j as α 0. We emak that an altenative definition of efficiency in the context of mechanism design is to find an allocation and payment ule so that, at the NEP, the sum of the payoff functions of the buyes is maximized. That is, such definition includes the payments of the buyes. Poposition 3.3 shows that the VCG-Kelly mechanism, using functions f s of the fom φ α, is asymptotically efficient in this altenative sense, in the limit as α, because the allocation maximizes the sum of the valuation functions, while the sum of payments conveges to zeo. IV. FUTURE WORK Futue wok includes exploing the space of suogate valuation functions futhe, investigation of dynamics fo aiving at NEPs, application to othe contexts, and implementation issues, such as inteaction with existing netwok potocols, such as TCP. APPENDIX The poofs of the popositions in this pape, including lemmas, ae given in this appendix. Lemma 1.1: The mapping x VCGK : W X C is continuous. Poof: Suppose w n w as n, such that w n W fo all n and w W. Let x n = x VCGK w n fo each n. It must be shown that x n x VCGK w. Since X C is compact, we assume x n x fo some x X C, and then it suffices to show that x = x VCGK w. Let Φ denote the function on W X C defined by Φw, x = R V w, x. Let x X C and ɛ > 0 be abitay. By petubing x slightly, we can obtain x ɛ X C, that has stictly positive coodinates, and satisfies Φw, x Φw, x ɛ + ɛ. We have Φw, x Φw, x ɛ + ɛ = lim n Φwn, x ɛ + ɛ lim sup Φw n, x n + ɛ n Φw, x + ɛ, whee the last inequality is a consequence of the fact that Φ is uppe semicontinuous. Since ɛ is abitay, if follows that Φw, x Φw, x. Since x X C is abitay, it follows that x = x VCGK w, which completes the poof of the lemma. Lemma 1.2: Let I be any subset of R which includes the special buyes. Then fo w 0 fo W I, both x VCGK w and µ VCGK w ae continuously diffeentiable functions of w : I ove the set 0, + I. Poof: The poof is given fo the special case I = R, which is sufficient, because buyes in R I can be ignoed. The optimality conditions can be witten as F x VCGK, µ VCGK, w = 0, whee w f F x, µ, w = x j µ j, R :j x C j, j J The function F is continuously diffeentiable and F x F D A T = µ A 0 w., 20 whee D = diagw f xvcgk. Obviously, D is positive definite. Fo any J-dimensional vecto α 0, A T α 0 because A has full ank because of the special buyes assumption. Then, α T AD 1 A T α = A T α T D 1 A T α > 0, because D is positive definite. Hence, AD 1 A T is positive definite. Thus, the matix Q defined by Q = AD 1 A T 1 exists and is also positive definite. It is then staightfowad to check that the matix on the ight hand side of 20 is nonsingula, and its invese is: H D 1 A T Q QAD 1 21 Q

6 6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 6, AUGUST 2007 whee H = D 1 D 1 A T QAD 1. Theefoe, by the implicit function theoem [16], the functions x VCGK w and µ VCGK w ae continuously diffeentiable, and o x VCGK µ VCGK x VCGK µ VCGK = F x 1 F F µ w = Hdiagf 22 = QAD 1 diagf. Lemma 1.3: Fix a buye R and the othe buyes bids w. Then the patial deivatives of x VCGK and λ VCGK with espect to w ae stictly positive. Poof: Since λ VCGK = A T µ VCGK, λ VCGK = AT QAD 1 diagf. 23 Let e = 0,, 1,, 0 T be the elementay vecto with -th enty 1. Then, the -th diagonal enty of A T QA satisfies e T AT QAe > 0, because Ae 0 and Q is positive definite. Notice that, in 23, the diagonal elements in λvcgk have the same sign as the diagonal elements in A T QA because D 1 diagf is a positive definite diagnal matix. Hence, λ VCGK > 0 fo any R. Let RB denote the ange of a matix B and N B the null space of B. We claim that multiplication by the symmetic matix G = D 1 2 A T QAD 1 2 is equivalent to othogonal pojection onto RD 1 2 A T. Indeed, on one hand if β RD 1 2 A T then β = D 1 2 A T α fo some vecto α, and Gβ = β, and on the othe hand if β RD 1 2 A T then β N AD 1 2 and Gβ = 0. Theefoe, multiplication by I G is equivalent to othogonal pojection onto R D 1 2 A T. Thus u T {I G}u 0 fo any vecto u R R with equality if and only if u RD 1 2 A T. Theefoe v T Hv 0 fo any vecto v R R with equality if and only if v RA T. The special buyes assumption implies that fo an appopiate odeing of the links, A has the stuctue A = I JJ : I JJ : Ã, which implies that e RA T. It follows that the diagonal enties of the matix H ae stictly positive, so that xvcgk > 0 fo any R. Remak 1.1: Fo any ate allocation x which satuates all the links and abitay vecto of Lagange multiplies µ, and vecto λ defined by λ = Aµ, µ j µ j C j λ s x s = µ j x s = s R s R j s j J s:j s x s = j J Theefoe, if x and x ae two ate allocation vectos that both satuate all the links, λ s x s x s = s R Lemma 1.4: Fo any buye and w fixed, / / x VCGK / = λ VCGK w, and the payoff function Π w, w of buye is stictly quasiconcave in w. A vecto w W is an NEP if and only if the following conditions hold: U xvcgk w ; w = µ VCGK j w ; w, if w > 0, j U 0 µ VCGK j 0; w, if w = 0. j Poof: The payoff function Π w ; w, which has the fom 13, is a continuously diffeentiable function of w by Lemma 1.2. The patial deivative of w ; w with espect to bid w is w ; w = s = s = λ VCGK w s f s xvcgk s λ VCGK s w xvcgk s w w xvcgk s w w xvcgk w. 25 The last equaliy above holds by 24. The fist pat of the lemma, identifying λ VCGK w as the maginal pice, is poven by 25. Also, Π w ; w = xvcgk w {U x VCGK w λ VCGK w}. x VCGK By Lemma 1.3, > 0 and U xvcgk w λ VCGK w is a stictly deceasing function of w. Theefoe, the payoff Π w ; w is a stictly quasiconcave function of w. Fo othe bids w s fixed, w is the best esponse of the buye if and only if the conditions given in the lemma hold. Poof of Poposition 3.1 establish the the poposition. Lemmas 1.1, 1.3, 1.4, and Poof of Poposition 3.2 It is shown that i x VCGK maps W NE into X, ii the mapping is sujective, and iii the mapping is injective. If w is an NEP, the conditions of Lemma 1.4 hold. Because x VCGK w = 0 if and only if w = 0, the vectos x VCGK and µ VCGK satisfy the efficiency conditions Theefoe, x VCGK w is efficient, so that x VCGK maps W NE into X. Fix an efficient allocation x. The optimality conditions ae satisfied, fo a vecto µ uniquely detemined by x. Let w = 0 if x = 0, and w = j µ j/f x if x > 0. Then conditions ae satisfied, so that x VCGK w = x and µ VCGK w = µ. Conditions and imply that the conditions of Lemma 1.4 ae also satisfied, so that w is an NEP. The equied mapping is sujective. Fix an efficient allocation x and suppose x VCGK w = x fo some NEP w. Then x and the associated Lagange multiplies µ VCGK w satisfy the conditions of Lemma 1.4, and the conditions The conditions of Lemma 1.4, unde the special buyes assumption, show that µ VCGK w is completely detemined by the choice of x, and then conditions show that w is also uniquely detemined by the choice of x. So the equied mapping is also injective, so that it is bijective, as was to be poved. Poof of Poposition 3.3 Due to the special buyes assumption, any allocation given up by buye has positive

7 YANG and HAJEK: VCG-KELLY MECHANISMS FOR ALLOCATION OF DIVISIBLE GOODS 7 value fo some othe buye, so that theefoe p NE function w NE > 0, and > 0. Motivated by the definition of the payment in 12, we let x = agmax ws NE f s x s, 26,x =0 The special buyes assumption implies that x and x both satuate all the links, so by Remak 1.1, λ s x s x s = λ x. 27 Compaing 15 and 18 shows that ws NE f sx s = λ s fo all s such that w s > 0. This, the fact the functions f s ae stictly concave, the fact x s x s fo at least one s, and 27 yield: w NE = = < = λ x w NE s f s x s f s x s λ sf s x s f s x s f sx s λ s x s x s 28 Thus p NE < λ. So we ve poved pne 0, λ. Next, suppose that f s x s = φ α x s fo all s, wee α > 0. Then 28 becomes α, w NE α = 1 1 α λ s x s x s x s α x s If α > 1 then 1 α < 0, and 29 yields α, w NE α 1 λ α 1 sx s if α > 1. Theefoe, lim α m α, w NE α = 0. By 27 and 29, = α, w NE α 1 1 α λ x + x λ α s x s s 1 x s 29 So, lim α 0 α, w NE α = λ x, which implies that lim α 0 p NE = λ. ACKNOWLEDGEMENT The authos wish to thank Pof. Steven R. Williams and the eviewes fo thei useful comments. This wok was suppoted in pat by the National Science Foundation unde gants NSF CNS and NSF ECS REFERENCES [1] F. P. Kelly, Chaging and ate contol fo elastic taffic, Euopean Tansactions on Telecommunications, vol. 8, pp , [2] F. P. Kelly, A. K. Maulloo, and D. Tan, Rate contol in communication netwoks:shadow pices, popotional fainess and stability, Jounal of the Opeational Reseach Society, vol. 49, pp , [3] R. Johai and J. N. Tsitsiklis, Efficiency loss in netwok esouce allocation game, Mathematics of Opeations Reseach, vol. 29, no. 3, pp , [4] W. Vickey, Countespeculation, auctions and competitive sealed tendes, Jounal of Finance, vol. 16, no. 1, pp. 8 37, [5] E. Clak, Multipat picing of public goods, Public Choice, vol. 2, pp , [6] T. Goves, Incentives in teams, Econometica, vol. 41, no. 4, pp , [7] R. T. Maheswaan and T. Başa, Social welfae of selfish agents; motivating efficiency fo divisible esouces, in Poceedings of 43d IEEE Confeence on Decision and Contol, Bahamas, Decembe [8] S. Yang and B. Hajek, Revenue and stablity of a mechanism fo efficient allocation of a divisible good, Pepint, [9] R. Johai and J. N. Tsitsiklis, Communication equiements of VCGlike mechanisms in convex envionments, in Poceedings of Alleton Confeence, [10] N. Semet, Maket mechanisms fo netwok esouce shaing. PhD thesis, Columbia Univesity, [11] T. Stoenescu and J. Ledyad, Implementation in nash equilibia of a ate allocation poblem in netwoks, in Poceedings of the 44th Annual Alleton Confeence on Communications, Contol, and Computing, Septembe [12] A. Dimakis, R. Jain, and J. Waland, Mechanisms fo efficient allocation in divisible capacity netwoks, in Poceedings of the 45th IEEE Confeence on Decision and Contol, Decembe [13] L. Ausubel, An efficient dynamic auction fo heteogeneous commodities, The Ameican Economic Review, vol. 96, pp , [14] B. Hajek and S. Yang, Stategic buyes in a sum bid game fo flat netwoks, in IMA Wokshop 6: Contol and Picing in Communication and Powe Netwoks, Univesity of Minnesota, Mach [15] J. Mo and J. Waland, Fai end-to-end window-based congestion contol, IEEE/ACM Tansactions on Netwoking, vol. 8, pp , Octobe [16] D. P. Betsekas, Nonlinea Pogamming. Athena Scientific, 2 ed., Sichao Yang eceived the B.E. and M.E. degees in Electonic Engineeing fom Shanghai Jiao Tong Univesity. Since 2001 he has been with the Univesity of Illinois at Ubana-Champaign, whee he obtained the M.S. degee in mathematics and is cuently pusuing the Ph.D. degee in electical and compute engineeing. His eseach inteests include communication and compute netwoks, mechanism design and auction algoithms, combinatoial optimization. Buce Hajek M 79-SM 84-F 89 eceived a B.S. in Mathematics and an M.S. in Electical Engineeing fom the Univesity of Illinois at Ubana-Champaign in 1976 and 1977, and a Ph. D. in Electical Engineeing fom the Univesity of Califonia at Bekeley in He is a Pofesso in the Depatment of Electical and Compute Engineeing and in the Coodinated Science Laboatoy at the Univesity of Illinois at Ubana-Champaign, whee he has been since His eseach inteests include communication and compute netwoks, stochastic systems, combinatoial and nonlinea optimization, and infomation theoy. He seved as Associate Edito fo Communication Netwoks and Compute Netwoks fo the IEEE Tansactions on Infomation Theoy , as Edito-in- Chief of the same Tansactions , and as Pesident of the IEEE Infomation Theoy Society D. Hajek was a winne of the 1973 USA Mathematical Olympiad, and he eceived the Eckman Awad of the Ameican Automatic Contol Council1982, an NSF Pesidential Young Investigatos Awad 1984, an Outstanding Pape Awad fom the IEEE Contol Systems Society He is a membe of the US National Academy of Engineeing and he eceived the 2003 IEEE Kobayashi Compute Communications Awad.