Local public good provisioning in networks: A Nash implementation mechanism

Transcription

1 Local publc good provsonng n networks: 1 A Nash mplementaton mechansm Shrutvandana Sharma and Demosthens Teneketzs Abstract In ths paper we study resource allocaton n decentralzed nformaton local publc good networks. A network s a local publc good network f each user s actons drectly affect the utlty of an arbtrary subset of network users. We consder networks where each user knows only that part of the network that ether affects or s affected by t. Furthermore, each user s utlty and acton space are ts prvate nformaton, and each user s a self utlty maxmzer. Ths network model s motvated by several applcatons ncludng wreless communcatons. For ths network model we formulate a decentralzed resource allocaton problem and develop a decentralzed resource allocaton mechansm game form) that possesses the followng propertes: ) All Nash equlbra of the game nduced by the mechansm result n allocatons that are optmal solutons of the correspondng centralzed resource allocaton problem Nash mplementaton). ) All users voluntarly partcpate n the allocaton process specfed by the mechansm ndvdual ratonalty). ) The mechansm results n budget balance at all Nash equlbra and off equlbrum. I. INTRODUCTION In networks ndvduals actons often nfluence the performance of ther drectly connected neghbors. Such an nfluence of ndvduals actons on ther neghbors performance can propagate through the network affectng the performance of the entre network. Examples nclude several real world networks. As an nstance, n a wreless cellular network, the transmsson of the base staton to a gven user an acton correspondng to ths user) creates nterference to the recepton of other users and affects ther performance. In an urban network, when a jursdcton nsttutes a polluton abatement program, the benefts also accrue to nearby communtes. The nfluence of neghbors s also observed n the spread of nformaton and nnovaton n socal and research networks. Networks wth above characterstcs are called local publc good networks. A local publc good network dffers from a typcal publc good system n that a local publc good alternatvely, the acton of an ndvdual) s accessble to and drectly nfluences the utltes of ndvduals

2 2 n a partcular neghborhood wthn a bg network. On the other hand a publc good s accessble to and drectly nfluences the utltes of all ndvduals n the system [1, Chapter 11]). Because of the localzed nteractons of ndvduals, n local publc good networks such as ones descrbed above) the nformaton about the network s often localzed;.e., the ndvduals are usually aware of only ther neghborhoods and not the entre network. In many stuatons the ndvduals also have some prvate nformaton about the network or ther own characterstcs that are not known to anybody else n the network. Furthermore, the ndvduals may also be selfsh who care only about ther own beneft n the network. Such a decentralzed nformaton local publc good network wth selfsh users gves rse to several research ssues. In the next secton we provde a lterature survey on pror research n local publc good networks. A. Lterature survey There exsts a large lterature on local publc goods wthn the context of local publc good provsonng by varous muncpaltes that follows the semnal work of [2]. These works manly consder network formaton problems n whch ndvduals choose where to locate based on ther knowledge of the revenue and expendture patterns on local publc goods) of varous muncpaltes. In ths paper we consder the problem of determnng the levels of local publc goods actons of network agents) for a gven network; thus, the problem addressed n ths paper s dstnctly dfferent from those n the above lterature. Recently, Bramoullé and Kranton [3] and Yuan [4] analyzed the nfluence of selfsh users behavor on the provson of local publc goods n networks wth fxed lnks among the users. The authors of [3] study a network model n whch each user s payoff equals ts beneft from the sum of efforts treated as local publc goods) of ts neghbors less a cost for exertng ts own effort. For concave beneft and lnear costs, the authors analyze Nash equlbra NE) of the game where each user s strategy s to choose ts effort level that maxmzes ts own payoff from the provsonng of local publc goods. The authors show that at such NE, specalzaton can occur n local publc goods provsonng. Specalzaton means that only a subset of ndvduals contrbute to the local publc goods and others free rde. The authors also show that specalzaton can beneft the socety as a whole because among all NE, the ones that are specalzed result n the hghest socal welfare sum of all users payoffs). However, t s shown n [3] that none of the NE of abovementoned game can result n a local publc goods provsonng that acheves the maxmum possble socal welfare. In [4] the work of [3] s extended to drected networks where the externalty effects of users efforts on each others payoffs can be undrectonal or bdrectonal. The authors of [4] nvestgate the relaton between the structure of drected networks and the exstence and

3 3 nature of Nash equlbra of users effort levels n those networks. For that matter they ntroduce a noton of ergodc stablty to study the nfluence of perturbaton of users equlbrum efforts on the stablty of NE. However, none of the NE of the game analyzed n [4] result n a local publc goods provsonng that acheves optmum socal welfare. In ths paper we consder a generalzaton of the network models nvestgated n [3], [4]. Specfcally, we consder a fxed network where the actons of each user drectly affect the utltes of an arbtrary subset of network users. In our model, each user s utlty from ts neghbors actons s an arbtrary concave functon of ts neghbors acton profle. Each user n our model knows only that part of the network that ether affects or s affected by t. Furthermore, each user s utlty and acton space are ts prvate nformaton, and each user s a self utlty maxmzer. Even though the network model we consder has smlartes wth those nvestgated n [3], [4], the problem of local publc goods provsonng we formulate n ths paper s dfferent from those n [3], [4]. Specfcally, we formulate a problem of local publc goods provsonng n the framework of mplementaton theory 1 and address questons such as How should the network users communcate so as to preserve ther prvate nformaton, yet make t possble to determne actons that acheve optmum socal welfare? How to provde ncentves to the selfsh users to take actons that optmze the socal welfare? How to make the selfsh users voluntarly partcpate n any acton determnaton mechansm that ams to optmze the socal welfare? In a nutshell, the pror work of [3], [4] analyzed specfc games, wth lnear cost functons, for local publc good provson, whereas our work focusses on desgnng a mechansm that can nduce, va nonlnear tax functons, approprate games among the network users so as to mplement the optmum socal welfare n NE. It s ths dfference n the tax functons that dstngushes our results from those of [3], [4]. Prevous works on mplementaton approach Nash mplementaton) for pure) publc goods can be found n [9], [10], [11], [12]. For our work, we obtaned nspraton from [10]. In [10] Hurwcz presents a Nash mplementaton mechansm that mplements the Lndahl allocaton optmum socal welfare) for a publc good economy. Hurwcz mechansm also possesses the propertes of ndvdual ratonalty.e. t nduces the selfsh users to voluntarly partcpate n the mechansm) and budget balance.e. t balances the flow of money n the system). A local publc good network can be thought of as a lmtng case of a publc good network, n whch the nfluence of each publc good tends to vansh on a subset of network users. However, takng the correspondng lmts n the Hurwcz mechansm does not result n 1 Refer to [5], [6], [7] and [8, Chapter 3] and for an ntroducton to mplementaton theory.

4 4 a local publc good provsonng mechansm wth all the orgnal propertes of the Hurwcz mechansm. In partcular, such a lmtng mechansm does not retan the budget balance property whch s very mportant to avod any scarcty/wastage of money. In ths paper we address the problem of desgnng a local publc good provsonng mechansm that possesses the desrable propertes of budget balance, ndvdual ratonalty, and Nash mplementaton of optmum socal welfare. The mechansm we develop s more general than Hurwcz mechansm; Hurwcz mechansm can be obtaned as a specal case of our mechansm by settng R = C j = N, j, N the specal case where all users actons affect all users utltes) n our mechansm. Our mechansm also provdes a more effcent way to acheve the propertes of Nash mplementaton, ndvdual ratonalty, and budget balance as t uses, n general, a much smaller message space than Hurwcz mechansm. To the best of our knowledge the resource allocaton problem and ts soluton that we present n ths paper s the frst attempt to analyze a local publc goods network model n the framework of mplementaton theory. Below we state our contrbutons. B. Contrbuton of the paper The key contrbutons of ths paper are: 1) The formulaton of a problem of local publc goods provsonng n the framework of mplementaton theory. 2) The specfcaton of a game form 2 decentralzed mechansm) for the above problem that, ) mplements n NE the optmal soluton of the correspondng centralzed local publc good provsonng problem; ) s ndvdually ratonal; 3 and ) results n budget balance at all NE and off equlbrum. The rest of the paper s organzed as follows. In Secton II-A we present the model of local publc good network. In Secton II-C we formulate the local publc good provsonng problem. In Secton III-A we present a game form for ths problem and dscuss ts propertes n Secton III-B. We conclude n Secton IV wth a dscusson on future drectons. Notaton used n the paper: We use bold font to represent vectors and normal font for scalars. We use bold uppercase letters to represent matrces. We represent the element of a vector by a subscrpt on the vector symbol, and the element of a matrx by double subscrpt on the matrx symbol. To denote the vector whose elements are all x such that S for some set S, we use the notaton x ) S and we abbrevate t as x S. We treat bold 0 as a zero vector of approprate sze whch s determned by the context. We use the notaton x, x /) to represent a vector of dmenson same as that of x, whose th 2 See [8, Chapter 3] and [7], [6], [5] for the defnton of game form. 3 Refer to [8, Chapter 3] and [7] for the defnton of ndvdual ratonalty and mplementaton n NE.

5 5 element s x and all other elements are the same as the correspondng elements of x. We represent a dagonal matrx of sze N N whose dagonal entres are elements of the vector x R N by dagx). II. THE LOCAL PUBLIC GOOD PROVISIONING PROBLEM In ths secton we present a model of local publc good network motvated by varous applcatons such as wreless communcaton [8, Chapter 5], socal and nformaton networks [4], [3]. We frst descrbe the components of the model and the assumptons we make on the propertes of the network. We then present a resource allocaton problem for ths model and formulate t as an optmzaton problem. A. The network model M) We consder a network consstng of N users and one network operator. We denote the set of users by N := {1, 2,..., N}. Each user N has to take an acton a A where A s the set that specfes user s feasble actons. In a real network, a user s actons can be consumpton/generaton of resources or decsons regardng varous tasks. We assume that, Assumpton 1: For all N, A s a convex and compact set n R that ncludes 0. 4 Furthermore, for each user N, the set A s ts prvate nformaton,.e. A s known only to user and nobody else n the network. Because of the users nteractons n the network, the actons taken by a user drectly affect the performance of other users n the network. Thus, the performance of the network s determned by the collectve actons of all users. We assume that the network s large-scale, therefore, every user s actons drectly affect only a subset of network users n N. Thus we can treat each user s acton as a local publc good. We depct the above feature by a drected graph as shown n Fg. 1. In the graph, an arrow from j to ndcates that user j affects user ; we represent the same n the text as j. We assume that for all N. Mathematcally, we denote the set of users that affect user by R := {k N k }. Smlarly, we denote the set of users that are affected by user j by C j := {k N j k}. We represent the nteractons of all network users together by a graph matrx G := [G j ] N N. The matrx G conssts of 0 s and 1 s, where G j = 1 represents that user s affected by user j,.e. j R and G j = 0 represents no nfluence of user j on user,.e. j / R. Note that G need not be a symmetrc matrx. Because, G = 1 for all N. We assume that, 4 In ths paper we assume the sets A, N, to be n R for smplcty. However, the decentralzed mechansm and the results we present n ths paper can be easly generalzed to the scenaro where for each N, A R n s a convex and compact set n hgher dmensonal space R n. Furthermore, each space R n can be of a dfferent dmenson n for dfferent N.

6 6 j I j = 5 C j5) = I jj = 4 C j4) = j 0 I pj = p j I hj = 1 = I j + 1 C j1) = h h k I kj = 2 C j2) = k Set R Set C j l I lj = 3 C j3) = l Set C j Fg. 1. A local publc good network depctng the Neghbor sets R and C j of users and j respectvely. Fg. 2. Illustraton of ndexng rule for set C j shown n Fg. 1. Index I rj of user r C j s ndcated on the arrow drected from j to r. The notaton to denote these ndces and to denote the user wth a partcular ndex s shown outsde the dashed boundary demarcatng the set C j. Assumpton 2: The sets R, C, N, are ndependent of the users acton profle a N := a k ) k N k N A k. Furthermore, for each N, C 3. We consder the condton C 3, N, so as to ensure constructon of a mechansm that s budget balanced at all possble allocatons, those that correspond to Nash equlbra as well as those that correspond to off-equlbrum messages. For examples of applcatons where Assumpton 2 holds, see Secton II-B and [4], [3]. We assume that, Assumpton 3: Each user N knows that the set of feasble actons A j of each of ts neghbors j R s a convex and compact subset of R that ncludes 0. The performance of a user that results from actons taken by the users affectng t s quantfed by a utlty functon. We denote the utlty of user N resultng from the acton profle a R := a k ) k R by u a R ). We assume that, Assumpton 4: For all N, the utlty functon u : R R R { } s concave n a R and u a R ) = for a / A. 5 The functon u s user s prvate nformaton. The assumptons that u s concave and s user s prvate nformaton are motvated by applcatons descrbed n Secton II-B and [4], [3]. The assumpton that u a R ) = for a / A captures the fact that an acton profle a R ) s of no sgnfcance to user f a / A. We assume that, Assumpton 5: Each network user N s selfsh, non-cooperatve, and strategc. Assumpton 5 mples that the users have an ncentve to msrepresent ther prvate nformaton, e.g. a user N may not want to report to other users or to the network operator ts true preference for the 5 Note that a s always an element of a R because and hence R.

7 7 users actons, f ths results n an acton profle n ts own favor. Each user N pays a tax t R to the network operator. Ths tax can be mposed for the followng reasons: ) For the use of the network by the users. ) To provde ncentves to the users to take actons that acheve a network-wde performance objectve. The tax s set accordng to the rules specfed by a mechansm and t can be ether postve or negatve for a user. Wth the flexblty of ether chargng a user postve tax) or payng compensaton/subsdy negatve tax) to a user, t s possble to nduce the users to behave n a way such that a network-wde performance objectve s acheved. For example, n a network wth lmted resources, we can set postve tax for the users that receve resources close to the amounts requested by them and we can pay compensaton to the users that receve resources that are not close to ther desrable ones. Thus, wth the avalable resources, we can satsfy all the users and nduce them to behave n a way that leads to a resource allocaton that s optmal accordng to a network-wde performance crteron. We assume that, Assumpton 6: The network operator does not have any utlty assocated wth the users actons or taxes. It does not derve any proft from the users taxes and acts lke an accountant that redstrbutes the tax among the users accordng to the specfcatons of the allocaton mechansm. Assumpton 6 mples that the tax s charged n a way such that t = 0. 1) N To descrbe the overall satsfacton of a user from the performance t receves from all users actons and the tax t pays for t, we defne an aggregate utlty functon u A a R, t ) : R R +1 R { } for each user N : t u A + u a R ), a R, t ) :=, f a A, a j R, j R \{}, otherwse. 2) Because u and A are user s prvate nformaton Assumptons 1 and 4), the aggregate utlty u A s also user s prvate nformaton. As stated n Assumpton 5, users are non-cooperatve and selfsh. Therefore, the users are self aggregate utlty maxmzers. In ths paper we restrct attenton to statc problems,.e. we assume, Assumpton 7: The set N of users, the graph G, users acton spaces A, N, and ther utlty functons u, N, are fxed n advance and they do not change durng the tme perod of nterest.

8 8 We also assume that, Assumpton 8: Every user N knows the set R of users that affect t as well as the set C of users that are affected by t. The network operator knows R and C for all N. In networks where the sets R and C are not known to the users beforehand, Assumpton 8 s stll reasonable because of the followng reason. As the graph G does not change durng the tme perod of nterest Assumpton 7), the nformaton about the neghbor sets R and C, N, can be passed to the respectve users by the network operator before the users determne ther actons. Alternatvely, the users can themselves determne the set of ther neghbors before determnng ther actons. 6 Thus, Assumpton 8 can hold true for the rest of the acton determnaton process. In the next secton we present some applcatons that motvate Model M). B. Applcatons 1) Applcaton A: Power allocaton n cellular networks: Consder a sngle cell downlnk wreless data network consstng of a Base Staton BS) and N moble users as shown n Fg. 3. The BS uses BS a N a 1 h 01 1 h 01 a1 a 2 h 02 h 0N 1 a N h 0N N N Ad 1 Ad 2 Ad 4 2 a 2 h 02 Fg. 3. A downlnk network wth N moble users and one base staton Ad 2 Ad 3 Ad 1 Ad 4 Ad 2 Ad 3 ) ) ) Fg. 4. Three dsplay ad clusters, each consstng of one man ad and two sub ads. Code Dvson Multple Access technology CDMA) to transmt data to the users and each moble user uses Mnmum Mean Square Error Mult-User Detector MMSE-MUD) recever to decode ts data. The sgnature codes used by the BS are not completely orthogonal as ths helps ncrease the capacty of the network. Because of non-orthogonal codes, each user experences nterference due to the BS transmssons ntended for other users. However, as the users n the cell are at dfferent dstances from the BS, and the power transmtted by the BS undergoes propagaton loss, not all transmssons by the BS create nterference to every user. For example, let us look at arcs 1 and N shown n Fg. 3 that are centered at 6 The exact method by whch the users get nformaton about ther neghbor sets n a real network depends on the network characterstcs.

9 9 the BS. Suppose the radus of arc 1 s much smaller than that of arc N. Then, the sgnal transmtted by the BS for users nsde crcle 1 that corresponds to arc 1) wll become neglgble when t reaches outsde users such as user N or user 2. On the other hand, the BS sgnals transmtted for user N and user 2 wll be receved wth sgnfcant power by the users nsde crcle 1. Ths asymmetrc nterference relaton between the moble users can be depcted n a graph smlar to one shown n Fg. 1. In the graph an arrow from j to would represent that the sgnal transmtted for user j also affects user. Note that snce the sgnal transmtted for user must reach, the assumpton made n Secton II-A holds n ths case. If the users do not move very fast n the network, the network topology can be assumed to be fxed for small tme perods. Therefore, f the BS transmts some plot sgnals to all network users, the users can fgure out whch sgnals are creatng nterference to ther sgnal recepton. Thus, each user would know ts nterferng) neghbor set as assumed n Assumpton 8. Note that f the power transmtted by the BS to the users change, t may result n a change n the set of nterferng neghbors of each user. Ths s dfferent from Assumpton 2 n Model M). However, f the transmsson power fluctuatons resultng from a power allocaton mechansm are not large, the set of nterferng neghbors can be assumed to be fxed, and ths can be approxmated by Assumpton 2. The Qualty of Servce QoS) that a user receves from decodng ts data s quantfed by a utlty functon. Due to nterference the utlty u ) of user, N, s a functon of the vector a R, where a j s the transmsson power used by the BS to transmt sgnals to user j, j N, and R s the set of users such that the sgnals transmtted by the BS to users n R also reach user. Note that n ths case all transmssons, n other words the actons a, N, are carred out by the BS unlke Model M) where each user N takes ts own acton a. However, as we dscuss below, the BS s only an agent whch executes the outcome of the mechansm that determnes these transmsson powers. Thus, we can embed the downlnk network scenaro nto Model M) by treatng each a as a decson correspondng to user, N, whch s executed by the BS for. Snce each user uses an MMSE-MUD recever, a measure of user s N ) utlty can be the negatve of the MMSE at the output of ts recever, 7.e., u a R ) = MMSE = mn E[ b z z T T y 2 ] = [ I + 2 S X R S ) 1] N, N. 3) R1 N 0 In 3) b s the transmtted data symbol for user, y s the output of user s matched flter generated from ts receved data, I s the dentty matrx of sze N N, N 0 /2 s the two sded power spectral 7 See [13] for the dervaton of 3).

10 10 densty PSD) of thermal nose, X R s the cross-correlaton matrx of sgnature waveforms correspondng to the users j R, and S := dags j ) j R ) s the dagonal matrx consstng of the sgnal ampltudes S j, j R, receved by user. S j s related to a j as Sj 2 = a j h 0, j R, where h 0 s the channel gan from the BS to user whch represents the power loss along ths path. As shown n [14], [7], the utlty functon gven by 3) s close to concave n a R. Thus, Assumpton 4 n Model M) can be thought of as an approxmaton to the downlnk network scenaro. Note that to compute user s utlty gven n 3), knowledge of N 0, X R, and h 0 s requred. The BS knows X R for each N as t selects the sgnature waveform for each user. On the other hand, user, N, knows the PSD N 0 of thermal nose and the channel gan h 0 as these can be measured only at the respectve recever. Consder a network where the moble users are selfsh and non cooperatve. Then, these users may not want to reveal ther measured values N 0 and h 0. On the other hand f the network operator that owns the BS does not have a utlty and s not selfsh, then, the BS can announce the sgnature waveforms t uses for each user. Thus, each user N would know ts correspondng cross correlaton matrx X R and consequently, ts utlty functon u. However, snce N 0 and h 0 are user s prvate nformaton, the utlty functon u s prvate nformaton of whch s smlar to Assumpton 4 n Model M). If the wreless channel condtons vary slowly compared to the tme perod of nterest, the channel gans and hence the users utlty functons can be assumed to be fxed. As mentoned earler, for slowly movng users the network topology and hence the set of nterferng neghbors can also be assumed to be fxed. These features are captured by Assumpton 7 n Model M). In the presence of lmted resources, the provson of desred QoS to all network users may not be possble. To manage the provson of QoS under such a stuaton the network operator BS) can charge tax to the users and offer them the followng tradeoff. It charges postve tax to the users that obtan a QoS close to ther desrable one, and compensates the loss n the QoS of other users by provdng a subsdy to them. Such a redstrbuton of money among users through the BS s possble under Assumpton 6 n Model M). 2) Applcaton B: Onlne advertsng: Consder an onlne guaranteed dsplay GD) ad system. In exstng GD systems, ndvdual advertsers sgn contracts wth web-publshers n whch web-publshers agree to serve wthn some gven tme perod) a fxed number of mpressons 8 of each ad for a lump sum payment. Here we consder an extenson of current GD systems n whch multple advertsers can 8 A dsplay nstance of an ad

11 11 form clusters as shown n Fg. 4 and contracts can be sgned for the number of mpressons of each cluster. Fgure 4 shows three dsplay ad clusters. In each ad cluster there s one man ad and two sub ads. For example n Fg. 4-), ad 1 s the man ad and ad 2 and ad 3 are the two sub ads. Suppose that the ad clusters are formed n a way so that n each cluster, the man ad creates a postve externalty 9 to the sub ads. For example, ad 1 can be a Honda ad, whereas ad 2 and ad 3 can be ads of local Honda dealer and local Honda mechanc. We call the ad cluster n whch ad appears as the man ad as cluster. 10 The arrangements of ads n clusters can be descrbed by a graph smlar to one shown n Fg. 1. In ths graph an arrow from j to would represent that ad appears n cluster j. For each mpresson of cluster, {1, 2, 4}, advertser pays some fxed prespecfed amount of money bd) b R + to the web publsher. Suppose the bd b s known only to advertser and the web publsher. Let a, {1, 2, 4}, denote the number of mpressons of clusters delvered to the users. Furthermore, let A max be the maxmum number of mpressons advertser can request for cluster,.e. 0 a A max. The constrant A max may arse due to the budget constrant of advertser, or due to the restrctons mposed by the web publsher. For these reasons A max may be prvate nformaton of advertser smlar to Assumpton 1) or prvate knowledge between advertser and the web publsher. Note that n the ad network, the number of mpressons a, {1, 2, 4}, can take only natural number values; therefore, the assumpton of convex acton sets A, N, n Assumpton 1 can be thought of as an approxmaton to ths case. Note also that n the above ad network example, no cluster s assocated wth advertser 3 or the web publsher.e. there s no cluster wth man ad 3 or an ad of the web publsher). Such a scenaro can be captured by Model M) by assocatng dummy acton varables a 3 and a 0 wth advertser 3 and web publsher respectvely, and assumng A max 3 = A max 0 = 0. Because of the way clusters are formed, each advertser obtans a non-negatve utlty from the mpressons of the clusters that t s part of. Thus we can represent the utltes of the four advertsers n the ad network of Fg. 4 as follows: u 1 a 1, a 2 ) = c 11 a 1 + c 12 a 2 b 1 a 1 u 2 a 1, a 2, a 4 ) = c 21 a 1 + c 22 a 2 + c 24 a 4 b 2 a 2 u 3 a 1, a 4 ) = c 31 a 1 + c 34 a 4 4) u 4 a 2, a 4 ) = c 42 a 2 + c 44 a 4 b 4 a 4. 9 See [1, Chapter 11] for the defnton of externalty. 10 We assume that there s at most one ad cluster n whch ad appears as the man ad; hence such a notaton s well defned.

12 12 In 4) c j R +,, j {1, 2, 3, 4} are non negatve real valued constants. The constant c j represents the value obtaned by advertser from each mpresson of cluster j. Suppose that for each j, c j s advertser s prvate nformaton. The term b a, {1, 2, 4} represents the loss n utlty/monetary value ncurred by advertser due to the prespecfed payment t makes to the web publsher. Because the web publsher receves payments from the advertsers, t also obtans a utlty as follows: u 0 a 1, a 2, a 4 ) = b 1 a 1 + b 2 a 2 + b 4 a 4. 5) Snce each bd b, {1, 2, 4}, s known to the web publsher, and none of the advertsers know all of these bds, the utlty functon u 0 s web publsher s prvate nformaton. Smlarly, snce c j for each j and the bd b are prvate nformaton of advertser, for each {1, 2, 4} the utlty functon u s advertser s prvate nformaton. These propertes of utlty functons along wth ther lnearty gven by 4) and 5) are modeled by Assumpton 4 n Model M). If we assume that the arrangements of ads n clusters and the bds b, {1, 2, 4}, of advertsers are predetermned, and do not change wth any decson regardng the mpressons delvery of varous clusters, then ths leads to assumptons 2 and 7 n Model M). As represented by 4), each advertser benefts from a number of other advertsers by beng part of ther ad clusters. For ths reason, n addton to makng a drect payment to the web publsher, each advertser should also make a payment to those advertsers that create postve externaltes to t. Furthermore, because of ther mutual payments the web publsher may offer dscounts to the advertsers n ther drect payments to her. These dscounts can ndrectly encourage advertsers to partcpate n the clustered GD ad scheme. However, one problem n the mplementaton of above type of payments/dscounts s that n a bg network, the advertsers may not have drect contracts wth all ther cluster sharng advertsers. Furthermore, f the advertsers and the web publsher are strategc and self utlty maxmzers, they would try to negotate payments so as to get cluster mpressons that maxmze ther respectve utltes. In such a scenaro, the abovementoned dstrbuton of money among advertsers and the web publsher can be facltated through a thrd party ad agency to whch all advertsers and the web publsher can subscrbe. The role of an ad agency can be mapped to that of the network operator n Model M). Havng dscussed varous applcatons that motvate Model M), we now go back to ths generc model and formulate a resource allocaton problem for t.

13 13 C. The decentralzed local publc good provsonng problem P D ) For the network model M) we wsh to develop a mechansm to determne the users acton and tax profles a N, t N ) := a 1, a 2,..., a N ), t 1, t 2,..., t N )). We want the mechansm to work under the decentralzed nformaton constrants of the model and to lead to a soluton to the followng centralzed problem. The centralzed problem P C ) max a N,t N ) D where, max a N,t N ) s.t. u a R ) N u A a R, t ) N t = 0 N D := {a N, t N ) R 2N a A N ; N t = 0} 6) 7) The centralzed optmzaton problem 6) s equvalent to 7) because for a N, t N ) / D, the objectve functon n 6) s negatve nfnty by 2). Thus D s the set of feasble solutons of Problem P C ). Snce by Assumpton 4, the objectve functon n 7) s concave n a N and the sets A, N, are convex and compact, there exsts an optmal acton profle a N for Problem P C). Furthermore, snce the objectve functon n 7) does not explctly depend on t N, an optmal soluton of Problem P C ) must be of the form a N, t N ), where t N s any feasble tax profle for Problem P C ),.e. a tax profle that satsfes 1). The solutons of Problem P C ) are deal acton and tax profles that we would lke to obtan. If there exsts an entty that has centralzed nformaton about the network,.e. t knows all the utlty functons u, N, and all acton spaces A, N, then that entty can compute the above deal profles by solvng Problem P C ). Therefore, we call the solutons of Problem P C ) optmal centralzed allocatons. In the network descrbed by Model M), there s no entty that knows perfectly all the parameters that descrbe Problem P C ) Assumptons 1 and 4). Therefore, we need to develop a mechansm that allows the network users to communcate wth one another and that leads to optmal solutons of Problem P C ). Snce a key assumpton n Model M) s that the users are strategc and non-cooperatve, the mechansm we develop must take nto account the users strategc behavor n ther communcaton wth one another. To address all of these ssues we take the approach of mplementaton theory [5] for the soluton of the decentralzed local publc good provsonng problem for Model M). Henceforth we call ths decentralzed allocaton

14 14 problem as Problem P D ). In the next secton we present a decentralzed mechansm game form) for local publc good provsonng that works under the constrants mposed by Model M) and acheves optmal centralzed allocatons. III. A DECENTRALIZED LOCAL PUBLIC GOOD PROVISIONING MECHANISM For Problem P D ), we want to develop a game form message space and outcome functon) that s ndvdually ratonal, budget balanced, and that mplements n Nash equlbra the goal correspondence defned by the soluton of Problem P C ). 11 Indvdual ratonalty guarantees voluntary partcpaton of the users n the allocaton process specfed by the game form, budget balance guarantees that there s no money left unclamed/unallocated at the end of the allocaton process.e. t ensures 1)), and mplementaton n NE guarantees that the allocatons correspondng to the set of NE of the game nduced by the game form are a subset of the optmal centralzed allocatons solutons of Problem P C )). We would lke to clarfy at ths pont the defnton of ndvdual ratonalty voluntary partcpaton) n the context of our problem. Note that n the network model M), the partcpaton/non-partcpaton of each user determnes the network structure and the set of local publc goods users actons) affectng the partcpatng users. To defne ndvdual ratonalty n ths settng we consder our mechansm to be consstng of two stages as dscussed n [15, Chapter 7]. In the frst stage, knowng the game form, each user makes a decson whether to partcpate n the game form or not. The users who decde not to partcpate are consdered out of the system. Those who decde to partcpate follow the game form to determne the levels of local publc goods n the network formed by them. 12 In such a two stage mechansm, ndvdual ratonalty mples the followng. If the network formed by the partcpatng users satsfes all the propertes of Model M), 13 then, at all NE of the game nduced by the game form among the partcpatng users, the utlty of each partcpatng user wll be at least as much as ts utlty wthout partcpaton.e. f t s out of the system). We would also lke to clarfy the ratonale behnd choosng NE as the soluton concept for our problem. Note that because of assumptons 1 and 4 n Model M), the envronment of our problem s one of ncomplete nformaton. Therefore one may speculate the use of Bayesan Nash or domnant strategy as 11 The defnton of game form, goal correspondence, ndvdual ratonalty, budget balance and mplementaton n Nash equlbra s gven n [8, Chapter 3]. 12 Ths network s a subgraph obtaned by removng the nodes correspondng to non-partcpatng users from the orgnal graph drected network) constructed by all the users n the system. 13 In partcular, the network formed by the partcpatng users must satsfy Assumpton 2 that there are at least three users affected by each local publc good n ths network. Note that all other assumptons of Model M) automatcally carry over to the network formed by any subset of the users n Model M).

15 15 approprate soluton concepts for our problem. However, snce the users n Model M) do not possess any pror belefs about the utlty functons and acton sets of other users, we cannot use Bayesan Nash as a soluton concept for Model M). Furthermore, because of mpossblty results for the exstence of non-parametrc effcent domnant strategy mechansms n classcal publc good envronments [16], we do not know f t s possble to desgn such mechansms for the local publc good envronment of Model M). The well known Vckrey-Clarke-Groves VCG) mechansms that acheve ncentve compatblty and effcency wth respect to non-numerare goods, do not guarantee budget balance [16]. Hence they are napproprate for our problem as budget balance s one of the desrable propertes n our problem. VCG mechansms are also unsutable for our problem because they are drect mechansms and any drect mechansm would requre nfnte message space to communcate the generc contnuous and concave) utlty functons of users n Model M). Because of all of above reasons, and the known exstence results for non-parametrc, ndvdually ratonal, budget-balanced Nash mplementaton mechansms for classcal prvate and publc goods envronments [16], we choose Nash as the soluton concept for our problem. We adopt Nash s orgnal mass acton nterpretaton of NE [17, page 21]. Implct n ths nterpretaton s the assumpton that the problems s envronment s stable, that s, t does not change before the agents reach ther equlbrum strateges. Ths assumpton s consstent wth our Assumpton 7. Nash s mass acton nterpretaton of NE has also been adopted n [16, pp ], [18, page 664], [7], and [19], [20]. Specfcally, by quotng [18], we nterpret our analyss as applyng to an unspecfed message exchange) process n whch users grope ther way to a statonary message and n whch the Nash property s a necessary condton for statonarty. We next construct a game form for the resource allocaton problem P D ) that acheves the abovementoned desrable propertes Nash mplementaton, ndvdual ratonalty, and budget balance. A. The game form In ths secton we present a game form for the local publc good provsonng problem presented n Secton II-C. We provde explct expressons of each of the components of the game form, the message space and the outcome functon. We assume that the game form s common knowledge among the users and the network operator.

16 16 The message space: Each user N sends to the network operator a message m R R R R + =: M of the followng form: m := ar, πr ); ar R R, πr R R +, 8) where, a User also sends the component a R := ak) k R ; πr := πk) k R, N. 9) k, π k ), k R, of ts message to ts neghbor k R. In ths message, a s the acton proposal for user k, k R k, by user, N. Smlarly, πk s the prce that user, N, proposes to pay for the acton of user k, k R. A detaled nterpretaton of these message elements s gven n Secton III-B. The outcome functon: After the users communcate ther messages to the network operator, ther actons and taxes are determned as follows. For each user N, the network operator determnes the acton â of user from the messages communcated by ts neghbors that are affected by t set C ),.e. from the message profle m C := m k ) k C : â m C ) = 1 k a C, N. 10) k C To determne the users taxes the network operator consders each set C j, j N, and assgns ndces 1, 2,..., C j n a cyclc order to the users n C j. Each ndex 1, 2,..., C j s assgned to an arbtrary but unque user C j. Once the ndces are assgned to the users n each set C j, they reman fxed throughout the tme perod of nterest. We denote the ndex of user assocated wth set C j by I j. The ndex I j {1, 2,..., C j } f C j, and I j = 0 f / C j. Snce for each set C j, each ndex 1, 2,..., C j s assgned to a unque user C j, therefore,, k C j such that k, I j I kj. Note also that for any user N, and any j, k R, the ndces I j and I k are not necessarly the same and are ndependent of each other. We denote the user wth ndex k {1, 2,..., C j } n set C j by C jk). Thus, C jij ) = for C j. The cyclc order ndexng means that, f I j = C j, then C jij +1) = C j1), C jij +2) = C j2), and so on. In Fg. 2 we llustrate the above ndexng rule for the set C j shown n Fg. 1. Based on the above ndexng, the users taxes ˆt, N, are determned as follows. ˆt m Cj ) j R ) = l j m Cj ) â j m Cj ) + j R C jij +1) C jij +1) πj a j πj a j ) C 2 jij +1) aj ) C 2 11) jij +2) a j

17 17 C jij +1) where, l j m Cj ) = πj C jij +2) π j, j R, N. 12) We would lke to emphasze here that the presence of the network operator s necessary for strategyproofness and mplementaton of the above game form. A detaled dscusson on the need and sgnfcance of the network operator can be found n [8]. The game form gven by 8) 12) and the users aggregate utlty functons n 2) nduce a game N M, â, ˆt ) N, {u A } N ). In ths game, the set of network users N are the players, the set of strateges of a user s ts message space M, and a user s payoff s ts utlty u A âj m )), Cj ˆt ) j R mcj ) j R) that t obtans at the allocaton determned by the communcated messages. We defne a NE of ths game as a message profle m N that has the followng property: N and m M, u A âj m C j ) ) ), ˆt ) m Cj ) j R u A âj m, m C j /) ) ), ˆt ) m, m C j /) j R. 13) As dscussed earler, NE n general descrbe strategc behavor of users n games of complete nformaton. Ths can be seen from 13) where, to defne a NE, t requres complete nformaton of all users aggregate utlty functons. However, the users n Model M) do not know each other s utltes; therefore, the game nduced by the game form 8) 12) and the users aggregate utlty functons 2) s not one of complete nformaton. Therefore, for our problem we adopt the NE nterpretaton of [18] and [16, Secton 4] as dscussed at the begnnng of Secton III. That s, we nterpret NE as the statonary messages of an unspecfed message exchange) process that are characterzed by the Nash property 13). In the next secton we show that the allocatons obtaned by the game form presented n 8) 12) at all NE message profles satsfyng 13)), are optmal centralzed allocatons. B. Propertes of the game form We begn ths secton wth an ntutve dscusson on how the game form presented n Secton III-A acheves optmal centralzed allocatons. We then formalze the results n Theorems 1 and 2. To understand how the proposed game form acheves optmal centralzed allocatons, let us look at the propertes of NE allocatons correspondng to ths game form. A NE of the game nduced by the game form 8) 12) and the users utlty functons 2) can be nterpreted as follows: Gven the users messages m k, k C, the outcome functon computes user s acton as 1/ C k C k a). Therefore, user s acton proposal a can be nterpreted as the ncrement that desres over the sum of other users acton proposals for, so as to brng ts allocated acton â to ts own desred value. Thus, f the acton

18 18 computed for based on ts neghbors proposals does not le n A, user can propose an approprate acton a and brng ts allocated acton wthn A. The flexblty of proposng any acton a R gves each user N the capablty to brng ts allocaton â wthn ts feasble set A by unlateral devaton. Therefore, at any NE, â A, N. By takng the sum of taxes n 11) t can further be seen, after some computatons, that the allocated tax profle ˆt ) N satsfes 1) even at off-ne messages). ) Thus, all NE allocatons â m C )) N, ˆt m Cj ) j R )) N le n D and hence are feasble solutons of Problem P C ). To see further propertes of NE allocatons, let us look at the tax functon n 11). The tax of user conssts of three types of terms. The type-1 term s l j m Cj ) â j m Cj ); t depends on all acton proposals for each of user s neghbors j R, and the prce proposals for each of these neghbors by users other than user. The type-2 term s πj C jij +1) 2; a j aj) ths term depends on a R as well as πr. Fnally, the type-3 term s the followng: C jij +1) πj C jij +1) aj C jij +2) 2; j) a ths term depends only on the messages of users other than. Snce πr does not affect the determnaton of user s acton, and affects only the type-2 term n ˆt, the NE strategy of user, N, that mnmzes ts tax s to propose for each j R, πj = 0 unless at the NE, C jij +1) aj = a j. Snce the type-2 and type-3 terms n the users tax are smlar across users, for each N and j R, all the users k C j choose the above strategy at NE. Therefore, the type-2 and type-3 terms vansh from every users tax ˆt, N, at all NE. Thus, the tax that each user N pays at a NE m N s of the form l j m C j ) â j m C j ). The NE term l j m C j ), N, j R, can therefore be nterpreted as the personalzed prce for user for the NE acton â j m C j ) of ts neghbor j. Note that at a NE, the personalzed prce for user s not controlled by s own message. The reducton of the users NE taxes nto the form l j m C j ) â j m C j ) mples that at a NE, each user N has a control over ts aggregate utlty only through ts acton proposal. 14 If all other users messages are fxed, each user has the capablty of shftng the allocated acton profle â R to ts desred value by proposng an approprate ar R R See the dscusson n the prevous paragraph). Therefore, the NE strategy of each user N s to propose an acton profle ar that results n an allocaton â R that maxmzes ts aggregate utlty. Thus, at a NE, each user maxmzes ts aggregate utlty for ts gven personalzed prces. By the constructon of the tax functon, the sum of the users tax s zero at all NE and off equlbra. Thus, the ndvdual aggregate utlty maxmzaton of the users also 14 Note that user s acton proposal determnes the actons of all the users j R ; thus, t affects user s utlty u âjm C j ) ) ) as well as ts tax l jm C j ) â jm C j ).

19 19 result n the maxmzaton of the sum of users aggregate utltes subject to the budget balance constrant whch s the objectve of Problem P C ). It s worth mentonng at ths pont the sgnfcance of type-2 and type-3 terms n the users tax. As explaned above, these two terms vansh at NE. However, f for some user N these terms are not present n ts tax ˆt, then, the prce proposal πr of user wll not affect ts tax and hence, ts aggregate utlty. In such a case, user can propose arbtrary prces πr because they would affect only other users NE prces. The presence of type-2 and type-3 terms n user s tax prevent such a behavor as they mpose a penalty on user f t proposes a hgh value of πr or f ts acton proposal for ts neghbors devates too much from other users proposals. Even though the presence of type-2 and type-3 terms n user s tax s necessary as explaned above, t s mportant that the NE prce l j m C j ), j R of user N s not affected by s own proposal πr. Ths s because, n such a case, user may nfluence ts own NE prce n an unfar manner and may not behave as a prce taker. To avod such a stuaton, the type-2 and type-3 terms are desgned n a way so that they vansh at NE. Thus, ths constructon nduces prce takng behavor n the users at NE and leads to optmal allocatons. The results that formally establsh the above propertes of the game form are summarzed n Theorems 1 and 2 below. Theorem 1: Let m N be a NE of the game nduced by the game form presented n Secton III-A and the ) users utlty functons 2). Let â N, ˆt N ) := â N m N ), ˆt N m N )) := â m C )) N, ˆt m C j ) j R )) N be the acton and tax profles at m N determned by the game form. Then, a) Each user N weakly prefers ts allocaton â R, ˆt ) to the ntal allocaton 0, 0). Mathematcally, u A ) ) â R, ˆt u A 0, 0, N. b) â N, ˆt N ) s an optmal soluton of Problem P C). Theorem 2: Let â N be an optmum acton profle correspondng to Problem P C). Then, a) There exst a set of personalzed prces l j, j R, N, such that â R = arg max lj â j + u â R ), N. â A j R â j R, \{} b) There exsts at least one NE m N of the game nduced by the game form presented n Secton III-A and the users utlty functons 2) such that, â N m N ) = â N. Furthermore, f ˆt := l jâ j, N,

20 20 the set of all NE m N = m ) N = a soluton of the followng set of condtons: R, π 1 a C k C = â, N, C jij +1) πj C ji j +2) πj = lj, j R, N, j π ) 2 a j C ji j +1) a j = 0, j R, N, R ) that result n â N, ˆt N ) s characterzed by the π j 0, j R, N. Because Theorem 1 s stated for an arbtrary NE m N of the game nduced by the game form of Secton III-A and the users utlty functons 2), the asserton of the theorem holds for all NE of ths game. Thus, part a) of Theorem 1 establshes that the game form presented n Secton III-A s ndvdually ratonal,.e., at any NE allocaton, the aggregate utlty of each user s at least as much as ts aggregate utlty before partcpatng n the game/allocaton process. Because of ths property of the game form, each user voluntarly partcpates n the allocaton process. Part b) of Theorem 1 asserts that all NE of the game nduced by the game form of Secton III-A and the users utlty functons 2) result n optmal centralzed allocatons solutons of Problem P C )). Thus the set of NE allocatons s a subset of the set of optmal centralzed allocatons. Ths establshes that the game form of Secton III-A mplements n NE the goal correspondence defned by the solutons of Problem P C ). Because of ths property, the above game form guarantees to provde an optmal centralzed allocaton rrespectve of whch NE s acheved n the game nduced by t. The asserton of Theorem 1 that establshes the above two propertes of the game form presented n Secton III-A s based on the assumpton that there exsts a NE of the game nduced by ths game form and the users utlty functons 2). However, Theorem 1 does not say anythng about the exstence of NE. Theorem 2 asserts that NE exst n the above game, and provdes condtons that characterze the set of all NE that result n optmal centralzed allocatons of the form â N, ˆt N ) = â N, ljâ j) N ), where â N s any optmal centralzed acton profle. In addton to the above, Theorem 2 also establshes the followng property of the game form. Snce the optmal acton profle â N n the statement of Theorem 2 s arbtrary, the theorem mples that the game form of Secton III-A can obtan each of the optmum acton profles of Problem P C ) through at least one of the NE of the nduced game. Ths establshes that the above game form s not based towards any partcular optmal centralzed acton profle. We present the proofs of Theorem 1 and Theorem 2 n Appendces I and II.

21 21 IV. FUTURE DIRECTIONS The problem formulaton and the soluton of the local publc goods provsonng problem presented n ths paper open up several new drectons for future research. Frst, the development of effcent mechansms that can compute NE s an mportant open problem. To address ths problem there can be two dfferent drectons. ) The development of algorthms that guarantee convergence to Nash equlbra of the games constructed n ths paper. ) The development of alternatve mechansms/game forms that lead to games wth dynamcally stable NE. Second, the network model we studed n ths paper assumed a gven set of users and a gven network topology. In many local publc good networks such as socal or research networks, the set of network users and the network topology must be determned as part of network objectve maxmzaton. These stuatons gve rse to nterestng admsson control and network formaton problems many of whch are open research problems. Fnally, n ths paper we focused on statc resource allocaton problem where the characterstcs of the local publc good network do not change wth tme. The development of mplementaton mechansms under dynamc stuatons, where the network characterstcs change durng the determnaton of resource allocaton, are open research problems. REFERENCES [1] A. Mas-Colell, M. D. Whnston, and J. R. Green, Mcroeconomc theory. Oxford Unversty Press, [2] C. M. Tebout, A pure theory of local expendtures, The Journal of Poltcal Economy, vol. 64, no. 5, pp , October [3] Y. Bramoullé and R. Kranton, Publc goods n networks, Journal of Economc Theory, vol. 135, pp , [4] K. chh Yuan, Publc goods n drected networks, preprnt. [5] M. Jackson, A crash course n mplementaton theory, n Socal choce and welfare, 2001, pp [6] T. Stoenescu and D. Teneketzs, Decentralzed resource allocaton mechansms n networks: realzaton and mplementaton, n Advances n Control, Communaton Networks, and Transportaton Systems, E. H. A. Brkh ruser, Ed., 2005, pp , n honor of Pravn Varaya. [7] S. Shrutvandana and D. Teneketzs, A game-theoretc approach to decentralzed optmal power allocaton for cellular networks, Telecommuncaton Systems journal, vol. 47, pp , [8] S. Sharma, A mechansm desgn approach to decentralzed resource allocaton n wreless and large-scale networks: Realzaton and mplementaton, Ph.D. dssertaton, Unversty of Mchgan, Ann Arbor, [9] T. Groves and J. Ledyard, Optmal allocaton of publc goods: A soluton to the free rder problem, Econometrca, vol. 45, pp , [10] L. Hurwcz, Outcome functons yeldng Walrasan and Lndahl allocatons at Nash equlbrum ponts, Revew of Economc studes, vol. 46, pp , [11] M. Walker, A smple ncentve compatble scheme for attanng Lndahl allocatons, Econometrca, vol. 49, pp , [12] Y. chen, A famly of supermodular mechansms mplementng Lndahl allocatons for quaslnear envronments, Economc Theory, vol. 19, no. 4, pp , 2002.