Sustaining Cooperation in Social Exchange Networks with Incomplete Global Information

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1 51st IEEE Conference on Decision and Control December 1-13, 212. Maui, Hawaii, USA Sustaining Cooperation in Social Exchange Networks with Incomplete Global Information Jie Xu and Mihaela van der Schaar Abstract In this paper, we build a rating system that shapes the incentives of selfish users in such a way that they find it in their self-interest to cooperate with each other in social exchange networks. A rating system consists of a policy, which provides differential service for users with different ratings and a rating update rule, which rewards or penalizes users depending on whether they cooperate or not when they are asked to provide service. A key novelty of this paper is that we consider social exchange networks where users interact based on only incomplete global information about the rating distribution of the users participating in the network. We rigorously formalize the design problem of the optimal rating system which needs to be solved by the network designer as a coupled two-level optimization: users incentive problem and the efficiency maximization problem. We study the system dynamics of the social rating when users only have incomplete information. We prove that the users optimal decision problems exhibit threshold properties and hence, users need to have sufficient trust in the other network users to decide to cooperate. We also show how the optimal design depends on specific environment parameters (e.g. the benefit and cost of providing service in these networks) and which of these rating systems are sustainable when users are self-interested. I. INTRODUCTION As the web has evolved, it has become increasingly social. People turn to the web to exchange ideas, data and services, as evidenced by the popularity of sites like Wikipedia, Bit- Torrent, Yahoo Answers and Yelp. While these systems, which we refer to as social exchange networks, differ in many ways, they share a common vulnerability to selfish behavior and free-riding. In order for these sites to thrive, participants must be properly motivated to contribute. Game-theoretical approaches are widely adopted to investigate incentive issues in systems formed by self-interested users. They can be classified as either pricing mechanisms or reciprocity mechanisms. Pricing mechanisms are appropriate in some settings, but do not make sense for applications like Yahoo Answers, Wikipedia, or Yelp, where much of the appeal is that the information is free. Under a reciprocity mechanism, a user is rewarded or punished based on its behavior in the system [18][1][19]. To measure good behavior, reciprocity mechanisms frequently associate a rating or reputation score with each other in the system. Depending on how a user s rating is generated, reciprocity-based protocols can be classified as direct reciprocity mechanisms [18], or indirect reciprocity mechanisms [1]. Direct reciprocity implies that the interaction between two users is influenced The authors are with the department of Electrical Engineering, University of California, Los Angeles jiexu@ucla.edu, mihaela@ee.ucla.edu only by their history of interactions with each other, and not by their interactions with other users. Though easy to implement, direct reciprocity requires frequent interactions between two users in order to establish accurate mutual ratings. This is restrictive in systems characterized by high churn, asymmetry of interests, or infrequent interactions between any pair of users, such as most peer production systems, online labor markets, and review sites. Protocols that are based on indirect reciprocity typically assign to each user a global rating [13] based on its past interactions with all other users in the system. A differential service scheme recommends actions (e.g., share a file with this user or do not share a file with this user ) based only on the ratings of users, and not on their entire history of interactions. Much of the existing work on reputation mechanisms is either concerned with practical implementation details [8][16] or on empirical studies [3][17]. There has also been some work analytically exploring the use of reputation mechanisms to combat moral hazard in a repeated games setting [4][5], including some that does not require the presence of a trusted centralized system [2]. This work typically considers one (or a few) long-lived seller with many short-lived buyers, which is not appropriate for social exchange systems where there are many interacting users playing the role of buyer or seller or both, contributing and seeking information. To rigorously capture the impact of various strategy and protocol design choices on social exchange systems, prior work [21] proposes a framework using social norms which were originally designed to sustain cooperation in a community with a large population of individuals participating in anonymous random matching games [9][14]. We build incentive schemes for social exchange networks based on a rating system. In social exchange networks, users often have incomplete knowledge of the global information, in particular, the rating distribution of participating users. For example, users observe the ratings of a limited number of other users on the website and form (probably biased) beliefs of the rating distribution. Users beliefs are heterogeneous since the observations of various users are different. Such belief heterogeneity affects users incentives to cooperate and causes a positive fraction of users to deviate. In contrast, standard equilibrium analysis [21][9] assumes complete global information: (1) users have homogenous and accurate knowledge about the reputation distribution; and (2) users believe that all users obey the social strategy. These assumptions hold only in an equilibrium where all users follow the social strategy. However, they do not hold /12/$ IEEE 5

2 in many practical systems where users optimize according to their beliefs and the system dynamics does not evolve to an equilibrium where all users cooperate. Therefore, users incomplete global information may lead to a different system design. Other works also consider incomplete information in reputation systems [11][7], but the incomplete information is about the payoffs which differs from our consideration. We propose a class of rating systems with finite rating labels, a policy of differential services and a maximal punishment rule. We prove that in this rating system class, a simple subclass which requires only binary rating labels is superior to others. We model the users heterogeneous beliefs of the rating distribution using a Bayesian belief model, which captures the feature that the observation depends on the current true rating distribution and that more observations lead to more accurate information about the rating distribution. This allows us to investigate the impact of information on the system performance. We prove that users cooperate only if their beliefs of the rating distribution are above a certain threshold, i.e., they need to have sufficient trust in the society. Using this result, we study the system dynamics and how the emerging equilibrium depends on the system design. The rest of this paper is organized as follows. Section II describes the basic setup and the rating system. Section II characterizes the equilibrium and the system dynamics. Section IV investigates the impact of punishment on the equilibrium performance. Numerical results are provided in Section V followed by conclusions in Section VI. A. Setup II. SYSTEM MODEL We consider a social exchange network where users can post tasks and provide solutions. We utilized the widely-used continuum model (mass 1), implicitly assuming that the user population is large and static. Time is divided into discrete periods. In each period, each user requests a task to be solved and a provider (from the same population) is assigned by the system to solve the task. We assume that the system is able to find the capable providers who can solve the task and randomly assigns one such provider to solve the task. We also assume that there is no price associated with the task and hence, the provider is the only strategic player which needs to decide whether or not to exert effort to solve the task. Denote the action space A = {C, D} where C stands for cooperate and D stands for defect. Upon accepting, the provider incurs a cost c to fulfill the task while the requester receives a benefit b. We assume that b > c > such that providing the service is socially valuable. This is a simple gift-giving game (see Fig. 1) in which the dominant strategy for the provider is not to provide service. Incentives can be provided if the provider is long-lived in the system and will also become a requester in the future. We assume that the users discount the future utility by a constant rate δ (, 1). M M Requester C Fig. 2. Fig. 1. M 1 M-1 Provide service Provider Not provide service b, -c, Gift-giving game.... D C D 1 The adopted rating system. B. Differential service rating system The system designer aims to design a rating system in such a way that the users have incentives to cooperate. A rating system κ is composed of a set of ratings Θ, a service policy σ, a rating update rule τ. The rating set is an ordered finite set Θ = {, 1, 2,..., M} with length M + 1. Each user is tagged with a rating θ Θ representing its social status. A higher rating indicates a better status. The service policy σ differentiates the quality of service for users with different ratings. This is done by providing different assignment probabilities. Denote the service policy as a mapping from the rating set to a probability space σ : Θ [, 1]. For a requester with a rating θ, the system assigns a capable provider with probability σ(θ) = θ. For a provider with a rating θ, the system assigns it to a task that it is able to solve also with probability σ(θ) = θ. Let i < j, i < j. Therefore, a higher rating user participates more in the social exchange system. As an extreme case, if σ(θ) =, the users with the rating θ are ostracized from the system. The rating update rule τ updates users ratings depending on their actions. Denote it as the mapping: τ : Θ A Θ, and τ(θ, C) = max{θ + 1, M}, τ(θ, D) =. If the provider provides the service when it is assigned to a task by the system, its rating increases to the next higher level until M. If the provider does not provide the service when it is assigned to a task by the system, its rating drops to the lowest rating. Therefore, the rating system rewards cooperative behaviors and maximally punishes defective behaviors. The rating system is illustrated in Fig. 2. C. Optimal structure of the rating system In the previous subsection, we introduce a family of rating systems with design parameters M and θ, θ Θ. In this subsection, we show that there is a simple structure of the optimal design within this family of rating systems. Optimality requires two conditions to be satisfied. Given the environment parameters δ, b, c 1) if some rating systems can be sustained (i.e. all users cooperate), then the optimal rating system can also be 1 C D 51

3 sustained. 2) the optimal rating system is more efficient (having more service provisions) than other rating systems. Our first result is that the optimal design only requires two rating labels. Proposition 1. Given the environment parameters b, c, δ, the optimal rating system requires only two rating labels and 1 = 1, [, 1] under the given rating system structure. Proposition 1 shows that two differential services are sufficient to generate the optimal performance under the maximal punishment rule. Furthermore, it significantly simplifies our analysis by allowing us to only focus on the rating systems with two labels and the only design parameter is. However, we show in the following that even this single parameter needs to be carefully designed, especially when users have incomplete information. In the remainder of this paper, we consider rating systems with two labels and simply write = for brevity if no confusions are likely to occur. D. Belief heterogeneity and trust Because users are far-sighted, their decisions on whether or not to provide service should depend on how their evaluation of the system status is, and on their gain from the system in the future. We assume users have beliefs of the rating distribution of the population in the system. Since we are considering a binary label rating system, the rating distribution can be fully described by the fraction of users with θ = 1. We define this fraction as the social rating ρ s and use it in the remainder of this paper. We consider the scenarios where users have incomplete information of this social rating. Specifically, users have inaccurate and heterogenous beliefs about it. In practical systems, the accurate value of ρ s is difficult to obtain unless users have full access to all users rating information. More realistic is the case that users form beliefs of the social rating based on their observations of a limited number of ratings of other users or their limited memory of the ratings of the users with whom they have interacted in the past. We assume that a user s belief ρ of the social rating follows a conditional distribution f ρ ρs (ρ) of the true social rating ρ s. This user-specific belief can also be interpreted as the user s own trust of the social exchange system. We make two assumptions on the belief distribution function: 1) f ρ ρs (ρ) has full support on [, 1]. That is, all beliefs are possible. 2) f ρ ρs (ρ) is continuous in ρ s. The true social rating ρ s has a continuous impact on users beliefs. Note that each user s belief changes over periods since they make new observations. E. Problem formulation The objective of the system designer is to design a rating system such that the service probability (i.e. the requester is assigned with a provider by the system and is served by the provider) is maximized. We will show later that the service probability corresponds to the social rating in steady state, and hence, the objective is simply to maximize ρ s. Because the optimal rating system has only two labels, the only design parameter is. However, finding the optimal is not easy. Because user beliefs are heterogenous, not all users cooperate. This belief heterogeneity causes two opposite effects. If is large, the incentive constraints are difficult to be satisfied since the punishment might be too mild. If is small, few users have high ratings. As a result, users will loose trust in the society and again, are more likely to defect. Therefore, the design problem is actually a coupled two-level optimization problem. The inner level problem is users incentive problem. For given and ρ s, we need to understand the users incentives to cooperate. Users decisions and the rating update rule induce dynamics in the rating distribution and hence, ρ s in the future. The outer level problem is the designer s problem. The designer needs to design a policy (i.e. ) such that the service probability in the steady state ρ s is maximized. III. EQUILIBRIUM AND SYSTEM DYNAMICS A. Equilibrium Suppose initially the social rating is ρ s. It induces the belief heterogeneity f ρ ρs among users. Users optimize their actions to maximize their utilities. According to the rating update rule τ, these users actions lead to a new social rating ρ + s in the next period. If the system is in an equilibrium state, we require that the social rating remains invariant, i.e. ρ + s = ρ. Formally, the equilibrium that we are interested in is a Bayesian-Nash equilibrium as follows. Definition 1. Given δ, b, c,, let ρ s be a social rating, f ρ ρs be the induced belief distribution due to incomplete information. We say that (ρ s, f ρ ρs ) constitutes an equilibrium if 1) Each user adopts the optimal action to maximize its long-term utility based on its belief ρ induced by f ρ ρs. 2) The social rating remains invariant, ρ + s = ρ s. The system designer wants to maximize the system efficiency E(, ρ s ) which is a function of, ρ s, E(, ρ s ) = ρ s q 1 + (1 ρ s )q, (1) where q is the probability that users cooperate. Therefore, the designer s problem is to find an optimal punishment rule opt such that the efficiency is maximized in the equilibrium. opt = arg max {E(, ρ s) : (ρ s, f ρ ρs ) is an equilibrium} (2) B. Users decision problem Because only users acting as providers are strategic, in order to determine the equilibrium for a given, we investigate a representative provider s decision problem assuming that it has a belief ρ. Depending on which action the provider takes, the rating transition follows the rating update rule. The provider will cooperate if and only if the long-term utility is larger than the utility obtained by deviating. We assume that the provider only trust users with θ = 1 and does not trust 52

4 users with θ = : the user believes that when it becomes a requester and meets another provider, the assigned provider will cooperate only if the provider s rating is θ = 1. As an extreme case, if ρ =, it believes that all users will not provide service when it requests and is assigned with capable providers. Therefore, the long-run utility is, V (1) = (ρb c) + δv (1) (3) V () = (1 )δv () + ((ρb c) + δv (1)) (4) To satisfy the incentive constraint c + δv (1) > δv () (5) Based on this, we derive a condition on users beliefs. That is, users cooperate if and only if their beliefs are above a threshold. c ρ ρ t = (6) bδ(1 ) We restate this result in the following proposition. Proposition 2. A provider cooperates if and only if its belief ρ of the social rating is above a threshold ρ t. We see that users incentive problem depends on the rating system design and the punishment parameter. Because ρ t is increasing in, decreasing in δ, b/c, incentive constraints are more likely to be satisfied for harsher punishment, users who are more patient and larger relative benefits. It is also obvious that if ρ t > 1, no user chooses to cooperate and hence, the rating system fails to work. Therefore, the system designer cannot choose too mild a punishment level. C. System dynamics Due to the threshold results of users trust in the social rating, the dynamics of ρ s becomes much simpler. The social rating ρ + s in the next period can be calculated as follows, ρ + s = (ρ s + (1 ρ s ))P ρs (ρ ρ t ) (7) where P ρs (ρ ρ t ) = 1 F ρ ρs (ρ t ). In practical systems, trembling hand errors may occur, e.g. users may accidently choose the wrong actions even if they do not intent to. Moreover, the rating updating process may also be subject to small processing errors which cause high rating users to have low ratings in the next period. Therefore, we are more interested in the equilibria which are robust to small disturbances. The following proposition provides the condition for the robust equilibria. Proposition 3. An equilibrium (ρ s, f ρ ρs ) is a robust equilibrium if and only if (ρ s ) = and d (ρs) dρ s < where (ρ s ) = (ρ s + (1 ρ s ))P ρs (ρ ρ t ) ρ s Based on the continuity of (ρ s ) and () >, (1) <, it is easy to see that the existence of the robust equilibrium is guaranteed and the best robust equilibrium is max{ρ s : (ρ s ) = }. With this, we propose a simple algorithm to compute the best robust equilibrium in Algorithm 1. The algorithm tries to find the solution of (ρ s ) = that is the closest to 1. If the step size φ is small enough, it is guaranteed ALGORITHM 1: Robust Equilibrium Computation Input: System parameters δ, b, c,. step size φ, error confidence ɛ Output: The social rating ρ s for the best robust equilibrium. ρ s = 1 ɛ. ; repeat Compute P ρs (ρ ρ t); (ρ s) = (ρ s + (1 ρ s))p ρs (ρ ρ t) ρ s; ρ s ρ s + φ (ρ s) until (ρ s) ɛ ; to find the best robust equilibrium within 1/(ɛφ) iterations due to the continuity of (ρ s ). Proposition 4. Given δ, b, c,, if satisfies (??), there exists at least one robust equilibrium with ρ s (, 1). Moreover, the best robust equilibrium is max{ρ s : (ρ s ) = }. In the steady state, the system efficiency (ρ s + (1 ρ s ))P ρs (ρ ρ t ) equals to the social rating ρ s since (ρ s ) =. Therefore, the design objective can simply be to maximize the social rating ρ s in the steady state. IV. OPTIMAL PUNISHMENT The threshold social rating belief ρ t is determined by the punishment. The harsher the punishment, the lower this threshold is and hence, incentive constraints are more likely to be satisfied. On the other hand, harsher punishments induce a lower social rating (since less agents are able to participate in the system to reestablish their ratings) which leads to fewer users have a belief that is higher than ρ t. Therefore, the tension between lowering the belief threshold and maintaining a high enough social rating needs to be considered. A. Belief distribution function We first introduce a family of belief distribution functions to model users belief heterogeneity due to incomplete information. Different models of belief heterogeneity will lead to different optimal designs. The belief functions that we propose here are the posterior beliefs based on observations. This is based on the fact that users trust in the society depends on their personal experiences. Suppose in each period, a user is able to observe K other users ratings and k [, K] of them have high ratings, then its belief ρ should be a function of K and k. In Bayesian statistics, the beta distribution B(k + 1, K k + 1) can be seen as the posterior probability of the parameter of a binomial distribution after observing k successes and K k failures. Therefore, we use the beta distribution to model the user s belief ρ when it observes k high ratings out of K total ratings. f k,k (ρ) = B(k + 1, K k + 1) Γ(K + 2) = Γ(k + 1)Γ(K k + 1) ρk (1 ρ) K k, where Γ( ) is the gamma function. Note that k is also a random variable, we need to take the expectation before getting the belief distribution function. (8) 53

5 Because k follows a binomial distribution with parameter ρ s, the belief distribution function thus also depends on ρ s, K ( ) K f ρ ρs (ρ) = E(f k,k (ρ)) = ρ k k s(1 ρ s ) K k f k,k (ρ) k= Note that in such a model, user s belief distribution is continuous and parameterized by K and ρ s. We make a few discussions about this particular belief distribution function. 1) If K =, f ρ ρs (ρ) = 1. Users have uniform beliefs of the social rating since they have no information. 2) If K, f ρ ρs (ρ) I(ρ ρ s ) where I( ) is the indicator function. Users asymptotically acquire infinite observations and they have almost accurate knowledge of the social rating. In this section, we will use this specific belief distribution function in our analysis and investigate the impact of incomplete information on the optimal design and performance. We regard this belief distribution as one of the reasonable characterization of users beliefs in practice but there could also be other forms of belief functions. First we establish an upper bound on the social rating (and hence, the system efficiency) in the robust equilibrium and see how the observation granularity affects the performance. Proposition 5. Given δ, λ, b, c, K, the robust equilibrium ρ s (1 ρ is bounded as follows, ρ s < K+1 t ) 1 (1 )(1 ρ K+1 t ). The above result shows that the upper bound depends on the observation granularity. If the system designer wants to achieve higher efficiency, it is necessary that users are able to make more observations to acquire more accurate social rating information. (Though having more observations may not be the sufficient condition.) In some systems, the number of observations can be designed by the designer. For example, the website designer may only allow users to access the ratings of a limited number of other users due to privacy concerns. Hence, the tradeoff between efficiency and privacy needs to be carefully considered. In this paper, we assume that the number of observations is exogenously determined. Determining the optimal punishment design for the general cases (i.e. for a general K) is difficult. Therefore, we analytically investigate a specific case where K = in the following and leave the general cases to numerical methods. B. Example: K = (constant belief distribution) We consider the simplest case K =, i.e. users belief distribution is uniform and independent of the exact social rating ρ s since they do not make observations. Therefore, f ρ ρs = 1 and F ρ ρs (ρ t ) = ρ t. Proposition 6. Given δ, b, c, if K =, the optimal design is = 1 2 (1 c bδ ), and the associated efficiency is ρ s = ( ) 2. bδ c bδ+c In this simple case, the optimal is increasing in both δ and b/c. Therefore, if the users are more patient or the relative benefits are larger, the system designer can use a 54 social rating ρ s K = K = 1 K = 2 K = Fig. 3. Optimal design and system efficiency for various K. milder punishment to achieve the optimal performance. This is consistent with our intuition since it is easier for users to cooperate for larger δ and b/c. As a result, the optimal efficiency is also increasing in δ and b/c. However, the relative benefits have larger impact on the optimal efficiency in the limiting sense. As b/c (, ρ ) s 1 while δ 1, 2. b c the optimal efficiency is at most b+c C. General cases The exact optimal design for the general cases is analytically intractable. However, based on the result for the special case K =, we show some properties of the optimal performance for the general cases. First, we study the performance for K = 1. Proposition 7. If bδ c > , then for K = 1, the optimal ( ) 2. efficiency ρ s is lower bounded by ρ bδ c s > bδ+c The above proposition provides a condition that having more information (K = 1) leads to higher efficiency than no information (K = ) using the proposed belief distribution function. We extend the above result to more general K and provide a sufficient condition when having more observations are better. Proposition 8. For given δ, b, c, K, the optimal design is and the corresponding belief threshold is ρ t, the optimal efficiency is ρ s. Suppose another observation granularity K > K is used, the optimal efficiency is increased if P ρ s (ρ ρ t K ) > P ρ s (ρ ρ t K) (9) V. SIMULATIONS In this section, we conduct simulations to show the performance of the rating system and the impact of the incomplete information on the design and the system performance. In Fig. 3, the optimal design is found for various observation capabilities. The environment parameters are fixed at b = 1, c = 1, δ =.9. Therefore, the maximal is.89 above which no cooperation can be sustained. For a given observation granularity K, the system performance depends

6 optimal optimal K = K = 1 K = K = K = 1 K = b/c δ Fig. 4. The impact of benefit-to-cost ratio on the optimal design. Fig. 5. The impact of discount factor on the optimal design. on the choices of. However, in all cases, the system is inefficient for too large or too small values of. This is because, if is too large (close to.89), incentive constraints to cooperate can hardly be satisfied; if is too small, punishments are too harsh to allow users to participate in the system. The exact optimal value of depends on how much users are able to observe and how accurate the incomplete information is. As K increases, the optimal efficiency also increase. However, we also note that having more observations does not necessarily lead to higher efficiency for all. Importantly, the efficiency curve approaches the one for K =, i.e. complete information, as K increases. When users have sufficient observations and hence, they have accurate enough information about the system, full efficiency can be achieved in the limit for any <.89. In this set of simulations, we investigate the impact of environment parameters on the system design and performance. In Fig. 4 and Fig. 5, the optimal punishment and the corresponding optimal efficiency are illustrated, respectively, for various benefit-to-cost ratios (Fix λ =.9). As the benefitto-cost ratio increases, the optimal efficiency also increases. This is because larger benefit-to-cost ratio gives more incentives for users to cooperate. However, the efficiency may be lower even when users make more observations. The optimal design does not have the monotonicity property as the system performance. The optimal needs to depend on the specific observation capability and environment parameters. VI. CONCLUSION In this paper, we designed the optimal rating system for social exchange systems where users have incomplete information of the social status. We first showed the optimal rating system exhibits a simple structure where only binary rating labels are required. Hence the design parameter set reduces to a single parameter, i.e. the punishment probability of users with low ratings. Next, we formalized the design problem as a coupled two-level optimization problem. The Baysian-Nash equilibrium is used as the solution concept because users optimize their action based on their own beliefs. We showed that users cooperate only when they have sufficient trust in the system (i.e. believe that sufficient users are cooperating). Then we proposed a family of belief distribution functions to model the belief heterogeneity and study the optimal punishment design problem analytically and numerically. We found that the optimal punishment should not be too harsh or too mild and needs to be carefully designed. Moreover, more observations do not necessarily lead to higher system efficiency. A sufficient condition is provided for when more observations are beneficial. REFERENCES [1] R. Alexander, The Biology of Moral Systems (Foundations of Human Behavior), Aldine Transaction, [2] S. Ba and P. A. Pavlou, Evidence of the effect of trust building technology in electronic markets: price premiums and buyer behavior, MIS Q. 26, , 22. [3] C. Dellarocas, Reputation mechanism design in online trading environments with pure moral hazard, Information Systems Research 16, 29-23, 25. [4] C. Dellarocas, How often should reputation mechanism update a trader s reputation profile, Information Systems Research, vol. 17, no. 3, , 26. [5] D. Fudenberg and E. Maskin, The folk theorem in repeated games with discounting or with incomplete information, Econometrica 54, 3, , [6] M. Kandori, Social norms and community enforcement, The Review of Economic Studies 59, 1, [7] D. M. Kreps and R. Wilson, Reputation and imperfect information, Journal of Economic Theory 27, 2, , [8] H. Masum and Y. Zhang, Manifesto for the reputation society, First Monday 9(7), 24. [9] M. Okuno-Fujiwara and A. Postlewaite, Social norms and random matching games, Games and Economic Behavior 9, 1, 79-19, [1] A. Ravoaja and A. Emmanuelle, Storm: A secure overlay for p2p reputation management, Proceedings of the First International Conference on Self-Adaptive and Self-Organizing Systems (SASO), , 27. [11] P. Resnick and R. Zeckhauser, Trust among strangers in internet transactions: Empirical analysis of ebay s reputation system, The Economics of the Internet and E-commerce 11, , 22. [12] R. L. Trivers, The evolution of reciprocal altruism, The Quarterly Review of Biology 46, 1, 35-57, [13] J. Xu, W. R. Zame and M. van der Schaar, Token Economy for Online Exchange Systems, to appear in AAMAS 12, 212. [14] G. Zacharia, A. Moukas and P. Maes, Collaborative reputation mechanisms for electronic marketplaces, Decision Support Systems 29, 4, , 2. [15] Y. Zhang, M. van der Schaar, Reputation-based incentive protocols in crowdsourcing applications, IEEE Infocom,