A fair cooperative content-sharing service

Computer Networks 57 (2013) 1955–1973

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Computer Networks journal homepage: www.elsevier.com/locate/comnet

A fair cooperative content-sharing service L. Militano a,⇑, A. Iera a, F. Scarcello b a b

University Mediterranea of Reggio Calabria, DIIES Department, Italy University della Calabria, Cosenza, DIMES Department, Italy

a r t i c l e

i n f o

Article history: Received 31 July 2012 Received in revised form 31 January 2013 Accepted 22 March 2013 Available online 8 April 2013 Keywords: Wireless cooperation Fairness Content sharing Nucleolus Game Theory Mediated P2P process

a b s t r a c t Wireless cooperative content sharing, based on the synergistic use of cellular and shortrange technologies, has recently gained much interest from academic and industrial communities. Besides energy saving and information transfer delay reduction, this paradigm can enable a significant reduction in cellular bandwidth usage, which also means monetary saving for users. In fact, such a cooperation is usually opposed by network and service providers, because it strongly reduces their potential profits. This paper deals with the provider perspective, too. In particular, a ‘‘mediated cooperative behavior’’ is proposed and analyzed within a scenario of short-range wireless file-sharing. The basic idea is that providers offer the possibility to users of cooperatively downloading contents, and increase their own profits because more users are attracted by such a service. Indeed, by participating in the devised service, users both avoid a (possibly expensive) stand-alone download of the desired product, and benefit of a special group-discount. The costs (and download tasks) distribution among users is based on a cooperative game theoretic model. Indeed, suitable solution concepts are applied to provide a fair solution, acceptable by all users, and hence to overcome the limitation of traditional optimization approaches to costs (and tasks) distribution. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Cooperation over short-range links among cellular devices is a paradigm, which has recently gained wide interest in the research community [1]. According to it, groups of users interested in a common content and in proximity to each other, might cluster together and exchange, over cost-free, energy efficient, and fast short-range links, a content downloaded through the costly cellular link. Advantages from the end-user point of view, in terms of energy efficiency, throughput enhancement, and cost reduction, are quite evident [2]. However, we are also interested in the network and service provider perspectives, describing a framework where ⇑ Corresponding author. Address: University Mediterranea of Reggio Calabria, DIIES Department, 89100 Reggio Calabria, Italy. Tel./fax: +39 0965 875276. E-mail addresses: [email protected] (L. Militano), antonio. [email protected] (A. Iera), [email protected] (F. Scarcello). 1389-1286/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comnet.2013.03.014

providers may have economic benefits, too. The wireless cooperation paradigm is conceptually very close to classic P2P and file sharing services, which show similar issues still requiring a solution. P2P file sharing services are nowadays highly successful among the younger generations and it is a common practice to share multimedia content such as music, videos, images, e-books, or similar, through P2P platforms to minimize the monetary cost of the content. Of course, whenever any form of content sharing violates copyright constraints, this behavior may cause large profit losses to content and network providers. The main reason that moves the end-user to go the way of illegality is the monetary service price, which is often judged to be too high. Network and service providers seem to be unable to find solutions to stop this phenomenon. In the authors’ opinion, a successful reaction for providers is to promote themselves as an active part of the system, by offering what we call a ‘‘mediated cooperative framework for file

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sharing service’’. The idea is that users in a group will be able to select a special P2P-option when they buy some product. This way, they get a group license that benefits of a high discount with respect to standard single-user licenses, and they are allowed to freely share the product within the P2P wireless network of the group. Moreover, users download only some parts of the whole product over the costly phone-link. Indeed, the service coordinator will compute and assign to users the fractions of the product to be downloaded by each of them, by taking into account download costs, time and the energy consumption. The main advantage for providers would derive from the increased chances to involve in the business those nodes (the majority) inherently oriented to ‘‘legality’’, but that avoid to access the provider downloading-service because prices are too high (at least with respect to their actual interest in that product). Indeed, in the proposed framework, users may save money because of the special group discount, so that the actual price to be paid could match the actual value/interest they assign to the product. Moreover, they have further pros in terms of energy/ throughput/download cost, as well as the additional security factor guaranteed by a downloading-service under the direct control of providers (instead of possible illegal and uncontrolled alternatives). Surely, to make this ‘‘mediated wireless cooperation’’ mechanism work properly, a key feature that the provider shall promote, to catch the interest of more users, is the implementation of a cost distribution among cooperating nodes, which is judged fair by all the players. This feature may be the only effective incentive to access the service since, in real environments, where nodes are rational and selfish players, it is difficult to reach such an agreement in cooperation. In particular, any proposed cost allocation should also be tested according to the stability property, guaranteeing that no (sub)group of users has an incentive to leave the cooperation group and to form a different coalition. Traditional optimization techniques show their limits in guaranteeing these requirement for all the participating members. Indeed, they only deal with the minimization of some global measure (total cost, energy, time, etc.), but provide no indication on how to divide such an optimal value among nodes while considering their individual or joint contribution to the mechanism. Therefore, the framework should consider solutions that guarantee fairness and stability in the cost distribution allowing the selection of the best suited coalition partition for the interested users. This way, the fare to be paid by each player (member of the group) takes into account his contribution to any possible coalition in terms of download costs of his terminal over the cellular link, possible agreements with the provider (e.g., frequent buyers), and so on. In particular, the service coordinator will act as an ‘‘impartial’’ entity guaranteeing that only solutions are applied that are judged fair, according to the proposed (publicly available) criterion. Moreover, since every player contributes to the process (at least allowing the group to obtain a higher multiple-license discount), the mechanism also avoids free riding, a typical behavior of pure peer-to-peer systems

where some nodes get benefits from the community without giving anything back. Based on the above considerations the main objectives of this paper are: (i) to propose a framework that gives a valid alternative to classic P2P systems, allowing the service providers to participate in the cooperation process and benefit from it as well; (ii) to include in the framework some proposals that guarantee fairness and stability in the cost distribution as an alternative to traditional optimization techniques; (iii) to model and analyze well-known solution concepts from cooperative Game Theory, to guarantee fairness in the cost distribution for the specific problem; and (iv) to perform an analysis on the monetary savings and profits for the involved entities under system and user constraints related to personal and technological parameters. The rest of this paper is structured as follows. Next section gives an overview of related work relevant to core issues and techniques under investigation in this research. A detailed description of the reference cooperative file sharing service, the user cost minimization step, and the game theoretic model for the problem are given in Section 3. A thorough performance evaluation is presented in Section 4, while final remarks are given in the conclusion section. 2. Related works From the literature, several research contributions are available, which propose improvements in the performance of remote content downloads [3]. Solutions are, for instance, based on bandwidth aggregation of multiple interfaces belonging to either the same device [4] or to different nearby devices [5]. Interesting contributions also address modeling and evaluation issues relevant to communication architectures that exploit the synergy between cellular and short-range systems. As an example, [6,7] show how cellular and short-range networks (WLAN, Ad hoc, and MANET) can be integrated to improve the performance levels. In [8] the attention has been given to the energy consumption and transfer delay benefits obtainable through this communication architecture. In contrast, issues such as end-user cost reduction due to bandwidth occupation decrease and network/content provider profit losses have attracted a little attention up to now. The focus of the present paper is to cover this lacking aspect, trying to introduce a service model able to attract the interest of network/content providers for the wireless cooperative content sharing. This is actually of utmost importance for the success of the framework. Cooperation over shortrange links and classic P2P show similar problems from the economics point of view for the network/content provider; thus, it is worth recalling P2P related researches about this issue to better highlight the differences among them. From the moment the first file sharing services appeared on the market, the providers started analyzing how to deal with such services [9], and considering either to cooperate with or to fight against them. There is much ongoing research activity trying to understand socio-economic aspects associated to file sharing services, like for

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BitTorrent in [10]. Moreover, in proposing content distribution services, counteractions to fight against unauthorized illegal content publishers represent an important part of the service model [11]. As claimed in [12], distribution cost is a key reason that prevents the content providers from making great amounts of content available for download in P2P content distribution networks. Furthermore, [12] points out that in classic P2P systems costs are pushed from the content distribution networks to the Internet Service Providers (ISPs). On the other hand, in wireless systems, user cooperation over ‘‘private’’ short range links reduces the overall income for the ISP. The framework considered in the present paper, instead, looks at the provider as a ‘‘promoter’’ or ‘‘mediator’’ of a P2P-like file sharing service, which controls and coordinates the service and, in doing so, obtains some benefits for itself. A way to reach this objective is to promote highly attractive (but monitored) cost reductions for users who cooperatively download some contents. On the other hand, a suitable game theoretic formulation allows us to achieve this goal while taking care of fairness and stability aspects, too. Game Theory is an analytical framework that attempts to analyze the behavior of rational entities with their own interests in reciprocal interactions [13]. Recently it has been applied also to various research fields relevant to wired and wireless communication networks [14]. Among cooperative game based contributions, the socalled coalitional problems are gaining large interest within the research community [15]. In [16] the focus is on vehicular networks, in [17] a coalitional problem is proposed to study fairness and cooperation gains in virtual MIMO systems, in [18] packet forwarding issues in ad hoc networks are addressed, while in [19] a task allocation problem is studied in a software system. Game theoretic models have been also applied to pricing schemes in cognitive wireless networks [20] or in heterogeneous wireless networks [21,22]. In a different study [23], the authors of the present paper also showed how cooperative Game Theory can be applied to provide a fair energy consumption cost-distribution in a cooperative cluster in which the introduced communication-systems constraints play a significant role. Moreover, a recent interesting contribution exploits Game Theoretical notions to deal with fairness in peer-assisted services for content delivery networks (for live streaming and similar highly resource-demanding contents) [24]. Indeed, such networks would benefit from a peer-to-peer architecture to reduce their operating costs, and the authors study a fluid Shapley value approach to provide a suitable incentive scheme for users cooperating in the content distribution process. P2P systems are often studied through non-cooperative game theoretic models to introduce incentives or to coordinate the system (for example [25] and [26]). The mediated service framework proposed in this paper is not self-organizing because it is controlled by the provider. At the same time, the users do not need any incentive to cooperate. Indeed, nodes are naturally incentivized to cooperate by the immediate monetary savings they experience, where the larger is the coalition (group) the larger is the discount (controlled by the provider). In fact, the mon-

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etary based economic-model is a point of strength of the cooperative process proposed in this paper, and cooperative Game Theory is the most natural framework to study agreements and interactions among such involved entities.

3. The cooperative file sharing framework With reference to the service framework introduced in Section 1, assume there is a content provider P that offers the P2P group-option to its clients, and plays the role of the Cooperation Server plotted in Fig. 1. In order to reach its goal of selling licenses of its product to all interested users (also, nodes), the service coordinator implements a policy aiming at minimizing the overall monetary cost for the users, while allowing a cost distribution that is perceived to be a fair and stable one. These are the compulsory conditions for the wireless cooperative process to be accepted by all parties. Actually no unique definition can be given for what a fair solution is, but several solution concepts have been proposed over the years in the field of Game Theory [27]. In this paper, we shall focus on two well-known and studied solution concepts; however, most of the paper is actually independent of the specific chosen notion. Let N be a group of users interested in downloading some product from this provider. Nodes start the procedure by contacting the Cooperation Server P and requesting the desired product with the P2P option. Each node sends information about its device interface identifier, cellular link throughput, cost per second of the cellular link (or alternatively the cost per unit of data), information related to the short-range link they cooperate through (e.g. Bluetooth or WLAN link). For every node i, the Server P knows the monetary cost cdi afforded to download the full product file from the provider, as well as the time ti required to complete such a download. Presenting a detailed cellular model is not among the objectives of the paper. Differently the aim is to analyze the impact the bandwidth costs have on the cooperative solutions. Therefore, in our performance evaluation study, we considered a wide set of cellular throughput values to assess how the cooperative scenarios are influenced by this parameter. Users may also specify some constraints over their contribution to the proposed P2P framework: any user i may impose a limit on the amount of data to be downloaded in the P2P application, expressed as a fraction 0 6 DownloadBoundi 6 1 of the desired product file, which of course entails a bound DownloadBoundi  cdi for the maximum download cost for player i. While the short-link technology is available for free to all users in N, we consider the energy consumption and the content transfer delay as further parameters of interest in the final product-sharing phase over this inexpensive link. In particular, any user i may define a TimeConstrainti P 0 to express the maximum time it is willing to spend to receive the wished content, in relation to the non-cooperative download. Similarly, any user i may define an EnergyConstrainti P 0 to express the maximum energy it is willing to spend, compared to the noncooperative download. In order to evaluate these constraints, the Cooperation Server P should be able to estimate

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Fig. 1. Wireless cooperative scenario.

the time and the energy consumption for the cooperative content download for every node. This requirement can be fulfilled providing the service provider with a specific model which deals both with the short-range link and the cellular data-exchange, as defined in Section 3.4. A suitable pricing function p(S) is defined by P for every non-empty group of users S # N, which may take into account frequent buyers, clients with high-feedbacks, and so on. To model the intended business strategy where large groups (buying many licenses) should be attracted, such a pricing function is required to be sub-additive, that is, p(S [ T) 6 p(S) + p(T), for every pairs of non-intersecting subgroups S and T. Therefore, adding users to any group always leads to a lower price-per-license, so that larger groups have higher discounts, as expected. After collecting information from nodes willing to cooperate, the Cooperation Server reports decisions to them about ‘‘how’’ to cooperate. This task consists in defining the cooperative coalitions and assigning the portion ai of content (file-share) that any node i in the cooperative framework shall download over its costly cellular link, and the total cost xi for player i to get the desired product. Note that this cost includes the network cost, and thus player i actually pays to the content provider xi  ai  cdi. Nodes will then proceed to download their assigned fileshare, and then they exchange these parts of the file over the (free) short-range link. In general, more copies of the file may be downloaded by nodes of the group, the only constraint is that the fractions downloaded by peers cover P at least a whole copy of the desired file. That is, i2N ai P 1 must hold, i.e. non-overlapping file fractions (may) sum to more than 1. Noteworthy, because of the group-discount modeled by the pricing function and the optimization performed to reduce the download cost (exploiting devices with lower bandwidth costs), the cost assigned to each node i will always be not higher than the cost that i would sustain for a stand-alone download (which is the sum of the content license cost p({i}) and the download cost cdi). In fact, the

precise cost is determined according to a game theoretical solution-concept, described in the subsequent section, that guarantees a fair cost allocation for all nodes and that is the key issue for the success of the proposed paradigm. In particular, the service coordinator needs to take into careful consideration the contribution of every node i in terms of price reduction and of device capability (bandwidth cost), to suitably evaluate how much discount it deserves. Note that, so far, we just considered a service provider that offers a file download service and directly partakes in a sort of ‘‘mediated’’ wireless cooperative data-exchange based on short-range links. We have not yet considered the possible role of the network provider (if it is distinct), that could oppose the proposed framework because it may reduce its profits on the costly cellular link. In a simple approach, one may just assume that the network provider has a mutual business agreement with the service provider that covers somehow such a loss. In fact, if the network provider has some control power over the proposed protocol, we could imagine that, in the business agreement, the network provider may require a lower bound on its profit per transaction, with respect to the case of stand-alone downloads for all users. Formally, we consider in the framework an additional parameter 0 6 GainConstraint 6 1, used to impose the constraint P P i2N ai  cdi P i2N cdi  GainConstraint over the total download-cost of the group. We believe that even low values of the profit-gain constraint are reasonable: if this situation is contrasted with typical P2P networks, where the network provider has no control at all, only a few users participate in the costly download phase, and thus it always suffers high profit losses. 3.1. Game-theory solution concepts for the P2P framework In this section, we recall the notions of Game Theory used in this paper, mapping their meaning to the reference problem and highlighting how they help towards our final objective. For more information on this subject, the reader

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is referred, e.g., to the book on Game Theory by Osborne [13]. It is natural to model the process of assigning a cost to each cooperating player (node) in a fair way as a coalitional cost-game G = hN, Ci with transferable utilities (TU), where N is the set of players and C : 2N ! R is a characteristic cost function that models the feasible cost for every coalition (set of players/nodes) S # N. In our case, the cost function models the fact that every coalition S has at its disposal the ability of each player i 2 S to download the file with some costs and some constraints, besides his contribution to determine the price p(S) of the desired product. For instance, player i may be a frequent buyer, and its presence in S may allow the coalition to get a higher discount. Of course, this may also happen for subgroups of players, so that some family S0 may attain some special price, and thus S0 # S entails that p(S) will be particularly cheap. Therefore, a fair price assignment should consider all possible contribution of players, trying to make everyone happy, and not only to meet the imposed constraints. Henceforth, components of any vector x 2 RjNj are oneto-one associated with players in N, so that xi denotes the component associated with player i 2 N. Moreover, for P any vector x 2 RjNj , we denote by x(S) the value i2S xi , where S # N is a coalition. A feasible payoff profile (or pre-imputation) of G is a vector x 2 RjNj such that x(N) = C(N). An imputation of G is a feasible cost profile x 2 RjNj such that xi 6 C({i}), for each i 2 N. This condition is usually called individual rationality. In our application, recall that every player i 2 N is associated with the two constants cdi and DownloadBoundi, which model the cost for player i to download the full desired file from the network, and the maximum amount of data that i would like to download in the P2P process (expressed as a fraction of the whole file). Thus, 0 6 DownloadBoundi 6 1, where 1 means that i sets no a priori limits on his possible download, and 0 means that i will not download anything from the costly-link (therefore this player will just participate in the sharing phase over the short-range link). Therefore, by individual rationality, the total cost for every player i should be not larger than C({i}) = p({i}) + DownloadBoundi  cd(i). The set of all imputations of the game G is denoted by X(G). An outcome for G is an imputation from X(G) that specifies the distribution of the cost to any player of the game. A typical requirement of a good outcome is to be ‘‘stable’’ with respect to the possibility that subsets of players find convenient to deviate from it, by forming alternative coalitions and starting a P2P process on their own, in order to attain lower costs. The set of such stable outcomes is known as the core of the game. Definition 1 (Core [28]). The core CðGÞ of a cost TU game G = hN, Ci is the set of all imputations1 x such that, for each coalition S # N, x(S) 6 C(S).

1 In the literature, feasible profiles are sometimes considered in place of imputations. In fact, it is easily checked that the two forms are equivalent as far as the definition of the core is concerned.

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In words, the core is the set of all imputations that satisfy the cost upper-bound of all coalitions according to the cost function C. We say that if an imputation associated with a coalition is in the core then the coalition is stable; otherwise, we say it is unstable. Indeed, if y R CðGÞ, there exists some coalition S such that y(S) > C(S). Therefore, players in the group S might leave the group N and buy the desired product on their own at the total cost C(S), which is less than what they were asked to pay according to the cost distribution y. In general, the core of a game may be empty as well as it may contain an infinite number of imputations. An important class of cost games where the core is always non-empty is the class of concave games (dually, the convex games for value games). A game G is said concave if, for every pair of coalitions S and T, C(S [ T) + C(S \ T) 6 C(S) + C(T). It can be shown that this holds if the cost function is submodular, that is, if C(T [ {i})  C(T) 6 C(S [ {i})  C(S), for each pair of coalitions S # T # Nn{i}. However, even if the core is not empty, it remains the problem of choosing an outcome out of possibly infinite many candidates belonging to the core. Thus, solution concepts associated with unique profiles are usually desirable in applications. In particular, the Shapley value [29] is one of the most used solution concepts in cost-sharing applications (see, e.g., [24]). Definition 2 (Shapley value [29]). The Shapley value of a cost TU game G = hN, Ci is the pre-imputation of G assigning to every player i 2 N the following cost

/i ðGÞ ¼

1 X jSj!ðjNj  jSj  1Þ!½CðS [ figÞ  CðSÞ: N! S # Nnfig

In words, the Shapley value assigns a cost to each player i taking into account his ‘‘average marginal contribution’’, where the average is computed over all different sequences according to which the grand coalition could be built up from the empty coalition. This solution concept has also a nice axiomatic characterization supporting its notion of fairness (it is the unique pre-imputation that satisfies the Symmetry, Dummy Player, and Additivity axioms). It is known that, in any concave game, the Shapley value belongs to the core and thus it is a stable imputation. However, in the general case the Shapley value may be outside the core, even if the core is not empty. Thus, in particular, the Shapley value is not necessarily an imputation, and thus it may also violate the individual rationality condition. Another approach to single out a fair outcome for TU games is based on the notion of Nucleolus, first introduced by Schmeidler [30], and based on the lexicographical minimization of the maximum unhappiness of coalitions. Although the Nucleolus in its current formulation has been defined in 1969, it was later discovered that its ability to divide in a fair way scarce resources among competing agents was at the basis of some (previously) mysterious rule in a Mishna of the Talmud attributed to Rabbi Nathan (about 1800 years ago), and in similar contested garment rules [31]. Formally, given a cost game G = hN, Ci, a coalition

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S # N and a cost allocation x, the excess (or unhappiness) P of S w.r.t x is equal to eðS; xÞ ¼ i2S xi  CðSÞ. For any imputation x, define h(x) as the vector where the various excesses of all coalitions (but the empty one) are arranged in nonincreasing order:

hðxÞ ¼ ðeðS1 ; xÞ; eðS2 ; xÞ; . . . ; eðS2jNj 1 ; xÞÞ: Let h(x)[i] denote the ith element of h(x). For a pair of imputations x and y, we say that h(x) is lexicographically smaller than h(y), denoted by h(x)  h(y), if there exists a positive integer q such that h(x)[i] = h(y)[i] for all i < q and h(x)[q] < h(y)[q]. Definition 3 (Nucleolus [30]). The Nucleolus N ðGÞ of a TU game G is the set N ðGÞ ¼ fx 2 XðGÞj 9 = y 2 XðGÞ s:t: hðyÞ  hðxÞg. Therefore, this solution concept first cares about players that are less satisfied, then it cares about the less satisfied among the remaining coalitions, and so on for the rest of the coalitions, with the same approach. Interestingly, there is a unique point that leads to the lexicographically minimum vector of excesses, that is, N ðGÞ is in fact a singleton. Moreover, this unique outcome belongs to the Core, whenever it is not empty, and usually offers a valid solution also for games with an empty core, so that the Nucleolus has been considered as one of the most interesting solutions to investigate [18]. A correct evaluation study on the effectiveness of Game Theory in designing ‘‘fair’’ cost allocation schemes in the envisaged scenario derives from the selection of an appropriate Game Theory solution concept to apply [27]. Different criteria have intrinsic differences in the fairness notion they want to promote. Although many of them might be applied to the wireless cooperation paradigm under study, the Nucleolus is preferred in this research, because minimizing the maximum unhappiness of players fits well the application at hands. Moreover, we formally prove in this paper that, for the class of games that best model our application scenario, the Nucleolus fulfills important game theoretic properties. In particular, it is shown that for such a class of games the core is always non-empty. It follows that for these games the Nucleolus is not only a fair solution, but also a stable one (in that it belongs to the core of the game), thus meeting two important requirements for the service. The Shapley value [29] has been considered and studied as well, but for the games associated with our P2P application they sometimes exhibit the undesired behavior of not being a stable allocation. 3.2. Cooperative-download cost games In this section, we describe a class of TU coalitional games that we call cooperative-download cost games (short: CDC games), defined by means of a suitable class of pricing functions determining the discount policy. Such games best model our problem of assigning costs to nodes participating in the mediated P2P file-sharing process in a fair (and stable) way. In particular, we show that these games have always a non-empty core, even if they are not

concave. As a consequence, the Nucleolus of any CDC game is always a stable imputation, while the Shapley value is not necessarily an imputation (it depends on the chosen pricing function). Formally, a CDC game G is a TU cost game hN, Ci, where N is the group of players interested in buying some product (or bundle of products) according to the mediated P2P download framework, and C is a cost function described below. First recall that the provider defines a suitable price function p(S) to determine how to charge the nodes of any coalition S for buying the jSj licenses for the product they are cooperatively downloading, and recall that we assume this function to be sub-additive, that is, for every pair of coalitions S and T with S \ T = ;, p(S [ T) 6 p(S) + p(T). In fact, this property formalizes the usual business logic ‘‘the more licenses the more discount’’. We are now ready to define the cost function of game G. With any coalition S # N, we associate a cost CðSÞ ¼ minfpðSÞ þ

X

ai  cdðiÞg

i2S

subject to :

X

ai P 1

i2S

0 6 ai 6 DownloadBoundi ; 8i 2 S ! X X ai  cdi P cdi  GainConstraint i2S

i2S

EnergyCoopi ðai Þ 6 EnergyNocoopi  EnergyConstraint i ; 8i 2 S TimeCoopi ðai Þ 6 ti  TimeConstraint i ; 8i 2 S ð1Þ

In any optimal solution of the linear program above, the ai control variable contains the (amount of) file fraction to be downloaded by player i, in order to minimize the total cost for the coalition S, while satisfying the problem constraints in (1). In particular, note that we require that the whole file can be reconstructed after the download of these fractions (in fact, the constraint just checks the sizes of the fractions, as the actual subdivision of the file is performed later by the Cooperation Server, respecting the information in the ai values). Also, every player does not download more data than it is allowed by the DownloadBoundi parameter, thus meeting the individual rationality constraint, and the GainConstraint of the network provider should be satisfied, as well. Finally, the data fractions to be downloaded are required to meet also users’ requirements in terms of energy consumption and time in cooperation which are limited by EnergyConstrainti and TimeConstrainti. We require that the linear program (1) has a solution at least for the case S = N. Indeed, in this case, the above optimization procedure provides the cost of the grand-coalition, and thus the total cost associated with any preimputation of the game. An unfeasible linear program would mean that the constraints imposed by players and network provider are too restrictive for the characteristics of the involved players, and the cooperative download involving the whole group of players N cannot be executed. In such a situation the Cooperation server will enter in a

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further procedure of coalition partition to find alternative coalitions involving subsets of the nodes for which the constraints are met and the cooperative download can be executed; details on this procedure are reported in Section 3.5. For any other coalition S  N different from the grand coalition, it is not necessarily required that a feasible solution is found by the above linear program. In fact, in these cases we can associate to the specific coalition the cost P CðSÞ ¼ pðSÞ þ i2S cdi , that is, the cost when the nodes in coalition S are not cooperating, i.e. the sum of the costs for players in S assuming that each of them downloads the whole file on her/his own. It could happen that some nodes have a ‘‘flat rate’’ for the bandwidth cost. This situation can easily be modeled by setting to zero the constant function cdi in the model. Whenever this happens, and if player i does not limit its maximum possible download, the cost definition procedure would assign to such a node the whole download process, with the consequence of having a zero download cost for the whole cooperating group, with the additional benefit of a reduced content price. The subsequent fair cost allocation will push the nodes with non-zero bandwidth costs to strongly reward the node with the ‘‘flat rate’’. This situation could even lead to the possibility of actually ‘‘earning money’’ in the process (in fact, note that such nodes are probably paying higher fees for their flat-rate network contracts). This is conceptually correct, and it should be faced by considering, e.g., some bonus-based mechanism. Noteworthy, in general, a similar situation could also show up for scenarios with high differences in the bandwidth costs among nodes and relatively low content prices. However, for the sake of simplicity, we will not deal with bonus/credits handling, as nothing changes in the theoretical framework. In particular, in the experiments presented in this paper we avoid the use of very low content prices with respect to download costs, and we rather focus on the more interesting situation where content and bandwidth cost levels are comparable. 3.2.1. Properties for the cooperative-download cost games Recall that a cost TU game is said sub-additive if its cost function is sub-additive, that is, for every pair of coalitions S and T with S \ T = ;, C(S [ T) 6 C(S) + C(T). For subadditive games we usually say that the grand-coalition forms. Indeed, in such games adding players to any coalition is always cost-effective, which entails that the grand-coalition is the most convenient configuration. Intuitively, we expect CDC games to have this property, since the business strategy of providers (‘‘the more licenses the more discount’’) is modeled by the choice of subadditive license (product) price functions. However, we still need a little proof of the property, because the price function provides only a part of the total cost of coalitions (which is also determined by the optimization problem solutions). Proposition 1. Any cooperative-download cost game hN, Ci is sub-additive.

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Proof. Let S and T be any pair of non-overlapping coalitions. We know that, by definition, p(S [ T) 6 p(S) + p(T), because the price of product licenses is such that larger coalitions get larger discounts. It remains to consider the download cost, determined for each player i by the assigned file-fraction ai. Assume first that for S and T download costs are defined according to the linear programming optimization (1), and consider any pair of  S and a  T for the linear programs for S feasible solutions a and T, respectively. It is straightforward to check that their union is a feasible solution for the linear program for S [ T. Indeed, all the player constraints are clearly satisfied by either solution, and the constraint P  P i2S[T ai  cdi P i2S[T cdi  GainConstraint is also satisfied by linearity, as the two corresponding constraints for S  S and a  T . Therefore, in any optimal and T are satisfied by a  oS[T for the latter program, the download-composolution a nent of the cost is at least as good as the sum of the down S and a T . load-cost components for a Finally, observe that the latter argument clearly holds as well if the download costs for S or/and T are defined in such a way that every player is assumed to download the whole file (that is, if either linear program is unfeasible). h We conclude the section by showing an important property of the class of CDC games we are most interested in, that is, the CDC games based on cardinality-discount cost-functions, called hereafter cardinality-discount CDC games. These games are based on pricing functions that depend only on the cardinality jSj of the given coalition, and have the following form:

pðSÞ ¼ price  jSj 

100  dðjSjÞ ; 100

ð2Þ

where price is the basic price of the product, that should be paid by any single buyer not involved in any cooperative download, and d() is a non-decreasing function over the naturals [0, N] which determines the (percentage) discount to be applied to the group S, given the number of users jSj belonging to it. No special assumption is required for the form of d(), but for its codomain [0,100] and the base cases d(0) = d(1) = 0 (no discount is given to the empty coalition or to single noncooperating nodes). Observe that the sub-additivity property holds for this class of functions, as required for legal pricing functions for CDC games. However it is worthwhile noting that these functions are not concave in general, and hence the cardinality-discount CDC games based on them are not concave, too. Indeed, whether or not this property holds depends on the choice of the specific discount function d(). Nevertheless, we are able to prove that, no matter on the choice of the discount function, these games (even if non-concave) have a non-empty core. Proposition 2. Every cardinality-discount CDC game has a non-empty core, and thus its Nucleolus is always a stable imputation.

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Proof. Let G = hN, Ci be a cardinality-discount CDC game, and let G0 be the modification of this game whose cost function is C0 (S) = C(S)  p(S), for every S # N. That is, the cost function of G0 is defined only by the linear optimization dealing with player constraints and the network provider constraint, and it is independent of the product price. Then, it is easy to see that G0 has a concave cost function, because C(T [ {i})  C(T) 6 C(S [ {i})  C(S), for each pair of coalitions S # T # Nn{i}. Indeed, such a newcomer player i adds the same constraints to both programs for S and T, but the larger coalition T is clearly more flexible in dealing with them. It follows that the core of G0 is not empty. Let x0 be any imputation in the core of G0 , and define a pre-imputation x00 for G such that x00i ¼ x0i þ pðNÞ=jNj, for every player i 2 N. That is, x00 is the same as x0 but now the price of the product licenses are taken into account: it is uniformly distributed among all players. Let S # N be any arbitrarily chosen coalition. We show that x00 (S) 6 C(S) holds, hence x00 belongs to the core of G. By construction, P  0 0 x00 ðSÞ ¼ i2S xi þ jSj  pðNÞ=jNj 6 C ðSÞ þ jSj  pðNÞ=jNj, where the latter inequality follows from the fact that x0 belongs to the core of G0 . Therefore, 00 x (S) 6 C(S)  p(S) + jSj  p(N)/jNj. To conclude, we just observe that (p(S) + jSj  p(N)/jNj) 6 0. Indeed, recall that in cardinality discount CDC games the price function has the form pðSÞ ¼ price  jSj  100dðjSjÞ for some non100 decreasing discount function d(). Therefore, we get pðNÞ=jNj ¼ price  100dðjNjÞ 6 pðSÞ=jSj ¼ price  100dðjSjÞ , because 100 100 the discount function d is non-decreasing with the coalition cardinality, and thus the maximum possible discount is assigned to the grand-coalition N. h Having a non-empty core and a stable Nucleolus is a remarkable property of cardinality-discount CDC games, as they seem quite appropriate to model the most frequent scenario for the proposed file sharing framework. On the contrary, the Shapley value is not necessarily in the core, if the game is not concave. 3.3. Cost of product licenses for the nodes in the coalition In this section, we conclude the description of the mediated P2P process, by focusing on the last phase, where the Cooperation Server actually assigns to every node i the part of the file to be downloaded. At this step, the Nucleolus x has already been computed, and thus every player i knows his total cost xi to get the product, comprising both the license cost and the download cost. Once the money issues are solved, we have the possibility to perform a final optimization step possibly focusing on different aspects. For instance, while meeting the constraints of the players and the total bandwidth cost to be paid to the network provider (C(N)  p(N)), the actual file shares to be downloaded may be assigned to nodes in such a way that the download time over the cellular link is minimized, in order to obtain the desired content in a shorter time. Other alternatives can be proposed as well, for instance one could minimize the total energy consumed over the cellular link, or relax the constraints on the total bandwidth cost to be paid to the network provider when service

and network provider are not distinct entities, and so on. The time optimization may be obtained by minimizing the maximum download-time over the participating nodes. To this end, the Cooperation Server may perform the following linear-programming optimization:

min t max subject to :

X

ai P 1

i2N

tmax P ai  t i ;

8i 2 N

0 6 ai 6 DownloadBoundi ; X ai cdi ¼ CðNÞ  pðNÞ

8i 2 N

i2N

EnergyCoopi ðai Þ 6 EnergyNocoopi  EnergyConstraint i ;

8i 2 N

TimeCoopi ðai Þ 6 t i  TimeConstrainti ;

8i 2 N ð3Þ

where tmax is the variable to be minimized, whose feasible (lowest) values are determined by the greatest product ai  ti, that is, by the slowest download. Then, from the values aoi ði 2 f1; . . . jNjgÞ of variables in any optimal solution of the linear program (3), the Cooperation Server computes the actual file parts, say bi, that any player i has to download over the costly link. The transaction ends when every player i has payed its license fee xi  aoi  cdi , and all parts have been downloaded. Note that the unique constraint P i2N ai cdi ¼ CðNÞ  pðNÞ suffices to deal with economic issues. For instance, it entails that the network provider’s GainConstraint is fulfilled, because it is so in the solution leading to the computation of C(N), which determines the network profit C(N)  p(N). Further constraints to be considered for this final optimization are the constraints on the energy consumption and the time in the cooperative content download as defined by the players. 3.4. Energy consumption and time constraints definition While the main focus for the proposed model is on the monetary costs related to the cooperative content download, the proposed framework foresees the possibility for the users to define energy consumption and time constraints when joining a cooperative content download. The Cooperation Server will thus compute the energy consumption and the time needed to receive the content in cooperation for the nodes. These will then be compared to the non-cooperative case to check whether the wished constraints are met for the nodes. The framework can work with any short-range link, but in order to have some realistic models and figures in the performance evaluation we need to focus the attention on a specific network, for example Bluetooth. While the interested reader can find details for the energy consumption model for the cellular-Bluetooth cooperative setting in [23], we will briefly report here the main findings needed in this paper. In a Bluetooth piconet, the number of nodes simultaneously active is limited to 8, where one node acts as master and the remaining nodes act as slaves. No direct slaveto-slave communications are possible and all transmis-

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sions go through the master. A round robin scheduling of the involved nodes is performed [32] and the master can perform broadcast communications to the slaves. The node playing the master role is a key factor in defining the energy consumption and the required time in cooperation. Consequently, when the Cooperation Server will perform the proposed linear optimization, it will also consider the different potential master–slave configurations. Let us define the energy when noncooperating, EnergyNocoopi, as the energy consumption for node i when downloading the whole content over its costly cellular link, and EnergyCoopi as the energy consumption for node i in cooperation. The definition of EnergyNocoopi is straightforPci ward: EnergyNocoopi ¼ Rc  X; where Pci and Rci are respeci tively the power consumption and the data rate on the cellular link for node i, and X is total content size (expressed in Kbyte). For what concerns the EnergyCoopi term, both cellular energy consumption and Bluetooth energy consumption have to be considered. In the reference architecture, nodes first download all data over the cellular link before sharing them over the short-range link. The energy consumption on the cellular link is simply computed as the mean cellular power consumption multiplied by the assigned file fraction and divided by the mean link throughput. Instead, a detailed analysis of the Bluetooth link is needed since, over the time, the number and type of packet transmissionsnreceptions of a node depend on several factors. Summarizing the results found in [23], the expression of the energy consumption in cooperation can be written as a function of the file fraction for each node i to be downloaded over the cellular link, ai, and on the role r played in the piconet (master or slave), as reported in Eq. (4). In Eq. (4), the first term is the energy consumption on the cellular link, the next three terms represent the energy consumption respectively in transmitting data, in receiving data, and during the idle time on the Bluetooth link and a final term measures the energy consumption for the GPS positioning of each node.

EnergyCoopi ðai Þ ¼ Edi ðai Þ þ Ebti ðr; ai Þ þ Ebr i ðr; ai Þ þ Ebii ðr; ai Þ þ Ep

ð4Þ

The Edi(ai) term is the energy consumption in downloading the assigned file-share ai over the cellular link (this term is not depending on the master or slave role in the Pci Bluetooth piconet). This is defined as Edi ðai Þ ¼ Rc  X  ai , i where X is the data size, Pci and Rci are the power consumption and the throughput on the cellular link for node i, respectively. The Ep term in Eq. (4), is the energy consumed by the nodes to gather their GPS positioning. For the further terms of Eq. (4) the values for the Bluetooth specific parameters are set according to the standard [32] and the exact definition is taken from the model presented in [23]. When looking instead at the time needed to receive the content, as introduced earlier in this section, ti is the time needed for node i to download the whole content over its cellular link. Then, let us define TimeCoopi as the time needed for node i to receive the content in cooperation.

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When cooperating the devices will first download the assigned file fractions over their cellular links and then share them over the Bluetooth link. During the first phase of cellular downloading, they can, in parallel, setup the Bluetooth piconet. Therefore, the time in cooperation is equal for all nodes i and can be computed as the sum of maximum time spent on the cellular link by the nodes, the time needed to distribute the data over the Bluetooth link, and a small contribution of time for the Cooperation Server to compute the solution:

TimeCoop ¼ t max þ TimeBT þ TimeServ er

ð5Þ

tmax is defined by the linear program in Eq. (3); the time on the Bluetooth link TimeBT is given by the number of Round Robin scheduling cycles needed to distribute the total data for a given packet payload (further details on this computation can be found in [8]); the time for the Cooperation Server to compute the solution TimeServer is here assumed equal to 19s as this is the worst case value for a computer with mean hardware capabilities when it is necessary to compute the optimization problem for all possible coalitions to find solutions meeting all the constraints for the problem. 3.5. Coalition partition definition As discussed earlier in this section, the different constraints set by the users and/or by the network may cause the linear optimization not to find a valid solution for the grand coalition. Moreover, if the Shapley value is the chosen cost-allocation solution, then the property of being outside the (non-empty) core may be considered unacceptable. To deal with these cases, the Cooperation Server may be equipped with a strategy to find alternative groups of players to cooperate successfully. The proposed policy is to exclude iteratively some of the nodes until a first coalition is found where all the requirements are met. If more alternatives are meeting the constraints, then a further policy should be introduced to select the preferred coalition. In this framework, we adopt the well-known concept of maximizing the social welfare [15], whereby the coalitions that maximize the total monetary savings (for themselves) are preferred. In more detail, if for the coalition involving N nodes a valid solution cannot be found, then smaller coalitions are considered. Whenever a coalition with the required properties is formed (clearly, it will be a maximal one), the process restarts with the excluded users, until a partition p of the players N in coalitions, also called a coalition structure [15], is found. Note that standard game theoretic solution concepts for coalitional structures, such as the core, consider stability conditions also on sets of players belonging to different elements of the partition p [33]. However, such inter-structure conditions lead to high computational costs that are not suitable for the practical applications we have in mind. Therefore, we adopt here a simplified approach where each element pi of the structure induces a distinct (sub) game which is analyzed separately from the others (being defined over disjoint sets of players).

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3.6. Remarks on practical implementation and computational analysis In this section we briefly discuss practical issues in a possible implementation of the proposed service. The Cooperation Server promotes the service and collects subscription of users interested in its product(s). The nodes registering to the service will agree to the term and conditions for the service. Product licenses will be active only after the cooperating process is ended. Thus, in case of a misbehavior of a node in cooperation, this can be easily detected, and (for instance) the license of such a node may be set to an invalid state by the provider. Note that a basic implementation of cooperative-downloading may be designed as an atomic transaction, as the involved group of users is intended to be static during the process. In this approach, if some node goes down during the process and is not able to recover within some reasonable time-out, then the process fails and a restart or some form of renegotiation is required (indeed other nodes may then be accepted, and if no further node comes in, then the price of the licenses may be higher for the resulting smaller group). For each service subscriber, a user profile will be defined collecting basic information provided when the device registers to the service. In general the Cooperation Server could be available on the Internet and accessible by the service subscribers from any location. The Cooperation Server will then wait for nodes to contact him providing all information needed for the service, including the geographic position of the node (e.g., determined through its GPS coordinates or any other positioning technique). A requirement for the nodes to be considered as part of a common content download, is that to be interested in the same content (possibly, a bundle of products) and in mutual coverage for a cooperative short-range link. Sample scenarios where these conditions are fulfilled can be groups of friends in aggregation places (such as a University campus) to exchange and download books, music or other contents. Once the users have provided the required information, the Cooperation Server notifies each device about: involved devices and how they are clustered, the masternslave role played by each node in the piconet (only in the analyzed case of using Bluetooth as short-range link), the exact data share to download through cellular links and the corresponding cost, and the content price that will be charged to the user account. All cooperating devices will then first download the assigned file fractions over their cellular links and then share them over the short-range link. During the first phase of cellular downloading, they can, in parallel, setup the Bluetooth piconet. Bluetooth limits the number of nodes simultaneously active on a piconet to 8; therefore, the Cooperation Server collects a maximum number of eight candidates requiring the service within a limited period of time, before computing the solutions. If a ninth device contacts it, then a new process is started. Obviously if less than eight nodes are interested, these could as well cooperate (for instance, a suitable timer can determine a time interval wherein the Cooperation Server collects the cooperation requests before starting the cooperation process). Finally, it is worth spending some words

concerning the computational cost of the presented framework. The Nucleolus computation is actually playing a minor role in the overall computational costs, once the cost function is known. In fact, the major contribution to the computational cost is given by the linear programming optimization step that is performed for every possible coalition, which means 2jNj  1 times. Moreover, for each coalition, a number of minimization problems proportional to the number of nodes k in the coalition is required to define the master and slave roles. The total number of minimiza  P N tion problems is: nmin ðNÞ ¼ Nk¼2 k . However, it is k worth mentioning that executing the introduced framework requires a few seconds, as tested with a basic hardware deployment; this order of magnitude is definitely acceptable for the proposed file sharing service.

4. Experimental evaluation A numerical evaluation of the model is conducted to observe the behavior of the envisaged paradigm under a wide range of system configurations. Main objective is to validate the ‘‘mediated wireless cooperation’’ by looking both at cost savings for the single nodes (compared to a standalone classical content download) and at providers’ profits, under different conditions. It will also clearly emerge that a game theoretic approach based on the Nucleolus is preferred to standard optimization criteria, as far as the fairness perceived by users is concerned. The results presented next are related to Bluetooth as short-range link, but the overall framework can equally work with any other short-range technology. We considered sample cases where cooperating nodes have different cellular throughput levels. All presented cases assume that nodes have a time-based billing agreement with the provider; we express this cost in terms of a given amount of generic Cost Units per second (CU). Note that from these measurements and from the size of the product file we may immediately compute the values cdi and ti characterizing the device features of any node i. Also the alternative billing policy foreseeing that nodes are charged on a data-amount basis has been investigated. As there is no conceptual difference, the results will refer to the first case only. In particular, the cost per second is set to 0.05 cost units, the file size is 100 Mbyte, and the basic content price price is equal to 400 cost units. Concerning the power consumption on both the cellular link and the Bluetooth interface, the values used in the present paper result from measurement campaigns (conducted on N95 Nokia smartphones). In particular, different values of power consumption have been measured for a device connected to a 2.5G or to a beyond 2.5G system (400 mW for 2.5G, 1400 mW for beyond 2.5G and we assume 250 kbps the maximum data rate value for a 2.5G connection). For the Bluetooth the adopted values are 178.2 mW, 108 mW and 59.4 mW in transmission, reception and idle time respectively. Finally, the energy consumed by the nodes to gather their GPS positioning is set according to results in [34], whereby the energy consumption for a Nokia N95 smartphone is equal to 13.32 J in the best case.

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Fig. 2. Content discount function for multi-license products.

It is assumed that the Cooperation Server chooses a price function of the type described in Section 3.2.1. More specifically, we next consider cardinality-discount CDC games whose discount functions are based on the functions proposed in [35], suitably adapted to the scope. Fig. 2 reports the discount proposed by the content provider w.r.t. to the normal basic price for different number of nodes. The plotted equation is given by

dq ðjSjÞ ¼ Maxdiscount 

exp½ðjSj  1Þ=q  1 exp½ðjSjmax  1Þ=q  1

ð6Þ

with q – Infinity where jSj and jSjmax are respectively the cardinality of the coalition and the maximum cardinality (when not differently stated, in our experiments this is set to 8), q determines the discount amount for either larger or smaller coalitions, Maxdiscount represents the highest discount proposed by the provider which is set here to 50. It is easy to see from the formula that the highest 50% discount is offered when the coalition size jSj is equal to jSjmax, while no discount is given to the noncooperating nodes when jSj is equal to 1. Based on the above discounts, we get the following class of cardinality-discount price-functions:

pq ðSÞ ¼ price  jSj 

100  dq ðjSjÞ 100

ð7Þ

We use these price functions for the experiments that illustrate the proposed framework. In particular, we focus on the case with q = 2, as this offers higher savings immediately, encouraging the users to join cooperating groups also of very small sizes. Interestingly, for positive q values and in particular for the chosen q = 2 value, sub-modularity does not hold for such a function, because the marginal discount obtained by larger coalitions is smaller than the marginal discount obtained by smaller coalitions (see

Fig. 2). Thus, the considered games are not concave, but from Proposition 2 we know that the core is always not empty. The Cooperation Server is made aware of the limit on the amount of data to be downloaded in the P2P application imposed by the nodes. In general each node could request different values for this parameter, but for simplicity in the presentation of the results and to better discuss the influence of this parameter, we assume all nodes having the same DownloadBoundi, hereafter just called DownloadBound. Several throughput distributions for the nodes can be considered. A straightforward test scenario is characterized by ‘‘homogeneous’’ nodes, i.e. nodes with the same cellular throughput level. In this case, if there are no further constraints, the solution found by the bandwidth-cost optimization procedure performed by the service coordinator assigns an equal share, hence an equal cost, to all nodes. Such a homogeneous cost allocation is in fact the Nucleolus of the associated game and it belongs to the core (in fact, it is also the Shapley value), and no further analysis and cost compensations are required. For any scenario different from the cited ‘‘homogeneous’’ one, the proposed game theoretic model proves that defining a fair cost allocation for the participating nodes is unavoidable. During the study, a wide range of scenarios with variable throughput distributions for the nodes have been considered. The results are also contrasted with the possible reasonable outputs of some alternative approach, which rely on an optimization only (i.e., not based on game theoretic concepts). The sample scenarios presented in the remaining part of the paper are selected to give the reader a broad overview of the proposed framework behavior. They are characterized by half of the nodes with a cellular throughput of T = 100 kbps, while the throughput of the remaining nodes equal to (T + d) kbps, with d variable. Clearly, a similar analysis can be repeated for different nodes configurations, obtaining similar plots.

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4.1. File-shares assignment, node costs, and service provider profit With reference to the output of the cooperative game, let us focus on the cost assignment to nodes, which is also equal to providers’ profit. The cost assigned to each node is a combination of the content license cost and the cost deriving from the download of the assigned file-share over the cellular links. Of course, these values directly depend on DownloadBound and node throughput values, while also the GainConstraint of the network provider influences the final results. These dependencies are investigated next, while it is assumed the nodes have not set stringent constraints on the energy consumption and time delay in cooperation. In Fig. 3 it is shown how the file-shares to be downloaded by nodes, hence their download costs, change with the parameter DownloadBound and the GainConstraint; this behavior is highlighted in sample cases in which d = 100 kbps. Recall that such file-shares are assigned by the final minimization step, as described in Section 3.3. Let us focus first on the case with GainConstraint (in short GC) set to zero and analyze the influence of the DownloadBound (in short DB), left side of the plot. To minimize the maximum download time according to linear program (3), the solution is to assign larger portions of data to be downloaded to the four nodes with higher cellular throughput (nodes 5–8 in the scenario). For any scenario with DownloadBound P 0.25 these four nodes will download 1/4 of the file each. When the nodes set more stringent constraints on the DownloadBound, also the other nodes are involved. For any value DownloadBound <0.125 (that is maximum 1/8 of the file) no feasible solution can be found, since the total number of nodes is 8. Note that similar considerations hold for different values of the total number of nodes in the coalition; obviously, these lead to different limiting values for the DownloadBound value. Moreover, it is worth commenting that in a more general system setting, where all values of nodes’ cellular throughput are different, again to minimize the maximum download time according to linear program (3), the solution is to assign a larger portion of file to download to the nodes

with higher cellular throughput as far as the DownloadBound constraint allows it. To assess the impact of the GainConstraint set by the network, we analyze the case where DownloadBound is set to 0.25. We can compare the scenario with GainConstraint not set (the third case in the left side of the plot in Fig. 3) and the scenario corresponding to the results reported on the right side of the plot in Fig. 3. Two main effects of an increase in the GainConstraint parameter emerge. First, the nodes with a less performing cellular link are now also involved in the content download over the cellular link. This is justified by the increase in the bandwidth income required by the network provider. When the GainConstraint increases even more, the second effect is that the cooperating cluster is forced to download more than one copy of the total file (e.g. two copies are downloaded when GC = 0.25). Next, we want to investigate on the costs repartition among the nodes in the considered scenarios. In Fig. 4 the cost for the content license and the content download is reported for the nodes in the sample scenarios with d = 100 kbps, GainConstraint = 0 and the DownloadBound set to one of the following values: 0, 0.125, 0.2, 0.25. In Fig. 5 instead, the same information is reported when GainConstraint assumes one of the following values 0, 0.125, 0.2, 0.25, while d = 100 kbps and DownloadBound = 0. What can be observed in both of the figures is that a node having higher costs for the content download, will pay a smaller contribution for the content license. This observation has a general validity also in a system setting with more differentiated values of cellular throughput for the nodes. It can also be observed that the cost for the content is often higher, for some nodes, than the cost given by the price function pq(S). The reason for this is that it has also to cover the costs for the node ’’more devoted to the content download’’. A numerical example is given to clarify the latter comment. Let us focus on a node i with cellular throughput 100 kbps and a node j with cellular throughput 200 kbps. For the specific bandwidth cost and file size settings (0.05 CU per second and 100 MB respectively), if nodes do not cooperate then they will be charged the following

Fig. 3. Influence of DownloadBound (DB) and GainConstraint (GC) on the file portions assigned for download to nodes; d is set to 100 kbps.

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Fig. 4. Cost distribution among the nodes according to the Nucleolus solution for different values of DownloadBound and d = 100 kbps, GainConstraint = 0.

Fig. 5. Cost distribution for the nodes according to the Nucleolus solution for different values of GainConstraint and d = 100 kbps, DownloadBound = 0.25.

basic costs: p(i) + cd(i) = 400 + 400 CU = 800 CU; p(j) + cd(j) = 400 + 200 CU = 600 CU. The nodes may decide to contribute to the cooperative download by fixing the upper bound on their possible downloads to 0.2, and expect to have the following maximum costs: pq(i) + DownloadBound  cd(i) = 200 + 80 = 280 CU; pq(j) + DownloadBound  cd(j) = 200 + 40 = 240 CU. This means an overall saving for the service equal to: Saving(i) = 520 CU (corresponding to a 65% saving) and Saving(j) = 360 CU (corresponding to a 60% saving). If we now observe the cost assigned to these nodes by Nucleolus (see Fig. 4), we notice a cost of 250 CU and of 200 CU for nodes i and j, respectively. These results lead to the following final savings w.r.t. to the non-cooperating case: Savingfair(i) = 550 CU (corresponding to a 69% saving) and Savingfair(j) = 400 CU (corresponding to a 67% saving). This simple computation demonstrates that the final savings meet the user

constraints, and this is done in a fair way, i.e. by considering the ‘‘merits’’ of each node. The next analysis presented in Fig. 6 shows the influence of the DownloadBound and the GainConstraint parameters on the total cost C(N), which equals the total providers’ profit. This value decreases with increasing values of the DownloadBound, since for lower DownloadBound values also nodes with lower bandwidth costs have to be involved in the download phase. On the other hand, it increases with the GainConstraint parameter. Interesting to observe is how two of the plots overlap completely (the case with DB = 0.25; GC = 0.125 and the case with DB = 0.125; GC = 0) even if the settings are different and the specific cost allocations to the nodes are different as shown in previous plots. Moreover, in Fig. 6, the influence of d value is also plotted. The decreasing trend when d is increasing is expected, because higher throughput values

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Fig. 6. Content and network provider profit in cooperation, variable d according to the Nucleolus solution.

mean faster content downloads and, thus, lower profits for the provider (as costs are assumed to be time-based). 4.2. Nucleolus vs. proportional cost distributions Previous plots testify to the good behavior of the Nucleolus in terms of cost savings, we next focus in more detail on fairness aspects, when this approach is compared with alternative solutions. In particular, one may wonder whether alternative reasonable approaches exist that are not based on game theoretic principles. We already observed that traditional optimization techniques aim at minimizing (or maximizing) some measure like the total cost, but typically do not care about fairness in the final cost assignment to the participating nodes. According to the specific application and objective, different optimal solutions can be proposed (e.g. for solutions oriented to delay in content distribution [8]). In our case, to compare such techniques with the Nucleolus solution, we next consider the following simple cost-distribution, that we call Optimization + Proportional Costs. Following a first step, where the optimal cost C(N) is computed according to the linear program presented in Eq. (1), this cost is then assigned to nodes, proportionally to the bandwidth costs, i.e. according to the cdi parameter Pcdi , for each i 2 N. for each node: xpr i ¼ CðNÞ  cd j2N

j

In Table 1 a comparison is presented among this proportional cost solution, the solution given by the linear program optimization, and the Nucleolus. Two sample cases are presented, but similar results are obtained for many other cases. In particular, the presented cases refer to scenarios where DownloadBound is set to 0.25 for all nodes, GainConstraint is set to zero, nodes 1–4 have cellular rate Rc1 = 100 kbps and nodes 5–8 have cellular rate Rc2 = (100 + d) kbps with d either equal to 100 or 1500. Besides the differences in the cost allocation, the experiment aims at observing whether the proposed allocation is stable according to the core and if it actually meets the cost/savings constraints set by the users. As it can be observed, the

optimal solution is not always a stable allocation, while the proportional cost allocation is not always meeting the constraints for the nodes. 4.3. Considerations on the Shapley value as alternative cost allocation method Note that our framework is rather independent from the particular game theoretic solution concept, as long as its notion of fairness fits well the proposed application. We thus briefly discuss the possible results obtainable with cost allocations following the Shapley value, a wellknown solution concept which is a valid alternative, and whose main properties have been described in Section 3.1. The numerical evaluation presented in the previous sections has been repeated by considering the Shapley value instead of the Nucleolus. In some experiments, the Shapley value is not a stable imputation for the coalition involving all the nodes, as it is outside the core and does not fulfill the individual rationality. This is not surprising, if we consider that our cost games are not concave for the chosen price function. However, as proved in Proposition 2, the core is not empty and hence the Nucleolus does not suffer from those drawbacks. Another observation is that, when using the Shapley value, the proposed framework requires a higher computational time. This varies with the different presented cases, but even increases up to 70% in the time delay with respect to the Nucleolus case have been reached. These considerations support our choice of focusing on the Nucleolus as the preferred solution concept for the proposed framework. When the Shapley value is not in the core for the game involving all the nodes, alternative coalitions are considered with a reduced number of cooperating nodes. The main consequence is that the savings introduced for the nodes are reduced. This is clearly plotted in Fig. 7, where the average cost savings per node when using the Shapley value or the Nucleolus are considered, in different scenarios with variable d and GainConstraint parameters, and DownloadBound = 0.25. In all the plotted cases, the Shapley

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Table 1 Comparison among optimal, proportional, and game theoretic cost allocations; DownloadBound is set to 0.25 for all nodes, GainConstraint is set to zero, nodes 1–4 have cellular rate Rc1 = 100 kbps and nodes 5–8 have cellular rate Rc2 = (100 + d) kbps. d = 100 kbps

Optimization

Proportional

Nucleolus

d = 1500 kbps

Nodes 1–4

Nodes 5–8

Nodes 1–4

Nodes 5–8

Cost per node (CU) Savings per node (CU) Constraints met Stable

200 600 Yes Yes

250 350 Yes Yes

200 600 Yes No

206.25 218.75 Yes No

Cost per node (CU) Savings per node (CU) Constraints met Stable

300 500 Yes Yes

150 450 Yes Yes

382.3 417.6 No Yes

23.9 401.1 Yes Yes

Cost per node (CU) Savings per node (CU) Constraints met Stable

250 550 Yes Yes

200 400 Yes Yes

250 550 Yes Yes

156.25 268.75 Yes Yes

Fig. 7. Shapley value vs. Nucleolus: average per node savings with variable d, DownloadBound = 0.25.

value does not guarantee that the grand coalition involving all 8 nodes is formed. In these cases, either 1, 2, 3 or 4 nodes are excluded from the coalition and, thus, lower savings are obtained when compared to the Nucleolus solutions. Only in a few cases a second coalition is formed that involves the excluded nodes. In particular, this happens for all cases with d P 1100 and GC = 0.2 and GC = 0.25, where always two coalitions of four nodes are formed. For the other scenarios, the GC = 0.2 and GC = 0.25 cases register a lower average cost saving w.r.t. the case with GC = 0.125. The reason for this is related to the coalition being formed and the nodes being excluded from cooperation. As an example, let us consider the d = 200 scenario. With GC = 0.125 a 6-nodes coalition is formed where two nodes with a higher cellular throughput and thus lower costs in non-cooperation are excluded; with GC = 0.2 a 5-nodes coalition is formed where now three nodes with low costs in non-cooperation are excluded; with GC = 0.25 a 6-nodes coalition is formed where this time two nodes are excluded with high costs in non-

cooperation. Similar considerations can be made for any other case. 4.4. Savings for different coalition sizes and discount function settings Results presented in previous sections have focused on the behavior of the grand coalition where fair cost allocations are found. Next, we first give the reader some insights on the benefits the nodes gain in joining larger coalitions instead of smaller sub-coalition. Then, we will show how the maximum coalition size set by the service provider in the discount function influences the benefits for cooperating users and provider. Fig. 8 shows the average per node savings obtained in joining coalitions of different sizes. The grand coalition size is jSj = 8, while also the sub-coalitions are considered with 2, 4, or 6 nodes (also these coalitions have the same cellular throughput combinations as for the grand coalition). The DownloadBound and the GainConstraint are not set for

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Fig. 8. Average per node savings for different coalition sizes jSj with variable d.

the plotted case for simplicity. As it clearly appears from the plots, and as expected, nodes have always higher monetary savings in joining larger coalitions. The subsequent analysis focuses on the influence that the maximum acceptable coalition size, set by the service provider in the definition of the discount function, has on the final monetary savings for the nodes and the overall profits for the service provider. This analysis has also the further objective to investigate on the possible trade-off between the service provider and the cooperating nodes interests. These are clearly conflicting like their objectives, with the service provider wishing to maximize the monetary incomes for the service, and the users wishing to minimize the service costs. Going into details, the service provider can decide to limit the maximum coalition size by tuning the value for jSjmax in Eq. (6). Changing this setting, the first effect is a modification in the discount func-

tion plotted in Fig. 2, with a maximum discount obtained at the corresponding jSjmax value. The jSjmax values we considered for this analysis are 2, 4, 6 or 8, while the choice in terms of q for the discount function is kept constant, namely q = 2. We consider some sample scenarios with either DownloadBound = 0 (see Fig. 9) or DownloadBound = 0.25 (see Fig. 10), where half of the nodes have a cellular throughput level Rc1 = 100 kbps and the second half have a cellular rate Rc2 = (100 + d) kbps with d either equal to 100 or 1500. As expected, the results in Fig. 9 show that the service provider profits increase with lower values of jSjmax, while the users savings increase with larger values of jSjmax. Moreover, a generally valid trade-off between the network provider and the users interest cannot be found. For instance, Fig. 9 shows that, when considering coalitions of only two nodes (i.e. with jSjmax = 2), the absolute values

Fig. 9. Average per node network profit and average saving per node, for different coalition sizes and jSjmax in Eq. (6) with q = 2; DownloadBound = 0.

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Fig. 10. Average per node network profit and average saving per node, for different coalition sizes and jSjmax in Eq. (6) with q = 2; DownloadBound = 0.25.

Table 2 Impact of the energy savings constraint on the cooperative content download; DownloadBound, GainConstraint and Time constraints are not set, nodes 1–4 have cellular rate Rc1 = 100 kbps and nodes 5–8 have cellular rate Rc2 = 200 kbps. Energy constraint

Cooperative coalition size Avg energy savings (J/Kbit) Avg monetary savings (CU) Max time delay (%)

0.2

0.4

0.6

0.8

1

– – – –

8 2.22 468 44

8 2.27 475 31

8 2.27 475 31

8 2.27 475 31

of the objective functions for the service provider and the users are the closest. By increasing the value of jSjmax, and thus the maximum coalition size, the two objective functions diverge. The observed trends for a variable value of jSjmax, suggest to use the case with jSjmax = 4 as a potential trade-off point. Nevertheless, when changing the DownloadBound value, like in Fig. 10, the situation completely changes and the same conclusions are not valid anymore. Thus, we can conclude that the discount function proposed by the service provider has definitely a key importance in the definition of the system performance. Unfortunately, it is not possible to design a generally valid function that guarantees a trade-off between service provider and user objectives in all potential scenarios.

4.5. Influence of the energy consumption and time constraints In the analysis presented so far, the focus has mainly been on monetary savings introduced by the proposed framework and the influence of the related constraints. In this section, the attention is put on the further constraints the nodes can set when joining the cooperative service, namely the energy consumption and the time constraints. To best highlight the impact of these two constraints, we present sample scenarios where the DownloadBound and the GainConstraint are not influencing the solutions. We consider only the Nucleolus solution for a sample scenario where nodes 1–4 have cellular rate Rc1 = 100 kbps and nodes 5–8 have cellular rate Rc2 = 200 kbps (a similar analysis can be repeated for other cases) and the energy consumption and time constraints are equal for all nodes, for simplicity. In Table 2 the values of the main performance indexes are reported for increasing values of the energy consumption constraint set by the users, while no time constraint is considered. In particular, energy consumption constraint values are considered in the range 0.2–1 (see the problem definition in Eq. (1)). As it clearly emerges from the results, the cooperative content download actually introduces energy savings for the nodes. This is not a surprising result, see e.g. [23]. What can be observed is that for energy constraints from 0.6 and above no difference in the final performances is obtained. Moreover, for an energy constraint

Table 3 Impact of the download time constraint on the cooperative content download; DownloadBound, GainConstraint and Energy constraints are not set, nodes 1–4 have cellular rate Rc1 = 100 kbps and nodes 5–8 have cellular rate Rc2 = 200 kbps. Time constraint

Cooperative coalition size Avg energy savings (J/Kbit) Avg monetary savings (CU) Max time delay (%)

0.4

0.6

0.8

1

1.2

1.4

1.6

– – –

4 2.6 460 42.5

4j2j2 2.6j199j199 460j181j181 42.5j26.6 j26.6

5j3 1.4j2.03 378j397 15j31

7 2.3 474 20

8 2.3 475 31

8 2.3 475 31

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of 0.2 the nodes will actually not cooperate, while in all other cases always the grand coalition of 8 nodes is formed and both energy and monetary savings are obtained. Finally, also the maximum time delay experienced by the nodes in cooperation is reported. What emerges is that a maximum delay of 44% is reached, which is experienced by the nodes with the fastest cellular throughput (the nodes with Rc2 = 200 kbps in the specific case). In Table 3 the same performance indexes are presented for variable values of the time constraint set by the users, while no energy constraints is set. Also for the time constraint, cases exist where actually a gain is obtained in cooperation (see the cases with negative delay values reported in the Table). For time constraint values equal to 0.4 and below no cooperative download can be activated due to the stringent constraints. In the other cases, instead, the main effect of setting the constraint is that smaller coalitions are formed and different average energy saving and monetary saving are obtained by the nodes in the coalitions. Finally, when the time constraint is not so stringent (from 1.4 and beyond) again the grand coalition of 8 nodes is formed and no influence of the time constraint on the results is observed.

5. Conclusions This paper uses the powerful mathematical framework of Game Theory to enable a mediated business model, according to which a provider promotes itself as a service coordinator for a cooperative file sharing service. Nodes cooperatively download a file of common interest over their cellular links and share it over a cost-free short-range link. The idea is to give the provider the opportunity to sell a larger number of licenses by proposing suitable groupdiscounts for content to be downloaded. On the other hand, nodes have significant cost savings when compared to a stand-alone download of the whole content, as they benefit both of a reduced cost for the content and of an optimal bandwidth cost-distribution over the cellular link. Moreover, they benefit from a ‘‘legal’’ service with the additional important guarantee that costs are fairly assigned to nodes and there are no free riders. Since fairness and stability of a solution is of utmost importance, the use of Game Theory to model the cost distribution problem proved to be an effective approach. Based on a coalitional transferable-utility cost game, the Nucleolus has been adopted as a valid solution to determine the fair cost allocations for the cooperative cluster. A number of properties of the proposed approach are shown and commented under a wide range of sample operational conditions.

Acknowledgments Special thanks go to Miguel Ángel Mirás Calvo and Estela Sánchez Rodríguez from University of Vigo and to Jean Derks from University of Maastricht for their important suggestions and the Matlab toolboxes they have kindly shared with us for the numerical evaluation campaign for this paper.

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Leonardo Militano is a research fellow at the University ‘‘Mediterranea’’ of Reggio Calabria, Italy. He received a Ms. degree in Telecommunications Engineering from the University of Reggio Calabria, in 2006 and his Ph.D in Telecommunications Engineering in 2010 at the same university. He has been a visiting Ph.D student at the Mobile Device group at University of Aalborg, Denmark. His major areas of research are wireless networks and user cooperation.

1973

Antonio Iera is a Full Professor of Telecommunications at the University ‘‘Mediterranea’’ of Reggio Calabria, Italy. He graduated in Computer Engineering in 1991 and received a Ph.D. degree from the University of Calabria. From 1994 to 1995 he has been with Siemens AG in Munich, Germany to participate to the RACE II ATDMA project under a CEC Fellowship Contract. Since 1997 he has been with the University Mediterranea, Reggio Calabria, where he currently holds the position of Head of the Department DIMET. His research interests include: new generation mobile and wireless systems, broadband satellite systems, Internet of Things.

Francesco Scarcello received the PhD degree in Computer Science from the University of Calabria in 1997. He is an associate professor of computer science (SSD ING-INF/05) at the University of Calabria. His research interests are computational complexity, Game Theory, graph and hypergraph theory, constraint satisfaction, logic programming, knowledge representation, non-monotonic reasoning, and database theory. He has extensively published in all these areas in leading conferences and journals. Professor Scarcello serves on program committees and as a reviewer for many international conferences and journals.