A Fair Cooperative Content-Sharing Service

L.Militanoa,∗, A.Ieraa, F.Scarcellob

aUniversity Mediterranea of Reggio Calabria, DIIES Department, ItalybUniversity della Calabria, Cosenza, DIMES Department, Italy

Abstract

Wireless cooperative content sharing, based on the synergistic use of cellularand short-range technologies, has recently gained much interest from aca-demic and industrial communities. Besides energy saving and informationtransfer delay reduction, this paradigm can enable a significant reduction incellular bandwidth usage, which also means monetary saving for users. Infact, such a cooperation is usually opposed by network and service providers,because it strongly reduces their potential profits. This paper deals withthe provider perspective, too. In particular, a ”mediated cooperative be-havior” is proposed and analyzed within a scenario of short-range wirelessfile-sharing. The basic idea is that providers offer the possibility to users ofcooperatively downloading contents, and increase their own profits becausemore users are attracted by such a service. Indeed, by participating in thedevised service, users both avoid a (possibly expensive) stand-alone down-load of the desired product, and benefit of a special group-discount. Thecosts (and download tasks) distribution among users is based on a cooper-ative game theoretic model. Indeed, suitable solution concepts are appliedto provide a fair solution, acceptable by all users, and hence to overcomethe limitation of traditional optimization approaches to costs (and tasks)distribution.

Keywords: Wireless Cooperation, Fairness, Content sharing, Nucleolus,Game Theory, mediated P2P process

∗Corresponding author. University Mediterranea of Reggio Calabria, DIIES Depart-ment, 89100 Reggio Calabria, Italy. Email: leonardo.militano@unirc.it. Telephone andfax: +39 0965 875276

Email addresses: leonardo.militano@unirc.it (L.Militano),antonio.iera@unirc.it (A.Iera), francesco.scarcello@unical.it (F.Scarcello)

Preprint submitted to Computer Networks April 2, 2013

1. Introduction

Cooperation over short-range links among cellular devices is a paradigm,which has recently gained wide interest in the research community [1]. Ac-cording to it, groups of users interested in a common content and in proximityto each other, might cluster together and exchange, over cost-free, energy ef-ficient, and fast short-range links, a content downloaded through the costlycellular link. Advantages from the end-user point of view, in terms of energyefficiency, throughput enhancement, and cost reduction, are quite evident [2].

However, we are also interested in the network and service provider per-spectives, describing a framework where providers may have economic ben-efits, too. The wireless cooperation paradigm is conceptually very close toclassic P2P and file sharing services, which show similar issues still requiring asolution. P2P file sharing services are nowadays highly successful among theyounger generations and it is a common practice to share multimedia contentsuch as music, videos, images, e-books, or similar, through P2P platforms tominimize the monetary cost of the content.

Of course, whenever any form of content sharing violates copyright con-straints, this behavior may cause large profit losses to content and networkproviders. The main reason that moves the end-user to go the way of ille-gality 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 stopthis phenomenon. In the authors’ opinion, a successful reaction for providersis to promote themselves as an active part of the system, by offering whatwe call a ”mediated cooperative framework for file sharing service”. The ideais that users in a group will be able to select a special P2P-option when theybuy some product. This way, they get a group license that benefits of a highdiscount with respect to standard single-user licenses, and they are allowedto freely share the product within the P2P wireless network of the group.Moreover, users download only some parts of the whole product over thecostly phone-link. Indeed, the service coordinator will compute and assignto users the fractions of the product to be downloaded by each of them, bytaking into account download costs, time and the energy consumption.

The main advantage for providers would derive from the increased chancesto involve in the business those nodes (the majority) inherently oriented to”legality”, but that avoid to access the provider downloading-service becauseprices are too high (at least with respect to their actual interest in that prod-uct). Indeed, in the proposed framework, users may save money because of

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the special group discount, so that the actual price to be paid could matchthe actual value/interest they assign to the product. Moreover, they havefurther pros in terms of energy/throughput/download cost, as well as the ad-ditional security factor guaranteed by a downloading-service under the directcontrol of providers (instead of possible illegal and uncontrolled alternatives).

Surely, to make this ”mediated wireless cooperation” mechanism workproperly, a key feature that the provider shall promote, to catch the interestof more users, is the implementation of a cost distribution among cooperatingnodes, which is judged fair by all the players. This feature may be the onlyeffective incentive to access the service since, in real environments, wherenodes are rational and selfish players, it is difficult to reach such an agreementin cooperation. In particular, any proposed cost allocation should also betested according to the stability property, guaranteeing that no (sub)groupof users has an incentive to leave the cooperation group and to form a differentcoalition.

Traditional optimization techniques show their limits in guaranteeingthese requirement for all the participating members. Indeed, they only dealwith the minimization of some global measure (total cost, energy, time, etc.),but provide no indication on how to divide such an optimal value amongnodes while considering their individual or joint contribution to the mech-anism. Therefore, the framework should consider solutions that guaranteefairness and stability in the cost distribution allowing the selection of thebest suited coalition partition for the interested users. This way, the fare tobe paid by each player (member of the group) takes into account his contri-bution to any possible coalition in terms of download costs of his terminalover the cellular link, possible agreements with the provider (e.g., frequentbuyers), and so on. In particular, the service coordinator will act as an ”im-partial” entity guaranteeing that only solutions are applied that are judgedfair, according to the proposed (publicly available) criterion. Moreover, sinceevery player contributes to the process (at least allowing the group to obtaina higher multiple-license discount), the mechanism also avoids free riding, atypical behavior of pure peer-to-peer systems where some nodes get benefitsfrom 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 sys-tems, allowing the service providers to participate in the cooperation processand benefit from it as well; (ii) to include in the framework some proposalsthat guarantee fairness and stability in the cost distribution as an alternative

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to traditional optimization techniques; (iii) to model and analyze well-knownsolution concepts from cooperative Game Theory, to guarantee fairness in thecost distribution for the specific problem; (iv) to perform an analysis on themonetary savings and profits for the involved entities under system and userconstraints related to personal and technological parameters.

The rest of this paper is structured as follows. Next section gives anoverview of related work relevant to core issues and techniques under inves-tigation in this research. A detailed description of the reference cooperativefile sharing service, the user cost minimization step, and the game theoreticmodel for the problem are given in section 3. A thorough performance evalu-ation is presented in section 4, while final remarks are given in the conclusionsection.

2. Related Works

From the literature, several research contributions are available, whichpropose improvements in the performance of remote content downloads [3].Solutions are, for instance, based on bandwidth aggregation of multiple in-terfaces belonging to either the same device [4] or to different nearby devices[5]. Interesting contributions also address modeling and evaluation issues rel-evant to communication architectures that exploit the synergy between cel-lular and short-range systems. As an example, [6] and [7] show how cellularand short-range networks (WLAN, Ad-hoc, and MANET) can be integratedto improve the performance levels. In [8] the attention has been given to theenergy consumption and transfer delay benefits obtainable through this com-munication architecture. In contrast, issues such as end-user cost reductiondue to bandwidth occupation decrease and network/content provider profitlosses have attracted a little attention up to now. The focus of the presentpaper is to cover this lacking aspect, trying to introduce a service model ableto attract the interest of network/content providers for the wireless cooper-ative content sharing. This is actually of utmost importance for the successof the framework. Cooperation over short-range links and classic P2P showsimilar problems from the economics point of view for the network/contentprovider; thus, it is worth recalling P2P related researches about this issueto 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 consid-ering either to cooperate with or to fight against them. There is much ongoing

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research activity trying to understand socio-economic aspects associated tofile sharing services, like for BitTorrent in [10]. Moreover, in proposing con-tent distribution services, counteractions to fight against unauthorized illegalcontent publishers represent an important part of the service model [11]. Asclaimed in [12], distribution cost is a key reason that prevents the contentproviders from making great amounts of content available for download inP2P content distribution networks. Furthermore, [12] points out that in clas-sic P2P systems costs are pushed from the content distribution networks tothe Internet Service Providers (ISP). On the other hand, in wireless systems,user cooperation over “private” short range links reduces the overall incomefor the ISP.

The framework considered in the present paper, instead, looks at theprovider as a “promoter” or “mediator” of a P2P-like file sharing service,which controls and coordinates the service and, in doing so, obtains somebenefits for itself. A way to reach this objective is to promote highly attrac-tive (but monitored) cost reductions for users who cooperatively downloadsome contents. On the other hand, a suitable game theoretic formulation al-lows us to achieve this goal while taking care of fairness and stability aspects,too.

Game theory is an analytical framework that attempts to analyze thebehavior of rational entities with their own interests in reciprocal interac-tions [13]. Recently it has been applied also to various research fields rele-vant to wired and wireless communication networks [14]. Among cooperativegame based contributions, the so-called coalitional problems are gaining largeinterest within the research community [15]. In [16] the focus is on vehic-ular networks, in [17] a coalitional problem is proposed to study fairnessand cooperation gains in virtual MIMO systems, in [18] packet forwardingissues in ad-hoc networks are addressed, while in [19] a task allocation prob-lem is studied in a software system. Game theoretic models have been alsoapplied to pricing schemes in cognitive wireless networks [20] or in hetero-geneous wireless networks [21]-[22]. In a different study [23], the authorsof the present paper also showed how cooperative Game Theory can be ap-plied to provide a fair energy consumption cost-distribution in a cooperativecluster in which the introduced communication-systems constraints play asignificant role. Moreover, a recent interesting contribution exploits GameTheoretical notions to deal with fairness in peer-assisted services for contentdelivery networks (for live streaming and similar highly resource-demandingcontents) [24]. Indeed, such networks would benefit from a peer-to-peer archi-

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tecture to reduce their operating costs, and the authors study a fluid Shapleyvalue approach to provide a suitable incentive scheme for users cooperatingin the content distribution process.

P2P systems are often studied through non-cooperative game theoreticmodels to introduce incentives or to coordinate the system (for example[25] and [26]). The mediated service framework proposed in this paper isnot self-organizing because it is controlled by the provider. At the sametime, the users do not need any incentive to cooperate. Indeed, nodes arenaturally incentivized to cooperate by the immediate monetary savings theyexperience, where the larger is the coalition (group) the larger is the discount(controlled by the provider). In fact, the monetary based economic-model isa point of strength of the cooperative process proposed in this paper, andcooperative Game Theory is the most natural framework to study agreementsand interactions among such involved entities.

Figure 1: Wireless cooperative scenario.

3. The Cooperative File sharing Framework

With reference to the service framework introduced in section 1, assumethere is a content provider P that offers the P2P group-option to its clients,and plays the role of the Cooperation Server coordinator plotted in Figure1. In order to reach its goal of selling licenses of its product to all interestedusers (also, nodes), the service coordinator implements a policy aiming at

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minimizing the overall monetary cost for the users, while allowing a costdistribution that is perceived to be a fair and stable one. These are thecompulsory conditions for the wireless cooperative process to be acceptedby all parties. Actually no unique definition can be given for what a fairsolution is, but several solution concepts have been proposed over the yearsin 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 actuallyindependent of the specific chosen notion.

Let N be a group of users interested in downloading some product fromthis provider. Nodes start the procedure by contacting the CooperationServer P and requesting the desired product with the P2P option. Each nodesends information about its device interface identifier, cellular link through-put, cost per second of the cellular link (or alternatively the cost per unitof 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 themonetary 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 adetailed cellular model is not among the objectives of the paper. Differentlythe aim is to analyze the impact the bandwidth costs have on the cooperativesolutions. Therefore, in our performance evaluation study, we considered awide set of cellular throughput values to assess how the cooperative scenariosare influenced by this parameter. Users may also specify some constraintsover their contribution to the proposed P2P framework: any user i may im-pose a limit on the amount of data to be downloaded in the P2P application,expressed as a fraction 0 ≤ DownloadBound i ≤ 1 of the desired productfile, which of course entails a bound DownloadBound i · cdi for the maximumdownload cost for player i. While the short-link technology is available forfree to all users in N , we consider the energy consumption and the contenttransfer delay as further parameters of interest in the final product-sharingphase over this inexpensive link. In particular, any user i may define aTimeConstraint i ≥ 0 to express the maximum time it is willing to spendto receive the wished content, in relation to the non-cooperative download.Similarly, any user i may define an EnergyConstraint i ≥ 0 to express themaximum energy it is willing to spend, compared to the non-cooperativedownload. In order to evaluate these constraints, the Cooperation Server Pshould be able to estimate the time and the energy consumption for the co-operative content download for every node. This requirement can be fulfilledproviding the service provider with a specific model which deals both with

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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, clientswith high-feedbacks, and so on. To model the intended business strategywhere large groups (buying many licenses) should be attracted, such a pricingfunction is required to be sub-additive, that is, p(S ∪ T ) ≤ p(S) + p(T ), forevery pairs of non-intersecting subgroups S and T . Therefore, adding usersto any group always leads to a lower price-per-license, so that larger groupshave higher discounts, as expected.

After collecting information from nodes willing to cooperate, the Coop-eration Server reports decisions to them about “how” to cooperate. Thistask consists in defining the cooperative coalitions and assigning the portionαi of content (file-share) that any node i in the cooperative framework shalldownload over its costly cellular link, and the total cost xi for player i to getthe desired product. Note that this cost includes the network cost, and thusplayer i actually pays to the content provider xi − αi · cdi. Nodes will thenproceed to download their assigned file-share, and then they exchange theseparts of the file over the (free) short-range link. In general, more copies of thefile may be downloaded by nodes of the group, the only constraint is that thefractions downloaded by peers cover at least a whole copy of the desired file.That is,

∑i∈N αi ≥ 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 func-

tion and the optimization performed to reduce the download cost (exploitingdevices with lower bandwidth costs), the cost assigned to each node i willalways be not higher than the cost that i would sustain for a stand-alonedownload (which is the sum of the content license cost p({i}) and the down-load cost cdi). In fact, the precise cost is determined according to a gametheoretical solution-concept, described in the subsequent section, that guar-antees a fair cost allocation for all nodes and that is the key issue for thesuccess of the proposed paradigm. In particular, the service coordinatorneeds to take into careful consideration the contribution of every node i interms of price reduction and of device capability (bandwidth cost), to suit-ably evaluate how much discount it deserves.

Note that, so far, we just considered a service provider that offers a filedownload service and directly partakes in a sort of “mediated” wireless coop-erative data-exchange based on short-range links. We have not yet consideredthe possible role of the network provider (if it is distinct), that could oppose

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the proposed framework because it may reduce its profits on the costly cellu-lar link. In a simple approach, one may just assume that the network providerhas a mutual business agreement with the service provider that covers some-how such a loss. In fact, if the network provider has some control power overthe proposed protocol, we could imagine that, in the business agreement, thenetwork provider may require a lower bound on its profit per transaction,with respect to the case of stand-alone downloads for all users. Formally, weconsider in the framework an additional parameter 0 ≤ GainConstraint ≤ 1,used to impose the constraint

∑i∈N αi · cdi ≥

∑i∈N cdi ·GainConstraint over

the total download-cost of the group. We believe that even low values of theprofit-gain constraint are reasonable: if this situation is contrasted with typ-ical P2P networks, where the network provider has no control at all, only afew users participate in the costly download phase, and thus it always suffershigh 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 theyhelp towards our final objective. For more information on this subject, thereader 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 cooperatingplayer (node) in a fair way as a coalitional cost-game G =< N,C > withtransferable utilities (TU), where N is the set of players and C : 2N → < isa 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 hasat its disposal the ability of each player i ∈ S to download the file with somecosts and some constraints, besides his contribution to determine the pricep(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. Ofcourse, this may also happen for subgroups of players, so that some familyS ′ may attain some special price, and thus S ′ ⊆ S entails that p(S) willbe particularly cheap. Therefore, a fair price assignment should consider allpossible contribution of players, trying to make everyone happy, and not onlyto meet the imposed constraints.

Henceforth, components of any vector x ∈ <|N | are one-to-one associatedwith players in N , so that xi denotes the component associated with player

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i ∈ N . Moreover, for any vector x ∈ <|N |, we denote by x(S) the value∑i∈S xi, where S ⊆ N is a coalition.A feasible payoff profile (or pre-imputation) of G is a vector x ∈ <|N | such

that x(N) = C(N). An imputation of G is a feasible cost profile x ∈ <|N |such that xi ≤ C({i}), for each i ∈ N . This condition is usually calledindividual rationality. In our application, recall that every player i ∈ N isassociated with the two constants cdi and DownloadBound i, which model thecost for player i to download the full desired file from the network, and themaximum amount of data that i would like to download in the P2P process(expressed as a fraction of the whole file). Thus, 0 ≤ DownloadBound i ≤ 1,where 1 means that i sets no a-priori limits on his possible download, and 0means that i will not download anything from the costly-link (therefore thisplayer will just participate in the sharing phase over the short-range link).Therefore, by individual rationality, the total cost for every player i shouldbe not larger than C({i}) = p({i}) + DownloadBound i · cd(i). The set of allimputations of the game G is denoted by X(G).

An outcome for G is an imputation from X(G) that specifies the distri-bution of the cost to any player of the game. A typical requirement of agood outcome is to be “stable” with respect to the possibility that subsets ofplayers find convenient to deviate from it, by forming alternative coalitionsand starting a P2P process on their own, in order to attain lower costs. Theset of such stable outcomes is known as the core of the game.

Definition 1 (Core [28]). The core C(G) of a cost TU game G = 〈N,C〉 is the set of all imputations1 x such that, for each coalition S ⊆ N ,x(S) ≤ C(S).

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 animputation associated with a coalition is in the core then the coalition isstable; otherwise, we say it is unstable. Indeed, if y /∈ C(G), there existssome coalition S such that y(S) > C(S). Therefore, players in the group Smight leave the group N and buy the desired product on their own at thetotal cost C(S), which is less than what they were asked to pay according tothe cost distribution y.

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

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In general, the core of a game may be empty as well as it may contain aninfinite number of imputations. An important class of cost games where thecore is always non-empty is the class of concave games (dually, the convexgames for value games). A game G is said concave if, for every pair ofcoalitions S and T , C(S∪T )+C(S∩T ) ≤ C(S)+C(T ). It can be shown thatthis holds if the cost function is submodular, that is, if C(T ∪ {i})−C(T ) ≤C(S ∪ {i})− C(S), for each pair of coalitions S ⊆ T ⊆ N \ {i}.

However, even if the core is not empty, it remains the problem of choosingan outcome out of possibly infinite many candidates belonging to the core.Thus, solution concepts associated with unique profiles are usually desirablein applications. In particular, the Shapley value [29] is one of the most usedsolution concepts in cost-sharing applications (see, e.g., [24]).

Definition 2 (Shapley value [29]). The Shapley value of a cost TU gameG = 〈N,C〉 is the pre-imputation of G assigning to every player i ∈ N thefollowing cost

φi(G) =1

N !

∑S⊆N\{i}

|S|!(|N | − |S| − 1)![C(S ∪ {i})− C(S)].

In words, the Shapley value assigns a cost to each player i taking into ac-count his “average marginal contribution”, where the average is computedover all different sequences according to which the grand coalition could bebuilt up from the empty coalition. This solution concept has also a niceaxiomatic characterization supporting its notion of fairness (it is the uniquepre-imputation that satisfies the Symmetry, Dummy Player, and Additivityaxioms). It is known that, in any concave game, the Shapley value belongsto the core and thus it is a stable imputation. However, in the general casethe 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, andthus it may also violate the individual rationality condition.

Another approach to single out a fair outcome for TU games is based onthe notion of Nucleolus, first introduced by Schmeidler [30], and based onthe 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 resourcesamong competing agents was at the basis of some (previously) mysteriousrule in a Mishna of the Talmud attributed to Rabbi Nathan (about 1800

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years ago), and in similar contested garment rules [31]. Formally, given acost game G =< N,C >, a coalition S ⊆ N and a cost allocation x, theexcess (or unhappiness) of S w.r.t x is equal to e(S, x) =

∑i∈S

xi − C(S). For

any imputation x, define θ(x) as the vector where the various excesses of allcoalitions (but the empty one) are arranged in non increasing order:

θ(x) = (e(S1, x), e(S2, x), . . . , e(S2|N|−1, x)).

Let θ(x)[i] denote the i-th element of θ(x).For a pair of imputations x and y, we say that θ(x) is lexicographically

smaller than θ(y), denoted by θ(x) ≺ θ(y), if there exists a positive integerq such that θ(x)[i] = θ(y)[i] for all i < q and θ(x)[q] < θ(y)[q].

Definition 3 (Nucleolus [30]). The Nucleolus N (G) of a TU game G isthe set N (G) = {x ∈ X(G) | @y ∈ X(G) s.t. θ(y) ≺ θ(x)}.

Therefore, this solution concept first cares about players that are less sat-isfied, then it cares about the less satisfied among the remaining coalitions,and so on for the rest of the coalitions, with the same approach. Interest-ingly, there is a unique point that leads to the lexicographically minimumvector of excesses, that is, N (G) is in fact a singleton. Moreover, this uniqueoutcome belongs to the Core, whenever it is not empty, and usually offers avalid solution also for games with an empty core, so that the Nucleolus hasbeen considered as one of the most interesting solutions to investigate [18].

A correct evaluation study on the effectiveness of Game Theory in de-signing ”fair” cost allocation schemes in the envisaged scenario derives fromthe selection of an appropriate Game Theory solution concept to apply [27].Different criteria have intrinsic differences in the fairness notion they wantto promote. Although many of them might be applied to the wireless co-operation paradigm under study, the Nucleolus is preferred in this research,because minimizing the maximum unhappiness of players fits well the appli-cation at hands. Moreover, we formally prove in this paper that, for the classof games that best model our application scenario, the Nucleolus fulfills im-portant game-theoretic properties. In particular, it is shown that for such aclass of games the core is always non-empty. It follows that for these gamesthe Nucleolus is not only a fair solution, but also a stable one (in that itbelongs to the core of the game), thus meeting two important requirementsfor the service. The Shapley value [29] has been considered and studied as

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well, but for the games associated with our P2P application they sometimesexhibit 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 callcooperative-download cost games (short: CDC games), defined by means ofa suitable class of pricing functions determining the discount policy. Suchgames best model our problem of assigning costs to nodes participating in themediated 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 arenot concave. As a consequence, the Nucleolus of any CDC game is always astable 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 〈N,C〉, where N is thegroup of players interested in buying some product (or bundle of products)according to the mediated P2P download framework, and C is a cost functiondescribed below.

First recall that the provider defines a suitable price function p(S) todetermine how to charge the nodes of any coalition S for buying the |S|licenses for the product they are cooperatively downloading, and recall thatwe assume this function to be sub-additive, that is, for every pair of coalitionsS and T with S ∩ T = ∅, p(S ∪ T ) ≤ p(S) + p(T ). In fact, this propertyformalizes the usual business logic ”the more licenses the more discount”.

We are now ready to define the cost function of game G. With anycoalition S ⊆ N , we associate a cost

C(S) = min

{p(S) +

∑i∈S

αi · cd(i)

}subject to :∑i∈S

αi ≥ 1

0 ≤ αi ≤ DownloadBoundi,∀i ∈ S∑i∈S

αi · cdi ≥ (∑i∈S

cdi) ·GainConstraint

EnergyCoopi(αi) ≤ EnergyNocoopi · EnergyConstrainti,∀i ∈ STimeCoopi(αi) ≤ ti · TimeConstrainti,∀i ∈ S (1)

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In any optimal solution of the linear program above, the αi control vari-able contains the (amount of) file fraction to be downloaded by player i, inorder to minimize the total cost for the coalition S, while satisfying the prob-lem constraints in (1). In particular, note that we require that the whole filecan be reconstructed after the download of these fractions (in fact, the con-straint just checks the sizes of the fractions, as the actual subdivision of thefile is performed later by the Cooperation Server, respecting the informationin the αi values). Also, every player does not download more data than it isallowed by the DownloadBoundi parameter, thus meeting the individual ra-tionality constraint, and the GainConstraint of the network provider shouldbe satisfied, as well. Finally, the data fractions to be downloaded are requiredto meet also users’ requirements in terms of energy consumption and time incooperation which are limited by EnergyConstrainti and TimeConstrainti.

We require that the linear program (1) has a solution at least for the caseS = N . Indeed, in this case, the above optimization procedure provides thecost of the grand-coalition, and thus the total cost associated with any pre-imputation of the game. An unfeasible linear program would mean that theconstraints imposed by players and network provider are too restrictive forthe characteristics of the involved players, and the cooperative download in-volving the whole group of players N cannot be executed. In such a situationthe Cooperation server will enter in a further procedure of coalition partitionto find alternative coalitions involving subsets of the nodes for which theconstraints are met and the cooperative download can be executed; detailson this procedure are reported in section 3.5.

For any other coalition S ⊂ N different from the grand coalition, it isnot necessarily required that a feasible solution is found by the above linearprogram. In fact, in these cases we can associate to the specific coalition thecost C(S) = p(S) +

∑i∈S 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 thateach 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 functioncdi in the model. Whenever this happens, and if player i does not limit itsmaximum possible download, the cost definition procedure would assign tosuch a node the whole download process, with the consequence of havinga zero download cost for the whole cooperating group, with the additionalbenefit of a reduced content price. The subsequent fair cost allocation willpush the nodes with non-zero bandwidth costs to strongly reward the node

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with the ”flat rate”. This situation could even lead to the possibility ofactually ”earning money” in the process (in fact, note that such nodes areprobably paying higher fees for their flat-rate network contracts). This isconceptually correct, and it should be faced by considering, e.g., some bonus-based mechanism. Noteworthy, in general, a similar situation could also showup for scenarios with high differences in the bandwidth costs among nodesand relatively low content prices. However, for the sake of simplicity, we willnot deal with bonus/credits handling, as nothing changes in the theoreticalframework. In particular, in the experiments presented in this paper we avoidthe use of very low content prices with respect to download costs, and werather focus on the more interesting situation where content and bandwidthcost 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 issub-additive, that is, for every pair of coalitions S and T with S ∩ T = ∅,C(S ∪ T ) ≤ C(S) + C(T ). For sub-additive games we usually say that thegrand-coalition forms. Indeed, in such games adding players to any coalitionis always cost-effective, which entails that the grand-coalition is the mostconvenient configuration. Intuitively, we expect CDC games to have thisproperty, since the business strategy of providers (”the more licenses themore discount”) is modeled by the choice of sub-additive license (product)price functions. However, we still need a little proof of the property, becausethe price function provides only a part of the total cost of coalitions (whichis also determined by the optimization problem solutions).

Proposition 1. Any cooperative-download cost game 〈N,C〉 is sub-additive.

Proof. Let S and T be any pair of non-overlapping coalitions. We knowthat, by definition, p(S ∪ T ) ≤ p(S) + p(T ), because the price of productlicenses is such that larger coalitions get larger discounts. It remains toconsider the download cost, determined for each player i by the assignedfile-fraction αi. Assume first that for S and T download costs are definedaccording to the linear programming optimization (1), and consider any pairof feasible solutions αS and αT for the linear programs for S and T , re-spectively. It is straightforward to check that their union is a feasible so-lution for the linear program for S ∪ T . Indeed, all the player constraintsare clearly satisfied by either solution, and the constraint

∑i∈S∪T αi · cdi ≥

15

(∑

i∈S∪T cdi) ·GainConstraint is also satisfied by linearity, as the two corre-sponding constraints for S and T are satisfied by αS and αT . Therefore, inany optimal solution αoS∪T for the latter program, the download-componentof the cost is at least as good as the sum of the download-cost componentsfor αS and αT .

Finally, observe that the latter argument clearly holds as well if the down-load costs for S or/and T are defined in such a way that every player isassumed to download the whole file (that is, if either linear program is un-feasible).

We conclude the section by showing an important property of the classof CDC games we are most interested in, that is, the CDC games basedon cardinality-discount cost-functions, called hereafter cardinality-discountCDC games. These games are based on pricing functions that depend onlyon the cardinality |S| of the given coalition, and have the following form:

p(S) = price · |S| · 100− d(|S|)100

, (2)

where price is the basic price of the product, that should be paid by any singlebuyer not involved in any cooperative download, and d(·) is a non-decreasingfunction over the naturals [0, N ] which determines the (percentage) discountto be applied to the group S, given the number of users |S| 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 emptycoalition or to single non cooperating 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 worth-while noting that these functions are not concave in general, and hence thecardinality-discount CDC games based on them are not concave, too. Indeed,whether or not this property holds depends on the choice of the specific dis-count function d(·). Nevertheless, we are able to prove that, no matter onthe choice of the discount function, these games (even if non-concave) havea non-empty core.

Proposition 2. Every cardinality-discount CDC game has a non-empty core,and thus its Nucleolus is always a stable imputation.

Proof. Let G = 〈N,C〉 be a cardinality-discount CDC game, and let G′ bethe modification of this game whose cost function is C ′(S) = C(S) − p(S),

16

for every S ⊆ N . That is, the cost function of G′ is defined only by thelinear optimization dealing with player constraints and the network providerconstraint, and it is independent of the product price. Then, it is easyto see that G′ has a concave cost function, because C(T ∪ {i}) − C(T ) ≤C(S ∪{i})−C(S), for each pair of coalitions S ⊆ T ⊆ N \ {i}. Indeed, sucha newcomer player i adds the same constraints to both programs for S andT , but the larger coalition T is clearly more flexible in dealing with them. Itfollows that the core of G′ is not empty.

Let x′ be any imputation in the core of G′, and define a pre-imputation x′′

for G such that x′′i = x′i+p(N)/|N |, for every player i ∈ N . That is, x′′ is thesame as x′ but now the price of the product licenses are taken into account:it is uniformly distributed among all players. Let S ⊆ N be any arbitrarilychosen coalition. We show that x′′(S) ≤ C(S) holds, hence x′′ belongs to thecore of G. By construction, x′′(S) = (

∑i∈S x

′i)+ |S| ·p(N)/|N | ≤ C ′(S)+ |S| ·

p(N)/|N |, where the latter inequality follows from the fact that x′ belongs tothe core of G′. Therefore, x′′(S) ≤ C(S)−p(S)+ |S| ·p(N)/|N |. To conclude,we just observe that (−p(S) + |S| · p(N)/|N |) ≤ 0. Indeed, recall that incardinality discount CDC games the price function has the form p(S) =

price ·|S|· 100−d(|S|)100

for some non-decreasing discount function d(·). Therefore,

we get p(N)/|N | = price · 100−d(|N |)100

≤ p(S)/|S| = price · 100−d(|S|)100

, becausethe discount function d is non-decreasing with the coalition cardinality, andthus the maximum possible discount is assigned to the grand-coalition N .

Having a non-empty core and a stable Nucleolus is a remarkable propertyof cardinality-discount CDC games, as they seem quite appropriate to modelthe most frequent scenario for the proposed file sharing framework. On thecontrary, the Shapley value is not necessarily in the core, if the game is notconcave.

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 assignsto every node i the part of the file to be downloaded. At this step, theNucleolus x has already been computed, and thus every player i knows histotal cost xi to get the product, comprising both the license cost and thedownload cost.

Once the money issues are solved, we have the possibility to perform afinal optimization step possibly focusing on different aspects. For instance,

17

while meeting the constraints of the players and the total bandwidth cost tobe paid to the network provider (C(N)− p(N)), the actual file shares to bedownloaded may be assigned to nodes in such a way that the download timeover the cellular link is minimized, in order to obtain the desired content ina shorter time. Other alternatives can be proposed as well, for instance onecould minimize the total energy consumed over the cellular link, or relax theconstraints on the total bandwidth cost to be paid to the network providerwhen service and network provider are not distinct entities, and so on. Thetime optimization may be obtained by minimizing the maximum download-time over the participating nodes. To this end, the Cooperation Server mayperform the following linear-programming optimization:

min tmax

subject to :∑i∈N

αi ≥ 1

tmax ≥ αi · ti,∀i ∈ N0 ≤ αi ≤ DownloadBoundi,∀i ∈ N∑

i∈N

αicdi = C(N)− p(N)

EnergyCoopi(αi) ≤ EnergyNocoopi · EnergyConstrainti,∀i ∈ NTimeCoopi(αi) ≤ ti · TimeConstrainti,∀i ∈ N (3)

where tmax is the variable to be minimized, whose feasible (lowest) val-ues are determined by the greatest product αi · ti, that is, by the slowestdownload. Then, from the values αoi (i ∈ {1, . . . |N |}) of variables in anyoptimal solution of the linear program (3), the Cooperation Server computesthe actual file parts, say βi, that any player i has to download over thecostly link. The transaction ends when every player i has payed its licensefee xi − αoi · cdi, and all parts have been downloaded. Note that the uniqueconstraint

∑i∈N

αicdi = C(N) − p(N) suffices to deal with economic issues.

For instance, it entails that the network provider’s GainConstraint is ful-filled, 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 tobe considered for this final optimization are the constraints on the energy

18

consumption and the time in the cooperative content download as definedby the players.

3.4. Energy Consumption and Time Constraints Definition

While the main focus for the proposed model is on the monetary costsrelated to the cooperative content download, the proposed framework fore-sees the possibility for the users to define energy consumption and timeconstraints when joining a cooperative content download. The CooperationServer will thus compute the energy consumption and the time needed toreceive the content in cooperation for the nodes. These will then be com-pared to the non-cooperative case to check whether the wished constraintsare 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 performanceevaluation we need to focus the attention on a specific network, for exampleBluetooth. While the interested reader can find details for the energy con-sumption model for the cellular-Bluetooth cooperative setting in [23], we willbriefly report here the main findings needed in this paper.

In a Bluetooth piconet, the number of nodes simultaneously active islimited to 8, where one node acts as master and the remaining nodes act asslaves. No direct slave-to-slave communications are possible and all transmis-sions go through the master. A round robin scheduling of the involved nodesis performed [32] and the master can perform broadcast communications tothe slaves. The node playing the master role is a key factor in defining theenergy 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 non cooperating, EnergyNocoopi, as theenergy consumption for node i when downloading the whole content overits costly cellular link, and EnergyCoopi as the energy consumption fornode i in cooperation. The definition of EnergyNocoopi is straightforward:EnergyNocoopi = Pci

Rci· X; where Pci and Rci are respectively the power

consumption and the data rate on the cellular link for node i, and X is to-tal content size (expressed in Kbyte). For what concerns the EnergyCoopiterm, both cellular energy consumption and Bluetooth energy consumptionhave to be considered. In the reference architecture, nodes first downloadall 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

19

cellular power consumption multiplied by the assigned file fraction and di-vided by the mean link throughput. Instead, a detailed analysis of the Blue-tooth link is needed since, over the time, the number and type of packettransmissions\receptions of a node depend on several factors. Summarizingthe results found in [23], the expression of the energy consumption in coop-eration can be written as a function of the file fraction for each node i to bedownloaded over the cellular link, αi, and on the role r played in the piconet(master or slave), as reported in equation (4). In equation (4), the first termis the energy consumption on the cellular link, the next three terms representthe energy consumption respectively in transmitting data, in receiving data,and during the idle time on the Bluetooth link and a final term measures theenergy consumption for the GPS positioning of each node.

EnergyCoopi(αi) = Edi(αi) + Ebti(r, αi) + Ebri(r, αi) + Ebii(r, αi) + Ep (4)

The Edi(αi) term is the energy consumption in downloading the assignedfile-share αi over the cellular link (this term is not depending on the masteror slave role in the Bluetooth piconet). This Edi(αi) = Pci

Rci·X · αi, where X

is the data size, Pci and Rci are the power consumption and the throughputon the cellular link for node i, respectively. The Ep term in equation (4), isthe energy consumed by the nodes to gather their GPS positioning. For thefurther terms of equation (4) the values for the Bluetooth specific parametersare set according to the standard [32] and the exact definition is taken fromthe model presented in [23].

When looking instead at the time needed to receive the content, as in-troduced earlier in this section, ti is the time needed for node i to downloadthe whole content over its cellular link. Then, let us define TimeCoopi asthe time needed for node i to receive the content in cooperation. Whencooperating the devices will first download the assigned file fractions overtheir cellular links and then share them over the Bluetooth link. During thefirst phase of cellular downloading, they can, in parallel, setup the Bluetoothpiconet. Therefore, the time in cooperation is equal for all nodes i and canbe computed as the sum of maximum time spent on the cellular link bythe nodes, the time needed to distribute the data over the Bluetooth link,and a small contribution of time for the Cooperation Server to compute thesolution:

20

TimeCoop = tmax + TimeBT + TimeServer (5)

tmax is defined by the linear program in equation (3); the time on theBluetooth link TimeBT is given by the number of Round Robin schedulingcycles needed to distribute the total data for a given packet payload (furtherdetails on this computation can be found in [8]); the time for the CooperationServer to compute the solution TimeServer is here assumed equal to 19s asthis is the worst case value for a computer with mean hardware capabilitieswhen it is necessary to compute the optimization problem for all possiblecoalitions 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 theusers and/or by the network may cause the linear optimization not to finda valid solution for the grand coalition. Moreover, if the Shapley value isthe 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 alternativegroups of players to cooperate successfully. The proposed policy is to excludeiteratively some of the nodes until a first coalition is found where all therequirements are met. If more alternatives are meeting the constraints, thena further policy should be introduced to select the preferred coalition. Inthis framework, we adopt the well-known concept of maximizing the socialwelfare [15], whereby the coalitions that maximize the total monetary savings(for themselves) are preferred. In more detail, if for the coalition involving Nnodes a valid solution cannot be found, then smaller coalitions are considered.Whenever a coalition with the required properties is formed (clearly, it willbe a maximal one), the process restarts with the excluded users, until apartition π of the players N in coalitions, also called a coalition structure [15],is found. Note that standard game-theoretic solution concepts for coalitionalstructures, such as the core, consider stability conditions also on sets ofplayers belonging to different elements of the partition π [33]. However,such inter-structure conditions lead to high computational costs that are notsuitable for the practical applications we have in mind. Therefore, we adopthere a simplified approach where each element πi of the structure inducesa distinct (sub)game which is analyzed separately from the others (beingdefined over disjoint sets of players).

21

3.6. Remarks on Practical Implementation and Computational AnalysisIn this section we briefly discuss practical issues in a possible implemen-

tation of the proposed service. The Cooperation Server promotes the serviceand collects subscription of users interested in its product(s). The nodesregistering to the service will agree to the term and conditions for the ser-vice. Product licenses will be active only after the cooperating process isended. Thus, in case of a misbehavior of a node in cooperation, this canbe easily detected, and (for instance) the license of such a node may be setto an invalid state by the provider. Note that a basic implementation ofcooperative-downloading may be designed as an atomic transaction, as theinvolved group of users is intended to be static during the process. In thisapproach, if some node goes down during the process and is not able to re-cover within some reasonable time-out, then the process fails and a restartor some form of renegotiation is required (indeed other nodes may then beaccepted, and if no further node comes in, then the price of the licenses maybe higher for the resulting smaller group).

For each service subscriber, a user profile will be defined collecting basicinformation provided when the device registers to the service. In generalthe Cooperation Server could be available on the Internet and accessible bythe service subscribers from any location. The Cooperation Server will thenwait for nodes to contact him providing all information needed for the service,including the geographic position of the node (e.g., determined through itsGPS coordinates or any other positioning technique). A requirement for thenodes to be considered as part of a common content download, is that tobe interested in the same content (possibly, a bundle of products) and inmutual coverage for a cooperative short-range link. Sample scenarios wherethese conditions are fulfilled can be groups of friends in aggregation places(such as a University campus) to exchange and download books, music orother contents.

Once the users have provided the required information, the CooperationServer notifies each device about: involved devices and how they are clus-tered, the master\slave role played by each node in the piconet (only in theanalyzed case of using Bluetooth as short-range link), the exact data share todownload through cellular links and the corresponding cost, and the contentprice that will be charged to the user account. All cooperating devices willthen first download the assigned file fractions over their cellular links andthen share them over the short-range link. During the first phase of cellulardownloading, they can, in parallel, setup the Bluetooth piconet. Bluetooth

22

limits the number of nodes simultaneously active on a piconet to 8; therefore,the Cooperation Server collects a maximum number of 8 candidates requiringthe 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 lessthan 8 nodes are interested, these could as well cooperate (for instance, asuitable timer can determine a time interval wherein the Cooperation Servercollects the cooperation requests before starting the cooperation process).

Finally, it is worth spending some words concerning the computationalcost of the presented framework. The Nucleolus computation is actuallyplaying a minor role in the overall computational costs, once the cost functionis known. In fact, the major contribution to the computational cost is givenby the linear programming optimization step that is performed for everypossible coalition, which means 2|N | − 1 times. Moreover, for each coalition,a number of minimization problems proportional to the number of nodes kin the coalition is required to define the master and slave roles. The totalnumber of minimization problems is: nmin(N) =

∑Nk=2 k

(Nk

). However, it

is worth mentioning that executing the introduced framework requires a fewseconds, as tested with a basic hardware deployment; this order of magnitudeis definitely acceptable for the proposed file sharing service.

4. Experimental Evaluation

A numerical evaluation of the model is conducted to observe the behaviorof the envisaged paradigm under a wide range of system configurations. Mainobjective is to validate the ”mediated wireless cooperation” by looking bothat cost savings for the single nodes (compared to a stand-alone classicalcontent download) and at providers’ profits, under different conditions. It willalso clearly emerge that a Game Theoretic approach based on the Nucleolusis preferred to standard optimization criteria, as far as the fairness perceivedby users is concerned. The results presented next are related to Bluetooth asshort-range link, but the overall framework can equally work with any othershort-range technology.

We considered sample cases where cooperating nodes have different cel-lular throughput levels. All presented cases assume that nodes have a time-based billing agreement with the provider; we express this cost in terms of agiven amount of generic Cost Units per second (CU). Note that from thesemeasurements and from the size of the product file we may immediatelycompute the values cdi and ti characterizing the device features of any node

23

i. Also the alternative billing policy foreseeing that nodes are charged ona data-amount basis has been investigated. As there is no conceptual dif-ference, the results will refer to the first case only. In particular, the costper second is set to 0, 05 cost units, the file size is 100 Mbyte, and the basiccontent price price is equal to 400 cost units. Concerning the power consump-tion on both the cellular link and the Bluetooth interface, the values usedin the present paper result from measurement campaigns (conducted on N95Nokia smartphones). In particular, different values of power consumptionhave been measured for a device connected to a 2.5G or to a beyond 2.5Gsystem (400mW for 2.5G, 1400mW for beyond 2.5G and we assume 250kbpsthe maximum data rate value for a 2.5G connection). For the Bluetooth theadopted values are 178.2mW, 108mW and 59.4mW in transmission, recep-tion and idle time respectively. Finally, the energy consumed by the nodesto gather their GPS positioning is set according to results in [34], wherebythe energy consumption for a Nokia N95 smartphone is equal to 13, 32J inthe best case.

It is assumed that the Cooperation Server chooses a price function ofthe type described in section 3.2.1. More specifically, we next considercardinality-discount CDC games whose discount functions are based on thefunctions proposed in [35], suitably adapted to the scope. Figure 2 reportsthe discount proposed by the content provider w.r.t. to the normal basicprice for different number of nodes. The plotted equation is given in (6):

dρ(|S|) = Maxdiscount ·exp[−(|S| − 1)/ρ]− 1

exp[−(|S|max − 1)/ρ]− 1

with ρ 6= Infinity (6)

where |S| and |S|max are respectively the cardinality of the coalitionand the maximum cardinality (when not differently stated, in our experi-ments this is set to 8), ρ determines the discount amount for either largeror smaller coalitions, Maxdiscount represents the highest discount proposedby the provider which is set here to 50. It is easy to see from the formulathat the highest 50% discount is offered when the coalition size |S| is equalto |S|max, while no discount is given to the non cooperating nodes when |S|is equal to 1.

Based on the above discounts, we get the following class of cardinality-

24

Figure 2: Content discount function for multi-license products.

discount price-functions:

pρ(S) = price · |S| · 100− dρ(|S|)100

(7)

We use these price functions for the experiments that illustrate the pro-posed framework. In particular, we focus on the case with ρ = 2, as this offershigher savings immediately, encouraging the users to join cooperating groupsalso of very small sizes. Interestingly, for positive ρ values and in particularfor the chosen ρ = 2 value, sub-modularity does not hold for such a function,because the marginal discount obtained by larger coalitions is smaller thanthe marginal discount obtained by smaller coalitions (see Figure 2). Thus,the considered games are not concave, but from Proposition 2 we know thatthe core is always not empty.

The Cooperation Server is made aware of the limit on the amount of datato be downloaded in the P2P application imposed by the nodes. In generaleach node could request different values for this parameter, but for simplicity

25

in the presentation of the results and to better discuss the influence of thisparameter, we assume all nodes having the same DownloadBoundi, hereafterjust called DownloadBound.

Several throughput distributions for the nodes can be considered. Astraightforward test scenario is characterized by ”homogeneous” nodes, i.e.nodes with the same cellular throughput level. In this case, if there are nofurther constraints, the solution found by the bandwidth-cost optimizationprocedure performed by the service coordinator assigns an equal share, hencean equal cost, to all nodes. Such a homogeneous cost allocation is in factthe Nucleolus of the associated game and it belongs to the core (in fact, it isalso the Shapley value), and no further analysis and cost compensations arerequired.

For any scenario different from the cited ”homogeneous” one, the pro-posed game theoretic model proves that defining a fair cost allocation forthe participating nodes is unavoidable. During the study, a wide range ofscenarios with variable throughput distributions for the nodes have been con-sidered. The results are also contrasted with the possible reasonable outputsof some alternative approach, which rely on an optimization only (i.e., notbased on game theoretic concepts). The sample scenarios presented in theremaining part of the paper are selected to give the reader a broad overviewof the proposed framework behavior. They are characterized by half of thenodes with a cellular throughput of T = 100 kbps, while the throughput ofthe remaining nodes equal to (T + δ) kbps, with δ variable. Clearly, a similaranalysis can be repeated for different nodes configurations, obtaining similarplots.

4.1. File-shares Assignment, Node Costs, and Service Provider Profit

With reference to the output of the cooperative game, let us focus on thecost assignment to nodes, which is also equal to providers’ profit. The costassigned to each node is a combination of the content license cost and thecost deriving from the download of the assigned file-share over the cellularlinks. Of course, these values directly depend on DownloadBound and nodethroughput values, while also the GainConstraint of the network providerinfluences the final results. These dependencies are investigated next, whileit is assumed the nodes have not set stringent constraints on the energyconsumption and time delay in cooperation.

In Figure 3 it is shown how the file-shares to be downloaded by nodes,hence their download costs, change with the parameter DownloadBound and

26

Figure 3: Influence of DownloadBound (DB) and GainConstraint (GC) on the file portionsassigned for download to nodes; δ is set to 100 kbps.

the GainConstraint; this behavior is highlighted in sample cases in which δ =100 kbps. Recall that such file-shares are assigned by the final minimizationstep, as described in section 3.3.

Let us focus first on the case with GainConstraint (in short GC) set tozero and analyze the influence of the DownloadBound (in short DB), leftside of the plot. To minimize the maximum download time according tolinear program (3), the solution is to assign larger portions of data to bedownloaded to the four nodes with higher cellular throughput (nodes 5-8in the scenario). For any scenario with DownloadBound≥ 0.25 these fournodes will download 1/4 of the file each. When the nodes set more stringentconstraints on the DownloadBound, also the other nodes are involved. Forany value DownloadBound < 0.125 (that is maximum 1/8 of the file) nofeasible solution can be found, since the total number of nodes is 8. Notethat similar considerations hold for different values of the total number ofnodes in the coalition; obviously, these lead to different limiting values forthe DownloadBound value. Moreover, it is worth commenting that in a moregeneral system setting, where all values of nodes’ cellular throughput aredifferent, again to minimize the maximum download time according to linearprogram (3), the solution is to assign a larger portion of file to download

27

Figure 4: Cost distribution among the nodes according to the Nucleolus solution fordifferent values of DownloadBound and δ = 100kbps, GainConstraint=0.

to the nodes with higher cellular throughput as far as the DownloadBoundconstraint allows it.

To assess the impact of the GainConstraint set by the network, we an-alyze the case where DownloadBound is set to 0.25. We can compare thescenario with GainConstraint not set (the third case in the left side of theplot in Figure 3) and the scenario corresponding to the results reported onthe right side of the plot in Figure 3. Two main effects of an increase in theGainConstraint parameter emerge. First, the nodes with a less performingcellular link are now also involved in the content download over the cellularlink. This is justified by the increase in the bandwidth income required bythe network provider. When the GainConstraint increases even more, thesecond effect is that the cooperating cluster is forced to download more thanone copy of the total file (e.g. 2 copies are downloaded when GC = 0.25).

Next, we want to investigate on the costs repartition among the nodesin the considered scenarios. In Figure 4 the cost for the content licenseand the content download is reported for the nodes in the sample scenarioswith δ = 100kbps, GainConstraint=0 and the DownloadBound set to oneof the following values: 0, 0.125, 0.2, 0.25. In Figure 5 instead, the sameinformation is reported when GainConstraint assumes one of the following

28

Figure 5: Cost distribution for the nodes according to the Nucleolus solution for differentvalues of GainConstraint and δ = 100kbps, DownloadBound=0.25.

values 0, 0.125, 0.2, 0.25, while δ = 100kbps and DownloadBound=0. Whatcan be observed in both of the Figures is that a node having higher costsfor the content download, will pay a smaller contribution for the contentlicense. This observation has a general validity also in a system setting withmore differentiated values of cellular throughput for the nodes. It can also beobserved that the cost for the content is often higher, for some nodes, thanthe cost given by the price function pρ(S). The reason for this is that it hasalso 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 focuson a node i with cellular throughput 100kbps and a node j with cellularthroughput 200kbps. For the specific bandwidth cost and file size settings(0,05 CU per second and 100 MB respectively), if nodes do not cooperatethen they will be charged the following basic costs: p(i) + cd(i) = 400 + 400CU = 800 CU; p(j)+cd(j) = 400+200 CU = 600 CU. The nodes may decideto contribute to the cooperative download by fixing the upper bound on theirpossible downloads to 0.2, and expect to have the following maximum costs:pρ(i)+DownloadBound·cd(i) = 200+80 = 280 CU; pρ(j)+DownloadBound·cd(j) = 200+40 = 240 CU. This means an overall saving for the service equalto: Saving(i) = 520 CU (corresponding to a 65% saving) and Saving(j)

29

Figure 6: Content and network provider profit in cooperation, variable δ according to theNucleolus solution.

= 360 CU (corresponding to a 60% saving). If we now observe the costassigned to these nodes by Nucleolus (see Figure 4), we notice a cost of 250CU and of 200 CU for nodes i and j, respectively. These results lead tothe 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 demonstratesthat the final savings meet the user constraints, and this is done in a fairway, i.e. by considering the ”merits” of each node.

The next analysis presented in Figure 6 shows the influence of theDownloadBoundand the GainConstraint parameters on the total cost C(N), which equalsthe total providers’ profit. This value decreases with increasing values ofthe DownloadBound, since for lower DownloadBound values also nodeswith lower bandwidth costs have to be involved in the download phase.On the other hand, it increases with GainConstraint parameter. Inter-esting to observe is how two of the plots overlap completely (the case withDB = 0.25;GC = 0.125 and the case with DB = 0.125;GC = 0) even ifthe settings are different and the specific cost allocations to the nodes aredifferent as shown in previous plots. Moreover, in Figure 6, the influence of δvalue is also plotted. The decreasing trend when δ is increasing is expected,

30

because higher throughput values 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 ofcost savings, we next focus in more detail on fairness aspects, when thisapproach is compared with alternative solutions.

In particular, one may wonder whether alternative reasonable approachesexist that are not based on game theoretic principles. We already observedthat traditional optimization techniques aims at minimizing (or maximizing)some measure like the total cost, but typically do not care about fairness inthe final cost assignment to the participating nodes. According to the specificapplication 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, wenext 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 equation (1), thiscost is then assigned to nodes, proportionally to the bandwidth costs, i.e.according to the cdi parameter for each node: xpri = C(N) · cdi∑

j∈Ncdj

, for each

i ∈ N .In Table 1 a comparison is presented among this proportional cost solu-

tion, the solution given by the linear program optimization, and the Nucle-olus. Two sample cases are presented, but similar results are obtained formany other cases. In particular, the presented cases refer to scenarios whereDownloadBound is set to 0.25 for all nodes, GainConstraint is set to zero,nodes 1-4 have cellular rate Rc1 = 100kbps and nodes 5-8 have cellular rateRc2 = (100 + δ)kbps with δ either equal to 100 or 1500. Besides the differ-ences in the cost allocation, the experiment aims at observing whether theproposed allocation is stable according to the core and if it actually meetsthe cost/savings constraints set by the users. As it can be observed, the op-timal solution is not always a stable allocation, while the proportional costallocation 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 pro-posed application. We thus briefly discuss the possible results obtainable with

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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 havecellular rate Rc1 = 100kbps and nodes 5-8 have cellular rate Rc2 = (100 + δ)kbps.

δ = 100kbps δ = 1500kbpsNodes 1-4 Nodes 5-8 Nodes 1-4 Nodes 5-8

Cost per node[CU] 200 250 200 206.25Savings per node[CU] 600 350 600 218.75

Optimization Constraints met yes yes yes yesStable yes yes no no

Cost per node[CU] 300 150 382.3 23.9Savings per node[CU] 500 450 417.6 401.1

Proportional Constraints met yes yes no yesStable yes yes yes yes

Cost per node[CU] 250 200 250 156.25Savings per node[CU] 550 400 550 268.75

Nucleolus Constraints met yes yes yes yesStable yes yes yes yes

cost allocations following the Shapley value, a well-known solution conceptwhich is a valid alternative, and whose main properties have been describedin section 3.1.

The numerical evaluation presented in the previous sections has beenrepeated by considering the Shapley value instead of the Nucleolus. In someexperiments, the Shapley value is not a stable imputation for the coalitioninvolving all the nodes, as it is outside the core and does not fulfill theindividual rationality. This is not surprising, if we consider that our costgames are not concave for the chosen price function. However, as provedin Proposition 2, the core is not empty and hence the Nucleolus does notsuffer from those drawbacks. Another observation is that, when using theShapley 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 thepreferred solution concept for the proposed framework.

When the Shapley value is not in the core for the game involving all thenodes, alternative coalitions are considered with a reduced number of coop-erating nodes. The main consequence is that the savings introduced for thenodes are reduced. This is clearly plotted in Figure 7, where the average cost

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Figure 7: Shapley-value vs. Nucleolus: average per node savings with variable δ,DownloadBound = 0.25.

savings per node when using the Shapley value or the Nucleolus are consid-ered, in different scenarios with variable δ and GainConstraint parameters,and DownloadBound = 0.25. In all the plotted cases, the Shapley valuedoes 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 excludednodes. In particular, this happens for all cases with δ ≥ 1100 and GC = 0.2and GC = 0.25, where always two coalitions of 4 nodes are formed. Forthe other scenarios, the GC = 0.2 and GC = 0.25 cases register a loweraverage cost saving w.r.t. the case with GC = 0.125. The reason for thisis related to the coalition being formed and the nodes being excluded fromcooperation. As an example, let us consider the δ = 200 scenario. WithGC = 0.125 a 6-nodes coalition is formed where two nodes with a highercellular throughput and thus lower costs in non-cooperation are excluded;with GC = 0.2 a 5-nodes coalition is formed where now three nodes withlow costs in non-cooperation are excluded; with GC = 0.25 a 6-nodes coali-tion is formed where this time two nodes are excluded with high costs innon-cooperation. Similar considerations can be made for any other case.

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Figure 8: Average per node savings for different coalition sizes |S| with variable δ.

4.4. Savings for Different Coalition Sizes and Discount Function Settings

Results presented in previous sections have focused on the behavior of thegrand coalition where fair cost allocations are found. Next, we first give thereader some insights on the benefits the nodes gain in joining larger coalitionsinstead of smaller sub-coalition. Then, we will show how the maximumcoalition size set by the service provider in the discount function influencesthe benefits for cooperating users and provider.

Figure 8 shows the average per node savings obtained in joining coalitionsof different sizes. The grand coalition size is |S| = 8, while also the sub-coalitions are considered with 2, 4, or 6 nodes (also these coalitions havethe same cellular throughput combinations as for the grand coalition). TheDownloadBound and the GainConstraint are not set for the plotted casefor simplicity. As it clearly appears from the plots, and as expected, nodeshave always higher monetary savings in joining larger coalitions.

The subsequent analysis focuses on the influence that the maximum ac-ceptable coalition size, set by the service provider in the definition of thediscount function, has on the final monetary savings for the nodes and theoverall profits for the service provider. This analysis has also the further ob-jective to investigate on the possible trade-off between the service providerand the cooperating nodes interests. These are clearly conflicting like their

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Figure 9: Average per node network profit and average saving per node, for differentcoalition sizes and |S|max in equation (6) with ρ = 2; DownloadBound = 0.

objectives, with the service provider wishing to maximize the monetary in-comes for the service, and the users wishing to minimize the service costs.Going into details, the service provider can decide to limit the maximumcoalition size by tuning the value for |S|max in equation (6). Changing thissetting, the first effect is a modification in the discount function plotted inFigure 2, with a maximum discount obtained at the corresponding |S|maxvalue. The emph|S|max values we considered for this analysis are 2, 4, 6 or8, while the choice in terms of ρ for the discount function is kept constant,namely ρ = 2.

We consider some sample scenarios with either DownloadBound = 0(see Figure 9) or DownloadBound = 0.25 (see Figure 10), where half ofthe nodes have a cellular throughput level Rc1 = 100kbps and the secondhalf have a cellular rate Rc2 = (100 + δ)kbps with δ either equal to 100 or1500. As expected, the results in Figure 9 show that the service providerprofits increase with lower values of |S|max, while the users savings increasewith larger values of |S|max. Moreover, a generally valid trade-off betweenthe network provider and the users interest cannot be found. For instance,Figure 9 shows that, when considering coalitions of only 2 nodes (i.e. with|S|max = 2), the absolute values of the objective functions for the service

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Figure 10: Average per node network profit and average saving per node, for differentcoalition sizes and |S|max in equation (6) with ρ = 2; DownloadBound = 0.25.

provider and the users are the closest. By increasing the value of |S|max, andthus the maximum coalition size, the two objective functions diverge. Theobserved trends for a variable value of |S|max, suggest to use the case with|S|max = 4 as a potential trade-off point. Nevertheless, when changing theDownloadBound value, like in Figure 10, the situation completely changesand the same conclusions are not valid anymore. Thus, we can conclude thatthe discount function proposed by the service provider has definitely a keyimportance in the definition of the system performance. Unfortunately, it isnot possible to design a generally valid function that guarantees a trade-offbetween 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 mone-tary savings introduced by the proposed framework and the influence of therelated constraints. In this section, the attention is put on the further con-straints the nodes can set when joining the cooperative service, namely theenergy consumption and the time constraints. To best highlight the impact ofthese two constraints, we present sample scenarios where the DownloadBoundand the GainConstraint are not influencing the solutions. We consider only

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the Nucleolus solution for a sample scenario where nodes 1-4 have cellularrate Rc1 = 100kbps and nodes 5-8 have cellular rate Rc2 = 200kbps (a similaranalysis can be repeated for other cases) and the energy consumption andtime constraints are equal for all nodes, for simplicity.

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 cellularrate Rc1 = 100kbps and nodes 5-8 have cellular rate Rc2 = 200kbps.

Energy constraint0.2 0.4 0.6 0.8 1

Cooperative coalition size - 8 8 8 8Avg energy savings [J/Kbit] - 2.22 2.27 2.27 2.27Avg monetary savings [CU] - 468 475 475 475

Max time delay [%] - 44 31 31 31

In Table 2 the values of the main performance indexes are reported for in-creasing values of the energy consumption constraint set by the users, whileno time constraint is considered. In particular, energy consumption con-straint values are considered in the range 0.2− 1 (see the problem definitionin equation (1)). As it clearly emerges from the results, the cooperative con-tent download actually introduces energy savings for the nodes. This is nota surprising result, see e.g. [23]. What can be observed is that for energyconstraints from 0.6 and above no difference in the final performances is ob-tained. Moreover, for an energy constraint of 0.2 the nodes will actually notcooperate, while in all other cases always the grand coalition of 8 nodes isformed and both energy and monetary savings are obtained. Finally, also themaximum time delay experienced by the nodes in cooperation is reported.What emerges is that a maximum delay of 44% is reached, which is expe-rienced by the nodes with the fastest cellular throughput (the nodes withRc2 = 200kbps in the specific case).

In Table 3 the same performance indexes are presented for variable valuesof 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 incooperation (see the cases with negative delay values reported in the Table).For time constraint values equal to 0.4 and below no cooperative downloadcan be activated due to the stringent constraints. In the other cases, instead,the main effect of setting the constraint is that smaller coalitions are formedand different average energy saving and monetary saving are obtained by thenodes in the coalitions. Finally, when the time constraint is not so stringent

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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 cel-lular rate Rc1 = 100kbps and nodes 5-8 have cellular rate Rc2 = 200kbps.

Time constraint0.4 0.6 0.8 1 1.2 1.4 1.6

Cooperative coalition size - 4 4 | 2 | 2 5 | 3 7 8 8Avg energy savings [J/Kbit] - 2.6 2.6 | 199 | 199 1.4 | 2.03 2.3 2.3 2.3Avg monetary savings [CU] - 460 460 | 181 | 181 378 | 397 474 475 475

Max time delay [%] - -42.5 -42.5 | -26.6 | -26.6 15 | -31 20 31 31

(from 1.4 and beyond) again the grand coalition of 8 nodes is formed and noinfluence of the time constraint on the results is observed.

5. Conclusions

This paper uses the powerful mathematical framework of Game Theoryto enable a mediated business model, according to which a provider promotesitself as a service coordinator for a cooperative file sharing service. Nodescooperatively download a file of common interest over their cellular links andshare it over a cost-free short-range link. The idea is to give the provider theopportunity to sell a larger number of licenses by proposing suitable group-discounts for content to be downloaded. On the other hand, nodes havesignificant cost savings when compared to a stand-alone download of thewhole content, as they benefit both of a reduced cost for the content and ofan optimal bandwidth cost-distribution over the cellular link. Moreover, theybenefit from a ”legal” service with the additional important guarantee thatcosts are fairly assigned to nodes and there are no free riders. Since fairnessand stability of a solution is of utmost importance, the use of Game Theoryto model the cost distribution problem proved to be an effective approach.Based on a coalitional transferable-utility cost game, the Nucleolus has beenadopted as a valid solution to determine the fair cost allocations for thecooperative cluster. A number of properties of the proposed approach areshown and commented under a wide range of sample operational conditions.

Acknowledgment

Special thanks go to Miguel Angel Miras Calvo and Estela Sanchez Rodrıguezfrom University of Vigo and to Jean Derks from University of Maastricht

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for their important suggestions and the Matlab toolboxes they have kindlyshared with us for the numerical evaluation campaign for this paper.

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