Wenyuan Tang & Rahul Jain Department of Electrical Engineering University of Southern California

Transcription

1 1 Herarchcal Aucton Mechansms for Network Resource Allocaton Wenyuan Tang & Rahul Jan Department of Electrcal Engneerng Unversty of Southern Calforna Abstract Motvated by allocaton of bandwdth, wreless spectrum and cloud computng servces n secondary network markets, we ntroduce a herarchcal aucton model for network resource allocaton. A Ter 1 provder owns a homogeneous network resource and holds an aucton to allocate ths resource among Ter 2 operators, who n turn allocate the acqured resource among Ter 3 enttes. The Ter 2 operators play the role of mddlemen, snce ther utltes for the resource depend on the revenues ganed from resale. We frst consder statc herarchcal aucton mechansms for ndvsble resources. We study a class of mechansms wheren each sub-mechansm s ether a frstprce or VCG aucton, and show that ncentve compatblty and effcency cannot be smultaneously acheved. We also brefly dscuss sequental auctons as well as the ncomplete nformaton settng. We then propose two VCGtype herarchcal mechansms for dvsble resources. The frst one s composed of sngle-sded auctons at each ter, whle the second one employs double-sded auctons at all ters except Ter 1. Both mechansms nduce an effcent Nash equlbrum. Keywords: Network economcs, mechansm desgn, auctons, herarchcal models, resource allocaton. I. INTRODUCTION As networks have become ncreasngly complex, so has the ownershp structure. Ths means that tradtonal models and allocaton mechansms used for resource exchange between prmary owners and end-users are no longer always relevant. Increasngly, there are mddlemen, operators who buy network resources from prmary owners and then sell them to end-users. Although mddlemen play an mportant role n the dstrbuton channel by matchng supply and demand, they also potentally cause neffcences n network resource allocaton. Consder the scenaro of bandwdth allocaton. Network bandwdth s prmarly owned by Ter 1 ISPs (Internet Servce Provders), who then sell t to varous Ter 2 ISPs. Ter 2 ISPs then sell t further to Ter 3 Manuscrpt receved 15 December 2011; revsed 1 June and 1 August Ths work was supported by the NSF grant IIS , CAREER award CNS and an IBM Faculty Award Intal results based on ths work were presented at the GameNets Conference ISPs, and so on. The presence of ISPs n the mddle stages can potentally skew network resource allocaton, and cause neffcences from a socal welfare pont of vew. Smlarly, n the case of wreless spectrum, prmary users that acqure spectrum from the FCC and lease some of t to secondary users also play the role of mddlemen n secondary spectrum markets. As another example, consder cloud computng servces by provders such as IBM, Google, Amazon and others for enterprse end-users (e.g., enterprses havng small computatonal or data center needs). Gartner [13] predcts that as cloud servces are more wdely adopted, there wll be cloud servce brokerages (e.g., Appro) that wll act as mddlemen between provders and end-users. Ths rases the key queston regardng what ncentve compatble or effcent herarchcal mechansms can be used n the presence of mddlemen, and whether these two can be acheved together at all. Auctons as mechansms for network resource allocaton have receved consderable attenton recently. Followng up on the network utlty model proposed by Kelly [10], Johar and Tstskls showed that the Kelly mechansm (wth per-lnk bds) can exhbt up to 25% effcency loss [8]. Ths led to a flurry of actvty n desgnng effcent network resource allocaton mechansms, ncludng the work of Maheshwaran and Basar [14], Johar and Tstskls [9], Yang and Hajek [19], Jan and Walrand [3], Ja and Canes [6] among others [1], [15]. Most of the work focused on sngle-sded auctons for dvsble resources, and s related to the approach of Lazar and Semret [12]. Double-sded network auctons for dvsble resources were developed n [3]. One of the very few to focus on ndvsble network resources s Jan and Varaya [4] whch proposed a Nash mplementaton combnatoral double aucton. Ths s also the only work known so far that presents an ncomplete nformaton analyss of combnatoral market mechansms [5]. All these mechansms ether nvolve network resource allocaton by an auctoneer among multple buyers, or resource exchange among multple buyers and sellers. Most of the proposed mechansms are Nash mplemen-

2 tatons,.e., n whch truth-tellng s a Nash equlbrum but not necessarly a domnant strategy equlbrum. In realty, however, markets for network resources often nvolve mddlemen. Often, they enable markets that do not exst due to nformaton asymmetres, but that can also potentally cause neffcences. However, models wth mddlemen have not been studed much, prmarly due to the dffculty of desgnng approprate mechansms. Even n economc and game theory lterature, the closest related model s one that nvolves a resale among the same set of players after an aucton, n whch the wnners can resell the acqured resources to the losers [2]. There s ndeed some game-theoretc work on network prcng n a general topology. [7] studed a network formaton game where the nodes wsh to form a graph to route traffc among themselves. [17] examned how transt and customer prces and qualty of servce are set n a 3-ter network. However, such work focused on the prcng equlbrum, and problems lke mechansm desgn were not studed. In ths paper, we consder a mult-ter settng. A Ter 1 provder owns a homogeneous network resource and holds an aucton to allocate ths resource among Ter 2 operators, who n turn allocate the acqured resource among Ter 3 enttes, and so on. Each end-user has a valuaton for the resource as a functon of acqured capacty, whle the mddlemen do not have any ntrnsc valuaton of the resource but a quas-valuaton whch depends on the revenue ganed from resale. Our goal s to desgn herarchcal aucton mechansms wth desrable propertes. We frst consder herarchcal mechansm desgn for ndvsble goods. We study a class of mechansms wheren each sub-mechansm s ether a frst-prce or VCG aucton, and show that ncentve compatblty and effcency cannot be acheved smultaneously by such herarchcal mechansms. Ths seems to foretell a more general mpossblty of achevng both ncentve compatblty and effcency n a herarchcal settng. We then study some representatve sequental herarchcal mechansms wth both complete and ncomplete nformaton settngs, and agan observe the dffculty of achevng ncentve compatblty and effcency smultaneously. When the network resource s dvsble, we propose two VCG-type mechansms that employ twodmensonal bds, one wth sngle-sded sub-mechansms at all ters, and one wth double-sded sub-mechansms at all ters except Ter 1. We show that both mechansms nduce an effcent Nash equlbrum. The paper s organzed as follows. We ntroduce the problem n Secton II. In Secton III, we study some herarchcal mechansms for ndvsble goods. Secton IV proposes two herarchcal mechansms for dvsble goods. Secton V concludes the paper. Ter 1: Socal Planner 0 C Γ 1 x 1 1 Ter 2: Mddlemen 2 x 2 Γ 2 Γ 2 3 x 3 x x 5 x 6 6 Ter 3: End-Users Fg. 1. An example of a 3-ter network wth N = 6. A. The Herarchcal Model II. PROBLEM STATEMENT Consder a Ter 1 provder (e.g., the FCC or Google) who owns C unts of a homogeneous network resource. Such a good can be dvsble (sold n arbtrary portons of the total amount) or ndvsble (sold n ntegral unts). Assume that there are K ters n the herarchcal network. The Ter 1 provder auctons off the resource among the Ter 2 enttes, referred to as the Ter 1 aucton. Each Ter 2 entty then auctons off the good acqured n the Ter 1 aucton to the Ter 3 enttes, referred to as the Ter 2 aucton, and n general at Ter k as the Ter k aucton (for 1 k < K). An example of a 3-ter network s shown n Fg. 1. We note that players other than the Ter 1 provder may own some of the resource. However, ths does not affect ther strategc consderatons, and hence s gnored. We consder the Ter 1 provder as the socal planner (ndexed by 0), who attempts to maxmze the socal welfare, whch we wll shortly defne. Ths assumpton s vald when the auctoneer s a governmental agency such as the FCC, and mght stll be reasonable even when the provder s a proft maxmzer (snce the two goals are not necessarly ncompatble). The enttes at other ters are strategc players (ndexed by = 1,..., N), among whch the Ter k (for 1 < k < K) enttes are regarded as the mddlemen, and the Ter K enttes as the end-users. The herarchcal model we consder s hghly stylzed, and each player can acqure the resource only from ts parent n the upper ter; ssues lke routng and peerng are not taken nto account. The stylzed model yelds concrete results that help us n ganng an nsght nto the problem. In fact, even n ths rather smplfed model, we obtan some negatve results, whch suggest the dffculty of herarchcal mechansm desgn n more general settngs. 2

3 B. The Mechansm Desgn Framework Let T () be the ter to whch player belongs, and ch() be the set of player s chldren. Denote the capacty acqured by player by x. Assume that each player has a quaslnear utlty functon u = v (x ) w, where v ( ) s hs valuaton functon and w s hs payment. When player s an end-user, v ( ) s ntrnsc; when player s a mddleman, v (x ) = π c (x ), where π s hs revenue ganed from resale and c ( ) s hs cost functon, snce mddlemen do not derve utltes from the resource but may ncur transacton costs. Denote x = (x 1,..., x N ). We defne the socal welfare S( ) as S(x) = v (x ) c (x ), :T ()=K :1<T ()<K the dfference between the aggregate valuaton derved by the end-users and the aggregate cost ncurred by the mddlemen. Ths s the aggregate socal surplus generated by an allocaton x. The socal planner s objectve s to acheve an effcent allocaton x = (x 1,..., x N ) that solves the socal welfare optmzaton (SWO) problem: maxmze S(x) j ch(0) j ch() x j C, x 0,. x j x, : 1 < T () < K, (1) The frst two constrants state that the total allocaton among the buyers n each aucton cannot exceed the allocaton acqured by ther parent. The thrd constrant s to ensure non-negatve allocatons. In addton, f the resource s ndvsble, x s should be ntegers. Snce the players are strategc and may msreport ther prvate nformaton, our goal s to desgn a mechansm that nduces an effcent allocaton maxmzng the socal welfare. Note that n a herarchcal settng, the socal planner specfes the mechansms to be used at all ters whch must then be used. Ths s qute reasonable when the socal planner s, say the government, and has the power to regulate the market. Denote the mechansm by Γ = (Γ 1,..., Γ K 1 ), n whch a common sub-mechansm Γ k (for 1 k < K) s employed n the Ter k auctons. An aucton (submechansm) s a sngle-sded aucton f buyers place bds and sellers do nothng; t s a double-sded aucton f buyers place buy-bds and sellers place sell-bds. In our model, there s always only one seller n each aucton. Before gong further, we provde a bref dscusson of the nature of the model and the dffcultes of the problem. 1) The players at dfferent ters may bd smultaneously or sequentally, whle the resources are always allocated from Ter 1 to Ter K. Ths suggests that one mght expect smlar results between the two cases under certan condtons. 2) One key dffculty s that mddlemen have no ntrnsc valuatons of the resource. We wll ntroduce the noton of quas-valuaton functons as ther types, whch are related to the revenues ganed from resale. As a result, mddlemen cannot have domnant strateges (see [16] for the defnton), and we wll ntroduce a weaker noton of a domnant strategy. 3) The herarchcal mechansm s decentralzed, wth multple auctons at each ter, and the socal planner holds only one of the N M +1 auctons. Ths makes the achevement of effcency even more dffcult. 4) For dvsble resources, t s mpossble for a player to report an arbtrary real-valued valuaton functon completely. Thus, we have to restrct the bd spaces to be fnte-dmensonal and focus on Nash mplementaton. III. HIERARCHICAL AUCTIONS FOR INDIVISIBLE RESOURCES When the resources are ndvsble, we study a class of mechansms wheren the common sub-mechansm at each ter s ether a frst-prce aucton or a VCG aucton. In a frst-prce aucton (denoted by F), the buyer wth the hghest bd wns the sngle unt good, and pays the amount of hs bd to the seller. In a secondprce aucton, the hghest bdder wns but pays only the second-hghest bd. A second-prce aucton gves buyers an ncentve to bd ther true value whle a frst-prce aucton does not. A generalzaton of the second-prce aucton to multple goods that mantans the ncentve to bd truthfully s known as the Vckrey-Clarke-Groves (VCG) aucton (denoted by V). The dea s that tems are assgned to maxmze the socal welfare; then each player pays the externalty mposed on the other players by hs partcpaton (see [16] for more detals). Wthout loss of generalty, we assume that mddlemen have no transacton cost, snce t can be drectly ncorporated nto the valuaton. 1 Before proceedng, we need to redefne some notons for the herarchcal settng. Defnton 1. A mddleman s quas-valuaton functon v : Z + R + specfes hs revenue from resale for each possble allocaton he may acqure, when all hs chldren report ther valuaton functons (for end-users) or quas-valuaton functons (for mddlemen) truthfully. 1 We wll consder cost functons for dvsble resources though ths elaboraton s stll not crucal. 3

4 The quas-valuaton v (x ) specfes the maxmum that a mddleman s wllng to pay for an allocaton x. Note that ths would depend on how much the mddleman s chldren are wllng to pay for such an allocaton. Gven an allocaton and a payment rule, the quas-valuaton functon s well defned. Also, note the backward-recursveness n the defnton. We now defne a domnant strategy as well as ncentve compatblty n ths new envronment, both of whch are weaker than the defntons n the standard settng. Defnton 2. Gven that all the players ch() report ther valuaton or quas-valuaton functons truthfully, a strategy s a herarchcal domnant strategy for player f t maxmzes hs payoff regardless of what the others play. A domnant strategy yelds the best payoff for a player regardless of what the others play. A player wll always play such a strategy f t exsts. Note that for an end-user, a strategy s a herarchcal domnant strategy f and only f t s a domnant strategy. Defnton 3. A herarchcal mechansm s ncentve compatble f t nduces a herarchcal domnant strategy equlbrum wheren all the players report ther valuaton or quas-valuaton functons truthfully. Such equlbrum strateges wll be referred to as truthtellng as a counterpart of the usual noton of truth-tellng n the standard settng [16]. We remnd the reader that a domnant strategy equlbrum s regarded as a strong soluton concept of a game because t s ndependent of the nformaton (or lack thereof) that a player may have about others. Thus, ncentve compatble mechansms are regarded as very desrable. We manly focus on statc auctons n a complete nformaton envronment. Later, we wll also consder sequental auctons and the ncomplete nformaton settng, and through a case study show how equlbra for these settngs can be derved. A. Statc Auctons wth Complete Informaton We frst study the herarchcal extenson of the frstprce aucton, whch we call the herarchcal frst-prce mechansm, n whch Γ 1 {F, V}, Γ 2 = = Γ K 1 = F,.e., the Ter 1 sub-mechansm s a frst-prce or VCG aucton, whle all the others are frst-prce auctons. We show the exstence of an effcent ɛ-nash equlbrum n ths mechansm. Proposton 1. Assume the socal planner and each mddleman have at least two chldren,.e., the outdegree of each non-termnal node s at least two. Suppose a sngle ndvsble good s to be allocated. 2 In the herarchcal frst-prce mechansm, there exsts an effcent ɛ-nash equlbrum. Proof: We construct such a strategy profle as follows. Suppose n the effcent allocaton, the sngle good s transferred n ths way: 1 = K, where T ( k ) = k for all 1 < k K. Consder the strategy profle: { vk, = b = 2,..., K, v K ɛ, otherwse. Note that v K s the valuaton (a scalar n ths case) of the wnnng end-user. Clearly, ths profle nduces the effcent allocaton that solves the SWO problem (1). It s also easy to check that t s an ɛ-nash equlbrum: no one can gan more than ɛ by unlaterally devatng from hs strategy. Ths proves the clam. Snce the standard frst-prce aucton s not ncentve compatble, the herarchcal extenson cannot be ether. Moreover, as long as a frst-prce aucton exsts as a submechansm, the entre herarchcal mechansm cannot be ncentve compatble, snce truth-tellng s a weakly domnated strategy (always achevng a zero utlty). We state ths as a proposton. Proposton 2. The herarchcal frst-prce mechansm s not ncentve compatble. More generally, for any herarchcal mechansm Γ, f there exsts some k such that Γ k = F, then Γ cannot be ncentve compatble. We now study the herarchcal extenson of the VCG aucton, whch we call the herarchcal VCG mechansm, n whch Γ 1 = = Γ K 1 = V,.e., each submechansm s a VCG aucton. We show the ncentve compatblty of ths mechansm. Proposton 3. Suppose multple unts of an ndvsble good are to be allocated. The herarchcal VCG mechansm s ncentve compatble. Proof: Accordng to Defnton 3, we need to show that truth-tellng s a herarchcal domnant strategy equlbrum. Consder the Ter K 1 aucton. Snce ths s a VCG aucton, truth-tellng s a domnant (and hence herarchcal domnant) strategy for each player wth T () = K. For the purpose of backward nducton, 3 assume each player wth T () = k reports truthfully. Then the quas-valuaton functon of each player wth T () = k 1 can be equvalently vewed as an ntrnsc valuaton functon by Defnton 1. It follows that truthtellng s agan a herarchcal domnant strategy for each 2 For smplcty and to understand the essence of problems that arse n desgn, we focus on allocatng a sngle unt. When multple unts present no addtonal complcatons, we consder them drectly. 3 Here backward refers to the network topology, whereas the game tself s stll statc. 4

5 player wth T () = k 1 by Defnton 2. Thus, all the players wll report truthfully. The fact that the herarchcal frst-prce mechansm s not ncentve compatble, but the herarchcal VCG mechansm s, s not surprsng snce the non-herarchcal frst-prce and VCG mechansms respectvely have these propertes. However, unlke the non-herarchcal VCG mechansm, effcency may not be acheved at the herarchcal domnant strategy equlbrum. We prove ths surprsng observaton n the followng proposton by provdng a counterexample. Proposton 4. The herarchcal domnant strategy equlbrum n the herarchcal VCG mechansm may not be effcent. Proof: We provde a non-trval counterexample. Consder the 3-ter network n Fg. 1 wth C = 5. Wth the notaton v = (v(1), v(2), v(3), v(4), v(5)), let the valuaton functons of the end-users be v 3 = (10, 18, 24, 28, 30), v 4 = (20, 25, 29, 32, 34), v 5 = (15, 24, 32, 39, 45), v 6 = (16, 20, 24, 27, 29). Gven Γ 2 = V, the quas-valuaton functons of the mddlemen are computed as v 1 = (10, 13, 15, 16, 15), v 2 = (15, 13, 16, 18, 19). Truth-tellng s a herarchcal domnant strategy equlbrum wth the allocaton (x 1, x 2, x 3, x 4, x 5, x 6) = (4, 1, 3, 1, 0, 1). However, the effcent allocaton derved by (1) s (x 1, x 2, x 3, x 4, x 5, x 6 ) = (2, 3, 1, 1, 2, 1). Thus, the herarchcal domnant strategy equlbrum s not effcent. As shown above, quas-valuaton functons are not monotone n general, whch suggests that t s very unlkely to be effcent for a herarchcal domnant strategy equlbrum n the herarchcal VCG mechansm. Moreover, though one may derve condtons on the valuaton functons of the end-users under whch effcency can be acheved for smple cases (e.g., 3-ter network wth a sngle unt), t s hard to obtan such condtons for general K-ter networks wth multple unts. A lmted mpossblty result follows mmedately when we restrct our attenton to frst-prce and VCG auctons as sub-mechansms. Theorem 1 (Herarchcal Impossblty). Suppose we allocate a sngle ndvsble good n a K-ter network (K 3). There does not exst an ncentve compatble herarchcal mechansm Γ wth Γ k {F, V} (for 1 k < K) whch nduces an effcent herarchcal domnant strategy equlbrum. Proof: By Proposton 2, ncentve compatblty cannot be acheved f there exsts some k such that Γ k = F. On the other hand, f Γ k = V for all k, effcency s not guaranteed n the herarchcal domnant strategy equlbrum by Proposton 4. Ths proves the clam. Our conjecture s that ths lmted mpossblty theorem foretells a more general mpossblty result n herarchcal mechansm desgn wth arbtrary submechansms at each ter. B. Sequental Auctons wth Complete Informaton In ths secton, we consder a settng where the resource s allocated herarchcally va a sequental aucton,.e., auctons at varous ters do not take place smultaneously but sequentally. Desgn of such sequental auctons requres the theory of dynamc mechansm desgn, whch s not well developed. Thus, we study some specfc dynamc mechansms to understand sequental herarchcal mechansm desgn and for smplcty, focus on a 3-ter network. We defne two types of sequental auctons. Top-down aucton (TD): In the frst stage, Ter 2 players bd smultaneously. In the second stage, after observng all the prevous bds, Ter 3 players bd smultaneously. Then the allocaton s realzed. Bottom-up aucton (BU): In the frst stage, Ter 3 players bd smultaneously whch are observed by all players. In the second stage, Ter 2 players bd smultaneously, and then the allocaton s realzed. Recall that a player s strategy n a game s a complete contngent plan that specfes how the player wll act n every contngency n whch he mght be called upon to move. In TD, each mddleman s strategy space s the same as that n the statc game (whch s dentcal to the acton space), whle each end-user s strategy must specfy one acton for each possble set of the mddlemen s bds (whch are observed when the endusers bd). In BU, however, each end-user s strategy space s the same as that n the statc game, whle each mddleman s strategy must specfy one acton for each possble set of the end-users bds. We can deduce the equlbra n the sequental auctons by a proper modfcaton of the equlbra n the statc auctons. Defnton 4. A strategy profle s = (s 1,..., s N ) n the sequental aucton s an adaptaton of the strategy (acton) profle a = (a 1,..., a N ) n the statc aucton f s ( ) a for all. That s, n an adaptaton, the strategy of each player s ndependent of the contngency. We have the followng proposton, the proof of whch s trval and thus omtted. Proposton 5. Assume the statc herarchcal aucton and the (TD or BU) sequental aucton employ the same sub-mechansms. For any Nash equlbrum n the statc 5

6 aucton, ts adaptaton s also a Nash equlbrum n the sequental aucton and nduces the same allocaton. Example 1. Consder VCG mechansms as submechansms n the 3-ter network wth v ( v ) as the valuatons (quas-valuatons) of the Ter 3 (Ter 2) players. A Nash equlbrum n the statc herarchcal aucton s { v, T () = 2, b = v, T () = 3. The adaptatons n TD and BU are { b T D v, T () = 2, = v, {b j } T (j)=2, T () = 3, { b BU v, {b = j } T (j)=3, T () = 2, v, T () = 3, whch are respectvely the Nash equlbra n the two sequental auctons. For dynamc games, however, subgame perfect Nash equlbra are more relevant. Recall that a strategy profle s a subgame perfect Nash equlbrum f t nduces a Nash equlbrum n every subgame of the orgnal game. We show that an adaptaton (whch s always a Nash equlbrum) may not be a subgame perfect Nash equlbrum. Example 2. Consder the same settng as n Example 1. It s easy to check that the adaptaton n TD s also a subgame perfect equlbrum. However, the adaptaton n BU s not. By a slght abuse of notaton, for each player who s a mddleman, let v ({b j } T (j)=3 ) be hs revenue functon (that depends on the end-users bds). Then, a subgame perfect equlbrum n BU s { v ({b b = j } T (j)=3 ), T () = 2, v, T () = 3. On the equlbrum path, we have v ({b j } T (j)=3 ) = v ({v j } T (j)=3 ) = v, the quas-valuaton functon. C. Sequental Frst-Prce Aucton wth Incomplete Informaton We now consder herarchcal auctons wth ncomplete nformaton. Specfcally, we nvestgate a natural extenson of frst-prce auctons, whch we call the sequental frst-prce aucton wth ncomplete nformaton. Note that varants of VCG auctons for the ncomplete nformaton settng would be trval due to the ncentve compatblty. Consder a 3-ter network wth a sngle ndvsble good to be allocated. There are two mddlemen, player 1 and player 2, wth N 1 and N 2 chldren respectvely. The end-users valuatons are drawn from a commonly known pror dstrbuton, and n partcular, are..d. random varables unformly dstrbuted on the nterval [0, 1]. The stages of the game are as follows: 1) Nature draws a valuaton v U[0, 1] for each end-user ndependently and reveals t to that player. 2) The end-users bd smultaneously, wth b = β (v ) for player. 3) Each mddleman learns the bds of hs own chldren, but not those of the others. 4) The mddlemen bd smultaneously, wth b 1 = β 1 (v 1 ) and b 2 = β 2 (v 2 ) respectvely. 5) The allocaton and the payments are determned accordng to the mechansm Γ 1 = Γ 2 = F. We look for a perfect Bayesan equlbrum of ths game. As we wll see, the problem can be converted nto an asymmetrc frst-prce aucton n the standard settng. 4 Let {1, 2}. It can be shown that the equlbrum strategy of player s chldren s β(v) = N 1 N v. The pror dstrbuton of player s revenue s F (v) = N v N 1 (N 1)v N, wth the assocated probablty densty functon f (v) = N (N 1)(1 v)v N 2. Whenever N 1 N 2, Γ 1 s a frst-prce aucton wth asymmetrc bdders,.e., F 1 ( ) F 2 ( ). In equlbrum, we have β 1 (0) = β 2 (0) = 0, β 1 (1) = β 2 (1). (2) Let φ := β 1. We obtan the frst-order condton for player : φ (b) = F (φ (b)) f (φ (b)) 1 φ j (b) b. (3) A soluton to the system of dfferental equatons (2)- (3) consttutes equlbrum strateges among the mddlemen. Whle a closed-form expresson s not avalable, we derve the propertes of the equlbrum strateges ndrectly. Assume that N 1 > N 2. It s easy to check that the dstrbuton F 1 domnates F 2 n terms of the reverse hazard rate,.e., f1(v) F > f2(v) 1(v) F 2(v) for all v (0, 1). In [11], t s proved that the weak player 2 bds more aggressvely than the strong player 1,.e., β 1 (v) < β 2 (v) for all v (0, 1). Clearly, ths result leads to the neffcency n the Ter 1 aucton, and therefore to the neffcency n the entre herarchcal mechansm. Agan, ths s a negatve result for herarchcal mechansm desgn: effcency s not guaranteed even when the end-users are symmetrc and frst-prce auctons are held everywhere. IV. HIERARCHICAL AUCTIONS FOR DIVISIBLE RESOURCES Often, network resources such as bandwdth and spectrum are avalable as (nfntely) dvsble resources. 4 Due to space constrants, we only provde the sketch of the dervaton. Readers can refer to [11] for detals of the approach. 6

7 We thus, now consder herarchcal mechansms for a dvsble resource. For smplcty of exposton, we focus on the 3-ter network as n Fg. 1, wth some change of notaton: let player be the th mddleman and player (, j) be the end-user who s the jth chld of player. 5 The valuaton functon v j ( ) of player (, j) s assumed to be strctly ncreasng, strctly concave and contnuously dfferentable on [0, ), wth v j (0) = 0. The cost functon c ( ) of player s assumed to be strctly ncreasng, strctly convex and contnuously dfferentable on [0, ), wth c (0) = 0. The payoff of player (, j) s u j = v j (x j ) w j, where w j s the payment made by player (, j). The payoff of player s u = π w c (x ), where π s player s revenue and w s the payment made by player. We defne the endogenous budget balance condton: π = j w j,. (4) The socal welfare optmzaton problem for the case of dvsble resources (DIV-OPT) s maxmze c (x ) v j (x j ) (,j) x C, [λ 0 ] x j x,, [λ ] j x, x j 0,, (, j), where λ 0 and λ s are the correspondng Lagrange multplers (lkewse for the followng). The soluton of the convex optmzaton problem s characterzed by the Karush-Kuhn-Tucker (KKT) condtons: (c (x ) + λ 0 λ )x = 0,, c (x ) + λ 0 λ 0,, (v j(x j ) λ )x j = 0, (, j), v j(x j ) λ 0, (, j), λ 0 ( x C) = 0, x C 0, λ ( j x j x ) = 0,, j x j x 0,, λ 0, λ, x, x j 0,, (, j). (5) Our objectve s to desgn a herarchcal mechansm that nduces an effcent allocaton as a soluton of the DIV-OPT problem, despte the strategc behavor of the players. Wth dvsble resources, however, t s mpossble for a player to report an arbtrary realvalued valuaton (or cost) functon completely. Thus, the 5 The results extend to the general K-ter network albet the notaton s more complcated. mechansm must ask each player to communcate an approxmaton to the functon from a fnte-dmensonal bd space, and domnant strategy mplementaton cannot be acheved here. Instead, we seek a statc Nash mplementaton n a complete nformaton settng. We propose two VCG-type mechansms, one sngle-sded and one double-sded, both of whch have two-dmensonal bds that specfy the unt prce and the quantty. Such bd spaces are natural and used n many practcal scenaros. A. Herarchcal Sngle-Sded VCG Mechansm We frst propose the herarchcal sngle-sded VCG (HSVCG) mechansm. In the Ter 1 aucton, player reports a bd b = (β, d ), where β s the bd prce and d s the maxmum quantty desred; n the th Ter 2 aucton, player (, j) reports a bd b j = (β j, d j ), where β j s the bd prce and d j s the maxmum quantty desred. The allocaton s then determned as follows. In the Ter 1 aucton, the allocaton x = ( x, ) s a soluton of the followng optmzaton problem (HSVCG-1): maxmze β x x C, [µ 0 ] x d,, [µ ] x 0,. (6) Let x = ( x l, l) denote the allocaton as a soluton of the above wth d = 0,.e., when player s not present. Then, the payment made by player s w = l β l ( x l x l ). In the th Ter 2 aucton (n whch x has been determned), the allocaton x = ( x j, j) s a soluton of the followng optmzaton problem (HSVCG-2): maxmze (β j β )x j j x j x, [ν ] j x j d j, j, [ν j ] x j 0, j. Let x j = ( x j k, k) denote the allocaton as a soluton of the above wth d j = 0,.e., when player (, j) s not present. Then, the payment made by player (, j) s w j = β ( x k x j k ) + k k j β k ( x j k x k), (7) and player s revenue equals exactly the total payment of hs chldren, as n (4). 7

8 The soluton of (6) s characterzed by the KKT condtons: (β µ 0 µ )x = 0,, β µ 0 µ 0,, µ 0 ( x C) = 0, x C 0, µ (x d ) = 0,, x d 0,, µ 0, µ, x 0,, and the soluton of (7) s characterzed by the KKT condtons (gven a fxed ): (β j β ν ν j )x j = 0, j, β j β ν ν j 0, j, ν ( j x j x ) = 0, j x j x 0, ν j (x j d j ) = 0, j, x j d j 0, j, ν, ν j, x j 0, j. Theorem 2. The HSVCG mechansm nduces an effcent Nash equlbrum. Proof: Let x = ((x, ), (x j, (, j))) be an effcent allocaton that solves (5). Then gven a fxed, v j (x j ) s are equal and v j (x j ) c (x ), for all j wth x j > 0. Consder the strategy profle: β = max j v j (x j ), d = x, βj = v j (x j ), d j = x j. It s easy to check that x s also a soluton of (6) and (7). It remans to show that the constructed strategy profle s a Nash equlbrum. 6 Consder player, a mddleman whose payoff s u = β x c (x ). Suppose he devates by changng hs bd to (β, d ) wth the resultng allocaton x. If t s possble for hm to be better off, there must be x x (gven hs chldren s demand fxed) and β β (otherwse he wll get a zero revenue). Then, we have u β x c (x ), and u u β (x x ) c (x ) + c (x c (x 0, )(x x ) ) c (x ) + c (x ) where the last nequalty follows from convexty and monotoncty. Thus, he has no ncentve to devate. Consder player (, j), an end-user whose payoff s u j = v j (x j ) v j (x j )x j. Suppose he devates by changng hs bd to (β j, d j ) wth the resultng 6 Due to space constrants, the KKT condtons used n the argument are not explctly stated. allocaton x j. If t s possble for hm to be better off, there must be β j βj (otherwse he wll get a zero allocaton). Then, u j v j (x j ) v j (x j )x j, and whether x j x j or x j < x j, we always have u j u j = v j (x j ) v j (x j )+v j(x j )(x j x j ) 0, where the nequalty follows from concavty and monotoncty. Thus, he has no ncentve to devate. B. Herarchcal Double-Sded VCG Mechansm In the HSVCG mechansm, all the sub-mechansms were sngle-sded auctons. We now propose the herarchcal double-sded VCG (HDVCG) mechansm n whch double-sded auctons are employed at all ters except Ter 1. It seems that ths mechansm provdes more freedom for a mddleman: besdes a buy-bd, he can place an addtonal sell-bd. Nevertheless, t turns out that the two mechansms are outcome-equvalent, n the sense that both mechansms nduce an effcent Nash equlbrum. We specfy the HDVCG mechansm for the 3-ter network. The Ter 1 aucton s sngle-sded: player reports a bd b = (β, d ), where β s the bd prce and d s the maxmum quantty desred. Each Ter 2 aucton s double-sded: n the th Ter 2 aucton, player reports a sell-bd a = (α, q ), where α s the sellbd prce and q s the maxmum quantty offered; player (, j) reports a buy-bd b j = (β j, d j ), where β j s the buy-bd prce and d j s the maxmum quantty desred. The allocaton s then determned as follows. In the Ter 1 aucton, the allocaton x = ( x, ) s a soluton of the followng optmzaton problem (HDVCG-1): 7 maxmze β x x C, [µ 0 ] x d,, [µ ] x 0,. (8) Let x = ( x l, l) denote the allocaton as a soluton of the above wth d = 0,.e., when player s not present. Then, the payment made by player s w = l β l ( x l x l ). In the th Ter 2 aucton (n whch x has been determned), the allocaton x = ( x j, j) s a soluton of the 7 Note that HDVCG-1 s dentcal to HSVCG-1. 8

9 followng optmzaton problem (HDVCG-2): maxmze (β j α )x j j x j mn{ x, q }, [ν ] j x j d j, j, [ν j ] x j 0, j. Let x j = ( x j k, k) denote the allocaton as a soluton of the above wth d j = 0,.e., when player (, j) s not present. Then, the payment made by player (, j) s w j = α ( x k x j k ) + k k j and player s revenue s π = j β j x j. β k ( x j k x k), (9) Note that unlke HSVCG, the endogenous budget balance s not necessarly acheved n HDVCG; however, as we wll see, t s ensured at the effcent Nash equlbrum we construct. The soluton of (8) s characterzed by the KKT condtons: (β µ 0 µ )x = 0,, β µ 0 µ 0,, µ 0 ( x C) = 0, x C 0, µ (x d ) = 0,, x d 0,, µ 0, µ, x 0,, and the soluton of (9) s characterzed by the KKT condtons (gven a fxed ): (β j α ν ν j )x j = 0, j, β j α ν ν j 0, j, ν ( j x j mn{x, q }) = 0, j x j mn{x, q } 0, ν j (x j d j ) = 0, j, x j d j 0, j, ν, ν j, x j 0, j. Theorem 3. The HDVCG mechansm nduces an effcent Nash equlbrum, at whch the endogenous budget balance s acheved. Proof: Let x = ((x, ), (x j, (, j))) be an effcent allocaton that solves (5). Consder the strategy profle: β = α = max j v j (x j ), d = q = x, βj = v j (x j ), d j = x j. It s easy to check that x s also a soluton of (8) and (9). Moreover, π = j v j (x j )x j = j w j for all. It remans to show that the constructed strategy profle s a Nash equlbrum. Consder player, a mddleman whose payoff s u = c (x ). Suppose he devates by changng hs bd to ((β, d ), (α, q )) wth the resultng allocaton x. If t s possble for hm to be better off, there must be x x (gven hs chldren s demand fxed) and α α (otherwse he wll get a zero revenue). Then, we have u α x c (x ), and α x u u α (x x ) c (x ) + c (x c (x 0. )(x x ) ) c (x ) + c (x ) Thus, he has no ncentve to devate. Consder player (, j), an end-user whose payoff s u j = v j (x j ) v j (x j )x j. Suppose he devates by changng hs bd to (β j, d j ) wth the resultng allocaton x j. If t s possble for hm to be better off, there must be β j βj (otherwse he wll get a zero allocaton). Then, u j = v j (x j ) v j (x j )x j, and whether x j x j or x j < x j, we always have u j u j = v j (x j ) v j (x j )+v j(x j )(x j x j ) 0. Thus, he has no ncentve to devate. V. CONCLUSIONS In ths paper, we ntroduced a herarchcal network resource allocaton model. We developed a general herarchcal mechansm desgn framework for such models. Such a model and framework s novel and ths paper s the frst work on mult-ter auctons to our best knowledge. When the resource s ndvsble, we nvestgated a class of mechansms wheren each sub-mechansm s ether a frst-prce or VCG aucton. We showed that the herarchcal mechansm wth a frst-prce or VCG aucton at Ter 1, and frst-prce auctons at all other ters s effcent but not ncentve compatble and surprsngly, the herarchcal VCG aucton mechansm s ncentve compatble but not necessarly effcent. Ths seems to foretell a more general mpossblty of achevng both ncentve compatblty and effcency n a herarchcal settng. We also studed some representatve mechansms for sequental auctons as well as the ncomplete nformaton settng, n whch smlar results can be obtaned. When the resource s dvsble, we propose two VCGtype mechansms. The HSVCG mechansm s composed of sngle-sded auctons at each ter, whle the HDVCG mechansm employs double-sded auctons at all ters except Ter 1. Both mechansms nduce an effcent Nash equlbrum. Moreover, the HSVCG mechansm 9

10 always acheves endogenous budget balance, whle that s ensured only at an effcent equlbrum n the HDVCG mechansm. We note that the mechansms we have desgned can easly be extended to the settng where there are endusers even at ntermedate ters. The key results wll reman unchanged for such a settng. Another natural queston s whether more general classes of ncentve compatble or effcent mechansms can be desgned than those wheren the sub-mechansms are ether frst-prce or VCG auctons. Indeed, ths s an mportant queston. But as we show for ndvsble resources, consderng just these two, leads to herarchcal mechansms that are ether effcent or ncentve compatble, but not both. We expect an mpossblty result whch clams the nonexstence of herarchcal mechansms that are both ncentve compatble and effcent. Provng such a conjecture requres new developments, whch we shall consder n future work. In future work, 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 allow for submechansm auctons wth multple sellers. Moreover, we may allow each Ter k player to partcpate n any of the Ter k 1 auctons as well. 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] H. CHENG AND G. TAN, Asymmetrc common-value auctons wth applcatons to prvate-value auctons wth resale, Economc Theory 45(1): , [3] R. JAIN AND J. WALRAND, An effcent Nash-mplementaton mechansm for network resource allocaton, Automatca 46(8): , [4] R. JAIN AND P. VARAIYA, A desgn for an asymptotcally effcent combnatoral Bayesan market: Generalzng the Satterthwate-Wllams mechansm, Intl. Conf. on Game Theory, [5] R. JAIN AND P. VARAIYA, An asymptotcally effcent mechansm for combnatoral network markets, submtted to Operatons Research, [6] P. JIA AND P. CAINES, Auctons on networks: Effcency, consensus, passvty, rates of convergence, Proc. CDC, [7] R. JOHARI, S. MANNOR AND J. TSITSIKLIS, A contract-based model for drected network formaton, Games and Economc Behavor 56(2): , [8] R. JOHARI AND J. TSITSIKLIS, Effcency loss n a network resource allocaton game, Mathematcs of Operatons Research 29(3): , [9] R. JOHARI AND J. TSITSIKLIS, Effcency of scalar parameterzed mechansms, Operatons Research 57(4): , [10] F. KELLY, Chargng and rate control for elastc traffc, Euro. Trans. on Telecommuncatons 8(1):33-37, [11] VIJAY KRISHNA, Aucton Theory, Second Edton, Academc Press, [12] 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, [13] B. LHEUREUX AND D. PLUMMER, Cloud Servces Brokerage Prolferates, Gartner Research, [Onlne] Avalable: [14] R. MAHESWARAN AND T. BASAR, Socal welfare of selfsh agents: Motvatng effcency for dvsble resources, Proc. CDC, [15] P. MAILLE AND B. TUFFIN, Mult-bd auctons for bandwdth allocaton n communcaton networks, Proc. INFOCOM, [16] A. MAS-COLELL, M. WHINSTON AND J. GREEN, Chapter 23, Mcroeconomc Theory, Oxford Unversty Press, [17] S. SHAKKOTTAI AND R. SRIKANT, Economcs of network prcng wth multple ISPs, IEEE/ACM Trans. on Networkng 14(6): , [18] W. VICKREY, Counterspeculaton, auctons, and sealed tenders, J. Fnance 16(1):8-37, [19] S. YANG AND B. HAJEK, VCG-Kelly mechansms for allocaton of dvsble resources: Adaptng VCG mechansms to onedmensonal sgnals, IEEE J. Selected Areas of Communcatons 25(6): , Wenyuan Tang s a USC Annenberg Fellow pursung a Ph.D. degree n Electrcal Engneerng at the Unversty of Southern Calforna. He receved a B.E. from Tsnghua Unversty n 2008, and an M.S. from the Unversty of Southern Calforna n 2010, both n Electrcal Engneerng. Hs research nterests nclude network economcs, game theory, mechansm desgn, and market desgn for smart grds. Rahul Jan s an assstant professor and the K. C. Dahlberg Early Career Char n the EE department at the Unversty of Southern Calforna. He receved hs PhD n EECS and an MA n Statstcs from the Unversty of Calforna, Berkeley, hs B.Tech from IIT Kanpur. He s wnner of numerous awards ncludng the NSF CAREER award, an IBM Faculty award and the ONR Young Investgator award. Hs research nterests span wreless communcatons, network economcs and game theory, queueng theory, power systems and stochastc control theory. 10