Designing Social Norm Based Incentive Schemes to Sustain Cooperation in a Large Community

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1 Designing Soial Norm Based Inentive Shemes to Sustain Cooperation in a arge Community Yu Zhang, Jaeo Par, Mihaela van der Shaar Eletrial Engineering Department, University of California, os Angeles yuzhang@ula.edu, {jaeo, mihaela}@ee.ula.edu Abstrat. Sustaining ooperation among self-interested agents is ritial for the proliferation of emerging networed ommunities, suh as the ommunities formed by soial networing servies. Providing inentives for ooperation in networed ommunities is partiularly hallenging beause of their uniue features: a large population of anonymous agents interating infreuently, having asymmetri interests, and dynamially joining and leaving the networ; networ operation errors; and low-ost identity whitewashing. In this paper, taing these features into onsideration, we propose a framewor for the design and analysis of a lass of inentive shemes based on soial norms. We first define the onept of sustainable soial norm under whih no agent has an inentive to deviate. We then formulate the problem of designing an optimal soial norm, whih selets a soial norm that maximizes overall soial welfare among all sustainable soial norms. Using the proposed framewor, we study the struture of optimal soial norms and the impats of punishment lengths and whitewashing on optimal soial norms. Our results show that optimal soial norms are apable of sustaining ooperation, with the amount of ooperation varying depending on the ommunity harateristis. Keywords: Inentive Shemes, Networed Communities, Reputation Shemes, Soial Norms Introdution Reent developments in tehnology have expanded the boundaries of ommunities in whih individuals interat with eah other. However, a large population and the anonymity of individuals in networ-based ommunities mae it diffiult to sustain ooperative behavior among self-interested individuals, with the so-alled free-riding behavior prevailing []. Hene, inentive shemes are needed to provide individuals with inentives for ooperation. The literature has proposed various inentive shemes. The popular forms of inentive devies used in many inentive shemes are payment and differential servie. Priing shemes use payments to reward and punish individuals for their behavior, whih in priniple an lead self-interested individuals to ahieve soial optimum by internalizing their external effets (see, for example, [8]). However, it is often laimed that priing shemes are impratial beause they reuire an aounting

2 infrastruture []. Differential servie shemes, on the other hand, reward and punish individuals by providing differential servies depending on their behavior. Suh inentive shemes are based on the priniple of reiproity and an be lassified into personal reiproation and soial reiproation. In personal reiproation shemes, individuals an identify eah other, and behavior toward an individual is based on their personal experiene with the individual. Personal reiproation is effetive in sustaining ooperation in a small ommunity where individuals interat freuently and an identify eah other, but it loses its fore in a large ommunity where anonymous individuals interat infreuently. In soial reiproation shemes, individuals obtain some information about other individuals (for example, rating) and deide their behavior toward an individual based on their information about that individual. Hene, an individual an be rewarded or punished by other individuals in the ommunity who have not had diret interation with it. Sine soial reiproation reuires neither observable identities nor freuent interations, it has a potential to form a basis of suessful inentive shemes for networ-based ommunities. As suh, this paper is devoted to the study of inentive shemes based on soial reiproation. Sustaining ooperation using soial reiproation has been investigated in the eonomis literature using the framewor of anonymous random mathing games and soial norm [7] and []. Eah individual is attahed a label indiating its reputation, status, et. whih ontains information about its past behavior, and individuals with different labels are treated differently by other individuals they interat with. However, [7] and [] have foused on obtaining the Fol Theorem by haraterizing the set of euilibrium payoffs that an be ahieved by using a soial norm based strategy when the disount fator is suffiiently lose to. Our wor, on the ontrary, addresses the problem of designing a soial norm given a disount fator and other parameters arising from pratial onsiderations. Speifially, our wor taes into aount the following features of networ-based ommunities. Asymmetry of interests. We allow the possibility of asymmetri interests by modelling the interation between a pair of individuals as a gift-giving game, instead of a prisoner s dilemma game, whih assumes mutual interests between a pair of individuals. Report errors. In a soial norm based inentive sheme, it is possible that the reputation of an individual is updated inorretly beause of errors in the report of individuals. Our model inorporates the possibility of report errors, whih allows us to analyze its impat on the design and performane, whereas most existing wors on reputation shemes [4][6] adopt an idealized assumption that reputations are always updated orretly. Dynami hange in the population. The members of a ommunity hange over time as individuals gain or lose interest in the servies provided by ommunity members. We model this feature by having a onstant fration of individuals leave and join the ommunity in every period to study the impat of population turnover on the design and performane. The remainder of this paper is organized as follows. In Setion, we desribe the repeated mathing game and inentive shemes based on a soial norm, and then formulate the problem of designing an optimal soial norm. In Setion 4, we provide analytial results about optimal soial norms. We onlude the paper in Setion 5.

3 Model We onsider an infinite-horizon disrete time model with a ontinuum of agents [4]. In a period, eah agent generates a servie reuest [9], whih is sent to another agent that an provide the reuested servie. Eah agent is eually liely to reeive the reuest from a partiular agent, and the mathing in eah period is independent. In a pair of mathed agents, the agent that reuests for a servie is alled a lient while the agent that reeives a servie reuest is alled a server. The interation between a pair of mathed agents is modelled as a gift-giving game [5], where the server has the binary hoie from the set = { FD, } of whether to fulfil, denoted as F, or deline the reuest, denoted as D, while the lient has no hoie. If the server fulfills the lient s reuest, the lient reeives a servie benefit of b > while the server suffers a servie ost of >. If the server delines the reuest, both agents reeive zero payoff. An agent plays the gift-giving game repeatedly with hanging partners until it leaves the ommunity. We assume that at the end of eah period a fration a Î [,] of agents in the urrent population leave and the same amount of new agents join the ommunity. We refer to a as the turnover rate [4]. Soial welfare in a time period is measured by the average payoff of the agents in that period. As we assume b >, soial welfare is maximized when all the servers hoose ation F in the gift-giving game they play, whih yields payoff b- to every agent. On the ontrary, ation D is the dominant strategy for the server in the gift-giving game, whih an be onsidered as the myopi euilibrium of the giftgiving game. When every server hooses its ation to maximize its urrent payoff myopially, an ineffiient outome arises where every agent reeives zero payoffs. In order to improve the ineffiieny of the myopi euilibrium, we use inentive shemes based on the idea of soial norms. A soial norm is defined as the rules that a group uses to regulate the behavior of members. We onsider a soial norm that onsists of a reputation sheme and a soial strategy, as in [7] and []. Formally, a reputation sheme determines the reputations of agents depending on their past ations as a server and is represented by two elements (, t ). is the set of reputations that an agent an hold, and t is the reputation update rule. After a server taes an ation, the lient sends a report about the ation of the server to the thirdparty devie or infrastruture that manages the reputations of agents, but the report is subjet to errors with a small probability e. That is, with probability e, D is reported when the server taes ation F, and vie versa. Assuming a binary set of reports, it is without loss of generality to restrit e in [,/ ]. We onsider a reputation update rule that updates the reputation of a server based only on the reputations of mathed agents and the reported ation of the server. Then, a reputation update rule an be represented by a mapping t :, where (,, ar ) t is the new reputation for a server with urrent reputation when it is mathed with a lient with reputation and its ation is reported as a R Î. A soial strategy presribes the ations that servers should tae depending on the reputations of the mathed agents and is represented by a mapping s :,

4 where s(, ) is the approved ation for a server with reputation that is mathed with a lient with reputation. To simplify our analysis, we impose the following restritions on reputation shemes we onsider: (i) is a nonempty finite set, i.e., = {,, ¼, } for some nonnegative integer ; (ii) t is defined by ìï min { +, } if ar = s(, ), t(,, ar ) = ï í () ï if ar ¹ s(, ). ïî Note that with the above restritions a nonnegative integer ompletely desribes a reputation sheme, and thus a soial norm an be represented by a pair = (, s). We all the reputation sheme determined by the maximum punishment reputation sheme with punishment length. In the maximum punishment reputation sheme with punishment length, there are total + reputations, and the initial reputation for new peers entering the networ is given as. If the reported ation of the server is the same as that speified by the soial strategy s, the server s reputation is inreased by while not exeeding. Otherwise, the server s reputation is set as. Below we summarize the seuene of events in a time period. Agents generate servie reuests and are mathed. Eah server observes the reputation of its lient and then determines its ation. Eah lient reports the ation of its server. The reputations of agents are updated, and eah agent observes its new reputation for the next period. A fration of agents leave the ommunity, and the same amount of new agents join the ommunity. As time passes, the reputations of agents are updated and agents leave and join the networ. Thus, the distribution of reputations evolves over time. In this paper, we use the stationary distribution in our analysis, whih will be written { h( )}, where h( ) be the fration of -agents in the total population at the beginning of an arbitrary period t and a -agent means an agent with reputation. { h( )} satisfies the following euality as h ( ) = ( -a) e, h () = ( -a)( -e) h ( - ) for -, () { } h ( ) = ( -a)( - e) h ( ) + h ( - ) + a. We now investigate the inentive of agents to follow a presribed soial strategy. Sine we onsider a non-ooperative senario, we need to he whether an agent an improve its long-term payoff by a unilateral deviation., be the ost suffered by a server with reputation that is mathed et s ( ) with a lient with reputation and follows a soial strategy s. Similarly, let

5 b s (, ) be the benefit reeived by a lient with reputation that is mathed with a server with reputation following a soial strategy s. Sine we onsider uniform random mathing, the expeted period payoff of a -agent under soial norm before it is mathed is given by v = h b, - h,. (3) ( ) å ( ) ( ) å ( ) ( ) s s Î Î To evaluate the long-term payoff of an agent, we use the disounted sum riterion in whih the long-term payoff of an agent is given by the expeted value of the sum of p be the transition disounted period payoffs from the urrent period. et ( ) probability that a -agent beomes a -agent in the next period under soial norm, the long-term payoff of an agent from the urrent period (before it is mathed) an be solved by the following reursive euations Î ( ) ( ) d ( ) ( ) v = v + å p v for Î, (4) where d = b( - a) is the weight that an agent puts on its future payoff. Now suppose that an agent deviates and uses a soial strategy s under soial norm. Sine the deviation of a single agent does not affet the stationary distribution, the expeted period payoff of a deviating -agent is given by v ( ) = h ( ) (, ) ( ), bs + h (, s s ). (5) et ( ) å Î å Î p s, be the transition probability that a -agent using soial strategy, s beomes a -agent in the next period under soial norm, when it is mathed with a lient with reputation. The long-term payoff of a deviating agent from the urrent period (before it is mathed) an be omputed by solving s, s, s, s, Î ( ) ( ) d ( ) ( ) v = v + å p v for Î.(6) In our model, a server deides whether to provide a servie or not after it is mathed with a lient and observes the reputation of the lient. Hene, we he the inentive for a server to follow a soial strategy at the point when it nows the reputation of the lient. Suppose that a server with reputation is mathed with a lient with reputation. When the server follows the soial strategy s presribed by soial norm, it reeives the long-term payoff - s (, ) + då p( ) v ( ), exluding the possible benefit as a lient. On the ontrary, when the server deviates to a soial strategy s, it reeives the longterm payoff - (, ) + d p (, ) v s å, s, s ( ), again exluding the possible benefit as a lient. By omparing these two payoffs, we an he whether a -agent has an inentive to deviate to s when it is mathed with a lient with reputation,

6 and define a soial norm is sustainable if no agent an gain from a unilateral deviation regardless of the reputation of the lient it is mathed with, i.e. - s (, ) + då p( ) v ( ) ³ (7) -, + d p, v for all s s, for all (, ) ( ) å s, ( ), s ( ). Thus, under a sustainable soial norm, agents follow the presribed soial strategy in their self-interest. Cheing whether a soial norm is sustainable using the above definition reuires omputing deviation gains from all possible soial strategies, whose omputation omplexity an be high for moderate values of. By employing the riterion of unimprovability in Marov deision theory [], we establish the one-shot deviation priniple for sustainable soial norms. For notation, let a be the ost suffered by a server that taes ation a, and let ( ) p, a, be the transition probability that -agent beomes a -agent in the next period under soial norm when it taes ation a to a lient with reputation. The values of, a (, ) p s ( ), by omparing a with (, ) p an be obtained in a similar way to obtain s.,, emma (One-shot Deviation Priniple). A soial norm is sustainable if and only if é ù s (, )- a d { p( ) -p, a (, )} v ( ) å (8) ê ë úû for all a Î, for all (, ). emma shows that if an agent annot gain by unilaterally deviating from s only in the urrent period and following s afterwards, it annot gain by swithing to any other soial strategy s either, and vie versa. First, onsider a pair of reputations (, ) suh that s(, ) = F. If the server with reputation serves the lient, it suffers the servie ost of in the urrent period while its reputation in the next period beomes min { +, } with probability ( - e) and with probability e. Thus, the expeted long-term payoff of a -agent when it provides a servie is given by V( F; F) =- + dé( - e) v ( min{ +, } ) + ev () ù ë û. (9) Similarly, if a -agent deviates and delines the servie reuest, the expeted longterm payoff of a -agent when it does not provide a servie is given by V( D; F) = dé( - e) v () + ev ( min{ +, } ) ù ë û. () The inentive onstraint that a -agent does not gain from a one-shot deviation is given by V ( F; F) ³ V ( D; F), whih an be expressed as,

7 d( - e) év ( min {, } ) v () ù ë + - û ³. () Now onsider a pair of reputations (, ) suh that ( ) s, = D. Using a similar argument as above, we an show that the inentive onstraint that a -agent does not gain from a one-shot deviation an be expressed as d( - e) év ( min {, } ) v () ù ë + - û ³-. () s, = F for some Note that () implies (), and thus for suh that ( ), we an he only the first inentive onstraint (). Therefore, a soial norm s, = F for some is sustainable if and only if () holds for all suh that ( ) and () holds for all suh that s(, ) = D for all. The left-hand side of the inentive onstraints () and () an be interpreted as the loss from punishment that soial norm applies to a -agent for not following the soial strategy. Therefore, in order to indue a -agent to provide a servie to some lients, the left-hand side should be at least as large as the servie ost, whih an be interpreted as the deviation gain. We use minî { d( - e) év ( min {, } ) v () ù ë + - û} to measure the strength of the inentive for ooperation under soial norm, where ooperation means providing the reuested servie in our ontext. Sine we assume that the ommunity operates at the stationary distribution of reputations, soial welfare under soial norm an be omputed by U = å h ( ) v ( ). (3) We assume that the ommunity operator aims to hoose a soial norm that maximizes soial welfare among sustainable soial norms. Then the problem of designing a soial norm an be formally expressed as maximize U = h ( ) v ( ) (, s) å subjet to d( - e) év ( min { +, } )- v () ù ³, " suh that $ suh that s(, ë û ) = F, d( - e) év ( min { +, } )- v () ù ³-, " suh that s(, ë û ) = D ". (4) A soial norm that solves the problem (4) is alled an optimal soial norm. 3 Analysis of optimal soial norms We first investigate whether there exists a sustainable soial norm, i.e., whether the design problem (4) has a feasible solution. Fix the punishment length and D onsider a soial strategy s where agents do not provide a servie at all, i.e., for

8 all (, ). Sine there is no servie provided in the ommunity when all the agents follow s, we have v (, s D ) () = for all. Hene, the relevant inentive D D onstraint () is satisfied for all, and the soial norm (, s ) is sustainable. This shows that the design problem (4) always has a feasible solution. Assuming that an optimal soial norm exists, let U be the optimal value of the design problem (4). In the following proposition, we study the properties of U. Proposition. (i) U b- ; (ii) b( -a)( -e) U = if > ; b - b( - a)( - 3e) (iii) U ³ [ -( -a) e] ( b- ) if b( -a)( - e) ; (iv) U < b- if b e > ; (v) U = b- if e = and b( - a) ; (vi) U = b- only if b b( - a) e = and. b - b( - a) Proof: See Appendix A. We obtain zero soial welfare at myopi euilibrium, without using a soial norm. Hene, we are interested in whether we an sustain a soial norm in whih agents ooperate in a positive proportion of mathes. In other words, we loo for onditions on the parameters (, bbae,,, ) that yield U >. In order to obtain analytial results, we onsider the design problem (4) with a fixed punishment length, alled DP. Note that DP has a feasible solution, D ( ) namely s, for any and that there are a finite number (total + ) of possible soial strategies given. Therefore, DP has an optimal solution for any. We use and s to denote the optimal value and the optimal soial strategy of DP, respetively. We first show that inreasing the punishment length annot derease the optimal value. Proposition. U ³ U for all and suh that ³. Proof: Choose an arbitrary. To prove the result, we will onstrut a soial strategy s + using punishment length + that is feasible and ahieves U. Define s + by ìï s (, ) for and, s (, ) for = + and, s (, + ) = ï í (5) s (, ) for and = +, ï s (, ) for = + and = +. ïî et (, s ) = and ( ) = +, s +. From (), we have h+ ( ) = h( ) for =,, - and h ( ) + h ( + ) = h ( ). Using this and (3), it is + +

9 straightforward to see that v ( ) v ( ) = for all =,, and v ( + ) = v ( ). Hene, we have that å + å + å + = = = - + = åh( ) v( ) + åh+ ( ) v( ) = = - = å h( ) v( ) + h( ) v( ) = U = U. = U = h ( ) v ( ) = h ( ) v ( ) + h ( ) v ( ) ( ) () ( ) () Using (5), we an show that v () - v () = v () - v () for all =,, and v + - v = v - v. By the definition of s +, the right-hand side of the relevant inentive onstraint (i.e., or - ) for eah =,, is the same both under s and under s +. Also, under s +, the right-hand side of the relevant inentive onstraint for = + is the same as that for =. Therefore, s + satisfies the relevant inentive onstraints for all =,, +. Proposition shows that U is non-dereasing in. Sine U b-, we have U = lim U = sup U. It may be the ase that the inentive onstraints eventually prevent the optimal value from inreasing with so that the supremum is attained by some finite. If the supremum is not attained, the protool designer an set an upper bound on based on pratial onsideration. Now we analyze the struture of optimal soial strategies given a punishment length. Proposition 3. Suppose that e > and a <. (i) If ( ˆ, ) ˆ, then s (, ) = F for all { ³ min ln / ln b, } b satisfies ³ -( ln - ln Y( a, e, ) ) ln b, where and ( ˆ, ) b (6) s = F for some ; (ii) If Î {,, - } ( -a) (-e) e-( -a) (-e) e Y ( ae,, ) = + ( -a) ( - e) e+ a s = F for some ˆ, then (, ) F s, = F for some ˆ, then s (, ) = F. Proof: To failitate the proof, we define u () by l u ( ) = å g v( min { + l, } ) for =,,. Then, by (5), we have l= v ()- v () = u ()- u () for all =,,. (7) s = ; (iii) If ( ˆ)

10 Suppose that ( ˆ, ) s = F for some ˆ. Then the relevant inentive onstraint for a -agent is d( - e) [ u () - u ()] ³. Suppose that s (, ) = D for some Î{,, -} suh that ³ ln / ln b. Consider a soial strategy s b defined by ìï s ( ) ( ) ( ), for, ¹,, s (, ) = ï í (8) ï F for (, ) = (, ). ïî et = (, s ) and = (, s ). Note that h t ( ) v ( ) v( ) t ( ) b - = h > sine e > and U - U = () ( )( b- ) >. Also, h h v () - v () =- < and ( ) ( ) a <. Thus, ìï [ v ()- v()] + g [ v -v ] + = ( -a) ( -e) eb [ b- ] for =, u ( ) - u ( ) =í ï ïï - g [ v ( ) v ( ) ] for,,, - = ¼ for = +, ¼,. î ï (9) Sine ³ ln / ln b, we have u ( ) - u ( ). Thus, b u () - u () ³ u () - u () for all =,,. Hene, s satisfies the inentive onstraints of DP, whih ontradits the optimality of s. This proves that s (, ) = F for all ln ³ b / ln b. Similar approahes an be used to prove s (, ) = F, (ii), and (iii). We an represent a soial strategy s as an ( + ) ( + ) matrix whose (, ij-entry ) is given by s( i-, j - ). Proposition 3 provides some strutures of an optimal soial strategy s in the first row and the last olumn of the matrix representation, but it does not fully haraterize the solution of DP. Here we aim to obtain the solution of DP for = and and analyze how it hanges with the parameters. We first begin with the ase of two reputations, i.e., =. In this ase, if s (, ) = F for some (, ), the relevant inentive onstraint to sustain = (, s) is d( - e) [ v () - v ()] ³. By Proposition 3(i) and (iii), if s (, ) = F for some (, ), then s (,) = s (,) = F, provided that e > and a <. Hene, among the total of 6 possible soial strategies, only four an be optimal soial strategies. These four soial strategies are

11 é D F ù é F F ù é D F ù é D D ù 3 D 4 D s =, s, s s, s s F F = D F = = = = D F D D.() êë úû êë úû êë úû êë úû The following proposition speifies the optimal soial strategy given the parameters. Proposition 4. Suppose that < (- ae ) < /. Then s ìï b( -a) ( - e) e s if <, b + b( - a) ( - e) e b( -a) ( - e) e b( -a)( - e)[ -( -a) e] s if <, = ï í + b( -a) (- e) e b -b(-a) (- e) e 3 b( -a)( - e)[ -( -a) e] s if < b( -a)( - e), -b(-a) (- e) e b 4 ïs if b( -a)( - e) < <. ïî b i i Proof: et (, s ) =, for i =,, 3, 4. We obtain that ( h ) ( h h ) U = - () ( b- ), U = - () () ( b-), () U 3 = ( -h ())( b- ), U 4 =. () Sine < (- ae ) < /, we have h() < h(). Thus, we have U > U > U 3 > U 4. Also, we obtain that v () - v () = h()( b-), v () - v () = b-h()( b-), (3) v 3() - v 3() = b, v 4() - v 4() =. Thus, we have v 3() - v 3() > v () - v () > v () - v () > v 4() - v 4(). By hoosing the soial strategy that yields the highest soial welfare among feasible ones, we obtain the result. Proposition 4 shows that the optimal soial strategy is determined by the ratio of the servie ost and benefit, / b. When / b is suffiiently small, the soial strategy s an be sustained, yielding the highest soial welfare among the four andidate soial strategies. As / b inreases, the optimal soial strategy hanges from s to s to s 3 4 and eventually to s. Figure shows the optimal soial strategies with = as varies. The parameters we use to obtain the results in the figures of this paper are set as follows unless otherwise stated: b =.8, a =., e =., and b =. Figure (a) plots the inentive for ooperation of the four soial strategies. We an find the region of in whih eah strategy is sustained by omparing the inentive for ooperation with the servie ost for 3 s, s, and s, and with - for s 4. The solid portion of the lines indiates that the strategy is sustained while the dashed portion indiates that the strategy is not

12 sustained. Figure (b) plots the soial welfare of the four andidate strategies, with solid and dashed portions having the same meanings. The triangle-mared line represents the optimal value, whih taes the maximum of the soial welfare of all sustained strategies. Next, we analyze the ase of three reputations, i.e. =. In Figure, we show the optimal value and the optimal soial strategy of DP as we vary. The optimal soial strategy s D hanges in the following order before beoming s as inreases: éf F Fù éd F Fù éd F Fù 3 s = D F F, s = F F F, s = D F F, F F F F F F F F F êë úû êë úû êë úû (4) éf F Fù éf F Fù éd F Fù éd F Fù s = F F F, s = D F F, s = F F F, s = D F F. D F F D F F D F F ê ú ê ú ê ú D F Fú ë û ë û ë û ëê úû B Note that s = s for small and 7 D s = s for large (but not too large to sustain ooperation), whih are onsistent with the disussion about Proposition 5. For the intermediate values of, only the elements in the first olumn hange in order to inrease the inentive for ooperation. We find that the order of the optimal soial strategies between s = s B and s 7 = s D depends on the ommunity s parameters (, b bae,, ). 5 Inentive for ooperation 4 3 servie ost

13 = 3 4 Soial welfare Figure. Performane of the four andidate soial strategies when = : (a) inentive for ooperation, and (b) soial welfare and the optimal soial strategy. = D 9 8 Optimal soial welfare Figure. Optimal soial welfare and the optimal soial strategy of DP. 4 Conlusions In this paper, we used the idea of soial norm to establish a rigorous mathematial framewor to analyze the inentive mehanisms based on indiret reiproity in networ-based ommunities. We proposed the idea of sustainable soial norm in whih no agent has the inentive to deliberately deviate. Based on this, we analyzed the optimal soial norm problem and uantified the effiieny loss in the soial norm as a trade-off to the inentive given to agents to follow it. We also showed that the optimal soial strategy has some ommon properties whih are independent from the

14 number of reputations in the soial norm. Our urrent analysis an be extended in different diretions, among whih we mention two. First, instead of implementing the maximum punishment reputation sheme, more ompliated reputation shemes an be used where the agent s reputation does not fall to diretly upon deviation. Seond, we grant the agents newly entering the networ the highest reputation in this paper. Nevertheless, this will provide agents with low reputations in the networ the inentive to repeatedly leave and rejoin the networ in the aim of obtaining a higher reputation and thus better long-term payoff, a phenomenon ommonly nown as whitewash. Hene, more sophistiated rules of reputation assignment an be designed to disourage the whitewash behavior. Referenes. E. Adar, B. A. Huberman: Free Riding on Gnutella, First Monday, vol. 5, No., Ot... C. Buragohain, D. Agrawal, and S. Suri: A Game-theoreti Framewor for Inentives in PP Systems, Pro. Int. Conf. Agent-to-Agent Computing, pp , Sep B. Cohen: Inentives Building Robustness in BitTorrent, Pro. PP Eon. Worshop, Bereley, CA, M. Feldman, K. ai, I. Stoia, J. Chuang: Robust Inentive Tehniues for Agent-to-Agent Networs, Pro. of the 5th ACM Conf. on Ele. Commere, Session 4, pp. -, P. Johnson, D. evine, and W. Pesendorfer: Evolution and Information in a Gift-Giving Game, J. Eon. Theory, vol., no., pp. -,. 6. S. Kamvar, M. T. Shlosser, H. G. Molina: The Eigentrust Algorithm for Reputation Management in PP Networs, Pro. th Int l Conf. on World Wide Web, pp , M. Kandori: Soial Norms and Community Enforement, Rev. Eonomi Studies, vol. 59, No., pp. 63-8, Jan J. K. MaKie-Mason and H. R. Varian: Priing ongestible networ resoures, IEEE J. Sel. Areas Commun., vol. 3, no. 7, pp. 4 49, Sep Massoulie, M. Vojnovi: Coupon Repliation Systems, IEEE/ACM Trans. on Networing, vol. 6, no. 3, pp , 5.. M. Ouno-Fujiwara and A. Postlewaite: Soial norms and random mathing games, Games Eon. Behavior, vol. 9, no., pp. 79 9, Apr P. Whittle: Optimization Over Time, New Yor: Wiley, 983. Appendix A: Proof of Proposition : D (i) U ³ follows by noting that (, s ) is feasible. Note that the objetive U = ( b - ) h h I( s, = F), where å, funtion an be rewritten as ( ) ( ) ( ) I is an indiator funtion. Hene, U b - for all, whih implies U b-. (ii) By (4), we an express v ( ) - v ( ) as

15 v () - v () = v () + d ( - e) v () + ev () -v -d - e v + ev [ ] () [( ) ( ) ()] [ ] = v() - v() + d( -e) v ( ) -v ( ). (5) Similarly, we have v ( ) - v ( ) = v ( ) - v ( ) + d( -e) [ v ( 3) -v ( )], [ ] v ( -) -v ( - ) = v( -) -v( - ) + d( -e) v ( ) -v ( - ), v ( ) -v ( - ) = v( ) -v( - ). (6) In general, for =, ¼,, - l v ( ) -v ( - ) = å g év( + l) - v( + l -) ù ë û, (7) l= where we define g = d( - e). Thus, we obtain v ( ) - v () - [ ] [ ] [ v ( ) v ( ) ] [ v ( ) v ( ) ] (8) = v () - v () + g v ( + ) - v () + + g v ( ) -v ( -) -+ - g g - l g év( min{ l, } ) v( l) ù å ë û, l= = + - for =, ¼,. Sine - v() b for all, we have v() - v() b + for all (, ). Hene, by (5), - g b + v () - v () ( b + ) -g -g for all =, ¼,, for all = (, s). Therefore, if d( - e)[( b + )/( - g)] <, or euivalently, / b > [ b( -a)( - e)]/[ -b( -a)( - 3 e)], then the inentive onstraint () annot be satisfied for any, for any soial norm (, s ). This s, = F, is not implies that any soial strategy s suh that ( ) for some ( ) D and (, ) feasible, and thus U =. D D (iii) For any, define a soial strategy s by s (, ) = D for = s = F for all > D, for all. In other words, with s eah agent (9)

16 delines the servie reuest of -agents while providing a servie to other agents. D Consider a soial norm = (, s ). Then v() =- h() and v() = b- h(). Hene, U = [ -( -ae ) ]( b- ) and v () - v () = b, and thus the inentive onstraint d( -e) ( v ( ) - v ( ) ) ³ is satisfied by the hypothesis / b b( -a)(- e). Sine there exists a feasible solution that ahieves U = [ -(-ae ) ]( b- ), we have U ³ [ -(-ae ) ]( b- ). (iv) Suppose, on the ontrary to the onlusion, that U = b-. If a =, then () annot be satisfied for any, for any, whih implies U =. Hene, it must be the ase that a <. et = (, s ) be an optimal soial norm that ahieves U = b-. Sine U = U = ( b- ) h ( ) h ( ) I( s (, ) = F) å, s should have (, ) F for all (, ), s =. However, under this soial strategy, all the agents are treated eually, and thus v () = = v ( ). Then s annot satisfy the relevant inentive onstraint () for all sine the left-hand side of () is zero, whih ontradits the optimality of (, s ). (v) The result an be obtained by ombining (i) and (iii). (vi) Suppose that U = b-, and let (, s ) be an optimal soial norm that ahieves U = b-. By (iv), we obtain e =. Then h () = for all - and h ( ) =. Hene, s should have s (, ) = F in order to attain U = b-. Sine v ( ) = b- and v ( ) ³- for all -, we have v ( ) - v ( ) b/( - g) by (5). If db/( - d) <, then the inentive onstraint for -agents, d [ v ( ) - v ()] ³, annot be satisfied. Therefore, we obtain / b d/( - d).