A Appendix

# On the Payoff Mechanisms in Peer-Assisted Services with Multiple Content Providers: Rationality and Fairness

## Abstract

This paper studies an incentive structure for cooperation and its stability in peer-assisted services when there exist multiple content providers, using a coalition game theoretic approach. We first consider a generalized coalition structure consisting of multiple providers with many assisting peers, where peers assist providers to reduce the operational cost in content distribution. To distribute the profit from cost reduction to players (i.e., providers and peers), we then establish a generalized formula for individual payoffs when a “Shapley-like” payoff mechanism is adopted. We show that the grand coalition is unstable, even when the operational cost functions are concave, which is in sharp contrast to the recently studied case of a single provider where the grand coalition is stable. We also show that irrespective of stability of the grand coalition, there always exist coalition structures which are not convergent to the grand coalition under a dynamic among coalition structures. Our results give us an incontestable fact that a provider does not tend to cooperate with other providers in peer-assisted services, and be separated from them. Three facets of the noncooperative (selfish) providers are illustrated; (i) underpaid peers, (ii) service monopoly, and (iii) oscillatory coalition structure. Lastly, we propose a stable payoff mechanism which improves fairness of profit-sharing by regulating the selfishness of the players as well as grants the content providers a limited right of realistic bargaining. Our study opens many new questions such as realistic and efficient incentive structures and the tradeoffs between fairness and individual providers’ competition in peer-assisted services.

## 1 Introduction

### 1.1 Motivation

The Internet is becoming more content-oriented, and the need of cost-effective and scalable distribution of contents has become the central role of the Internet. Uncoordinated peer-to-peer (P2P) systems, e.g., BitTorrent, have been successful in distributing contents, but the rights of the content owners are not protected well, and most of the P2P contents are in fact illegal. In its response, a new type of service, called peer-assisted service, has received significant attention these days. In peer-assisted services, users commit a part of their resources to assist content providers in content distribution with objective of enjoying both scalability/efficiency in P2P systems and controllability in client-server systems. Examples of application of peer-assisted services include nano data center [1] and IPTV [2], where high potential of operational cost reduction was observed. For instance, there are now 1.8 million IPTV subscribers in South Korea, and the financial sectors forecast that by 2014 the IPTV subscribers is expected to be 106 million [3]. However, it is clear that most users will not just “donate” their resources to content providers. Thus, the key factor to the success of peer-assisted services is how to (economically) incentivize users to commit their valuable resources and participate in the service.

One of nice mathematical tools to study incentive-compatibility of peer-assisted services is the coalition game theory which covers how payoffs should be distributed and whether such a payoff scheme can be executed by rational individuals or not. In peer-assisted services, the “symbiosis” between providers and peers are sustained when (i) the offered payoff scheme guarantees fair assessment of players’ contribution under a provider-peer coalition and (ii) each individual has no incentive to exit from the coalition. In the coalition game theory, the notions of Shapley value and the core have been popularly applied to address (i) and (ii), respectively, when the entire players cooperate, referred to as the grand coalition. A recent paper by Misra et al.[4] demonstrates that the Shapley value approach is a promising payoff mechanism to provide right incentives for cooperation in a single-provider peer-assisted service.

However, in practice, the Internet consists of multiple content providers, even if only giant providers are counted. In the multi-provider setting, users and providers are coupled in a more complex manner, thus the model becomes much more challenging and even the cooperative game theoretic framework itself is unclear, e.g., definition of the worth of a coalition. Also, the results and their implications in the multi-provider setting may experience drastic changes, compared to the single-provider case.

The grand coalition is expected to be the “best” coalition in the peer-assisted service with multiple providers in that it provides the highest aggregate payoff. To illustrate, see an example in Fig. 1 with two providers (Google TV and iTunes) and a large number of peers. Consider two cooperation types: (i) separated, where there exists a fixed partition of peers for each provider, and (ii) coalescent, where each peer is possible to assist any provider. In the separated case, a candidate payoff scheme is based on the Shapley value in each disconnected coalition. In the coalescent case, the Shapley value is also a candidate payoff scheme after a worth function of the grand coalition is defined, where a reasonable worth function1 can be the total optimal profit, maximized over all combinations of peer partitions to each provider. Consequently, the total payoff for the coalescent case exceeds that for the separated case, unless the two partitions of both cases are equivalent. Shapley value is defined by a few agreeable axioms, one of which is efficiency2, meaning that the every cent of coalition worth is distributed to players. Since smaller worth is shared out among players in the separated case, at least one individual is underpaid as compared with the coalescent case. Thus, providers and users are recommended to form the grand coalition and be paid off based on the Shapley values.

However, it is still questionable whether peers are willing to stay in the grand coalition and thus the consequent Shapley-value based payoff mechanism is desirable in the multi-provider setting. In this paper, we anatomize incentive structures in peer-assisted services with multiple content providers and focus on stability issues from two different angles: stability at equilibrium of Shapley value and convergence to the equilibrium. We show that the Shapley payoff scheme may lead to unstable coalition structure, and propose a different notion of payoff distribution scheme, value, under which peers and providers stay in the stable coalition as well as better fairness is guaranteed.

### 1.2 Related Work

The research on incentive structure in the P2P systems (e.g., BitTorrent) has been studied extensively. To incapacitate free-riders in P2P systems, who only download contents but upload nothing, from behaving selfishly, a number of incentive mechanisms suitable for distribution of copy-free contents have been proposed (See [5] and references therein), using game theoretic approaches. Alternative approaches to exploit the potential of the P2P systems for reducing the distribution (or operational) costs of the copyrighted contents have been recently adopted by [4, 1]. To the best of our knowledge, the work by Misra et al.[4] is the first to study the profit-sharing mechanism (payoff mechanism) of peer-assisted services.

Coalition game theory has been applied to model diverse networking behaviors, where the main focus in most cases (e.g., [4]) was to study the stability of a specific equilibrium i.e., the grand coalition in connection with the notion of core. Recently, Saad et al.[6, 7], discussed the stability and dynamics of endogenous formation of general coalition structures. In particular, [7] introduced a coalition game model for self-organizing agents (e.g., unmanned aerial vehicles) collecting data from arbitrarily located tasks in wireless networks and proved the stability of the proposed algorithm by using hedonic preference (and dominance). In this paper, we use the stability notion by Hart and Kurz [8] (see also [9]) to study the dynamics of coalition structures in peer-assisted services. The stability notion in [8] is based on the preferences of any arbitrary coalition while the hedonic coalition games are based on the preferences of individuals. Other subtle differences are described in [10].

### 1.3 Main Contributions and Organization

We summarize our main contributions as follows:

1. Following the preliminaries in Section 2, in Section 3, we describe and propose the cooperative game theoretic framework of the peer-assisted service with multiple providers. After defining a worth function that is provably the unique feasible worth function satisfying two essential properties, i.e., feasibility and superadditivity of a coalition game, we provide a closed-form formula of the Shapley value for a general coalition with multiple providers and peers, where we take a fluid-limit approximation for mathematical tractability. This is a non-trivial generalization of the Shapley value for the single-provider case in [4]. In fact, our formula in Theorem 3.2 establishes the general Shapley value for distinguished multiple atomic players and infinitesimal players in the context of the Aumann-Shapley (A-S) prices [11] in coalition game theory.

2. In Section 4, we discuss in various ways that the Shapley payoff regime cannot incentivize rational players to form the grand coalition, implying that fair profit-sharing and opportunism of players cannot stand together. First, we prove that the Shapley value for the multiple-provider case is not in the core under mild conditions, e.g., each provider’s cost function is concave. This is in stark contrast to the single-provider case where the concave cost function stabilizes the equilibrium. Second, we study the dynamic formation of coalitions in peer-assisted services by introducing the notion of stability defined by the seminal work of Hart and Kurz [8]. Finally, we show that, if we adopt a Shapley-like payoff mechanism, called Aumann-Drèze value, irrespective of stability of the grand coalition, there always exist initial states which do not converge to the grand coalition.

3. In Section 5, we present three examples stating the problems of the non-cooperative peer-assisted service: (i) the peers are underpaid compared to their Shapley payoffs, (ii) a provider paying the highest dividend to peers monopolizes all peers, and (iii) Shapley value for each coalition gives rise to an oscillatory behavior of coalition structures. These examples suggest that the system with the separated providers may be even unstable as well as unfair in a peer-assisted service market.

4. In Section 6, as a partial solution to the problems of Shapley-like payoffs (i.e., Shapley and Aumann-Drèze), we propose an alternative payoff scheme, called value [12]. This payoff mechanism is relatively fair in the sense that players, at the least, apportion the difference between the coalition worth and the sum of their fair shares, i.e., Shapley payoffs, and stabilizes the whole system. It is also practical in the sense that providers are granted a limited right of bargaining. That is, a provider may award an extra bonus to peers by cutting her dividend, competing with other providers in a fair way. More importantly, we show that authorities can effectively avoid unjust rivalries between providers by implementing a simplistic measure.

After presenting a practical example of peer-assisted services with multiple providers in delay-tolerant networks in Section 7, we conclude this paper.

## 2 Preliminaries

Since this paper investigates a multi-provider case, where a peer can choose any provider to assist, we start this section by defining a coalition game with a peer partition (i.e., a coalition structure) and introducing the payoff mechanism thereof.

### 2.1 Game with Coalition Structure

A game with coalition structure is a triple where is a player set and ( is the set of all subsets of ) is a worth function, . is called the worth of a coalition . is called a coalition structure for ; it is a partition of where denotes the coalition containing player . For your reference, a coalition structure can be regarded as a set of disjoint coalitions. The grand coalition is the partition . For instance3, a partition of is and the grand coalition is . is the set of all partitions of . For notational simplicity, a game without coalition structure is denoted by . A value of player is an operator that assigns a payoff to player . We define for all .

To conduct the equilibrium analysis of coalition games, the notion of core has been extensively used to study the stability of grand coalition :

###### Definition (Core)

The core of a game is defined by:

 Misplaced \cr

If a payoff vector lies in the core, no player in has an incentive to split off to form another coalition because the worth of the coalition , , is no more than the payoff sum . Note that the definition of the core hypothesizes that the grand coalition is already formed ex-ante. We can see the core as an analog of Nash equilibrium from noncooperative games. Precisely speaking, it should be viewed as an analog of strong Nash equilibrium where no arbitrary coalition of players can create worth which is larger than what they receive in the grand coalition. If a payoff vector lies in the core, then the grand coalition is stable with respect to any collusion to break the grand coalition.

### 2.2 Shapley Value and Aumann-Drèze Value

On the premise that the player set is not partitioned, i.e., , the Shapley value, denoted by (not ), is popularly used as a fair distribution of the grand coalition’s worth to individual players, defined by:

 φi(N,v)=∑S⊆N∖{i}|S|!(|N|−|S|−1)!|N|!(v(S∪{i})−v(S)). (1)

Shapley [13] gives the following interpretation: “(i) Starting with a single member, the coalition adds one player at a time until everybody has been admitted. (ii) The order in which players are to join is determined by chance, with all arrangements equally probable. (iii) Each player, on his admission, demands and is promised the amount which his adherence contributes to the value of the coalition.” The Shapley value quantifies the above that is axiomatized (see Section 2.3) and has been treated as a worth distribution scheme. The beauty of the Shapley value lies in that the payoff “summarizes” in one number all the possibilities of each player’s contribution in every coalition structure.

Given a coalition structure , one can obtain the Aumann-Drèze value (A-D value) [14] of player , also denoted by , by taking , which is the coalition containing player , to be the player set and by computing the Shapley value of player of the reduced game . It is easy to see that the A-D value can be construed as a direct extension of the Shapley value to a game with coalition structure. Note that both Shapley value and A-D value are denoted by because the only difference is the underlying coalition structure .

### 2.3 Axiomatic Characterizations of Values

We provide here an axiomatic characterization of the Shapley value [13].

.

###### Axiom (Coalition Restricted Symmetry, CS)

If and for all , then .

For all coalition functions , and , .

###### Axiom (Null Player, NP)

If for all , then .

Recall that the basic premise of the Shapley value is that the player set is not partitioned, i.e., . It is well-known [12, 13] that the Shapley value, defined in (1), is uniquely characterized by CE, CS, ADD and NP for . The A-D value is also uniquely characterized by CE, CS, ADD and NP (Axioms 2.3-2.3), but in this case for arbitrary coalition structure [14]. In the literature, e.g., [6, 15], the A-D value has been used to analyze the static games where a coalition structure is exogenously given.

###### Definition (Coalition Independent, CI)

If , and , then .

From the definition of the A-D value, the payoff of player in coalition is affected neither by the player set nor by coalitions , . Note that only contains the player . Thus, it is easy to prove that the A-D value is coalition independent. From CI of the A-D value, in order to decide the payoffs of a game with general coalition structure , it suffices to decide the payoffs of players within each coalition, say , without considering other coalitions , . In other words, once we decide the payoffs of a coalition , the payoffs remain unchanged even though other coalitions, , , vary. Thus, for any given coalition structure , any coalition is just two-fold in terms of the number of providers in : (i) one provider or (ii) two or more providers, as depicted in Fig. 1.

Yet another reason why CI attracts our attention is that it enables us to define the stability of a game with coalition structure in the following simplistic way:

###### Definition (Stable Coalition Structure [8])

We say that a coalition structure blocks , where , , with respect to if and only if there exists some such that for all . In this case, we also say that blocks . If there does not exist any which blocks , is called stable.

Due to CI of the A-D value, all stability notions defined by the seminal work of Hart and Kurz [8] coincide with the above simplistic definition, as discussed by Tutic [9]. Definition 2.3 can be intuitively interpreted that, if there exists any subset of players who improve their payoffs away from the current coalition structure, they will form a new coalition . In other words, if a coalition structure has any blocking coalition , some rational players will break to increase their payoffs. The basic premise here is that players are not clairvoyant, i.e., they are interested only in improving their instant payoffs in a myopic way. If a payoff vector lies in the core, the grand coalition is stable in the sense of Definition 2.3, but the converse is not necessarily true (see Fig. 2).

### 2.4 Comparison with Other Values

In a particular category of games, called voting games or simple games, Banzhaf value as well as the Shapley value (also known as Shapley-Shubik index in this context) has been used in the literature (See, e.g., [16] and references therein). While the Shapley value has been extensively studied in many papers, there are no similar results for the Banzhaf value. For instance, the Shapley value is proven to lie in the core for a special type of games, called convex games, whereas there is no equivalent result for the Banzhaf value. Moreover, the Banzhaf value violates the efficiency axiom CE in Section 2.3 for a certain coalition structure , leading to inefficient sharing of the grand coalition worth.

As compared with Aumann-Drèze value, a new value, referred to as Owen value (See, e.g., [15, Chapter 8.8] or [17, Chapter XII]) has emerged based on an alternative viewpoint on coalition, where a coalition forms not to share the coalition worth, but only to maximize their bargaining power with regard to division of the worth of the grand coalition. In other words, players form a labor union (coalition) to obtain a better bargaining position leading to a larger payoff, implying that the coalition efficiency axiom CE is also violated. A delicate premise of this approach is that players must form the grand coalition, the worth of which is in fact the largest worth in superadditive games (See Definition 3.1), and bargain with each other at the same time. Also, in the context of P2P systems, whether it is more reasonable to nullify CE so that a portion of a worth of a coalition (peers and providers) becomes transferrable to other coalitions , , remains an open economic question.

## 3 Coalition Game in Peer-Assisted Services

In this section, we first define a coalition game in a peer-assisted service with multiple content providers by classifying the types of coalition structures as separated, where a coalition includes only one provider, and coalescent, where a coalition is allowed to include more than one providers (see Fig. 1). To define the coalition game, we will define a worth function of an arbitrary coalition for such two cases.

### 3.1 Worth Function in Peer-Assisted Services

Assume that players are divided into two sets, the set of content providers , and the set of peers , i.e., . We also assume that the peers are homogeneous, e.g., the same computing powers, disk cache sizes, and upload bandwidths. Later, we discuss that our results can be readily extended to nonhomogeneous peers. The set of peers assisting providers is denoted by where , i.e., the fraction of assisting peers. We define the worth of a coalition to be the amount of cost reduction due to cooperative distribution of the contents by the players in in both separated and coalescent cases.

Separated case: Denote by the operational cost of a provider when the coalition consists of a single provider and assisting peers. Since the operational cost cannot be negative and may decrease with the number of assisting peers, we assume the following to simplify the exposition:

• Assumption: is non-increasing in for all .

Note that from the homogeneity assumption of peers, the cost function depends only on the fraction of assisting peers. Then, we define the worth function for a coalition having a single provider as:

 ^v(S):=Ωηp(0)−Ωηp(x(S)) (2)

where corresponds to the cost when there are no assisting peers. For a coalition with no provider, we simply have For notational simplicity, is henceforth denoted by unless confusion arises.

Coalescent case: In contrast to the separated case, where a coalition includes a single provider, the worth for the coalescent case is not clear yet, since depending on which peers assist which providers the amount of cost reduction may differ. One of reasonable definitions would be the maximum worth out of all peer partitions, i.e., the worth for the coalescent case is defined by: for a coalition with at least two providers,

 v(S):=max{∑C∈P^v(C)∣∣P∈P(S)  s.t. |Z∩C|=1, ∀C∈P}, (3)

and for a coalition with at most one provider. The definition above implies that we view a coalition containing more than one provider as the most productive coalition whose worth is maximized by choosing the optimal partition among all possible partitions of . Note that (3) is consistent with the definition (2) for , i.e., for .

Five remarks are in order. First, as opposed to [4] where ( is the subscription fee paid by any peer), we simply assume that . Note that, as discussed in [15, Chapter 2.2.1], it is no loss of generality to assume that, initially, each provider has earned no money. In our context, this means that it does not matter how much fraction of peers is subscribing to each provider because each peer has already paid the subscription fee to providers ex-ante.

Second, may not be decreasing because, for example, electricity expense of the computers and the maintenance cost of the hard disks of peers may exceed the cost reduction due to peers’ assistance in content distribution, e.g., Annualized Failure Rate (AFR) of hard disk drives is over 8.6% for three-year old ones [18].

Third, the worth function in peer-assisted services can reflect the diversity of peers. It is not difficult to extend our result to the case where peers belong to distinct classes. For example, peers may be distinguished by different upload bandwidths and different hard disk cache sizes. A point at issue for the multiple provider case is whether peers who are not subscribing to the content of a provider may be allowed to assist the provider or not. On the assumption that the content is ciphered and not decipherable by the peers who do not know its password which is given only to the subscribers, providers will allow those peers to assist the content distribution. Otherwise, we can easily reflect this issue by dividing the peers into a number of classes where each class is a set of peers subscribing to a certain content.

Fourth, it should be clearly understood that our worth function (3) does not encompass more than just the peer-partition optimization. That is, we speculate that cooperation among providers might lead to further expenses cut by optimizing their network resources. We recognize the lack of this ‘added bonus’ to be the major weakness in our model.

Lastly, it should be noted that the worth function in (3) is selected in order to satisfy two properties. First of all, it follows from the definition of in (3) that no other coalition function can be greater than , i.e., because is the total cost reduction that is maximized over all possible peer partitions to each provider.

###### Definition (Feasibility)

For all worth function , we have for all .

The second property, superadditivity, is one of the most elementary properties, which ensures that the core is nonempty by appealing to Bondareva-Shapley Theorem [15, Theorem 3.1.4].

A worth is superadditive if .

The following lemma holds by the fact that a feasible worth function cannot be greater than (3), i.e., the largest worth.

###### Lemma

When the worth for the separated case is given by (2), for the coalescent case, there exists a unique worth function that is both superadditive and feasible, given by (3).

###### Proof

Suppose we have a superadditive worth . Firstly, it follows directly from the assumption (the worth function for the separate case is (2)) that if includes one provider. (i) Feasibility: It follows from the definition of feasibility that we have because is the maximum over all possible partitions . (ii) Superadditivity: In the meantime, since is superadditive, it must satisfy for all disjoint , . This in turn implies for all such that . The right-hand side should coincide with for some such that for all (See (3)), where is the peer partition which maximizes . Therefore, we have . Combining this with uniquely determines .

In light of this lemma, we can restate that our objective in this paper is to analyze the incentive structure of peer-assisted services when the worth of coalition is feasible and superadditive. This objective in turn implies the form of worth function in (3).

### 3.2 Fluid Aumann-Drèze Value for Multi-Provider Coalitions

So far we have defined the worth of coalitions. Now let us distribute the worth to the players for a given coalition structure . Recall that the payoffs of players in a coalition are independent from other coalitions by the definition of A-D payoff. Pick a coalition without loss of generality, and denote the set of providers in by . With slight notational abuse, the set of peers assisting is denoted by . Once we find the A-D payoff for a coalition consisting of arbitrary provider set and assisting peer set , the payoffs for the separated and coalescent cases in Fig. 1 follow from the substitutions and , respectively. In light of our discussion in Section 2.2, it is more reasonable to call a Shapley-like payoff mechanism ‘A-D payoff’ and ‘Shapley payoff’ respectively for the partitioned and non-partitioned games and 4.

Fluid Limit: We adopt the limit axioms for a large population of users to overcome the computational hardness of the A-D payoffs:

 limη→∞˜Ωηp(⋅)=˜Ωp(⋅)   where ˜Ωηp(⋅)=1ηΩηp(⋅) (4)

which is the asymptotic operational cost per peer in the system with a large number of peers. We drop superscript from notations to denote their limits as . From the assumption , we have . To avoid trivial cases, we also assume is not constant in the interval for any . We also introduce the payoff of each provider per user, defined as . We now derive the fluid limit equations of the payoffs, shown in Fig. 3, which can be obtained as . The proof of the following theorem is given in Appendix A.1.

###### Theorem (A-D Payoff for Multiple Providers)

As , the A-D payoffs of providers and peers under an arbitrary coalition converge to (FluidAD1) in Fig. 3 where and . Note that .

The following corollaries are immediate as special cases of Theorem 3.2, which we will use in Section 5.

###### Corollary (A-D Payoff for Single Provider)

As , the A-D payoffs of providers and peers who belong to a single-provider coalition, i.e., , converge to:

 ⎧⎪⎨⎪⎩˜φ{p}p(x)=˜Ωp(0)−∫10M{p}Ω(ux)du,˜φ{p}n(x)=−∫10udM{p}Ωdx(ux)du,for n∈¯H. (5)

###### Corollary (A-D Payoff for Dual Providers)

As , the A-D payoffs of providers and peers who belong to a dual-provider coalition, i.e., , converge to (FluidAD2).

Note that our A-D payoff formula in Theorem 3.2 generalizes the formula in Misra et al.[4, Theorem 4.3] (i.e., ). It also establishes the A-D values for distinguished multiple atomic players (the providers) and infinitesimal players (the peers), in the context of the Aumann-Shapley (A-S) prices [11] in coalition game theory.

Our formula for the peers is interpreted as follows: Take the second line of (FluidAD2) as an example. Recall the definition of the Shapley value (1). The payoff of peer is the marginal cost reduction that is averaged over all equally probable arrangements, i.e., the orders of players. It is also implied by (1) that the expectation of the marginal cost is computed under the assumption that the events and for are equally probable, i.e., . Therefore, in our context of infinite player game in Theorem 3.2, for every values of along the interval , the subset contains fraction of the peers. More importantly, the probability that each provider is a member of is simply because the size of peers in , , is infinite as so that the size of is not affected by whether a provider belongs to or not. Therefore, the marginal cost reduction of each peer on the condition that both providers are contained in becomes . Likewise, the marginal cost reduction of each peer on the condition that only one provider is in the coalition is .

## 4 Instability of the Grand Coalition

In this section, we study the stability of the grand coalition to see if rational players are willing to form the grand coalition, only under which they can be paid their respective fair Shapley payoffs. The key message of this section is that the rational behavior of the providers makes the Shapley value approach unworkable because the major premise of the Shapley value, the grand coalition, is not formed in the multi-provider games.

### 4.1 Stability of the Grand Coalition

Guaranteeing the stability of a payoff vector has been an important topic in coalition game theory. For the single-provider case, , it was shown in [4, Theorem 4.2] that, if the cost function is decreasing and concave, the Shapley incentive structure lies in the core of the game. What if for ? Is the grand coalition stable for the multi-provider case? Prior to addressing this question, we first define the following:

###### Definition (Noncontributing Provider)

A provider is called noncontributing if .

To understand this better, note that the above expression is equivalent to the following:

 ∑i∈Z˜Ωi(0)−MZΩ(1)=∑i∈Z∖{p}˜Ωi(0)−MZ∖{p}Ω(1) (6)

which implies that there is no difference in the total cost reduction, irrespective of whether the provider is in the provider set or not. Interestingly, if all cost functions are concave, there exists at least one noncontributing provider.

###### Lemma

Suppose . If is concave for all , there exist noncontributing providers.

To prove this, recall the definition of :

 MZΩ(x)=miny∈Y(x)∑i∈Z˜Ωi(yi)
 where Y(x):={(y1,⋯,y|Z|) ∣∣ ∑i∈Zyi≤x, yi≥0}.

Since the summation of concave functions is concave and the minimum of a concave function over a convex feasible region is an extreme point of as shown in [19, Theorem 3.4.7], we can see that the solutions of the above minimization are the extreme points of , which in turn imply for providers in . Note that the condition is necessary here.

We are ready to state the following theorem, a direct consequence of Theorem 3.2. Its proof is in Appendix A.2.

###### Theorem (Shapley Payoff Not in the Core)

If there exists a noncontributing provider, the Shapley payoff for the game does not lie in the core.

It follows from Lemma 4.1 that, if all operational cost functions are concave and , the Shapley payoff does not lie in the core. This result appears to be in good agreement with our usual intuition. If there is a provider who does not contribute to the coalition at all in the sense of (6) and is still being paid due to her potential for imaginary contribution assessed by the Shapley formula (1), which is not actually exploited in the current coalition, other players may improve their payoff sum by expelling the noncontributing provider.

The condition plays an essential role in the theorem. For , the concavity of the cost functions leads to the Shapley value not lying in the core, whereas, for the case , the concavity of the cost function is proven to make the Shapley incentive structure lie in the core [4, Theorem 4.2].

### 4.2 Convergence to the Grand Coalition

The notion of the core lends itself to the stability analysis of the grand coalition on the assumption that the players are already in the equilibrium, i.e., the grand coalition. However, Theorem 4.1 still leaves further questions unanswered. In particular, for the non-concave cost functions, it is unclear if the Shapley value is not in the core, which is still an open problem. We rather argue here that, whether the Shapley value lies in the core or not, the grand coalition is unlikely to occur by showing that the grand coalition is not a global attractor under some conditions.

To study the convergence of a game with coalition structure to the grand coalition, let us recall Definition 2.3. It is interesting that, though the notion of stability was not used in [4], one main argument of this work was that the system with one provider would converge to a full sharing mode, i.e., the grand coalition, hinting the importance of the following convergence result with multiple providers. The proof of the following theorem is given in Appendix A.3.

###### Theorem (A-D Payoff Doesn’t Lead to Grand Coalition)

Suppose and is not constant in the interval for any where . The following holds for all and .

• The A-D payoff of provider in coalition is larger than that in all coalition for .

• The A-D payoff of peer in coalition is smaller than that in all coalition for .

In plain words, a provider, who is in cooperation with a peer set, will receive the highest dividend when she cooperates only with the peers excluding other providers whereas each peer wants to cooperate with as many as possible providers. It is surprising that, for the multiple provider case, i.e., , each provider benefits from forming a single-provider coalition whether the cost function is concave or not. There is no positive incentives for providers to cooperate with each other under the implementation of A-D payoffs. On the contrary, a peer always looses by leaving the grand coalition.

Upon the condition that each provider begins with a single-provider coalition with a sufficiently large number of peers, one cannot reach the grand coalition because some single-provider coalitions are already stable in the sense of the stability in Definition 2.3. That is, the grand coalition is not the global attractor. For instance, take as the current coalition structure where all peers are possessed by provider . Then it follows from Theorem 4.2 that players cannot make any transition from to where is any superset of because provider will not agree to do so.

## 5 Critique of A-D Payoff for Separate Providers

The discussion so far has focused on the stability of the grand coalition. The result in Theorem 4.1 suggests that if there is a noncontributing (free-riding) provider, which is true even for concave cost functions for multiple providers, the grand coalition will not be formed. The situation is aggravated by Theorem 4.2, stating that single-provider coalitions (i.e., the separated case) will persist if providers are rational. We now illustrate the weak points of the A-D payoff under the single-provider coalitions with three representative examples.

### 5.1 Unfairness and Monopoly

###### Example (Unfairness)

Suppose that there are two providers, i.e., , with and , both of which are decreasing and convex. All values are shown in Fig. 4 as functions of . In line with Theorem 4.2, provider is paid more than her Shapley value, whereas peers are paid less than theirs.

We can see that each peer will be paid () when he is contained by the coalition and the payoff decreases with the number of peers in this coalition. On the other hand, provider wants to be assisted by as many peers as possible because is increasing in . If it is possible for to prevent other peers from joining the coalition, he can get . However, it is more likely in real systems that no peer can kick out other peers, as discussed in [4, Section 5.1] as well. Thus, will be assisted by fraction of peers, which is the unique solution of while will be assisted by fraction of peers.

###### Example (Monopoly)

Consider a two-provider system with and , both of which are decreasing and concave. Similar to Example 5.1, we can obtain , , and . All values including the Shapley values are shown in Fig. 5. Not to mention unfairness in line with Example 5.1 and Theorem 4.2, provider monopolizes the whole peer-assisted services. No provider has an incentive to cooperate with other provider. It can be seen that all peers will assist provider because for . Appealing to Definition 2.3, if the providers are initially separated, the coalition structure will converge to the service monopoly by . In line with Lemma 4.1 and Theorem 4.1, even if the grand coalition is supposed to be the initial condition, it is not stable in the sense of the core. The noncontributing provider (Definition 4.1) in this example is .

### 5.2 Instability of A-D Payoff Mechanism

The last example illustrates the A-D payoff can even induce an analog of the limit cycle in nonlinear systems, i.e., a closed trajectory having the property that other trajectories spirals into it as time approaches infinity.

###### Example (Oscillation)

Let us consider a game with two providers and two peers where . If , and assist the content distribution of , the reduction of the distribution cost is respectively 10$, 9$ and 11$per month. However, the hard disk maintenance cost incurred from a peer is 5$. In the meantime, if , and assist the content distribution of , the reduction of the distribution cost is respectively 6$, 3$ and 13$per month. In this case, the hard disk maintenance cost incurred from a peer is supposed to be 2$ due to smaller contents of as opposed to those of .

For simplicity, we omit the computation of the A-D payoffs for all coalition structures and stability analysis (see Appendix of [20] and Table 1 in [20] for details). We first observe that the Shapley payoff of this example does not lie in the core. As time tends to infinity, the A-D payoff exhibits an oscillation of the partition consisting of the four recurrent coalition structures as shown in Fig. 6, where, for notational simplicity, we adopt a simplified expression for coalitional structure : a coalition is denoted by and each singleton set is denoted by . The evolution of coalition structure is governed by a simple rule: if there exist blocking coalitions (See Definition 2.3), then arbitrary one of them will be formed.

Let us begin with the partition . Player could have achieved the maximum payoff if he had formed a coalition only with . However, player will remain in the current coalition because he does not improve away from the current coalition. Instead, Player breaks the coalition so that and can form coalition for their benefit. As soon as the coalition is broken, betrays to increase his payoff by colluding with . It is not clear how this behavior will be in large-scale systems, as reported in the literature [9].

## 6 A Fair, Bargaining, and Stable Payoff Mechanism for Peer-Assisted Services

The key messages from the examples in Section 5 imply that the A-D value of the separate case gives rise to unfairness, monopoly, and even oscillation. Also, it turns out that some players’ coalition worth exceeds their Shapley payoffs which they are paid in the grand coalition (Theorem 4.1). Thus, the Shapley payoff scheme does not seem to be executable in practice because it is impossible to make all players happy, unequivocally. That being said, the fairness of profit-sharing and the opportunism of players are difficult to stand together. Then, it is more reasonable to come up with a compromising payoff mechanism that (i) forces players to apportion the difference between the coalition worth and the sum of their fair shares, (ii) grant providers a limited right of bargaining, and (iii) stabilize the whole system. We will use a slightly different notion of payoff mechanism, called value, originally proposed by Casajus [12].

### 6.1 An Axiomatic Characterization of χ Value

The value is characterized by a similar set of axioms used for the A-D value. The only difference is that (i) NP is weakened to GNP, causing a deficiency in axiomatic characterization, which is made up by WSP:

###### Axiom (Grand Coalition Null Player, GNP)

If for all , then .

###### Axiom (Weighted Splitting, WSP)

If is finer than (i.e., , ) and ,

 ϕi(N,v,P)−ϕi(N,v,P′)wi=ϕj(N,v,P)−ϕj(N,v,P′)wj.

The cornerstone of value is the very observation that, as the grand coalition is broken into two or more coalitions, player now has another option to ally with other coalitions than and this outside option must be assessed. To allow the assessment of the outside options, it is inevitable to weaken NP (See Section 2.3) to GNP, by satisfying only which, a player may receive positive payoff so far as he contributes to the worth of the grand coalition, even though he does not to that of the current coalition, i.e., NP. In the end, it is all about how to valuate the outside option, the value’s choice of which is to stick to the Shapley value by equally dividing the difference between the coalition worth and the sum of Shapley values, i.e., WSP for .

Recalling the definition in Section 2.1, we present the following theorem (see [12, 21] for the proof):

###### Theorem (χ Value)

The value is uniquely characterized by CE, CS, ADD, GNP and WSP as follows:

 Missing or unrecognized delimiter for \left (7)

where is Shapley value of player for non-partitioned game .

### 6.2 Fluid χ Value for Multi-Provider Coalitions

Recall , , and . To compute the payoff for the multiple provider case, we first establish in the following theorem5 a fluid value in line with the analysis in Section 3.2 with the limit axioms:

###### Theorem (χ Payoff for Multiple Providers)

As tends to infinity, the payoffs of providers and peers under an arbitrary coalition converge to (FluidChi) in Fig. 7 where the Shapley payoffs are given in (FluidAD1) in Fig. 3.

To intuitively interpret value, it is crucial to know the roles of Axiom WSP and its weights . In our context, because of fairness between peers, it is more reasonable to set for . It does not make sense to differentiate payoffs between peers due to the peer-homogeneity assumption in Section 3.1. On the contrary, we will clarify in Sections 6.3 and 6.4 why the weights of providers , do not necessarily have to be . The essential difference between A-D value and value lies in WSP.

Interpretation of WSP: It implies that, if peer loses, say , when the coalition structure changes, e.g., from the grand coalition to a finer coalition structure , the provider will lose . There are two implications of this weighted splitting. First, since the payoff of each player is computed based on the baseline, i.e., the Shapley value, and the surplus or deficit incurred by formation of the coalition are equally distributed for , value leads to a fair share of the profit. Secondly, now a provider may bargain with peers over the dividend rate by setting to any positive number. We elaborate on these two implications in the following subsections.

### 6.3 Fairness: Surplus-Sharing

On the basis of the first implication of WSP, value is fairer than A-D value in the following sense:

###### Definition (Surplus-Sharing)

A value of game is surplus-sharing if the following condition holds: if the coalition worth of coalition is greater than, equal to, or less than the sum of Shapley values of players in , i.e., , then the payoff of player is greater than, equal to, or less than the Shapley value of player , respectively, i.e., , for all and for all .

Since we proved in Theorem 4.2 that, for , the payoff of provider in coalition exceeds her Shapley value and that of peer is smaller than his, it is clear from this definition that A-D value is not surplus-sharing for , whereas value is surplus-sharing for any , e.g., see (7) and (FluidChi). For reference, both A-D and values are surplus-sharing if .

The corresponding payoffs of Examples 5.1 and 5.1 for , , are shown in Figs. 8 and 9. As was the case of the A-D payoffs in Examples 5.1 and 5.1, the grand coalitions are not stable. However, due to the surplus-sharing property of the payoff, whenever the coalition worth is larger than the Shapley sum of players in the coalition, all players in the coalition are paid more and vice versa. For instance, we can see from Fig. 8 that if the coalition is formed by provider and fraction of peers, all members of the coalition are paid more than their respective Shapley payoffs.

As shown in Fig. 9, the monopoly phenomenon of Example 5.1 for the case of A-D payoff is still observed for the case of value. Regarding Example 5.1, as shown in Fig. 8, payoff even induces the monopoly by , which did not exist for the case of A-D payoff.

### 6.4 Bargaining over the Dividend Rate

Another implication of WSP is that a provider bargains with peers over the division of the profit and loss by setting to a nonnegative real value. For instance, consider the case when the coalition worth exceeds the Shapley sum of players in the coalition, e.g.,