IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 6, NO. 7, SEPTEMBER 008 95 Managng a Peer-to-Peer Data Storage System n a Selfsh Socety Patrck Mallé and László Toka Abstract We compare two possble mechansms to manage a peer-to-peer storage system, where partcpants can store data onlne on the dsks of peers n order to ncrease data avalablty and accessblty. Due to the lack of ncentves for peers to contrbute to the servce, we suggest that ether each peer s use of the servce be lmted to her contrbuton level symmetrc schemes), or that storage space be bought from and sold to peers by a system operator that seeks to maxmze proft. Usng a noncooperatve game model to take nto account user selfshness, we study those mechansms wth respect to the socal welfare performance measure, and gve necessary and suffcent condtons for one scheme to socally outperform the other. Index Terms Peer-to-peer networks, game theory, ncentves, prcng. I. INTRODUCTION THE DIGITAL SOCIETY that has been soarng snce the creaton of the Internet mples that all knds of dgtal documents are now lkely to be created, accessed, and modfed from several types of devces. Therefore, an approprate system for storng the data of a user should offer varous servces, such as versonng, ease of access, protecton aganst devce falures, and short transfer tme to a gven devce. In that context, the possblty of storng data onlne appears as a promsng soluton. Indeed, havng access to the Internet becomes easer and easer, wth the multplcaton of WF hotspots, the development of WMAX and thrd generaton wreless networks, and the appearance of other access modes, such as mult-hop networks that work n an ad-hoc fashon to reach an access pont. Let us also hghlght the hgh rse of avalable transmsson rates n access networks, whch renders transfer tmes reasonable, even for large fles. Fnally, onlne storage systems are able to cope wth document versonng, and to protect data not only aganst user devce falures but also aganst dsk falures, through the use of data replcates stored on dfferent dsks. For those reasons, many companes now propose onlne data storage servces, most of them offerng a gven storage capacty between and 5 ggabytes) for free, wth the Manuscrpt receved July 3, 007; revsed March 5, 008. Ths work has been partally supported by the TELECOM Insttute project DsParSe. P. Mallé s wth the Department of Network, Securty and Multmeda, TELECOM Bretagne,, rue de la Châtagnerae, 35576 Cesson-Sévgné Cedex, FRANCE e-mal: patrck.malle@telecom-bretagne.eu). L. Toka s wth the Department of Network and Securty, EURECOM, 9 route des Crêtes, 06560 Sopha-Antpols Cedex, FRANCE and wth the Department of Telecommuncatons and Meda Informatcs, Budapest Unversty of Technology and Economcs, Magyar Tudósok krt., 7 Budapest, HUNGARY e-mal: toka@eurecom.fr). Dgtal Object Identfer 0.09/JSAC.008.08096. 0733-876/08/$5.00 c 008 IEEE possblty of extendng that quota to a hgher value for a fxed prce per year the prce per year per ggabyte beng of the order of $). However, whle creatng such a storage servce mples ownng huge memory capactes and affordng the assocated energy and warehouse costs, one can magne usng the smaller but numerous storage spaces of the servce users themselves, as s done n peer-to-peer fle sharng systems. In a peer-to-peer storage system, the partcpants are at the same tme the provders and the users of the servce: each partcpant offers some memory capacty possbly from multple locatons n the network: part of her dsk space at home, storng devce devoted to the servce,...) to provde the servce to the others, and benefts from storng her own data onto the system. The added value of the servce then comes from the protecton aganst falures provded by the system, from the ease of data access, from the versonng management that may be ncluded, and from the dfference n the amount of data stored nto the system versus offered to the servce. An onlne storage servce s valuable only f data are avalable: therefore to cope wth dsk falures and wth partcpants dsconnectng ther dsk from the system, data replcates must be spread over several suffcently relable) peers to guarantee that data are not lost and are almost always avalable; the data replcaton rate then depends on the relablty of the partcpants. To work properly, a peer-to-peer storage network therefore needs that partcpants offer a suffcent part of ther dsk space to the system, and reman onlne often enough. However, both of those requrements mply costs or at least constrants) for partcpants, who may be reluctant to devote some of ther storage capacty to the system nstead of usng t for ther own needs. In ths paper, we consder that users behave selfshly,.e. are only senstve to the qualty of servce they experence, regardless of the effects of ther actons on the other users. The framework of noncooperatve Game Theory [] s therefore partcularly well-suted to study the nteractons among peers. For a peer-to-peer storage system, t s clear that wthout any reward for contrbutng partcpants, selfshly behavng users wll only beneft from the servce wthout provdng any part of t. In other words, the only Nash equlbrum of the noncooperatve game s the stuaton where the system actually does not exst due to the lack of offerng peers. The work presented n ths paper focuses on the ncentves that can be used to make partcpants contrbute to the system,.e. the changes that can be brought to the game to modfy ts Such a behavor, called free-rdng [], also appears n peer-to-peer fle sharng networks, and s problematc for the survval of those altrusm-based networks. 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96 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 6, NO. 7, SEPTEMBER 008 Nash equlbra. Whle the economc aspects of peer-to-peer fle sharng networks have already been extensvely studed see [3], [], [5], [6] and references theren), there are to our knowledge no references on the economcs of peer-to-peer storage networks. Now, the economc models developed for peer-to-peer fle sharng systems do not apply to peer-to-peer storage servces: n fle sharng systems, when a peer provdes some fles to the communty, she adds value to the system for all users snce they all can access the data she proposes; n that sense the resource offered to the system s a publc good. On the contrary, n a peer-to-peer storage system the memory space offered by a peer s a prvate good: t can be shared among dfferent users but each part s then devoted to only one user. Therefore the economc mplcatons of those systems are necessarly dfferent. The exstng lterature on peer-to-peer storage systems manly focuses on securty, relablty and techncal feasblty ssues [7], [8], [9], whereas the ncentve aspect receved lttle attenton. Only solutons that do not mply fnancal transactons are consdered n current works, therefore to create some ncentves to partcpate, the counter payment for provdng servce s usually the servce n queston as well. Ths approach fnally leads to a scheme where every peer should contrbute to the system n terms of servce at least as much as she benefts from others [0], []. We call such a mechansm mposng the contrbuton of each peer to equal her use of the system a symmetrc scheme. In ths paper, we also nvestgate solutons based on monetary exchanges: users can buy storage space for a fxed unt prce, and sell ther own memory space to the system at another unt prce. It s known from economc theory that when those unt prces are fxed by the supply and demand curves as n a perfect market []), then user selfsh choces lead to a socally effcent stuaton. However, t s more lkely here that the system be managed by a proft-maxmzng entty that fxes prces so as to maxmze revenue. That entty then acts as the leader of a Stackelberg game []. The man queston addressed n ths paper s whether t s socally better to mpose a symmetrc scheme or to let a proftmaxmzng monopoly set prces. The performance measure we consder s socal welfare,.e. the total value that the system has for all partcpants. Under some assumptons on the peers utlty functons, we derve a necessary and suffcent condton for symmetry-based systems to outperform revenueorented management. We obtan that user heterogenety tends to favor prcng-based schemes that are more flexble, and above a gven user heterogenety threshold even a monopolymanaged system wll be socally better than a system mposng symmetry. The results presented here are a generalzaton of our prelmnary work [3] that dd not consder ncentves to stay onlne and where the only source of heterogenety came from the prce senstvtes. Ths paper s organzed as follows. Secton II ntroduces the model we consder for user preferences, and for the two ncentve mechansms studed n ths paper, namely symmetrybased and proft orented prce-based schemes. In Secton III we defne the socal welfare performance measure and compute ts value for those two types of schemes. We compare them n Secton IV to determne the management scheme that s best suted to the socety, and present our conclusons n Secton V. II. MODEL A. Content avalablty management and assocated costs In a peer-to-peer storage system the avalablty of the stored data s consdered as the most mportant factor n user s apprecaton. As the storage dsks are users property, there are no drect means to guarantee that a gven user dsk storng a specfc fle wll be onlne 00% of the tme. To ensure data avalablty, the system can ntroduce several tools, such as data replcaton and codng []. We suppose here that the system detectng that a peer has gone offlne trggers a recovery of the data stored n that peer from the replcas n the system, and a new storage of those data nto other peers. Lkewse, when a peer comes back onlne, then new data wll be transferred nto her offered storage space, ndependently what and whose data she was storng before. Such a scheme s purely reactve actons are taken when a user departure s detected). One could also magne usng proactve approaches, or a combnaton of both, to smoothe the ncurred traffc [5]. Ths data protecton mechansm mples data transfers, and therefore nonmonetary costs due to resource consumpton CPU, bandwdth utlzaton, etc.). A peer s concerned by those data transfers n two stuatons: when she comes back onlne after an offlne perod new data load), and when other peers enter and leave the system upload traffc f user stores replcates of the leavng user s data, download traffc whenuser has to store more data). The mean data transfer assocated to the frst stuaton s thus proportonal to the amount of capacty C she offers to the system, and to the mean number of onlne-offlne cycles per unt of tme: denotng by t on resp. t off ) the mean duraton of onlne resp. offlne) perods of user, the correspondng mean amount of data transferred s then proportonal to C /t on + t off ). The mean amount of data transferred to and from user per unt of tme n the second stuaton s proportonal to the weghted by the offered capacty) mean μ of peer status changes per unt of tme. Ths term appears only at those peers who offer storage space proportonally to ther offered capacty snce the probablty that user be concerned by a peer s departure s proportonal to C ), and only durng the tme they are onlne t s therefore also proportonal to the mean avalablty of user, π := t on /ton + t off )). Consequently, the transfer cost perceved by user for offerng capacty C wth the mean avalablty π expresses C π δ /t on + γ μ), where δ and γ are parameters that reflect the user characterstcs such as senstvty, access bandwdth, or hardware profle. B. User preferences We descrbe the preferences of a user n the user set denoted by I by a utlty functon, that reflects the beneft of usng the servce by storng an amount C s of data n the Actually peer should only be senstve to the status change rate of all other peers but hers. However we consder here a system wth a very large number of users, so that takng the mean of the change rates over all partcpants but one s equvalent to consderng all partcpants. Authorzed lcensed use lmted to: Unversty of Houston. Downloaded on Aprl 0, 009 at 6:38 from IEEE Xplore. Restrctons apply.
MAILLÉ and TOKA: MANAGING A PEER-TO-PEER DATA STORAGE SYSTEM IN A SELFISH SOCIETY 97 system, the cost of offerng storage space C o := π C for other users, and the monetary transactons, f any. We suggest to use a separable addtve functon. Defnton : The utlty U of a user I s of the form U C s,c,t on,t off ),ɛ = V C s ) O C π ) C π δ /t on + γ μ) } {{ } :=P C,t on,toff ) ɛ, ) where V C s ) s user s valuaton of the storage servce,.e. the prce she s wllng to pay to store an amount C s of data n the system 3. We assume that V ) s postve, contnuously dfferentable, ncreasng and concave n ts argument, and that V 0) = 0 no servce yelds no value). P C,t on,toff ) s the overall non-monetary cost of user for offerng capacty C to the system wth mean onlne and offlne duratons respectvely equal to t on and t off,.e. wth avalablty π =. It conssts of two dstnct costs: ton t on +toff an opportunty cost O C π ) of offerng storage capacty for other users durng onlne perods) nstead of usng t for her own needs,whereo ) s assumed postve, contnuously dfferentable, ncreasng and strctly convex, and such that O 0) = 0 no contrbuton brngs no cost); + γ μ) due to the data protecton mechansm mplemented by the system as descrbed n the prevous subsecton. data transfer costs C π δ /t on ɛ s the monetary prce pad by user. Thsterms0 n case of a symmetrc scheme, and otherwse equals the prce dfference between the charge for storng her data nto the system and the remuneraton for offerng her dsk space. Remark that we mplctly say that the storage space necessary to safely store some data n the system equals the sze of those data. Ths s done wthout loss of generalty, takng nto account the redundancy factor r added by the system n users cost functon: a user consdered to offer space to store an amount C of data actually devotes more of her dsk space rc ) to the servce. Lkewse, prces are then per unt of protected data. C. Incentve schemes for cooperaton Users selfshly choose strateges that maxmze ther utlty. We assume here that apart from C s and C o, each user can also decde about her behavor related to avalablty. In ths subsecton, we descrbe the two types of ncentve mechansms that we ntend to compare n ths paper. Both schemes may mply the exstence of a central authorty or clearance servce to supervse the peers behavor and/or manage payments: as the model ams to gve hnts for commercal applcatons, we do not try to avod such a centralzed system control. 3 We assume here that data replcaton ensures a gven avalablty, so that ths avalablty does not appear n the utlty functon. We mplctly assume here that the opportunty cost depends only on the mean capacty offered over tme, snce durng offlne perods the user can use the dsk space for other purposes than the servce. ) Symmetrc schemes: We follow here the deas suggested n the lterature for schemes wthout prcng. As evoked n the ntroducton, the prncple of those schemes s that users are nvted to contrbute to, at least as much as they take from, the other users. The avalablty of the peer s therefore checked e.g. at randomly chosen tmes) to ensure that C o = π C exceeds the peer s servce use C s. We assume n ths paper that ths verfcaton s techncally feasble. Determnng whether and how t can be done remans an actve topc of research and s beyond the scope of ths paper, snce we only focus here on ncentves. ) Payment-based schemes: We consder a smple payment-based mechansm where users can buy storage space n the system for a unt prce p s per byte and per unt of tme) and sell some of ther tme-average avalable) dsk capacty for a unt prce p o. The possbly negatve) amount that user s charged s then ɛ = p s C s po C o. In ths paper, we assume that prces are set by the system operator so as to maxmze her revenue, knowng a pror the reactons of the users. The operator can thus drve the outcome of the game to the most proftable stuaton for herself, and n ths sense, she acts as the leader of a Stackelberg or leader-follower) game []. In a real mplementaton of the mechansm, the operator may not perfectly know the user reactons, but an teratve tâtonnement of prces can converge to those proft-maxmzng prces. D. User behavor related to avalablty In the game we study, a peer I has four strategc varables, namely her offered C and stored C s capactes, and her mean onlne t on and offlne t off perod duratons. Equvalently, we can also consder that the four strategc varables are C s,co,ton, and toff. From ), when C s and C o are fxed, the utlty of each user s ncreasng n t on,so t on wll be set by user to a maxmum value. We denote by t on that maxmum value, whch s only lmted by uncontrolled events power black-out, accdents, hardware falures, etc) that may force the user off the network. Notce that ths selfsh decson s proftable to the whole network: longer onlne perods mean fewer data protecton transfers and therefore smaller costs for the system the parameter μ n ) beng small). Remark also that snce t off does not appear n ), there reman only two decson varables, namely C s and Co that equals C t on on / t + t off )). From now we wll therefore wrte P C o) nstead of P C,t on,toff ),and wll also use the notaton p mn := δ / t on + γ μ, ) so that the transfer costs smply wrte C o pmn. E. User supply and demand functons Supply and demand functons are classcally used n economcs [], and are respectvely derved from the valuaton of consumers and cost functons of provders. Notce however the partcularty here that peers can be consumers and provders at the same tme. Authorzed lcensed use lmted to: Unversty of Houston. Downloaded on Aprl 0, 009 at 6:38 from IEEE Xplore. Restrctons apply.
98 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 6, NO. 7, SEPTEMBER 008 quantty C o C d p) s p) utlty U Consequently, the total supply functon S := I s s a pecewse affne) nondecreasng convex functon on the nterval [mn p mn, max p mn ], andsaffne on [max p mn, + ). Lkewse, the total demand functon D := I d s nonncreasng, affne on [0, mn p max ] and convex on [mn p max, max p max ], as llustrated n Fgure dsplayed n subsecton III-B. C s slope a p mn p p o p s unt prce slope b p max Fg.. Reactons to prces and utlty of a user Iunder Assumpton A. Defnton : For a user I, we call supply functon resp. demand functon) the functon s ) resp. d )) such that for all p R +, s p) := nf{q 0:P q) p}, d p) := nf{q 0:V q) p}, where g stands for the dervatve functon of g, and wth the conventon nf =+. For a gven p 0, s p) resp. d p)) s the amount of storage capacty that user would choose to sell resp. buy) f she were pad resp. charged) a unt prce p for t. For the sake of smplcty, our man results n the followng consder a partcular form of supply and demand functons descrbed n the assumpton below. Assumpton A: For all I, the supply and demand functons of user are affne. More precsely, there exst nonnegatve values a,b, and p max such that s p) = a [p p mn ] +, 3) d p) = b [p max p] +, ) where p mn that max p mn s gven n ), x + := max0,x), and we assume < mn p max. Ths actually corresponds to quadratc functons for the valuaton and opportunty cost wth denotng the mn): O C o )= a C o, V C s )= b Cs b p max ) ) + b p max C s b p max ). Under Assumpton A, a user s entrely descrbed by four parameters see Fgure ): two prce thresholds, namely p mn and p max, that respectvely represent the mnmum value of the unt prce p o such that user sells some of her dsk space and the maxmum value of the unt prce p s such that she buys some storage space, two prce senstvtes a and b, that respectvely correspond to the ncrease of sold capacty wth the unt prce p o p mn the unt prce p s p max and the decrease of bought storage space wth. p III. SOCIAL WELFARE PERFORMANCE OF INCENTIVE MECHANISMS In ths secton we ntroduce the performance measure used n ths paper to compare ncentve schemes, and study ts value for the socal optmum and the outcomes of the two ncentve schemes that are the object of ths paper. Defnton 3: We call socal welfare or welfare) and denote by W the sum of the utltes of all agents n the system: W := V C s ) P C o ). 5) I Notce that no prces appear n 5), snce all system agents are consdered, ncludng the operator that receves or gves payments, f any, and whose utlty s her revenue. The operator beng a member of the socety, all money t exchanges wth the users stays wthn the system and therefore does not nfluence socal welfare. A. Optmal value of socal welfare The optmal stuaton n terms of socal welfare) that the system can attan corresponds to the maxmzaton problem max C s,c o I V C s) P C o ), subject to the feasblty constrants C o 0, Cs 0 for and Co Cs. Ths classcal convex optmzaton problem can be solved by the Lagrangan method: f p and C are the unque) solutons of the demand-supply equaton C := s p )= d p ), 6) then the maxmum socal welfare s attaned when C s = d p ),andc o = s p ) for all I. Under Assumpton A, the optmal socal welfare W s then W = b p max p ) a p ) p mn. 7) Ths maxmal value W as well as the so-called shadow prce p are llustrated n Fgure dsplayed n Subsecton III-B. Remark that ths optmal stuaton can be attaned wth a payment-based scheme where p o = p s = p. B. Performance of symmetrc schemes Under a symmetry-based management scheme, each user chooses C o and C s so as to maxmze V C s) P C o), subject to C o C s.asp ) s ncreasng n C s,tsn each user s best nterest to choose a strategy wth C o = Cs. User then maxmzes her utlty at the pont C s = Co = C where V C )=P C ), as llustrated n Fgure. Under Assumpton A, ths corresponds to every user exchangng capacty C = ab a +b p max p mn ) at the vrtual unt prce Authorzed lcensed use lmted to: Unversty of Houston. Downloaded on Aprl 0, 009 at 6:38 from IEEE Xplore. Restrctons apply.
MAILLÉ and TOKA: MANAGING A PEER-TO-PEER DATA STORAGE SYSTEM IN A SELFISH SOCIETY 99 quantty C C / Dp) = P d p) Sp) = P s p) 00 00 0000000 0000000 Maxmal welfare W Operator s proft User welfare 00000 000000000 00000 000000000 00000 000000000 000000000 00000 000000000 00000 000000000 max p mn ) p o p p s mn p max ) unt prce Fg.. Total supply S and demand D functons, maxmum socal welfare and surplus repartton wth a revenue-drven monopoly under Assumpton A. p quantty Dp) Sp) C max p mn ) p mn p max ) unt prce Fg. 3. Illustraton of the proof of Proposton. p apmn +b := p max a +b p. Compared to the socally optmal stuaton, each and every user loses p p ) a + b ) of utlty f max p mn p mn p max. In that case, the welfare loss of the system s W W sym = a + b p p ). 8) Remark that p P = a+b)p s then the weghted mean P of p a+b, therefore the loss of welfare only depends on the heterogenety of users p. In partcular, n the case when all users have the same p, then symmetrc management schemes maxmze socal welfare. C. Performance of proft-orented prcng schemes We now study a prcng mechansm where the system operator strves to extract the maxmum proft out of the busness by playng on prces p s and p o. Knowng that each user wll sell s p o ) and buy d p s ), the operator faces the followng maxmzaton problem. max p s d p s ) p ) o s p o ), 9) p s,p o subject to p s 0, p o 0 and the feasblty constrant s p o ) d p s ). Let us examne the best choces for such a proft-drven monopoly. Fgure plots two curves: the total supply S = s and the total demand D = d as functons of the unt prce p. Frst remark that p o and p s must be chosen such that Sp o )=Dp s ): otherwse t s always possble for the operator to decrease p o f Sp o ) >Dp s )) or ncrease p s f Sp o ) <Dp s )) to strctly mprove ts revenue. The operator revenue wth such prces s then the area of the rectangle dsplayed n the left hand sde of Fgure, embedded wthn a zone whose area s the maxmum value of socal welfare. Whle p o > max p mn and p s < mn p max,the largest revenue s attaned when Sp o )=Dp s ) = C /. However we are not guaranteed that such p o and p s ndeed verfy p o > max p mn and p s < mn p max, nor are we assured that such a choce yelds the maxmum revenue that maxmum mght actually be attaned wth p o < max p mn or p s > mn p max ). To be able to predct the choces of the proft-orented monopoly, we therefore make the followng assumpton regardng user prce thresholds, that fxes those two ponts. Assumpton B: The repartton of prce thresholds p mn and p max s such that max p mn p +mn p mn, 0) mn p max p +max p max, ) P apmn +b p max where p = P a+b from 6). Moreover user profle values a resp. b )ofallusers Iare ndependent and dentcally dstrbuted, and a and b are ndependent. Remark that ths straghtforwardly mply that max p mn p mn p max, so under Assumptons A and B, 8) holds as notced n the prevous subsecton. We can now quantfy the performance of prcng mechansms desgned to maxmze revenue. Proposton : Under Assumptons A and B, a proftorented prcng yelds a socal welfare W mon such that wth C gven n 6)): W W mon = 8 C a + ) b. ) Proof: We frst establsh that the monopoly chooses the proft-maxmzng unt prces p o and p s such that Sp o )= Dp s ) = C /, wherep s the welfare-maxmzng prce gven n 6). To do so, we compute an upper bound of the revenue that can be attaned when choosng the prces n the nonlnear part of S or D. Snce both functons are convex, we can upper bound them by ther cords on [mn p mn, max p mn ] for the supply functon, and on [mn p max, max p max ] for the demand functon. We extend these segments untl the vertcal lne p = p to form two trangles wth the abscssa axs). Under Assumpton B, the largest rectangle embedded n each trangle s ndeed embedded n the trangle formed by extendng the affne parts of S and D, as llustrated n Fgure 3. Authorzed lcensed use lmted to: Unversty of Houston. Downloaded on Aprl 0, 009 at 6:38 from IEEE Xplore. Restrctons apply.
300 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 6, NO. 7, SEPTEMBER 008 Therefore the sum of ther areas s smaller than the revenue yelded by the prces verfyng Sp o )=Dp s )=Sp )/, whch are thus the proft-maxmzng prces. As llustrated n Fgure, the dfference W W mon s then smply the area of the hatched trangle above the horzontal lne C /, whch gves ). IV. WHICH MANAGEMENT TO PREFER? In ths secton we compare the outcomes of the two practcal schemes,.e symmetrc and payment-based schemes. From 8) and ) we mmedately have the followng result. Proposton : Under Assumptons A and B, symmetrc schemes socally outperform proft-orented prcng mechansms f and only f C a + ) b a + b )p p ), = apmn +b p max a +b. where C and p are gven n 6), and p That condton s equvalent to p ) ) α p mn β p max p ω p p ), 3) wth the weghts for all I: α := P a, β a := P b,and b ω := P a+b. Proof: a+b Relaton 3) comes after some algebra, usng the equaltes p = ω p and C = a p p mn )= b p max p ). Proposton combnes the four user heterogenety factors, namely the prce thresholds p mn,p max and prce senstvtes a,b, to determne the best mechansm n terms of socal welfare. Whereas the rght-hand term of 3) s the varance of the p wth weghts w, the left-hand term s hard to nterpret. We thus suggest to have a look at the partcular cases where user heterogenety les entrely on prces senstvtes resp. on prce thresholds). A. Homogeneous prce thresholds We consder here that users only dffer by ther prce senstvtes a and b. That smplfed model has been studed n a prevous work [3], we therefore recall the man results and refer the nterested reader to [3] for detals. Assumpton C: All users Ihave the same prce thresholds p mn and p max. Wthout loss of generalty va a change of abscssa n Fgure ), we can therefore assume that I, p mn =0 and p max = p max. Notce that under Assumptons A and C, Assumpton B always holds. It can then be proved see [3]) that ) [ ] W sym = P + P a b W, a + b W mon = 3 W. Ths yelds the followng comparson whch can also be drectly obtaned from Proposton after some algebra). Proposton 3: Under Assumptons A and C, symmetrc schemes socally outperform proft-orented prcng mechansms f and only f a + ) b a + 3 b. ) Moreover, f the couples a,b ) are ndependently chosen for all users and dentcally dstrbuted, then when the number of users tends to nfnty, ) wrtes E[fa, b)] fe[a], E[b]) 3, wth f :x, y) /x +/y. 5) Snce the functon f s strctly concave, from Jensen s nequalty the left-hand term of 5) s always smaller than, and decreases as the dsperson of a, b) ncreases. Remark that when a,b) are determnstc then the left-hand term of 5) equals and symmetrc schemes are better than proftorented ones, as we remarked n subsecton III-B. Let us have a look at 5) for two smple examples of dstrbutons for a,b), assumng that a and b are ndependent varables. Unform dstrbuton. Ifa resp. b) s unformly dstrbuted over [0,a max ] resp. [0,b max ]), E[fa, b)] fe[a], E[b]) = 3 a max + b max ) a max + b max ) a max ln + b max ) b max ln + a max ). 6) b max a max a max b max Ths expresson s mnmum when a max = b max, n whch case t equals 8 ln))/3 0.8. Consequently nequalty 5) always holds. Exponental dstrbuton. If a resp. b) follows an exponental dstrbutons wth parameter μ a resp. μ b ),.e. Pa >x)=e µa x, then we obtan after some calculaton E[fa, b)] fe[a], E[b]) 3 α μ a μ b α, where α 0.79 s the smallest postve root of x +x x) 3 x +x lnx)) 3/. In that case, ether a symmetrc or a proft orented mechansm s socally preferable dependng on the relatve values of μ a and μ b. B. Homogeneous prce senstvtes We now consder the case where the prce thresholds p mn can be user specfc, but the prce senstvtes a and p max and b are dentcal for every user. Assumpton D: All users have the same prce senstvty of supply resp. demand),.e. I, a = a and b = b. Moreover, the couples p mn,p max dentcally dstrbuted among users, and p mn of p max for all I. ) are ndependent and s ndependent In that case, we establsh that one mechansm s always preferable to the other. Proposton : Under Assumptons A, B and D, management mechansms based on symmetry are always socally better than proft-orented prcng mechansms. Proof: Assumpton D mples that for all, the weghts α, β and ω ntroduced n 3) all equal n,wheren s the Authorzed lcensed use lmted to: Unversty of Houston. Downloaded on Aprl 0, 009 at 6:38 from IEEE Xplore. Restrctons apply.
MAILLÉ and TOKA: MANAGING A PEER-TO-PEER DATA STORAGE SYSTEM IN A SELFISH SOCIETY 30 number of users. Moreover we have p = a pmn +b p max a+b,where p mn P := pmn n and p max P := pmax n.sowhenn tends to nfnty, 3) s equvalent to ab p max p mn) a Varp mn )+b Varp max ), where Var denotes the varance, and where we used the ndependence assumpton of p max and p mn to develop the rght-hand term. Snce the varance of a real varable wth support length y s always smaller than y /, and usng 0) Varp mn ) p max p mn ) p p mn ), where the last nequalty comes from p mn max p mn p. Lkewse, Varp max ) p max p ) /. Therefore by replacng the optmal shadow prce p by a p mn + b p max )/a + b) and applyng the nequalty a + b) ab, weget a Varp mn )+b Varp max ) ab pmax p mn ), Therefore Relaton 3) s always satsfed and symmetrc schemes always outperform proft-maxmzng prcng schemes. V. CONCLUSIONS AND FUTURE WORK In ths work we have addressed the problem of user ncentves n a peer-to-peer storage system. Usng a game theoretcal model to descrbe selfsh reactons of all system actors users and the operator), we have studed and compared the outcomes of two possble managng schemes, namely symmetry-based and proft orented payment-based. Not only the sze of the offered storage space was targeted wth ncentves, but as the avalablty and relablty are partcularly mportant ssues n storage systems, the model also amed to reduce churn. By comparng the socal welfare level at the outcome n the two cases, under some assumptons on user preferences we exhbted a necessary and suffcent condton for a type of management to be preferable to the other: t appears that proft orented payment-based schemes may be socally better than symmetrc ones under some specfc crcumstances, namely f the heterogenety among user profles s hgh. There are dfferent ways to extend the results we have obtaned. Frst of all, n realty the perceved utlty of a user should not only depend on the amount of stored data and the assocated avalablty, but also on the rapdty to access those data. Therefore the avalable bandwdth of a storage space offerer should be taken nto account n addton to the amount of space proposed. Another nterestng drecton would be to consder demand and supply functons that are not affne, but can have any form, or eventually to carry out experments to estmate the form of those functons. Fnally, snce a more complete and realstc model may not be solvable analytcally, a smulaton testbed could be bult n order to study the behavour of a peer-to-peer storage system n a more complex settng and eventually exhbt other phenomena that are not captured by our model. REFERENCES [] D. Fudenberg and J. Trole, Game Theory. MIT Press, Cambrdge, Massachusetts, 99. [] E. Adar and B. Huberman, Free rdng on Gnutella, Frst Monday, vol. 5, no. 0, Oct 000. [3] P. Antonads, C. Courcoubets, and R. Mason, Comparng economc ncentves n peer-to-peer networks, Computer Networks, vol. 6, no., pp. 33 6, 00. [] C. Courcoubets and R. Weber, Incentves for large peer-to-peer systems, IEEE JSAC, vol., no. 5, pp. 03 050, May 006. [5] P. Golle, K. Leyton-Brown, I. Mronov, and M. Lllbrdge, Incentves for sharng n peer-to-peer networks, n Proc. 3rd ACM conference on Electronc Commerce EC 0), Tampa, Florda, USA, Oct 00, pp. 6 67. [6] K. La, M. Feldman, I. Stoca, and J. Chuang, Incentves for cooperaton n peer-to-peer networks, n Proc. st Workshop on Economcs of Peer-to-Peer Systems PPECON 03), Berkeley, CA, USA, Jun 003. [7] C. Batten, K. Barr, A. Saraf, and S. Treptn, pstore: A secure peerto-peer backup system, MIT Laboratory for Computer Scence, Tech. Rep. MIT-LCS-TM-63, Dec 00. [8] P. Druschel and A. Rowstron, PAST: A large-scale, persstent peerto-peer storage utlty, n HotOS VIII, Schloss Elmau, Germany, May 00, pp. 75 80. [9] M. Lllbrdge, S. Elnkety, A. Brrell, M. Burrows, and M. Isard, A cooperatve nternet backup scheme, n Proc. st Workshop on Economcs of Peer-to-Peer Systems PPECON 03), Berkeley, CA, USA, Jun 003. [0] L. Cox and B. Noble, Samsara: Honor among theves n peer-topeer storage, n Proc. 9th ACM Symposum on Operatng Systems Prncples SOSP 03), Bolton Landng, NY, Oct 003. [] B. Stefansson, A. Thods, A. Ghods, and S. Hard, MyradStore, Swedsh Insttute of Computer Scence, Tech. Rep. T006:09, May 006. [] D. Besanko and R. R. Braeutgam, Mcroeconomcs. Wley, 005. [3] L. Toka and P. Mallé, Managng a peer-to-peer backup system: Does mposed farness socally outperform a revenue-drven monopoly? n Proc. th Internatonal Workshop on Grd Economcs and Busness Models Gecon 007), Aug 007. [] R. Rodrgues and B. Lskov, Hgh avalablty n DHTs: Erasure codng vs.replcaton, n Proc. th Internatonal Workshop on Peer-to-Peer Systems IPTPS), Ithaca, USA, Feb 005. [5] A. Dumnuco, E. Bersack, and T. En-Najjary, Proactve replcaton n dstrbuted storage systems usng machne avalablty estmaton, n Proc. 3rd Internatonal Conference on emergng Networkng EXperments and Technologes CoNEXT), New York, USA, Dec 007. Patrck Mallé graduated from Ecole polytechnque and TELECOM ParsTech, France, n 000 and 00, respectvely. He has been an assstant professor at the Network, Securty, Multmeda department of TELECOM Bretagne snce 00, where he obtaned hs PhD n appled mathematcs n 005. Hs research focuses on game theory appled to networks: resource prcng, routng games, consequences of user selfshness on network performance. László Toka graduated n 007 at Budapest Unversty of Technology and Economcs and receved hs MSc degree n Telecommuncatons. He also obtaned the engneer dploma of Telecom Bretagne and Eurecom n France. After graduatng he enrolled as a PhD canddate at Telecom Pars and at Budapest Unversty of Technology and Economcs. Hs research nterests are on economc modelng of dstrbuted systems, wth specal focus on ncentves and mechansm desgn. Authorzed lcensed use lmted to: Unversty of Houston. Downloaded on Aprl 0, 009 at 6:38 from IEEE Xplore. Restrctons apply.