Stability of Alliances between Service Providers

—Three service providers in competition, try to optimize their quality of service / content level and their service access price. But, they have to deal with uncertainty on the consumers’ preferences. To reduce their uncertainty, they have the opportunity to buy information and to build alliances. We determine the Shapley value which is a fair way to allocate the grand coalition’s revenue between the service providers. Then, we identify the values of β (consumers’ sensitivity coefﬁcient to the quality of service / contents) for which allocating the grand coalition’s revenue using the Shapley value guarantees the system stability. For other values of β , we prove that it is possible for the regulator to impose a per-period interest rate maximizing the market coverage under equal allocation rules.


I. INTRODUCTION
Alliance definition We speak about alliances 1 when firms on a market agree with one another to realize profits which are superior to the standard profits that they should receive under competition.The standard profits are those obtained at the non cooperative Nash equilibrium where each firm tries to maximize selfishly his utility 2 .
Alliances can be explicit when firms agree explicitly with one another on prices, quantities, production capacities, investments, etc., via contracts, for instance.Alliances are said tacit provided this is not the case.
Under competition, firms maximize their profits.But, sometimes, firms realize that coordination might increase their joint profit.Hence, firms on a market have natural incentives to agree together in order to increase their market power and their profit.Competition between firms can then be compared with a prisonner dilemma: • A firm chooses his strategy in order to maximize his profit but, does not care about the effect of his decision on the other firms.• In an alliance, firms take into account how their own decisions affect the others' profits.There exists of course various forms of alliances: joint price determination, joint quantity setting, geographic allocation, etc. [2].
Alliances do not emerge on every market since collusion may be forbidden (competition policy) and since the firms in the alliance might have incentives to cheat, deviating from the cooperative equilibrium.Firms may for instance, deviate unilaterally from the collusion agreement by proposing a PRISM, Université de Versailles Saint-Quentin, 45, avenue des Etats-Unis, 78035 Versailles, France.E-mail: Helene.Le-Cadre@prism.uvsq.fr.
This work has been funded by the european FP 7 project ETICS (Economics and Technologies for Inter-Carrier Services). 1 The term of collusion exists also in the literature; these two terms will be used undistinctly in the article. 2 As usual in game theory, firms and service providers will be associated with male or female players.
price inferior to the expected one in the hope to capture the larger part of the market demand.If each firm forecasts that the others might cheat, cooperation is not imaginable anymore.Punishment mechanisms should then be implemented to prevent the firms in the alliance from cheating.Courts punish explicit accords whose objectives are clearly to decrease the competition.Heavy sanctions have been applied to international accords on the vitamin market (855 millions of euros), on lysine and citric acid (200 millions of dollars and imprisonment years) [2] and more recently, on the memory chip market (331 millions of euros).
Alliance instability Cases of collusion (explicit or tacit) have been reported in the telecommunications literature, both in the long distance market [10] and in the mobile sector [7].In practice, alliances are difficult to build since firms might suspect that their partner would defect and then, adopt tit for tat strategy which, in a short time-scale, does not give incentives to firms for cooperation.Besides, it is difficult to deal with the selfish tendency of the alliance partners.These latter might for instance, enter a learning race where the firm who learns the quickest wins and then, leaves the alliance.Furthermore, alliances may be highly unstable due to changes in customers' demand, political relation and alliance management.de Man et al. propose a robust framework based on a win-win strategy to guarantee the long-term alliance stability in the airline industry [3].However, reputation phenomenon and guarantee provisioning 3 might give firms incentives to stay in the alliance [5].

Cooperation in supply chains
The different relations between the economic actors can be modeled by a supply chain i.e., a complex network containing a large number of entities who sometimes compete and sometimes cooperate to fulfill customers' needs.Selfish behaviors generating a loss of efficiency, decisions should be centralized in order to maximize the global profit.Hence, building alliances appears as a successful strategy in modern supply chain networks.Cooperation implies then, a better exploitation, benefits from large economies of scope, the decrease of total costs and the increase of total savings.The problem is to identify properly which coalitions can be expected to form and how the alliance members should share the pie i.e., the total benefit to guarantee the stability of the formed alliance [1].tives to leave it.Various allocations of the grand coalition's revenue rules exist such as the Shapley value which guarantees a fair sharing, the nucleolus which provides an allocation that minimizes the dissatisfaction of the players from the allocation that they can receive, the proportional allocation which shares the total revenue depending on the players' intial investment costs, the equal allocation which gives an equal proportion to each player, etc. [1], [11], [15].

Article originality and organization
The article aims at studying the alliances that might emerge between three service providers in competition, who have to cope with uncertainty on the consumers' preferences.The originality of our approach is to incorporate uncertainty about the consumers' preferences in the game between the operators.Besides, to our knowledge, this is the first article proposing solutions to deal with alliance instability in the telecommunications framework and modelling explicitly information exchange / investment between the operators.
The article is organized as follows.We start by introducing the two level game between the service providers in Section II.Then, in Section III, we compute the Shapley value of the cooperative game which enables us to allocate fairly the grand coalition's revenue between the three service providers.Depending on the consumers' sensitivity to the quality of service / contents (β), we determine then, whether the Shapley value belongs to the core of the game i.e., guarantees the alliance stability.If the grand coalition is stable then, the regulatory authority's advices would be without effect on the providers since these latter would have no incentives at all to leave the alliance where the total revenue is shared according to the Shapley value.Finally, in Section IV, we consider the values of β for which the Shapley value is not in the core.Having no guarantee on the system's dynamic behavior, the regulator should intervene to assure the consumers' welfare.We prove that it is possible for the regulator to impose a per-period interest rate maximizing the market coverage under equal allocation of the coalitions' revenues.We conclude in Section V.

Notations
The main notations used throughout the article are stored in the table below.

II. GAME MODEL AND INFORMATION SHARING
Description of the providers' utilities We consider three service providers: A 1 , A 2 and A 3 .They can belong to various categories.It can be content providers, Internet access providers, or TV channels, etc.We assume that these three firms are interconnected via the Internet backbone.Besides, they have the opportunity to exchange information 4 (cf.Figure 1).The information can be either bought from other providers or the providers can choose to invest together to gather data through a common data-base.Description of the relations between the three service providers.
Consider firm A i , i = 1, 2, 3, his utility is of the form with p i the acess price to the firm i's network / service for the consumers 5 , n i the total number of clients for firm A i .α i represents the level of information collected / bought by firm A i while I(.) is the cost of information acquisition.
The quality of service (QoS) / content level q i can be seen as a function of firm A i 's information level i.e., α i = ϑ(q i ) with ϑ : R + → R + an invertible function on R + .Furthermore, we assume that Ioϑ is convex on R + .In economics, Schlee [16], Johnson and Myatt [6] use convex marginal cost functions, in different contexts.
The quality of service / content level is tightly related to the information level acquired by the firm.Indeed, if firm A i manages to extract the most pertinent information about the network topologies or more generally, about the consumers' preferences, he will be able to optimise q i more easily.Of course the consumers' perception of firm A i will by turns depend on the firm's information level since, in this case, the firm's information level conditions his quality of service / content level.To give an illustration, in Figure 2, the firms' information levels are expressed as linear and quadratic (with stars) functions of the quality of service / content level; we have set: α i = 4q i and α i = 1 2 q 2 i (with stars).Furthermore, we assume that the firm is limited in the improvement of his quality of service / content level by the access network capacity; thus: The idea that the more information the provider acquires, the easier it is for him to optimize his quality of service / content level can be found again in the concept of entropy which has been introduced by Shannon.The entropy is a mathematical function which corresponds to the quantity of information being contained or delievered by a source (the consumers, for instance) [14].The more information is received by the receptor (provider A i ), the more the entropy (or uncertainty) about the message (consumers' preferences) decreases, due to this information gain.
In practice, service providers reserve bandwidth on transit providers' networks who route their traffics.These transit operators have approximately identical marginal costs.It is then essential for the service providers to book the optimal bandwidth volume on transit providers' networks using the acquired information to infer consumers' preferences.Indeed, service providers will have to pay for the quantity of data transfered.Besides, it is essential for service providers to reserve the exact quantity of bandwidth; otherwise transit providers punish them either by evening out their traffic if the reserved bandwidth is insufficient or by overbooking and deprioritarizing if there is a waste.It might then altered the quality of service / content level perceived by the consumers.Game description for firm A i The three service providers play simultaneously the two level game described below.Then, depending on their choices, the consumers subscribe to a service or report their decision.The two levels in the game result from the difference of timing between the access price determination and the contract of bandwidth reservation.
(1) A i sets his quality of service / content level q i (2) A i chooses the access price to his service p i Description of the consumers' choice model We suppose that the consumers have the opportunity to choose their service provider or to not subscribe to any service.In order to make their choice, they should compute the opportunity costs [13] associated with each service provider.For firm A i , the opportunity cost is c i = p i +βq i with −1 ≤ β ≤ 06 coefficient characterizing consumers' sensitivity to the quality of service / content level.Now, the consumer choice model description requires the introduction of intrinsic utilities [4] for the consumers, associated with each provider, being independent of the opportunity costs.Consumers' intrinsic utility for firm A i will be denoted U i .Furthermore, we assume that the consumers have an a priori preference for the firms which is characterized by the following order U 1 ≤ U 2 ≤ U 3 .The order is arbitrary and it can be modified.We assume that 0 ≤ c 1 (.) < c 2 (.) < c 3 (.)< 1.This hypothesis guarantees the existence of positive market shares for each of the three firms.An approching choice model for the consumers has already been detailed in [9].
Then, we introduce the maximum admissible opportunity cost for consumer k, k = 1, 2, ..., N : Ξ k , whose realization ξ k is generated according to the uniform density on [0; 1]7 .This choice of a uniform density to model the consumers' intrinsic utility is motivated by the assumption that the firms have a priori no information on the consumers' preferences.Necessity to introduce a maximum admissible opportunity cost results from the following observation: a consumer will refuse to subscribe to the provider's service or will report his purchase to a later date for the follwing reasons, either the access price is too high, either the quality of service level is not sufficient, or the contents are not enough diversified compared to what he expects.Consumer k's utility for firm A i , i = 1, 2, 3 is The main assumptions related to the providers' game and to the consumer choice model are listed below.
• firms are limited in their investment in the quality of service / contents by the capacity of their access networks i.e., q i ≤ q max i , i = 1, 2, 3.
Then, we can compute the number of clients for each firm.Let n 0 be the total number of consumers who choose to not subscribe to any service, n i , the total number of clients for firm A i and N , the total number of consumers on the market.

Lemma 1. Consumers are distributed between the three service providers according to the following rule
• n 0 = N p 1 + βq 1 do not subscribe to any service,

III. SHARING MECHANISM OF THE GRAND COALITION'S REVENUE AND STABILITY
The game contains 2 3 −1 = 7 distinct coalitions.We assume that the game is with transferable utility i.e., the total revenue associated with each coalition can be freely shared between the coalition members.It implicitly means that the coalition members can freely give, receive or even, burn money [11].Games with transferable utility require the introduction of a characteristic function ν : 2 3 − 1 → R which associates a global revenue with each coalition.Function ν valuates on a coalition, measures its worth [8].
To valuate the characteristic function on each coalition, we have chosen a representation by defensive equilibria [11].
Let S be the set of the 7 possible coalitions and s ∈ S a given coalition.We note σ opt s = (p opt i , q opt i ) i∈s with p opt i , q opt i ∈ R 2 + , ∀i ∈ s, the strategy on prices, quality of service / content level chosen by the coalition s members and σ opt S−s = (p opt j , q opt j ) j∈S−s with p opt j , q opt j ∈ R 2 + , ∀j ∈ S−s, the strategy selected by the firms who do not belong to coalition s.For each coalition s ∈ S, we have to solve the following system to determine the providers' optimal strategies Using coalition s optimal strategy expression, we obtain the value of the characteristic function ν on coalition s:

. For each possible coalition, the two level game between the three providers admits a unique pure Nash equilibrium on prices and quality of service / content level. The characterisitc function is then defined uniquely on each possible coalition.
Proof of Theorem 2 In order to fix the ideas, we assume in the rest of the article that I(.) is the identity application and suppose that the information level is either quadratic in the QoS / content level i.e., α i = ϑ(q i ) = q 2 i 2 (example 1), or that it is linear in the QoS / content level i.e., α i = ϑ(q i ) = q i (example 2).However, the proof can be generalized to far more complex functions satisfying the hypotheses of Section II.
For each possible coalition, we have to solve for every provider in the coalition, a two level game in price and quality of service / content.Going backward, we start by determining the optimal prices for each provider in the coalition by differentiating the coalition's utility with respect to the prices.Then, we substitute the optimal prices in the coalition's utility and differentiate it once more, with respect to the quality of service / content level.Finally, we obtain the optimal quality of service / content level and the optimal prices for each provider in the coalition.The resulting strategies form a Nash equilibrium for the players since none of them has incentives to deviate from it unilaterally.Details about the analytical determination of the Nash equilibrium and its unicity are provided in Appendix A. The characteristic function ν is also evaluated on each coalition.
A cooperative game is an interactive decision model based on the behavior of groups of players or coalitions.In a cooperative game, one of the most difficult problem to solve is the sharing of the coalition's total revenue between its members.Shapley has proposed a fair sharing rule for a n player cooperative game.Another solution concept should require that the coalition members had no incentives to deviate to increase their revenue.Such a solution concept should guarantee the system stability.The set of the allocations of the grand coalition's revenue which should guarantee the system stability is the core of the cooperative game; formally, it is the set of the global revenue allocations

Computation of the Shapley value
The core of a cooperative game can be empty or very large.It explains partly, why this notion is so difficult to apply to predict the players' behavior.An alternative approach might be to identify a unique mapping φ : S → R 3 such that for the cooperative game defined by the characterisitc function ν, the expected revenue of each provider i is φ i (ν).
Shapley has approched this problem axiomatically by defining a solution that prescribes a single payoff for each player which is the average of all marginal contributions of that player to each coalition he is member of.It satisfies four main axioms: efficiency i.e., the payoffs must add up to ν({1, 2, ..., n}) which means that all the grand coalition's surplus is allocated; symmetry i.e., if two players are substitutable because they contribute the same to each coalition the solution should treat them equally; additivity i.e., the solution to the sum of two games with transferable utility must be the sum of what it awards to each of the two games; dummy player i.e., if a player contributes to nothing to every coalition the solution should pay him nothing.[11]) There exists exactly one application φ : S → R satisfying the four axioms cited above.In a cooperative game with n players and transferable utility, this function satisfies the following equation for each player i and for any characteristic function ν

Theorem 3. (Shapley
Formula in Theorem 3 can be interpreted the following way.Suppose that we plan to assemble the grand coalition in a room but, the door to the room is only large enough for one player to enter at a time, so the players randomly line up in a queue at the door.The are n! different ways that the players might be ordered in the queue.For any set s that does not contain player i, there are |s|!(n − |s| − 1)! different ways to order the players so that s is the set of players who are ahead of player i in the queue.Thus, if the various orderings are equally likely, is the probability that, when player i enters the room, he will find the coalition s there ahead of him.If i finds s ahead of him when he enters then, his marginal contribution to the worth of the coalition in the room when he enters is ν(s∪{i})−ν(s) .Thus, under this story of randomly ordered entry, the Shapley value of any player is his expected marginal contribution when he enters.
For each of the firms, we can compute the Shapley value which corresponds to a fair sharing of the grand coalition's total benefit between the firms.
Analytical expressions of Shapley value allocations for each provider when the information level is quadratic (respectively linear) in the quality of service / content level are detailed in Appendix B.
Influence of β on the system stability with Shapley value as sharing rule Presently, we want to characterize the influence of the consumers' sensitivity coefficient to the quality of service / content level (β) on the mechanism of fair revenue sharing for the grand coalition.
We have represented the allocations given by the Shapley value for providers A 1 , A 2 , A 3 , for different number of consumers on the market (N = 5, N = 10, N = 50, N = 120) as functions of β ∈ [−1; 0].Each provider's allocation issued from the Shapley value are plotted on Figures 3 and 4 with information cost function being quadratic and linear in the quality of service / content level, respectively.
We observe that for small sensitivity coefficients8 , firms do not invest very much in quality of service / content and prefer proposing small access prices.In this case, we observe in Figure 3, that A 3 dominates the market.Indeed, access prices being small, providers' revenues are small and hardly compensate the information cost which is quadratic in the quality of service / content level.Thus, it is the favorite firm (A 3 ) which starts with a certain advantage and dominates the market.For an information cost linear in the quality of service / content, we note in Figure 4, that compensation is less difficult and that A 1 and A 3 alternately dominate the market.Indeed, A 1 is a priori the firm who differentiates himself the most from the others by proposing the smallest prices while A 3 is the favorite provider for the consumers.
For intermediate sensitivity coefficients, we observe on Figures 3 and 4, oscillations on the providers' revenues.Indeed, these latter are involved in a price war.
Finally, when the sensitivity coefficient is high, providers invest massively in the quality of service / content.The problem is then that the providers' access networks are limited in capacity.Thus, they cannot invest more than a fixed quantity q max i , i = 1, 2, 3.For a large number of consumers, each provider invests at the maximum of his capacity and tries to differentiate with the access prices.At the end, the game converges when β decreases towards a perfect competition situation where the access prices are fixed such that the firms' profits are near 1 or even null 9 .Sufficient conditions guaranteeing the core existence To determine the core of the game, we have to solve the following system of equations then, if the grand coalition's total revenue is shared according to the Shapley value, the grand coalition is stable i.e., no provider has incentives to leave it.

Proof of Propositon 4
To start, we want to determine if there exists β values for which the Shapley value is in the core.To perform this point, we draw the 7 functions of β, each one of them describing System (C) equality / inequalities.These functions are Suppose that the information level is quadratic in the QoS / content level.In Figure 5, we have plotted these 7 functions of β.We observe that the constraints defining the core of the game are satisfied if, and only if, β ∈ [−1; −0.78] with an information level quadratic in the quality of service / content investment level.In Figure 6, we have plotted these 7 functions of β assuming that the information level is this time, linear in the QoS / content level.We observe that if β ∈ [−1; −0.49] then, the Shapley value satisfies the constraints characterizing the core of the game.If the grand coalition is stable then, the regulatory authority's advices would not be followed by the providers.Indeed, the Shapley value being in the core of the game, the providers would rather follow this fair way to share the grand coalition's profit than deviating from it by following the regualtor's advices and risking to lose money.However, if the Shapley value is not in the core of the game, the providers' behaviors might be unpredictable and the regulator should intervene to control the system.

IV. INTERVENTION OF THE REGULATORY AUTHORITY TO MAXIMIZE THE CONSUMERS' WELFARE
In order to fix the ideas, we assume in the rest of the article that I(α i ) = q 2 i 2 .We have seen in Section III that in this case, if β ∈ [−1; −0.78], the Shapley value is in the core of the game.Thus, if the three firms agree to implement this sharing mechanism, the grand coalition should be stable and none of the firms would deviate.
But, if β > −0.78, the System of equations (C) being impossible to solve analytically, we cannot conclude on the existence of a core for the game.The considered ecosystem might then be highly unstable, penalizing heavily the firms and the consumers.Then, the regulatory authority should intervene.His aim is to increase the consumers' welfare while guaranteeing a maximal coverage of the market.In practice, it means that the regulatory authority will try to determine the system organization minimizing n 0 = N p 1 + βq 1 i.e., the total number of consumers who choose to not subscribe to any service, as a function of β and N .
Contrary to Section III, for each coalition, the regulatory authority being unbiased, divides equally its total revenue between its members.This is the simplest allocation rule [1].It means that the regulatory authority does not want to take into account the providers' worth in the coalition to allocate the total revenue.Besides, since β > −0.78, the Shapley value is not in the core anymore; hence the providers would most certainly deviate from this sharing rule if the regulator imposed it.Consequentely, it is not a solution for the regulator to impose the Shapley value as sharing rule in this section.
Determination of the domain of definition for (β, N ) depending on the proposed services In Figure 7, we have  depicted which system organization should maximize the market coverage i.e., minimize the number of consumers without subscription (n 0 = N p 1 + βq 1 ) as a function of β and N .The different possible system organizations are: complete competition (M 0 ), cooperation between A 1 and A 2 solely (M 1 ), cooperation between A 1 and A 3 (M 2 ), cooperation between A 2 and A 3 solely (M 3 ) and cooperation between the three operators (M 4 ).
The applications that we consider in this article, are highly sensitive to the quality of service like for instance, video over demand, music downloadings, interactive multiplayer games, etc.; hence we assume that β < −0.7.Besides, the total number of consumers on the market being rather large, Figure 7 tells that the regulatory authority should break all the emerging alliances (M 0 ) to guarantee perfect competition between the providers and maximize the market converage.Suppose that the providers cannot communicate (to avoid cheating or reputation phenomenon for instance) and that they play simultaneously the two-level game described in Section II.Initially, they are all members of a common alliance.The no-communication assumption means that a two-player coalition might be formed only if both players decide at the same instant to form a sub-coalition without prior arrangement.Furthermore, border situations might appear where one provider continues to play as if he were in an alliance although the other players have already defected.But of course, the observation of the resulting payoffs gives indications to the providers about their allies' past choices 10 .
In Tables I and II, we have listed all the joint possible actions for the providers.In fact, each provider is confronted to a prisonner dilemma: either he tries to cooperate or, he defects.Contrary to the classical version of the dilemma [18], we have to cope here with three players in competition.To be more precise, we describe the game setting.If the three providers choose to cooperate, they form a grand coalition and their access prices are p * i , i = 1, 2, 3 as defined in Appendix A. If only two of the providers accept to cooperate (for instance A i and A j ), they form (secretly) a coalition against A k , i, j, k = 1, 2, 3, i = j, i = k, j = k who, being not informed of this alliance, continues to play as under total competition.The access prices are denoted p AiAj i and p AiAj j (resp.p N k ) for A i and A j (resp.A k ) as defined in Appendix A.
In Tables I and II, (p N 1 , p N 2 , p N 3 ) is a Nash equilibrium if, and only if, the providers have no incentives to leave this equilibrium to form coalitions. Formally, the three following inequalities should be satisfied ) is a Nash equilibrium for the three providers on the domain of definition D if,and only if, N ≥ 14 100 + 500.10 3 β 2 .

Proof of Proposition 5
In Figure 8, we have depicted  the areas in β and N where the Nash equilibrium conditions for (p N 1 , p N 2 , p N 3 ) are satisfied for each provider.We observe that these areas can be delimited by second order polynomial equations.For firm A 1 , the area guaranteed that (p N 1 , p N 2 , p N 3 ) is a Nash equilibrium is delimited by the equation N = 15 600 + 93 750β 2 .Identically, for firm A 2 , we get: N = 15 400 + 156 250β 2 and for firm A 3 , we obtain: N = 14 100 + 500.10 3 β 2 .In Figure 9, we have pictured the three resulting curves as functions of β and N as well as the domain of definition D. We then get numerically that (p N 1 , p N 2 , p N 3 ) is a Nash equilibrium for the three providers if, and only if (β, N ) lie in the area delimited by the polynomial equation in β: N = 14 100 + 500.10 3 β 2 .
Controled dynamic evolution from cooperation to total competition As already stated, initially, the three providers form a common alliance.We assume that the conditions obtained in Proposition 5 are satisfied and that two providers cannot simultaneously leave the alliance; besides, the total number of consumers on the market is fixed and known by all the providers.The cooperative game introduced in Section III is now repeated under an infinite horizon and discrete time steps.In a dynamic framework, we need to introduce a discount factor δ ∈]0; 1[ which enables us to valuate a future profit, today [2], [8], [18].Three evolution scenarios may raise at time instant T > 0 11 , A 3 deviates at 11 T , is fixed arbitrarily.time instant T .Then at time instant T + 1 12 , three cases might appear: either A 1 and A 2 continue to cooperate in the rest of the game; either A 1 deviates; or A 2 breaks the alliances he had with A 1 .If A 1 or A 2 breaks the alliance with the other provider, cooperation cannot be envisaged anymore in the rest of the game since the providers' prices stabilize forever in the Nash equilibrium , A 2 leaves the grand coaltion at time instant T .Then at time instant T +1, three cases are possible: either A 1 and A 3 continue to cooperate in the rest of the game; either A 1 breaks the alliance he had with A 3 ; or A 3 breaks the alliance he had with A 1 .In these two latter cases, no more cooperation can emerge in the rest of the game.
, A 1 leaves the grand coalition at time instant T .Then at time instant T + 1, three cases can be envisaged: either A 2 and A 3 continue to cooperate in the rest of the game; either A 2 breaks the alliance he had with A 3 ; or A 3 breaks the alliance he had with A 2 .In these two latter cases, competition becomes total forever.
The main hypotheses of these section are recalled below.
• two providers cannot simultaneously leave the alliance, • contrary to Section III, for each coalition, the regulatory authority divides equally its total revenue between its members, • δ ∈]0; 1[ is the discount factor of the repeated game.

Study of Scenario 1
We let U * ij be the coalition {A i , A j } maximum revenue when both A i and A j cooperate; it is simply defined as the maximized sum of A i and A j 's utilities.Similarly, U * denote the grand coalition maximum revenue when the three providers cooperate; it coincide with the maximized sum of the three providers utilities.If then, A 3 deviates at time instant T and then, A 1 and A 2 cooperate for the rest of the game.If then, A 3 deviates at time instant T and then, A 1 deviates at time instant T + 1 and no cooperation cannot be envisaged anymore for the rest of the game.If then, A 3 leaves the grand coalition at time instant T and then, A 2 deviates at time instant T + 1 and the providers do not cooperate anymore in the rest of the game.Proof of Proposition 6 A 3 leaves the grand coalition at time instant T if the profit he expects to receive from this deviation is superior to the profit that he would receive if he continues to cooperate with A 1 and A 2 .In order to valuate A 3 's worth of deviation, this latter should distinguish between three cases has deviated punish him by forming a coalition.In this case, A 3 deviates in spite of his rival's threats if, and only if punishment .
• At time instant T +1, A 1 understands that A 3 has deviated and to punish him refuses to cooperate once more with A 2 .A 3 deviates in spite of A 1 's threat if, and only if punishment .
• At time instant T +1, A 2 understands that A 3 has deviated and to punish him refuses to cooperate once more with A 1 .A 3 deviates in spite of A 2 's threat if, and only if If then, A 2 leaves the grand coalition and then, A 3 deviates and no more cooperation can emerge during the rest of the game.Proof of Proposition 7 A 2 deviates at time instant T from the grand coalition if the profit he receives from this deviation is superior to the profit that he should have received if he had continued to cooperate with A 1 and A 3 .In the same spirit as in the proof of Proposition 6, we need to distinguish between three cases.
• A 2 leaves the grand coalition at time instant T then at time instant T + 1, A 1 and A 3 realize it and decide to cooperate.A 2 chooses to deviate in spite of {A 1 , A 2 }'s threat if, and only if punishment .
• A 2 leaves the grand coalition at time instant T and then at time instant T +1, A 1 realizing it, refuses to cooperate once more with A 3 .The three providers are then in competition for the rest of the game.A 2 deviates in spite of A 1 's threat if and only if ). • A 2 leaves the grand coalition at time instant T and then at time instant T +1, A 3 realizing it, refuses to cooperate once more with A 1 .The three providers are then in competition for the rest of the game.A 2 deviates in spite of A 3 's threat if, and only if . At the bottom of Figure 11, we have pictured F 3 .F 3 > 0 for β ∈ [−0.66; −0.56[ i.e., A 2 has no incentives to leave the grand coalition for these values of β.At the top of Figure 11, we observe F 0 − F 1 (left) and F 0 − F 2 (right).We infer that if A 2 leaves the grand coaltion, A 1 and A 3 will never cooperate.Figure 12, we have depicted F 4 (left) and F 5 (right).We infer the system evolution towards complete competition as function of β and of δ.
then, A 1 deviates and then, A 2 and A 3 continue to cooperate during the rest of the game.If then, A 1 deviates and then, A 2 refuses to cooperate with A 3 only and leaves the coalition.The three providers are in competition for the rest of the game.If then, A 1 deviates and then, A 3 breaks the alliance with A 2 .The three providers are in competition during the rest of the game.

Proof of Proposition 8
In the same spirit as in the proofs of Propositions 6 and 7, A 1 deviates from the grand coalition if the profit he receives from this deviation is superior to the profit he would have received if he had continued to cooperate with A 2 and A 3 .Three cases should be distinguised.The arguments are the same as thos detailed in the previous propositions; hence it will not be detailed. Let At the botton of Figure 13, we have pictured F 3 .F 3 > 0 for β ∈ [−0.56; −0.43] if, and only if A 1 leaves the grand coalition eventhough he thinks that A 2 and A 3 will punish him by forming a coalition.Let ).On figure 14, we observe F 4 (left) and F 5 (right).Depending on β and δ values, we determine the system dynamic evolution.Proposition 9.For the regulatory authority, it is sufficient to choose an interest rate smaller than 53% to maximize the market coverage and a fortiori, the consumer welfare.
Proof of Proposition 9 Using Proposition 6, 7 and 8, we could infer the system's behavior as function of β and δ values.
Using Proposition 6 Cooperation cannot be envisaged in the rest of the game.The discount factor can be expressed as a function of the interest rate: δ = 1 1+r with r > 0 the per-period interest rate imposed by the regulatory authority [2].It is equivalent with r = 1 δ − 1.rate r, should be smaller than 53% to insure that competition is total between the three service providers.

V. CONCLUSIONS
In this article, we have studied the game that arises between three service providers in competition, who have the opportunity to cooperate in order to increase their profits.We prove that if the consumers' sensitivity to the quality of service / content coefficient (β) is inferior to −0.78 then, the Shapley value is a fair way to share the grand coalition revenue between the service providers and belongs to the core of the game i.e., provided the three service providers agree on this sharing rule the game remains stable.But if β > −0.78, the regulatory authority should intervene since the system behavior might be highly unstable penalizing the consumers and the weakest providers.We show that under equal allocations of the coalitions' revenues, if the regulatory authority imposes a per-period interest rate smaller than 53%, the consumer welfare is guaranteed and the market coverage is maximized.
Extensions of the article might consider other sharing rules such as the nucleolus or the proportional fairness criteria [15].The game resolution can also be extended to deal with far many service providers in interaction and / or various sources of uncertainty.To perform this point, simulation and stochastic processes should be introduced [17].

A Proofs
We just recall some basic notations that could be useful to understand the following proofs.Under complete competition, provider i's optimal decision will be denoted p N i for the access price and q N i for the optimal QoS / content level.Provider i's maximized utility is U N i .When firms A i and A j form a coalition against A k , the firms optimal decisions will be denoted p for the prices and q AiAj i , q AiAj j , q AiAj k for the QoS / content levels.Then, the coalition maximized utility is denoted U * ij while A k maximized utility is U * k .Finally, when all the providers enter a grand coalition, its optimal utility is denoted U * .
Computation of ν(A 1 ), ν(A 2 ), ν(A 3 ) In this case, the three providers are in competition and no cooperation can emerge.By differentiation of each provider's utility, we obtain that the optimal access prices for A 1 , A 2 et A 3 are of the form By substitution of the price in the providers' utilities, we obtain Then, to get the optimal quality of service / content q * i , it is sufficient to differentiate U i with respect to q i and to solve the associated linear system.
As previously mentioned, in order to fix the ideas, we consider two examples.In the first example, the information level is quadratic in the quality of service / content level i.e., 2α i and the information cost function I(.) is the identity.In the second example, the information level is linear in the quality of service / content level i.e., α i = q i while the information cost function I(.) is still the identity.
Example 1. Information level quadratic in the quality of service / content level In this case, the maximized utilities for each firm A i , i = 1, 2, 3 can be written as The optimal quality of service / content levels for each firm are Finally, by substitution of the optimal prices in the previous QoS / content level, we obtain the access price final expressions Example 2. Information level quadratic in the quality of service / content level The maximized utilities for each of the three providers are identical The quality of service / content levels are obtained Finally, the optimal access prices for each firm are In this case, firms A 1 and A 2 form an alliance against firm A 3 who remains alone.The optimal prices p 1 and p 2 , are obtained by differentiation of the alliance's utility U 1 + U 2 .A 3 's optimal access price p 3 , is obtained by differentiation of U 3 .We get By substitution of the access prices, the provider A 3 and the alliance utilities become Example 1. Information level quadratic in the quality of service / content level The maximized utilities for alliance {A 1 , A 2 } and its rival A 3 are The optimal quality of service / content levels are of the form Finally, the optimal access prices for the partners in the alliance {A 1 , A 2 } and for A 3 are Example 2. Information level linear in the quality of service / content level The maximized utilities for coalition {A 1 , A 2 } against A 3 are The optimal quality of service / content levels are obtained Finally, the optimal prices become Firms A 1 and A 3 form a coalition against firm A 2 who is alone.By differentitation of {A 1 , A 3 }'s utilities and of firm A 2 's utility, we obtain the optimal prices for the three providers Then, substituting these optimal prices, coalition {A Example 1. Information level quadratic in the quality of service / content level The maximized utilities for coalition The optimal quality of service / content levels are Finally, the optimal prices for the providers take the following expression Example 2. Information level linear in the quality of service / content level The maximized utilities for coalition {A 1 , A 3 } and provider A 2 are The optimal quality of service / content levels are Computation of ν(A 2 , A 3 ) In this case, firms A 2 and A 3 form a coalition against firm A 1 .The optimal prices are obatined by differentiation of coalition {A 2 , A 3 }'s utility and of firm A 1 's utility p 1 = 1 6 β(q 3 + q 2 − 3q 1 ) + 1 By substitution of the optimal prices, coalition {A 2 , A 3 }'s utility and A 1 's utility can be expressed as functions of quality of service / content levels q i , i = 1, 2, 3 exclusively; which gives U 1 = 1 36 ((β 2 q 2 3 + (2β 2 q 2 − 6β 2 q 1 + 2β)q 3 + β 2 q 2 2 + (2β − 6β 2 q 1 )q 2 + 9β 2 q 2 1 − 6βq 1 + 1)N − 36Ioϑ(q 1 )) U 2 + U 3 = 1 3 ((β 2 q 2 3 + (−β 2 q 2 − β)q 3 + β 2 q 2 2 − βq 2 + 1)N − 3Ioϑ(q 3 ) − 3Ioϑ(q 2 ))

Example 1. Information level quadratic in the quality of service / content level
The maximized utility for coalition {A 2 , A 3 } and for firm The optimal quality of service / content levels are The optimal quality of service / content levels are Finally, the optimal access prices are Computation of ν(A 1 , A 2 , A 3 ) In this case, the three firms are in the grand coalition.The optimal access prices for the grand coalition are obtained by differentiation of U 1 +U 2 +U 3 p 1 = 1 4 β(q 3 + q 2 − 3q 1 ) + 1 p 2 = − 1 2 β(−q 3 + q 2 + q 1 ) − 1 β(q 3 + q 2 + q 1 ) − 3 By substitution, the grand coalition's utility takes the form (3β 2 q 2 3 + (−2β 2 q 2 − 2β 2 q 1 − 2β)q 3 + 3β 2 q 2 2 + (−2β 2 q 1 − 2β)q 2 + 3β 2 q 2 1 − 2βq 1 + 3)N − 8 i=1,2,3 Ioϑ(q i ) Example 1. Information level quadratic in the quality of service / content level The grand coalition's maximized utility is The optimal quality of service / content levels are identical for the three providers in the grand coalition The optimal access prices are identical for firm A 1 and A 2

B Computation of the Shapley value allocations
We determine the Shapley value of our game for two categories of information level functions.The Shapley value gives us a fair manner to share the grand coalition's total revenue between the three providers.

Fig. 2 .
Fig. 2. Quality of service / content level as a function of firm Ai's information level.

Fig. 3 .
Fig. 3. Shapley value allocations for A1, A2, A3 as functions of β with information quadratic in the quality of service / content level.

Fig. 4 .
Fig. 4. Shapley value allocations for A1, A2, A3 as functions of β with information linear in the quality of service / content level.
The relation between N and β < −0.7 delimiting the area where no alliance should emerge can be interpolated by a second order polynomial equation in β and N : N = 1 100 + 20 833(0.8+β) 2 .Now and in the rest of the article, we assume to be in the domain of definition D, characterized by the relations D = {β > −0.78}∩{N ≥ 1 100+20 833(0.8+β) 2 } .The regulatory authority should then intervene to guarantee perfect competition between the providers.

Fig. 7 .
Fig. 7. System organization minimizing the total number of consumers without subscription as function of β and N .

Fig. 8 .
Fig. 8. Conditions for each provider on β and N guaranteeing the existence of a Nash equilibrium in (p N 1 , p N 2 , p N 3 ).

Fig. 9 .
Fig. 9. Conditions on β and N guaranteeing the existence of a Nash equilibrium in (p N 1 , p N 2 , p N 3 ) on the domain of definition.

Fig. 10 .
Fig. 10.Conditions on β and δ under which A3 leaves the grand coalition and no cooperation between the providers can emerge, N = 450.10 3 .

TABLE I POSSIBLE
INTERACTIONS BETWEEN THE PROVIDERS WHILE A1 REFUSES TO COOPERATE.

TABLE II POSSIBLE
INTERACTIONS BETWEEN THE PROVIDERS WHILE A1 AGREES TO COOPERATE.
Proposition 6.If provider A 3 leaves the grand coalition then, there does not exist any value of β ∈ [−0.78; 0] and δ ∈]0; 1[ guaranteeing that providers A 1 and A 2 would cooperate.
Proposition 7. If β ∈ [−0.78; −0.57[ then, provider A 2 does not deviate provided he thinks that providers A 1 and A 3 are going to cooperate.In this case, the three providers stay in the grand coalition.

TABLE III AREA
OF β AND δ WHERE SCENARIOS HOLD.Comparing Propositions 6, 7 and 8 results (cf.Table III), we observe that cooperation might emerge if, and only if,