Hierarchical Auctions for Network Resource Allocation

Transcription

1 Herarchcal Auctons for Network Resource Allocaton Wenyuan Tang and Rahul Jan Department of Electrcal Engneerng Unversty of Southern Calforna Abstract. Motvated by allocaton of cloud computng servces, bandwdth and wreless spectrum n secondary network markets, we ntroduce a herarchcal aucton model for network resource allocaton. The Ter 1 provder owns all of the resource, who holds an aucton n whch the Ter 2 provders partcpate. Each of the Ter 2 provders then holds an aucton to allocate the acqured resource among the Ter 3 users. The Ter 2 provders play the role of mddlemen, snce ther utlty for the resource depends entrely on the payment that they receve by sellng t. We frst consder the case of ndvsble resource. We study a class of mechansms where each sub-mechansm s ether a frst-prce or a second-prce aucton, and show that ncentve compatblty and effcency cannot be smultaneously acheved. We then consder the resource to be dvsble and propose the herarchcal network second-prce mechansm n whch there exsts an effcent Nash equlbrum wth endogenous strong budget balance. Keywords: Network economcs, mechansm desgn, auctons, herarchcal models. 1 Introducton As networks have become ncreasngly complex, so has the ownershp structure. Ths means that the tradtonal models and resource allocaton mechansms that are used for resource exchange between the prmary owners and the end-users are no longer always relevant. Increasngly, there are mddlemen, operators who buy network resources from the prmary owners and then sell them to the end-users. Ths potentally causes neffcences n network resource allocaton. Consder the case of bandwdth allocaton. The network bandwdth s prmarly owned by a Ter 1 ISP (Internet Servce Provder) or carrer, who then sells t to varous Ter 2 ISPs. The Ter 2 ISPs then sell t further ether to corporate customers or to the Ter 3 ISPs, who provde servce drectly to consumers. The presence of the Ter 2 ISPs can potentally skew the network resource allocaton, and cause t to be neffcent from a socal welfare pont of vew. Another case The second author s research on ths proect s supported by the NSF CAREER award CNS and an IBM Faculty Award. R. Jan and R. Kannan (Eds.): GameNets 2011, LNICST 75, pp , c Insttute for Computer Scences, Socal Informatcs and Telecommuncatons Engneerng 2012

2 12 W. Tang and R. Jan n pont, s the emergng market of cloud computng servces. Provders such as IBM, Google, Amazon and others are provdng cloud computng servces whch end-users (e.g., enterprses havng small computatonal or data center needs) can buy. Of course, the dstrbuton channel for these servces s lkely to nvolve mddlemen. Ths rases the key queston what herarchcal mechansms can be used n the presence of mddlemen that are ncentve compatble and/or effcent. Auctons as mechansms for network resource allocaton have receved lots of attenton recently. Followng-up on the network utlty model proposed by Kelly [12], Johar and Tstskls showed that the Kelly mechansm can have up to 25% effcency loss [10]. Ths led to a flurry of actvty n desgnng effcent network resource allocaton mechansms, ncludng the work of Maheshwaran and Basar [15], Johar and Tstskls [11], Yang and Haek [20], Jan and Walrand [5], Ja and Canes [8] among others [2,16]. Most of the work focused on onesded auctons for dvsble resources, and s related to the approach of Lazar and Semret [13]. Double-sded network auctons for dvsble resources were developed n [5]. The only work to focus on ndvsble network resources s Jan and Varaya [6] whch proposed a Nash mplementaton combnatoral double aucton. Ths s also the only work known so far that presents ncomplete nformaton analyss of combnatoral market mechansms [7]. All those mechansms ether nvolve network resource allocaton by an auctoneer among multple buyers, or network resource exchange among multple buyers and sellers. Most of the proposed mechansms are Nash mplementatons,.e., truth tellng s a Nash equlbrum but not necessarly a domnant strategy equlbrum, and have ether unque Nash equlbra whch are effcent, or at least one that s. In realty, however, markets for network resources often have mddlemen operators, and effcency can be rather hard, f not mpossble to acheve wth ther presence. Unfortunately, models wth mddlemen have not been studed at all, prmarly due to the dffculty of desgnng approprate mechansms. Even n the economc and game theory lterature, the closest related aucton models are those that nvolve a resale after an aucton. That s, there s only a sngle ter aucton, and the wnners can then resell the resources acqured n the aucton [3]. There s ndeed some game-theoretc work on network prcng n a more general topology. Johar, Mannor and Tstskls [9] studed a network game where the nodes of the network wsh to form a graph to route traffc between themselves, and they characterzed connected lnk stable equlbra. Shakkotta and Srkant [18] examned how transt and customer prces and qualty of servce are set n a network consstng of multple ISPs, where a 3-ter herarchcal model s proposed. However, such work only focused on the prcng equlbrum, and ssues lke mechansm desgn and auctons were never studed. In ths paper, we consder a mult-ter settng. We consder a homogeneous network resource. Ths could be bandwdth, wreless spectrum or cloud computng servce, all owned by a sngle entty, the Ter 1 provder. He conducts an aucton to allocate the resource among the Ter 2 operators. The Ter 2 operators then further allocate the network resource they have acqured n the aucton

3 Herarchcal Auctons for Network Resource Allocaton 13 among the Ter 3 enttes, who may be the end-users. Each Ter 3 user has a valuaton for the resource, whch s strctly ncreasng and concave wth respect to the capacty. On the other hand, the Ter 2 enttes are more lke mddlemen. They do not have any ntrnsc valuaton for the network resource but a quasvaluaton whch depends on the revenue that they wll acqure by sellng t off n an aucton. Our goal s to desgn a herarchcal aucton mechansm that specfes one sub-mechansm for each ter. We develop a general herarchcal mechansm desgn framework and consder the settng where all auctons are conducted smultaneously. Admttedly, ths does not fully meet the realty (where auctons n dfferent ters may take place one after another), but provdes nsghts nto the problem from a theoretcal pont of vew. We frst consder the resource to be ndvsble. We nvestgate a class of mechansms where each sub-mechansm s ether a frst-prce or a second-prce aucton. We show that the all-ter second-prce aucton mechansm s ncentve compatble but not effcent,.e., socal-welfare maxmzng. Ths s a surprsng observaton and the only known nstance of ts type nvolvng the VCG/secondprce mechansm [19]. We then show that the herarchcal mechansm wth a frstprce or a second-prce sub-mechansm at Ter 1, and frst-prce sub-mechansms at all other ters s ndeed effcent but not ncentve compatble. When the resource s dvsble, t s mpossble for bdders to report ther arbtrary real-valued valuaton functons. They are thus asked to report a twodmensonal bd sgnal,.e., a per-unt bd prce and the maxmum quantty that they want to buy/sell. We note that whle the Ter 1 sub-mechansm s a snglesded aucton, the sub-mechansms at all lower ters are double-sded auctons. In ths framework, we propose a herarchcal mechansm wth a VCG-type aucton at each ter. We show that n ths herarchcal mechansm, there exsts an effcent Nash equlbrum that s strongly budget-balanced at all ters except the top ter, where a sngle-sded aucton s conducted. 2 Model and Problem Statement 2.1 The Herarchcal Model Consder a Ter 1 provder who owns a homogeneous network resource, say bandwdth. Such an entty could be ether a carrer (e.g., AT&T), the FCC, or a cloud servce provder such as IBM. The Ter 1 provder auctons C unts of the resource among Ter 2 operators va a sngle-sded aucton. We wll refer to ths as the Ter 1 aucton. Each of the Ter 2 operators then auctons off the resource acqured n the Ter 1 aucton to the Ter 3 operators. These wll be referred to as the Ter 2 auctons, and n general at Ter k as the Ter k auctons. We wll assume that there are K ters. The only Ter 1 provder wll be consdered as the socal planner, and the Ter K enttes are the end-users or the consumers, whle operators at all other ters wll be consdered as mddlemen. Ter k (1 <k K) operators can partcpate n and acqure the resource only from ther Ter k 1 parent. That s, each mddleman has exclusve access to hs chldren, and there s only one seller n each aucton. Ths smplfes the network topology. Extenson

4 14 W. Tang and R. Jan (e.g., allowng competton among sellers) s possble but would be much more complcated. An example of the network topology s shown n Fg. 1. We now ntroduce the notaton to ease further dscusson. Let the nodes n the tree network be numbered =0, 1,...,N wth M termnalnodes,wherenode0attherootof the tree s the socal planner. Let T () be a functon that specfes the ter to whch node belongs. The ter of node 0 s consdered as Ter 1. By parent(), we shall denote the parent of node n the tree network, and by chldren(), we shall denote the set of the chldren of node. Each node represents a player. To avod cumbersomeness, we shall use the redundant notaton P (k) that s at Ter k. Let the capacty acqured by P (k) k 1 aucton), and the capacty that node offers denoted by y (k) k aucton). Note that chldren() x(k+1) y (k) Ter 1: Socal Planner for node denoted by x (k) (n the Ter (n the Ter x (k),fork =1,..., K 1. P (1) 0 C x (2) P (2) 1 Ter 2: Mddlemen P (2) 1 2 x (2) 2 P (3) 3 x (3) P (3) 4 P (3) 5 P (3) 3 x (3) 4 x (3) 5 6 x (3) 6 Ter 3: End-Users Fg. 1. An example of a 3-ter network 2.2 The Mechansm Desgn Framework We now descrbe the mechansm desgn framework. We assume that each player P (k) has a quas-lnear utlty functon u (k) (x, w )=v (k) (x) w,wherev (k) (x) s the valuaton of player P (k) when he s allocated a capacty x, andw s the payment made to hs parent. Typcally, for the mddlemen P (k) (k =2,...,K 1), v (k) (x) =w c (k) (x), where w s the revenue from resellng and c (k) (x) s the cost functon, snce they do not derve any utlty from the allocaton but may ncur a transacton cost. We defne the socal welfare to be the total utlty derved by the end-users mnus the total cost ncurred by the mddlemen,.e., S(x) = v (K) (x (K) ) c (k) (x (k) ) :T ()=K 2 k K 1 :T ()=k

5 Herarchcal Auctons for Network Resource Allocaton 15 where x =(x 1,...,x N ). The socal planner s obectve s to realze an (allocatvely) effcent allocaton x that maxmzes the socal welfare, and solves HN-OPT : max S(x) (1) s.t. x (2) C, chldren(0) x (k+1) x (k), (, k) :2 k K 1, chldren() x (k) 0, (, k) :2 k K. The frst constrant follows because n the Ter 1 aucton, the auctoneer (player 0) allocates the total capacty C among the Ter 2 players. The second constrant follows from the fact that the total allocaton among the buyers n the Ter k aucton cannot be greater than the allocaton receved by P (k) from the Ter k 1 aucton. The thrd constrant s requred to ensure non-negatve allocatons. Furthermore, we could consder the resource to be ndvsble and let the x s to be ntegral, or consder t to be dvsble and allow the x s to be real. The socal planner cannot acheve the obectve (.e., socal-welfare maxmzng) by hmself as he does not know the valuaton and cost functons of the end-users and the mddlemen respectvely. Thus, a decentralzed mplementaton s necessary. However, the strategc players are selfsh and may msreport ther nformaton. Furthermore, n the herarchcal model, the mechansm s dstrbuted wth multple auctons at each ter. Ths makes the achevement of the socal welfare maxmzaton even more dffcult. Our goal thus s an ncentve mechansm Γ that s composed of varous submechansms (Γ (k), =0, 1,...,N M,k = T ()). Each sub-mechansm (.e., aucton) Γ (k) sconductedateachnode ofthe tree, exceptthem leaf (termnal) nodes. Note that the aucton Γ (k) nvolves player P (k) as a seller, and the players chldren() as the buyers. Note that node 0 acts only as an seller and the termnal nodes only act as buyers, whereas the mddlemen P (k) (2 k K 1) act as buyers n the Ter k 1 aucton, and as sellers n the Ter k aucton. Generally, each Γ (k) can be dfferent, though we consder the settng for whch Γ (k) = Γ (k),.e., a common sub-mechansm s used at each Ter k. Stll, ths s a smplfcaton but subect to extenson. Snce the mddlemen have no ntrnsc valuaton for the resource tself, we defne the noton of quas-valuaton functons for the mddlemen. Let X denote the allocaton space, whch s Z + when the resource s ndvsble and R + when the resource s dvsble. Defnton 1. A quas-valuaton functon of player P (k) s a functon v (k) : X R + that specfes the revenue he receves n the aucton Γ (k) from hs chldren for each possble allocaton, when all the players chldren() report ther valuaton functons (for end-users) or quas-valuaton functons (for mddlemen) truthfully.

6 16 W. Tang and R. Jan Note the backward-recursveness n the defnton of quas-valuaton functons. They can be easly computed by the players n complete nformaton settngs. We now see the role of such functons n defnng herarchcal ncentve compatble and Nash mplementaton mechansms. Defnton 2. The (drect) herarchcal mechansm Γ =(Γ (1),...,Γ (K 1) ) s ncentve compatble (or strategy-proof) f there s a domnant strategy equlbrum wheren all the end-users report ther valuaton functons and all the mddlemen report ther quas-valuaton functons, truthfully. Such equlbrum strateges wll be referred to as truth tellng as a counterpart to standard notons of truth-tellng n non-herarchcal mechansms [17]. We now defne the noton of effcency n herarchcal mechansms. Defnton 3. The (drect) herarchcal mechansm Γ =(Γ (1),...,Γ (K 1) ) s effcent f there s an equlbrum that maxmzes the socal welfare n the optmzaton problem HN-OPT (1). We study smultaneous herarchcal mechansms, n whch all sub-mechansm auctons take place smultaneously (whch are modeled as a normal form game). Thus, usual notons of Nash equlbrum shall be studed [1,4]. 3 Herarchcal Auctons for Indvsble Resources When the resource s ndvsble, we present a class of herarchcal mechansms Γ =(Γ (1),...,Γ (K 1) ) wheren a common sub-mechansm s used at each ter, and each such sub-mechansm s ether a frst-prce aucton (denoted by F) or a second-prce aucton (denoted by S),.e., Γ (k) {F, S}. We nvestgate the effcency and ncentve compatblty of such herarchcal mechansm desgns. We frst consder the case where there s only a sngle unt to be allocated,.e., C = 1. Here, we assume that the mddlemen have no transacton costs,.e., c (k) (x) = 0. We note that the ntroducton of transacton costs would be trval n the case of ndvsble resources, and t can be easly extended f desred. Let b (k) denote the buy-bd of player who s at Ter k, andx (k) the unt he acqures (n the Ter k 1 aucton as defned). Recall that there are N M + 1 auctons that are conducted smultaneously, though some aucton outcomes cannot be fulflled snce there s only a sngle ndvsble unt. Ths, however, s not unreasonable snce there s really a sngle wnner among the end-users. The mddlemen that connect ths end-user to the root wll also be purported to be wnners. Theorem 1. Assume each player except the end-users has at least two chldren. Suppose a sngle ndvsble unt s to be allocated through a herarchcal aucton mechansm ˆΓ such that ˆΓ (1) {F, S}, ˆΓ (2) = = ˆΓ (K 1) = F. Then,there exsts an ɛ-nash equlbrum whch s effcent.

7 Herarchcal Auctons for Network Resource Allocaton 17 (K 1) Proof. We prove by constructon. Consder a Ter K 1aucton ˆΓ.Fndthe player that has the hghest valuaton n that aucton,.e., arg max chldren() v (K). Defne the bds of the player as follows b (K) b (K) = v(k), = v (K) ɛ, chldren(),,.e., player bds truthfully, whle all others n that aucton bd ust a bt below. Consder a Ter k aucton Γ (k) (1 <k<k 1). As before, fnd a player arg max chldren() v (k+1), and defne the bds of the players n ths aucton as b (k+1) b (k+1) = v(k+1), = v (k+1) ɛ, chldren(),. Now, consder the Ter 1 aucton ˆΓ (1).Fnd a player arg max chldren(0) v (2). If ˆΓ (1) = F, defne the bds of players n ths aucton as b (2) = v(2), b (2) = v (2) ɛ, chldren(0),. Otherwse, f ˆΓ (1) = S, defne the bds of players n ths aucton as b (2) = v (2), chldren(0). It s obvous that such bds nduce the effcent allocaton. We argue that these bds consttute an ɛ-nash equlbrum. Note that every player gets a non-negatve payoff n such a bd profle. Consder a player P (K). If he s a wnner, he has no ncentve to ncrease hs = v (K), and he has no ncentve to decrease hs bd snce there wth parent() =parent( ) whose bd s b (K) ɛ. Ifhes aloser,wehaveb (K) >v (K) ɛ. Clearly, he has no ncentve to ether ncrease or decrease hs bd. Consder a player P (k) (2 <k<k). If he s a wnner, he has no ncentve bd snce b (K) exsts a player P (K) to ncrease hs bd snce b (k) snce there exsts a player P (k) = v (k), and he has no ncentve to decrease hs bd wth parent() =parent( ) whose bd s b (k) ɛ. If he s a loser, we have b (k) > v (k) ɛ. Clearly, he has no ncentve to ether ncrease or decrease hs bd. It s also easy to verfy that such bds are the best responses of the Ter 2 players for ˆΓ (1) = F and ˆΓ (1) = S respectvely. Ths proves the clam.

8 18 W. Tang and R. Jan The followng example shows that the above mechansm ˆΓ acheves effcency but s not ncentve compatble. Proposton 1. The herarchcal mechansm ˆΓ s effcent but not ncentve compatble. Proof. We prove by provdng a counter example. Assume the network topology s as n Fg. 1,.e., there are two Ter 2 players P (2) 1 (wth hs Ter 3 chldren P (3) 3, P (3) 4 )andp (2) 2 (wth hs Ter 3 chldren P (3) 5, P (3) 6 ). Let the valuaton functons of the Ter 3 players be v (3) 3 =2,v (3) 4 =3,v (3) 5 =1,v (3) 6 =4. Snce ˆΓ (2) = F, the quas-valuaton functons of the Ter 2 players can be easly computed to be v (2) 1 =3, v (2) 2 =4. However, truth tellng s not an equlbrum n ths aucton. Rather, t s easy to verfy that an ɛ-nash equlbrum s (b (2) 1,b(2) 2 (b (3) 3,b(3) 4,b(3) 5,b(3) 6 )=(4 ɛ, 4), )=(3 ɛ, 3, 4 ɛ, 4). The correspondng equlbrum allocaton s (x (2) 1,x(2) 2 (x (3) 3,x(3) 4,x(3) 5,x(3) 6 )=(0, 1), )=(0, 0, 0, 1), whch s exactly the effcent allocaton. Thus, ths mechansm s effcent but not ncentve compatble. We now ntroduce a natural herarchcal extenson of the second-prce or VCG aucton mechansm. Theorem 2. Suppose multple unts of an ndvsble resource are to be allocated through a herarchcal aucton mechansm Γ such that Γ (1) = = Γ (K 1) = S (whch we shall call the second-prce herarchcal aucton). Then, the mechansm s ncentve-compatble. Proof. We argue by backward nducton that truth tellng s a domnant strategy equlbrum. Consder the Ter K 1 aucton, whch s a second-prce submechansm. The end-user P (K) (T () = K) wll bd truthfully, no matter how the other players bd and what capacty hs parent s allocated, snce that s hs domnant strategy n a second-prce aucton. Ths s the fundamental property of the VCG mechansm.

9 Herarchcal Auctons for Network Resource Allocaton 19 Gven that all the P (K) s report truthfully, the quas-valuaton functons of the players P (K 1) s are true. Furthermore, the Ter K 2 aucton s agan a VCG mechansm n whch truth-tellng s a domnant strategy equlbrum. Now, we can argue by backward nducton. Assumng the Ter k +1 players n the Ter k auctons have true quas-valuaton functons, they wll bd truthfully. So, the quas-valuaton functons of the Ter k players wll be true as well. Snce ths s true for k = K 1, t s true for all k = K 1,...,1. Hence, all the players bd truthfully, and the herarchcal mechansm s ncentve-compatble. The second-prce herarchcal aucton mechansm, as can be expected, has truth tellng by each player as a domnant strategy equlbrum. The surprse s that unlke non-herarchcal settngs, effcency may not be acheved. Proposton 2. The second-prce herarchcal mechansm Γ s not necessarly effcent. Proof. We prove by provdng a counter example. Let C = 5 be allocated by the second-prce herarchcal mechansm n a 3-ter network as n Fg. 1. Let the valuaton functons of the Ter 3 players be (v (3) 3 (x),x =1, 2, 3, 4, 5) = (10, 18, 24, 28, 30), (v (3) 4 (x),x =1, 2, 3, 4, 5) = (20, 25, 29, 32, 34), (v (3) 5 (x),x =1, 2, 3, 4, 5) = (15, 24, 32, 39, 45), (v (3) 6 (x),x =1, 2, 3, 4, 5) = (16, 20, 24, 27, 29). Accordng to (1), the effcent allocaton s (x (2) 1,x (2) 2 )=(2, 3), (x (3) 3,x (3) 4,x (3) 5,x (3) 6 )=(1, 1, 2, 1). Snce Γ (2) = S, the quas-valuaton functons of the Ter 2 players can be easly computed to be ( v (2) 1 (x),x =1, 2, 3, 4, 5) = (10, 13, 15, 16, 15), ( v (2) 2 (x),x =1, 2, 3, 4, 5) = (15, 13, 16, 18, 19). In the mechansm Γ, truth tellng s a Nash equlbrum as we have already proved. Thus, the correspondng equlbrum allocaton s (x (2) 1,x(2) 2 (x (3) 3,x(3) 4,x(3) 5,x(3) 6 )=(4, 1), )=(3, 1, 0, 1), whch however, s dfferent from the effcent allocaton. Thus, n the case of multple unts of an ndvsble resource, ths herarchcal mechansm s ncentve-compatble but not effcent.

10 20 W. Tang and R. Jan An even greater surprse s the followng mpossblty result f we restrct our attenton to frst-prce and second-prce sub-mechansms. Theorem 3 (Herarchcal Impossblty). Suppose we allocate a sngle unt of the ndvsble resource through a herarchcal aucton mechansm Γ such that Γ (k) {F, S} (for all k =1,...,K 1 and K 3). Then, there exsts no such herarchcal mechansm whch s both ncentve-compatble and effcent. Proof. As we have already seen n Proposton 1 that ncentve compatblty s not guaranteed f there exsts a k such that Γ (k) = F. We have also seen n Proposton 2 that effcency s not guaranteed f there exsts a k such that Γ (k) = S. Thus, f the choces of the Γ (k) s are restrcted to the two alternatves (F or S), ncentve compatblty and effcency cannot be smultaneously acheved. Our conecture s that ths lmted mpossblty theorem foretells a more general mpossblty result for herarchcal mechansm desgn. 4 Herarchcal Auctons for Dvsble Resources We now consder the resource to be dvsble, and propose a herarchcal aucton mechansm. We wll now consder the settng where the mddlemen have transacton costs as well. Whle the Ter 1 aucton wll reman sngle-sded, Ter 2 through Ter K 1 auctons wll be double-sded,.e., n such auctons buyers wll make buy-bds, and sellers wll make sell-bds. For smplcty of exposton, we wll only consder a 3-ter network as n Fg. 1. Also, we drop the superscrpts and adopt a more concse notaton here,.e., denote the th Ter 2 player by P and the th chld of P by P (Ter 3 player). The notatons of valuaton functons, bds, etc. are changed correspondngly. We wll assume that the valuaton functons of the end-users, v (x )are strctly ncreasng and concave, and smooth, wth v (0) = 0. The cost functons of the mddlemen, c (x ) are assumed to be strctly ncreasng and convex, and smooth, wth c (0) = 0. The end-user s payoff s u = v (x ) w,wherew s the payment made by player P to player P. A mddleman P has a utlty u = w w c (x ), where w s P s revenue from resellng and w s the payment made by P to player 0. In ths settng the socal welfare optmzaton problem s as followng DIV-OPT : max v (x ) c (x ) (2), s.t. x C, [λ 0 ] x x,, [λ ] x 0, x 0,, (, ).

11 Herarchcal Auctons for Network Resource Allocaton 21 Here, λ 0 and λ s are the Lagrange multplers of the correspondng constrants above. The above s a convex optmzaton problem, and a soluton exsts, whch s characterzed by the KKT condtons [14] (c (x )+λ 0 λ )x =0,, c (x )+λ 0 λ 0,, (v (x ) λ )x =0, (, ), (3) v (x ) λ 0, (, ), λ 0 ( x C) =0, λ ( x x )=0,. Our obectve s to desgn a herarchcal mechansm to allocate the dvsble resource that acheves the socal welfare optmum despte the strategc behavor of the players. An mportant ssue n the context of dvsble resources s that t s mpossble for a bdder to communcate a complete arbtrary real-valued valuaton functon. Thus, the bdders must communcate an approxmaton to t from a fnte-dmensonal bd space. Herarchcal Network Second-Prce Mechansm We now propose the herarchcal network second-prce (HNSP) mechansm Γ that can be used to allocate a dvsble resource n a mult-ter network. We take a 3-ter network as an example. The mechansm Γ =( Γ (1), Γ (2) )scomposedof two sub-mechansms Γ (1) and Γ (2). The sub-mechansm Γ (1) employed at Ter 1 s a sngle-sded VCG-type aucton mechansm n whch Ter 2 players (the mddlemen) report bds b =(β,d )whereβ s nterpreted to be the per-unt (2) bd prce, and d as the maxmum quantty wanted. The sub-mechansm Γ employed at Ter 2 s a double-sded VCG-type aucton mechansm where Ter 2 players report sell-bds a =(α,q )whereα s the per-unt sell-bd prce and q s the maxmum quantty offered for sale, whle the Ter 3 players (end-users) report buy-bds b =(β,d )whereβ s the per-unt buy-bd prce and d s the maxmum quantty wanted. Once the bds are receved n all the auctons, the aucton outcomes are determned as follows. In the Ter 1 aucton Γ (1), the allocaton ˆx s a soluton of the optmzaton problem HNSP-1 : max β x (4) s.t. x C, [λ 0] x d,, [μ ] x 0,.

12 22 W. Tang and R. Jan Let ˆx denote the allocaton as a soluton of the above wth d = 0,.e., when the player P (a mddleman) s not present. Then, the payment made by P s w = β (ˆx ˆx ), (5) whch s the externalty that P mposes on all the other players (n the Ter 1 aucton) by hs partcpaton. Let λ 0 and μ s be the Lagrange multplers of the correspondng constrants. Then, the soluton of HNSP-1 s characterzed by the KKT condtons (β λ 0 μ )x =0,, β λ 0 μ 0,, (6) λ 0 ( x C) =0, μ (x d )=0,. In the Ter 2 aucton Γ (2), the mddleman s the seller and hs chldren (the end-users) are the buyers. The sub-mechansm Γ (2) s a VCG-type double-sded aucton,.e., both the seller and the buyers place bds, and the allocaton ( x, ỹ) s a soluton of the optmzaton problem HNSP-2 : max β x α y (7) s.t. x y, [λ ] x d,, [μ ] y q, [ν ] x 0,, y 0. Let ( x, ỹ ) denote the allocaton as a soluton of the above wth d =0. Then, the payment made by player P s w = k β k ( x k x k) α (ỹ ỹ ), (8) and the payment receved by player P s w = β x. (9) These transfers are the externaltes that the players mpose on the others through ther partcpaton. Let λ s, μ s and ν s be the Lagrange multplers correspondng to the constrants n the HNSP-2 above. Then, the soluton s characterzed by the followng KKT condtons

13 Herarchcal Auctons for Network Resource Allocaton 23 (β λ μ )x =0,, β λ μ 0,, (α λ + ν )y =0, α λ + ν 0, (10) λ ( x y )=0, μ (x d )=0, ν (y q )=0. Ths completes the defnton of the HNSP mechansm. We now show the exstence of an effcent Nash equlbrum n the smultaneous herarchcal network second-prce mechansm by constructon. Moreover, we show that the Ter 2 sub-mechansm Γ (2) acheves endogenous strong budget balance at ths equlbrum,.e., the payment receved by each mddleman equals the total payments made by hs chldren. Theorem 4. In the HNSP mechansm Γ, there exsts an effcent Nash equlbrum wth endogenous strong budget balance. Proof. Let x be an effcent allocaton correspondng to the problem DIV- OPT n (2). Then, there exst Lagrange multplers λ 0 and λ s that satsfy the KKT condtons (3). Consder the bd profle d = q = x, d = x, β = v (x ) c (x ), α = c (x )+λ 0,andβ = v (x ). Frst, we prove that the bd profle nduces the effcent allocaton. Let λ 0 = λ 0, λ = λ, μ = β λ and μ = ν = 0. Then the KKT condtons (6) and (10) are equvalent to the KKT condtons (3). Ths mples x s also a soluton of the problem HNSP-1 n (4) and the problem HNSP-2 n (7) wth these bds. Now, we prove that the strategy profle s a Nash equlbrum. Consder an end-user P wth bd (β,d ). Hs payoff at the effcent allocaton s u = v (x ) w = v (x ) α x. Then, gven the bds of others, f he changes hs bd to decrease hs allocaton x by a δ>0(whenx > 0), then note that the allocatons of buyers P k (k ) do not change but seller P sells less. Hs new payoff s u = v (x ) α x = v (x δ) α (x δ). Thus, hs payoff changes by u u = α δ + v (x δ) v (x =(c (x ) )+λ 0 )δ + v (x δ) v (x ) = λ δ + v (x δ) v (x ) = v (x )δ + v (x δ) v (x ) < 0. Thus, hs payoff wll decrease. Now suppose he changes hs bd to ncrease hs allocaton x by a δ>0, then note that the allocaton of player P does not

14 24 W. Tang and R. Jan change but that of some players P k (k ) decrease. Hs new payoff s u = v (x + δ) k β k(x k x k ) α x.thus, u u = k k β k (x k x k )+v (x + δ) v (x ) λ (x k x k )+v (x + δ) v (x ) = λ δ + v (x + δ) v (x ) = λ δ + v (x + δ) v (x ) v (x )δ + v (x + δ) v (x ) < 0. Thus, hs payoff wll decrease agan. Therefore, the best response of an end-user P s to bd (β,d ), and he has no ncentve to devate. Consder a mddleman P wth bd (β,d ) n Ter 1 aucton and bd (α,q )n Ter 2 aucton. Hs payoff at the effcent allocaton s u = β x c (x ). Then, gven the bds of others, f he changes hs bd to ncrease hs allocaton x by a δ>0, hs payoff wll be u = β x c (x + δ) w <u. That s, hs revenue remans the same, whle hs cost and hs payment to player 0 ncrease. Thus, he has no ncentve to ncrease hs allocaton. Now, suppose he changes hs bd to decrease hs allocaton x by a δ>0 (when x > 0). Hs payment to player 0 does not change but the payment he receves changes. Hs new payoff s u = β x c (x δ). Thus, = u u β x β x c (x δ)+c (x ) = β x λ x c (x δ)+c (x ) = λ λ x = λ δ c (x = λ δ c (x λ x c (x δ)+c (x ) (x x ) c (x δ)+c (x δ)+c (x ) ) δ)+c (x ) = (c (x )+λ 0 )δ c (x c (x )δ c (x < 0. δ)+c (x δ)+c (x ) )

15 Herarchcal Auctons for Network Resource Allocaton 25 Thus, hs payoff wll decrease. Snce hs payoff wll decrease by devaton n ether drecton, bddng (β,d )and(α,q ) s hs best response. Ths mples that the constructed bd profle s a Nash equlbrum n the HNSP mechansm and yelds an effcent outcome. We now prove that there s endogenous strong budget balance at ths Nash equlbrum. Note that w = α x = λ x w (for all ), whch s what we wanted to prove. = β x.so w = β x = Remark 1. We can easly check that each end-user and each mddleman has a non-negatve payoff at the Nash equlbrum constructed above. Remark 2. We also note that the HNSP mechansm can be easly extended to the general mult-ter model wheren the Ter 1 sub-mechansm Γ (1) s a VCGtype sngle-sded mechansm, whle sub-mechansms at all lower ters, Γ (2),..., Γ (K 1) are VCG-type double-sded mechansms. Lkewse, we can then establsh the exstence of an effcent Nash equlbrum wth endogenous strong budget balance. 5 Concluson In ths paper, we ntroduced a herarchcal aucton model for network settngs wth mult-ter structures. We developed a general herarchcal mechansm desgn framework. Such a model s nnovatve and ths paper s the frst work on multter auctons to our knowledge. When the resource s ndvsble, we nvestgated a class of mechansms where each sub-mechansm s ether a frst- or a second-prce aucton. We showed that the herarchcal mechansm wth a frst- or a second-prce sub-mechansm at Ter 1, and frst-prce sub-mechansms at all other ters s effcent but not ncentve-compatble and surprsngly, the all-ter second-prce aucton mechansm s ncentve-compatble but not effcent. Ths seems to fortell a more general mpossblty of achevng ncentve compatblty and effcency at the same tme n a herarchcal settng. When the resource s dvsble, we proposed the herarchcal network secondprce mechansm, where the Ter 1 sub-mechansm s a sngle-sded VCG-type aucton and the sub-mechansm at all lower ters s a VCG-type double-sded aucton. We showed that n ths herarchcal mechansm, there exsts an effcent Nash equlbrum wth endogenous strong budget balance. As part of future work, we ntend to study more general classes of mechansms than those where the sub-mechansms are ether frst- or second-prce auctons. We wll also consder the Stackelberg aucton settng, wheren the auctons at varous ters are conducted one after another. We wll also consder more general network topologes wheren there may be more than one resource (e.g., bandwdth on multple lnks, or bandwdth, storage and computaton), and also allow for sub-mechansm auctons wth multple sellers.

16 26 W. Tang and R. Jan References 1. Basar, T., Olsder, G.J.: Dynamc Non-cooperatve Game Theory, 2nd edn. SIAM classcs n appled mathematcs (1999) 2. Btsak, M., Stamouls, G., Courcoubets, C.: A new strategy for bddng n the network-wde progressve second prce aucton for bandwdth. In: Proc. CoNEXT (2005) 3. Cheng, H.H., Tan, G.: Assymetrc common-value auctons wth applcatons to prvate-value auctons wth resale. Economc Theory (2009) 4. Fudenberg, D., Trole, J.: Game Theory, ch. 8. MIT Press (1991) 5. Jan, R., Walrand, J.: An effcent Nash-mplementaton mechansm for network resource allocaton. Automatca 46, (2010) 6. Jan, R., Varaya, P.: A desgn for an asymptotcally effcent combnatoral Bayesan market: Generalzng the Satterthwate-Wllams mechansm. In: Internatonal Conf. on Game Theory, Stony Brook (July 2007) 7. Jan, R., Varaya, P.: The combnatoral sellers bd double aucton: An asymptotcally effcent market mechansm, workng paper (2010) 8. Ja, P., Canes, P.: Auctons on networks: Effcency, consensus, passvty, rates of convergence. In: Proc. IEEE Control and Decson Conf. (CDC) (December 2009) 9. Johar, R., Mannor, S., Tstskls, J.: A contract-based model for drected network formaton. Games and Economc Behavor 56, (2006) 10. Johar, R., Tstskls, J.: Effcency loss n a network resource allocaton game. Mathematcs of Operatons Research 29(3), (2004) 11. Johar, R., Tstskls, J.: Effcency of scaler parameterzed mechansms. Operatons Research 57(4), (2009) 12. Kelly, F.: Chargng and rate control for elastc traffc. Euro. Trans. on Telecommuncatons 8(1), (1997) 13. Lazar, A., Semret, N.: Desgn and analyss of the progressve second prce aucton for network bandwdth sharng. Telecommuncaton Systems - Specal ssue on Network Economcs (1999) 14. Luenberger, D.: Optmzaton by vector space methods. John Wley and Sons (1969) 15. Maheswaran, R., Basar, T.: Socal welfare of selfsh agents: Motvatng effcency for dvsble resources. In: Proc. CDC (2004) 16. Malle, P., Tuffn, B.: Mult-bd auctons for bandwdth allocaton n communcaton networks. In: Proc. IEEE INFOCOM (2004) 17. Mas-Colell, A., Whnston, M., Green, J.: Mcroeconomc Theory, ch. 23. Oxford Unversty Press (1995) 18. Shakkotta, S., Srkant, R.: Economcs of network prcng wth multple ISPs. IEEE/ACM Transactons on Networkng 14(6), (2006) 19. Vckrey, W.: Counterspeculaton, auctons, and sealed tenders. J. Fnance 16, 8 37 (1961) 20. Yang, S., Haek, B.: VCG-Kelly Mechansms for allocaton of dvsble resources: Adaptng VCG mechansms to one-dmensonal sgnals. IEEE J. Selected Areas of Communcatons 25, (2007)