Distributed Mechanism Design for Unicast Transmission

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1 Dstrbuted Mechansm Desgn for Uncast Transmsson Nasmeh Heydarben Department of Electrcal and Computer Engneerng Unversty of Mchgan, Ann Arbor, MI, USA Achlleas Anastasopoulos Department of Electrcal and Computer Engneerng Unversty of Mchgan, Ann Arbor, MI, USA Abstract The standard (Hurwtz-Reter) Mechansm Desgn framework requres that agents broadcast ther messages to a central authorty that subsequently determnes allocaton (and tax/subsdes) for each user. We consder a settng where agents can only communcate messages to ther neghbors defned by a gven communcaton graph. As a result, allocaton and tax functons for each user can only depend on local neghborhood messages. Ths gves rse to a new, dstrbuted, class of mechansms. In ths paper we propose such a mechansm for the problem of rate allocaton over a network wth uncast transmsson. The proposed mechansm s dstrbuted, t fully mplements the optmal allocaton n Nash equlbra (e, there are no extraneous equlbra), s ndvdually ratonal and weak budget balanced. The message space dmenson of the proposed mechansm grows lnearly wth the number of agents n the network. Index Terms mechansm desgn, rate allocaton, decentralzed optmzaton, strategc users, Nash equlbrum I. INTRODUCTION In networks wth a large number of heterogeneous agents, determnng socal welfare maxmzng allocaton of goods and servces s sgnfcant and complcated because agents have prvate valuaton functons over ther allocated goods and they may not be wllng to share these functons wth a central authorty. Moreover, t may not even be possble to transmt these functons due to the communcaton overhead. Furthermore, the agents may be strategc and wsh to maxmze ther beneft and so they could choose to le whenever t s proftable. Mechansm desgn s an approprate framework to deal wth large networks of strategc agents. It s a tool that helps to decentralze the process of reachng a desrable global goal wthout the need for prvate nformaton of agents. It has been used n varous research areas, such as prvate/publc goods allocatons [1] [3], rate and resource allocatons [4] [7], data securty on server farms [8], etc. The standard (Hurwtz-Reter) mechansm desgn framework requres that agents broadcast ther messages to a central authorty that subsequently determnes allocaton (and tax/subsdes) for each user. Ths can cause a communcaton overhead n large networks even when the message space of the desgned mechansm s small. Furthermore, there may be some communcaton constrants for message transmsson. To allevate ths problem, we consder a settng where agents can only communcate messages to ther neghbors defned by a gven communcaton graph. As a result, allocaton and tax functons for each user can only depend on local neghborhood messages. Ths gves rse to a new, dstrbuted, class of mechansms. We call ths framework, Dstrbuted Mechansm Desgn (DMD). An alternatve vewpont of DMD s that t tres to augment the area of dstrbuted optmzaton (see [9], [10]) wth the requred tools to handle strategc agents. In [11] the authors ntroduced a dstrbuted mechansm for the problem of Walrasan and Lndhal allocaton. In ths paper, we propose a dstrbuted mechansm for the problem of rate allocaton over a data communcaton network wth uncast transmsson. Non-dstrbuted mechansms for effcent allocaton n the uncast transmsson network have been proposed n [12], [13]. Our contrbutons n ths paper are as follows. The proposed mechansm s dstrbuted, t fully mplements the optmal allocaton n Nash equlbra (e, there are no extraneous equlbra), and t has message space dmenson that grows lnearly wth the number of agents n the network. Furthermore, ndvdual ratonalty and weak budget balance are satsfed at NE. In order to acheve all these goals, a number of technques have been used. For nstance, we have utlzed an dea smlar to the radal allocaton [3], [4], [7] to acheve feasblty at NE. Fnally, summary messages [14, Ch. 4] have been used whch enable the message dmensonalty of each agent to grow lnearly wth the sze of her neghborhood unlke the mechansm proposed n [11] n whch message dmensonalty grows wth the sze of the whole network. The structure of the paper s as follows. In Secton II, the model and problem formulaton are dscussed. Secton III presents the desgned dstrbuted mechansm after clarfyng all of the assumptons and features of message communcaton constrants. In Secton IV, we characterze the propertes of the mechansm. In Secton V, two alternatve mechansms are dscussed. We conclude n Secton VI wth some comments on message dmensonalty. II. MODEL A par of transmtter and recever n the network consttutes an agent. There are N strategc agents n the network that are denoted by N = {1,..., N}. Data s communcated va a uncast transmsson network. In the uncast transmsson, a separate data stream wth a specfc data rate s

2 transmtted from each agent s transmtter to ts recever and these data streams pass through network lnks denoted by L = {1,..., L} and each lnk l L has capacty c l. The data stream of agent passes through lnks n the set L L wth L = L. Also, the agents usng any lnk l, or n other words the agents for whom ther data stream s passng through lnk l, are denoted by N l wth N l = N l. We assume N l 2 that s at least two agents use any lnk l. Ths assumpton s made so that there s competton between agents for usng any lnk. The data rate of agent s x and the vector of allocated rates s denoted by x = (x 1, x 2,..., x N ). Agent has a valuaton functon v (x ) that s a functon of the data rate she receves. The desgner s goal s to determne x to maxmze the summaton of all of the agents valuaton functons whle the network lnks capacty constrants are satsfed. Therefore, we can formulze ths goal by the followng optmzaton problem, max x v (x ) N (1a) s.t. x 0 N (1b) and x j c l l L. (1c) j N l We mpose the followng assumptons on the valuaton functons. Frst, v (.) V 0, where V 0 s the set of strctly concave, twce dfferentable, monotoncally ncreasng, R + R functons wth contnuous second dervatves. Second, for all N, v (0) s fnte whch mples, due to concavty, that v (x) s fnte for all x R +. Generally, problem (1) can model any scenaro of network utlty maxmzaton wth lnear nequalty constrants and uncast transmsson s just an example of such scenaros. A. Necessary and Suffcent Optmalty Condtons In order to characterze the soluton of problem (1) we use dual varables λ and wrte the KKT condtons for ths problem. Snce the valuaton functons are concave and all of the constrants of problem (1) are affne, problem (1) s a convex optmzaton problem and so KKT condtons are necessary and suffcent. These condtons at the optmal pont (x, λ ) are (a) Prmal Feasblty: x satsfes (1b) and (1c). (b) Dual Feasblty: λ l 0 l L (c) Complmentary Slackness: λ l (c l x j ) = 0 j N l l L (2) (d) Statonarty: v (x ) = l L λ l f x > 0 (3) v (x ) l L λ l f x = 0 (4) III. DISTRIBUTED MECHANISM The mechansm s a sealed bd aucton n whch each agent N has a message space M and allocaton and tax functons that are denoted by ˆx (.) and ˆt (.) respectvely. We can characterze the mechansm completely by specfyng the tuple (M, (ˆx 1 (.), ˆx 2 (.),..., ˆx N (.)), (ˆt 1 (.), ˆt 2 (.),..., ˆt N (.))), where M = M 1 M 2... M N. A. Message Network The agents have access to each other s messages va a network that s modeled by a connected undrected graph. The vertces correspond to the agents and an edge between agent to j means that agent can lsten to agent j s messages and vce versa. The absence of lnk between two agents s due to communcaton constrants between them. We call ths network message network. We emphasze that ths s a dfferent network from the one related to problem (1). Ths network relates to message transmsson that enables the decentralzed soluton of problem (1). In that sense, the message network s relevant even for the more general allocaton problems modeled by problem (1). We further consder an arbtrary spannng tree on the graph of the message network and denote t by G = (N, E). In the followng we only consder communcaton over ths spannng tree and therefore, we call ths the communcaton graph. For all N, N () s defned as the the set of neghbors of agent n the G and n(, j) s the neghborng agent of agent on the shortest path from to j. We denote the sze of the set N () by N () = N(). For each lnk l L, N l () denotes the set of agents n N () usng lnk l. The sze of ths set s denoted by N l () = N l (). For each agent N, the functon Φ() randomly chooses one neghbor of agent. We can defne the set I = {k N () : Φ(k) = } based on the functon Φ(). For the sake of smplcty of exposton, we frst mpose the followng assumpton on the communcaton graph. Assumpton 1: For each lnk l, the subgraph consstng of agents N l s a connected graph. Ths assumpton wll be relaxed n Secton V and two alternatve mechansms wll be dscussed to avod mposng ths assumpton on the graph. B. Message Components Agent announces the message m = (y, n, q, p ), where y R + s a proxy for her demand and n =, j N (), l L) R L N() + conssts of components n j,l, each of whch s a proxy for the sum of demands of the agents k on lnk l wth n(, k) = j. Further, the message q = (q j,l, j L I I, l L) R + s a vector of components q j,l, each of whch s a proxy for the demand of neghborng agent j I. Ths message has been consdered to be used wherever somethng needs not to be determned by agent and yet t needs the value of y at NE. Fnally, the message p = (p l, l

3 L L ) R + s the prce that agent thnks every agent usng each lnk l L should pay. We defne y l as y l = C. Allocaton Functons { y f l L 0 oth. In order to have prmal feasblty for the optmalty condtons, we must defne the allocaton functon n a way that the allocatons ˆx (m) are feasble at Nash equlbra. Wth ths goal n mnd, we utlze an dea smlar to the radal allocaton [7]. The allocaton functon s defned as (5) ˆx (m) = r y (6) where r s the radal allocaton factor and s defned as where r l s and f l s r l = f l = q,l Φ() + r = mn l L rl (7) { c l f f l f l > 0 f f l = 0 (8) (y l j + j ). (9) The quantty f l s defned n ths way to act as a proxy for the sum of demands of agents on lnk l at NE. D. Tax Functons The tax functons are ˆt (m) = ˆt l L l (m) and for each component ˆt l (m) we have two cases. For l L we have ˆt l (m) = p l ˆx (m) + y l j j )2 + j I (q j,l y l j) 2 + (p l p l ) 2 + (p l p l ) p l (c l r f l ) 2 and p l s defned as p l = 1 N l () Second, for l / L, ˆt l (m) = yj l j N l () (10) p l j. (11) j )2 + (q j,l yj) l 2. j I (12) Ths mechansm nduces a game G s = (N, M 1... M N, (û 1,..., û N )) where the utlty functons are û (m) = v (ˆx (m)) ˆt (m). IV. MECHANISM PROPERTIES Let s denote the set of all Nash equlbra of the game G by M and each NE s denoted by m = ( m 1,..., m N ) and for each m we have m = (ỹ, ñ, q, p ). Theorem 1: (Full Implementaton, Indvdual Ratonalty and Weak Budget Balance) For all m M allocaton vector ˆx( m) s effcent;.e. ˆx( m) = x where x s the soluton of problem (1). Furthermore, ndvdual ratonalty s satsfed for all agents at all NE. Also, we have weak budget balance at NE whch means ˆt N 0. Accordng to Theorem 1, the allocaton vector s unque for all of the Nash equlbra of the game G, due to unqueness of x. In order to prove Theorem 1, we frst provde a number of lemmas. Lemma 2: (Concavty) The functon û (m, m ) s twce dfferentable and strctly concave w.r.t. m. Proof: It s obvous from the defnton of the functon û (m, m ) and assumptons on the valuaton functons that û (m, m ) s twce dfferentable w.r.t. m. To prove concavty, we calculate the Hessan matrx, H, of û (m, m ) w.r.t. m and show that t s a negatve defnte matrx. As t was defned, û (m, m ) = v (r y ) ˆt (y, n, q, p, m ). Snce the cross dervatves of û (m, m ) w.r.t. dfferent components of m are zero, all of the non-dagonal elements of H are zero. Hence, we calculate the dagonal elements. = 2 v (y) ( y ) 2 2 = r 2 v (ˆx ) ( ˆx ) 2 2 and snce We have 2 û (m) ( y ) 2 v (ˆx ) s strctly concave w.r.t. x, 2 v (ˆx ) 2 û (m) ( y ) 2 ( ˆx ) < 0 and therefore, 2 < 0 (note that r s not dependent to any of agent s messages.). We can also see that 2û (m) 2 û (m) = 2 and = 2 ( q j,l ) 2 ( n j,l ) 2 and 2 û (m) = 2 Snce all of the dagonal elements of H ( p l are negatve )2 and non-dagonal elements are zero, matrx H s negatve defnte and hence û (y, n, q, p, m ) s strctly concave w.r.t. m. Lemma 3: At any NE, m M we have q j,l = ỹ l j, j I, l L. (13) Proof: Accordng to Lemma 2, the utlty functons are strctly concave w.r.t. the messages and so the best response functon of agent at each m s unque and s determned by settng the gradent of utlty functon w.r.t. m to zero, f possble, and f t s not possble for each of the elements of m, the best response would be the upper boundary message for postve dervatve and lower boundary message for negatve dervatve. Snce the message spaces do not have upper boundares, n order to have a best response, the dervatve has to be negatve and the best response would be the lower boundary message whch n our message space t would be zero. The elements of gradent can all be set to zero except the dervatve w.r.t. y that may not get to zero. Hence, f at NE ỹ > 0 or equvalently ˆx ( m) > 0, the gradent of û (m, m )

4 s zero and otherwse, f ỹ = 0 or equvalently ˆx ( m) = 0, the dervatve of û (m, m ) w.r.t y s negatve and other elements of gradent are zero. Hence, by settng the dervatve of û (m, m ) w.r.t. elements of m ncludng q j,l to zero we have û (m, m ) q j,l = 0 2(q j,l y l j) = 0 q j,l = ỹ l j N, j I, l L. Snce ỹj l 0 the above equaton can always hold and the lemma s proved. As mentoned earler, each message component q j,l can be used as a proxy for yj l to act lke ỹl j at NE and yet, t s not determned by agent. Lemma 4: At any NE of game G, ñ j,l = ỹ l j + Ths mples that at any NE, ñ j,l = k N,n(,j)=k ñ k,l j. (14) ỹ l k. (15) Proof: Just lke the proof of Lemma 3, we set the dervatve of û (m, m ) w.r.t. each n j,l to zero û (m, m ) n j,l = 0 2 yj l j ) = 0 n j,l = yj l + j N, j N (), l L Usng a smlar argument as the one used n [14, p. 131] t can be shown that n j,l = k N,n(,j)=k yl k. Lemma 5: (Prmal Feasblty) At any NE of game G, the allocaton vector ˆx( m) s feasble. Proof: Accordng to Lemmas 3 and 4, at any NE we have and so at any NE we have N l ˆx ( m) = = f l = j N ỹ l j c l r ỹ ỹ N l j N ỹl j N l c l j N ỹl j N ỹ l = c l l L, whch mples the allocaton at NE s feasble. Lemma 6: (Dual Feasblty) At any NE of game G, p l 0 and also p l = pl, N, l L. Proof: It s obvous by the defnton of the message spaces that p l 0. To show the next part, we frst derve the followng equaton, Suppose t s not correct, p l = p l l L, L. l L, L : p l p l. Then there exsts an agent j N l : p l j > pl j (as an example we could consder the agent j wth the hghest p l j over all of the agents and f we have multple choces, at least one of them wll work.) and we show that agent j has a proftable devaton to p l j = p l j = pl j ɛ. We can wrte û j (ỹ j, ñ j, q j, p l j, p l j, m j) û j (ỹ j, ñ j, q j, p j, m j ) = (ɛ) 2 + ɛ p l j(c l r j fj) l 2 = ɛ( }{{} ɛ + p l j(c l r j fj) l 2 ) > 0, }{{} >0 0 (16) therefore, we must have p l = pl. As a result of ths equalty and because of the connectvty of G, t s obvous that p l = pl j,, j N and we denote ths common prce by p l. Lemma 7: (Complementary Slackness) At any NE of game G, p l (c l ˆx ) = 0 l L (17) N l Proof: By settng the dervatve of the utlty functon w.r.t. p l to zero we have û (m, m ) p l = 0 2(p l p l ) + p l (c l r f l ) 2 = 0 }{{} =0,Due to lemma 6 p l (c l r f l ) 2 = 0 p l (c l r ỹj) l = 0 p l (c l N l ˆx ) = 0. j N Lemma 8: (Statonarty) At any NE of game G, the followng constrants are satsfed, v (ˆx ) = l L p l f ˆx > 0 (18) v (ˆx ) l L p l f ˆx = 0. (19) Proof: Accordng to the explanatons at the begnnng of the proof of Lemma 3, f ˆx > 0 we have û (m, m ) = 0 ( û (m, m ) ) dx = 0 y x dy and f ˆx = 0, (v (x ) l L p l )r = 0 v (x ) = l L û (m, m ) 0 ( û (m, m ) ) dx 0 y x dy (v (x ) l L p l )r 0 v (x ) l L Note that at NE, r > 0 and ths s due to the fact that ˆx = 0 can never happen at NE. p l, p l.

5 Lemma 9: (Indvdual Ratonalty and Weak Budget Balance) At any NE, m, of the game G, ndvdual ratonalty s satsfed,.e. v (ˆx ( m)) ˆt ( m) v (0) 0 N. (20) Also the sum of taxes satsfes ˆt ( m) 0 (21) N that s weak budget balance. Proof: Frst consder the weak budget balance equaton. At NE, we can wrte ˆt ( m) = ˆx ( m) l L p l and hence ˆt N ( m) 0. Next, consder the ndvdual ratonalty part, t s obvous for ˆx ( m) = 0. For ˆx ( m) > 0, we defne the functon u as u (x) = v (x) x l L p l. Snce u (x) s concave w.r.t. x and u (ˆx ( m)) = 0 and u (y) > 0 for 0 < y < ˆx ( m), we can conclude u (y) > u (0) and snce u (0) = v (0) and u (ˆx ( m)) = v (ˆx( m)) ˆt ( m), t s obvous that v (ˆx( m)) ˆt ( m) v (0). Lemma 10: There exsts a NE for the game G. Proof: Ths s proved by showng that any allocaton equal to the soluton of problem (1) s a NE of the game G. Frst note that snce the valuaton functons are monotoncally ncreasng, the more the allocatons are the better t s for all of the agents. So the soluton of problem (1) s always on the boundares of feasble regon. In order to prove the exstence of NE n the game G, we show that any canddate message m wth allocaton ˆx(m) = x and n, q and p satsfyng lemmas 4, 3 and 6 and p l = p l = λ l N l s a NE. Frst we show that such messages exst. Obvously, we can set the messages n, q and p so that the mentoned crcumstances are satsfed. The demand vector y could be any multple of x so that the allocaton ˆx(m) would be ˆx(m) = x. Ths s because of the fact that each message y s projected to the ˆx(m) that s on the boundary of feasble regon and s a scaled verson of y. Now by consderng such message we must show that ths s a NE. If ˆx (m) > 0, we proved that the gradent of û w.r.t. m s zero for all and so every agent s best respondng to others messages and t s a NE. If ˆx (m) = 0 the gradent of utlty functon û (m, m ) w.r.t. all elements of m except y s zero for all and so those messages are best responses to m. Also, we know that û y y=0 < 0 and therefore, agent can not proft by ncreasng y from zero to some postve value and hence, ths s a NE. Proof of Theorem 1: By usng all of the lemmas n ths secton, we can now prove Theorem 1. Consder any m M, the allocaton vector, ˆx( m) and the varables p l satsfy all of the KKT condtons f ˆx( m) are used as the prmal varables and p = { p 1,..., p L } are used as the dual varables. Therefore, ˆx( m) = x for any m M and so, full mplementaton s satsfed too. Furthermore, Lemma 9 proves ndvdual ratonalty and weak budget balance. V. RELAXING THE ASSUMPTIONS ON MESSAGE NETWORK As mentoned n Secton III, for the sake of smplcty of exposton, we have assumed that for every lnk l, the subgraph consstng of agents N l s a connected graph (Assumpton 1). In order to relax ths assumpton, we wll present two solutons separately One s n the expense of larger message space and the other s n the expense of more complcated mechansm but less ncrement n the message space dmensons. Both of these solutons wll be explaned n ths secton and the resultng changes n the mechansm wll be dscussed. A. Frst Soluton The frst soluton s to modfy the defnton of message element p as p = (p l, l L). Ths s extendng the prce message announced by each agent to be for every lnk n the network nstead of only lnks that agent s usng. Hence, the defnton of p l n (11) s changed as follows p l = 1 p l N() j. (22) Ths necesstates a modfcaton n the tax functon defnton for the case of l / L as ˆt l (m) = yj l j )2 + (q j,l yj) l 2 j I + (p l p l ) 2 + (p l p l ) p l (c l r f l ) 2 (23) By these changes, all of the results n Secton IV are vald by exchangng every term ncludng p l, l L wth p l, l L. Hence, the mechansm s qute smlar to the orgnal one except that n the message elements, we have L prce messages nstead of L ones for each agent. The basc dea behnd ths soluton s as follows. The assumpton of connectedness of the subgraph of agents usng lnk l (Assumpton 1) was requred n order for all users n ths subgraph to be able to come to a consensus on the common prce p l for usng the lnk l. By lettng each agent announcng prces for all lnks (not just the ones she s nvolved) ths assumpton s not needed anymore as ths nformaton can propagate through the entre graph (whch s assumed to be connected). B. Second Soluton In ths soluton, there s no need for every agent to quote a prce for every lnk that she s not usng. For every lnk l, we consder a connected subgraph G l = (N l, E l ) ncludng all agents N l plus the mnmum number of agents that do not use lnk l and are requred to make the subgraph connected. We call t lnk l s subgraph. The mportant property of ths subgraph s that t should be connected so that t can play the role that the whole graph plays n the mechansm of Secton V-A. Snce the graph G s connected, the subgraph G l exsts. For each agent the set of lnks l / L whch N l are denoted by L wth L = L. Now, we redefne the message

6 component p as p = (p l, l L L ), that s each agent quotes a prce for any lnk she s usng and also for the lnks whch she s on ther subgraph. We defne the set N l () as N l () = { N () N l } (24) Just lke prevous notatons, we denote the sze of ths set by N l () = N l (). Next, the defnton of p l s changed to: p l = 1 p l j (25) N l () j N l () The tax functon should be modfed as follows. It s the same for l L. For l / L L, the tax functon s the same as the l / L case for the orgnal mechansm. For l L, we defne the tax functon as ˆt l (m) = y l j j )2 + (q j,l yj) l 2 j I + (p l p l ) 2 + (p l p l ) p l (c l r f l ) 2 (26) Ths soluton results n the same lemmas that we had n Secton IV for the orgnal mechansm. We should just exchange every term ncludng p l, l L wth p l, l L L. VI. CONCLUSION In ths paper, we consder the problem of rate allocaton between strategc agents n uncast transmsson. The mportant new feature of ths settng s that messages are not broadcasted (as n the standard MD lterature) but agents have to obey the message communcaton constrants mposed by an underlyng message communcaton network. The dmensonalty of the agent s message n the frst case (wth smplfyng assumpton on the message network) s M = 1 + N()L + I L + L. Snce the functon Φ() chooses one agent j, the average sze of the sets I would be 1. Hence, the average sze of each agent s message s 1 + E N (N())L + L + E N (L ) and by denotng E N (N()) and E N (L ) by N and L respectvely, we can have E N M = N(1 + L( N + 1) + L). (27) [4] R. T. Maheswaran and T. Basar, Socal welfare of selfsh agents: motvatng effcency for dvsble resources, n Decson and Control, CDC. 43rd IEEE Conference on, vol. 2. IEEE, 2004, pp [5] R. Jan and J. Walrand, An effcent Nash-mplementaton mechansm for network resource allocaton, Automatca, vol. 46, no. 8, pp , [6] A. Kakhbod and D. Teneketzs, Correcton to An effcent game form for mult-rate multcast servce provsonng, IEEE Journal on Selected Areas n Communcatons, vol. 31, no. 7, pp , [7] A. Snha and A. Anastasopoulos, Mechansm desgn for resource allocaton n networks wth ntergroup competton and ntragroup sharng, IEEE Trans. on Control of Network Systems, vol. PP, no. 99, pp. 1 1, 2017, (prepublcaton). [8] Y. Lu, A. Sarab, J. Zhang, P. Naghzadeh, M. Karr, M. Baley, and M. Lu, Cloudy wth a chance of breach: Forecastng cyber securty ncdents. n USENIX Securty Symposum, 2015, pp [9] A. Nedc and A. Ozdaglar, Dstrbuted subgradent methods for multagent optmzaton, IEEE Transactons on Automatc Control, vol. 54, no. 1, pp , [10] S. Boyd, N. Parkh, E. Chu, B. Peleato, J. Ecksten, et al., Dstrbuted optmzaton and statstcal learnng va the alternatng drecton method of multplers, Foundatons and Trends n Machne Learnng, vol. 3, no. 1, pp , [11] A. Snha and A. Anastasopoulos, A dstrbuted mechansm for publc goods allocaton wth dynamc learnng guarantees, n NetEcon 2017: The 12th Workshop on the Economcs of Networks, Systems and Computaton, Boston, MA, June [12] A. Kakhbod and D. Teneketzs, Correcton to An effcent game form for uncast servce provsonng [feb ], IEEE Transactons on Automatc Control, vol. 60, no. 2, pp , Feb [13] A. Snha and A. Anastasopoulos, A practcal mechansm for network utlty maxmzaton for uncast flows on the Internet, n Proc. Internatonal Conf. Communcatons, June 2015, pp [14] A. Snha, Mechansm desgn wth allocatve, nformatonal and learnng constrants, Ph.D. dssertaton, Unversty of Mchgan, Clearly, the dmensonalty of message space grows lnearly wth N. For the second mechansm, the length of message s E N M = N(1 + L( N + 2)) whch s larger than the frst case because of havng L nstead of L. Ths s due to the prce message expanson. For the thrd mechansm, the length of message s E N m = N(1 + L( N + 1) + L + L) where we denote L = E N L. REFERENCES [1] L. Hurwcz, Outcome functons yeldng Walrasan and Lndahl allocatons at Nash equlbrum ponts, The Revew of Economc Studes, vol. 46, no. 2, pp , [2] T. Groves and J. Ledyard, Optmal allocaton of publc goods: A soluton to the free rder problem, Econometrca: Journal of the Econometrc Socety, pp , [3] S. Yang and B. Hajek, Revenue and stablty of a mechansm for effcent allocaton of a dvsble good, preprnt, 2005.