Cooperative Game Theory Framework for Energy Efficient Policies in Wireless Networks Matteo Sereno Dipartimento di Informatica, Universit`a di Torino, Corso Svizzera 185, Torino email:
[email protected] Preliminary Draft
Abstract—In this paper we focus the attention on the energyaware cooperative management of cellular access networks that offer service over the same area. In particular, we propose an approach based on the cooperative game theory to address issues such as stability of the cooperation and sharing of the benefits derived by cooperative behaviors.
I. I NTRODUCTION Energy and environmental implications are driven the ICT market and the researcher communities to study energy-aware and energy-efficient solutions. In particular, due to the enormous diffusion of cellular accesses, one of the most interesting field concerns the development of energy efficient solutions for wireless and cellular networks. One of the main motivation of this interest is that the energy consumed by such huge number of access networks is largely wasted in the periods of time when the number of users served by the wireless access network is low. In other words, these access networks are dimensioned based on the peak hour traffic, so that when traffic load decreases (e.g., during the night) they are overdimensioned. In general the same area is normally covered by several competing wireless (or cellular) network providers whose access networks are tailored with respect to the peak hour traffic. When traffic is low, the networks become over-dimensioned, and it may happen that only few of them (or even one) can serve the traffic of entire area. In this manner, if the providers cooperate, some access networks are switched off, and the service can be provided by the networks that remain active. In the literature there are many studies that investigate this approach (see for instance [1] for cellular networks, and [4] for WLANs). Most of the previous proposals (see for instance [2] and [1]) focus the attention on the evaluation of the energy saving that can be obtained by switching off some of the access networks and on the switching strategies to increase the energy saving. In this paper we study a different issue concerning the cooperation among the providers. Rather than the performance evaluation of switching strategies, we shift the attention on the structure of the cooperation itself. Cooperation that must be considered as a novel networking paradigm that can improve
the performance of wireless communication networks and/or the energy efficiency. In particular, we will address questions such as: Why the providers should cooperate ? What is the cooperative structures ? We address efficiency and stability (also called consistency) of the coalition, and allocation problem of the coalition benefits in equitable and fair manner among the players of the coalition. In the paper we present a game theory framework to describe the interaction among the network providers that cooperate by using only a few (or even one) access networks. In particular, we use the cooperative game theory that mainly deals with the formation of cooperative groups, i.e., coalitions, that allow the cooperating player to strengthen their positions in a given game. The reminder of this paper is organized as follows. Section II provides the system description and introduces the basic ideas of the cooperative game theory with transferable utility. Section III introduces the basic idea of the core. That is a is the set of allocations which guarantees that no group of players has an incentive to leave the coalition to form another coalition. In Section IV we discuss a set of allocation methods pointing out pros and cons of each technique. Finally VI concludes the paper outlying possible future directions. II. P ROBLEM F ORMULATION A. System Description We consider an area served by n providers, whose access networks fully cover the area. The set of providers is denoted by N = {1, 2, . . . , n}. We assume that each access network is dimensioned according to a peak traffic demand of the provider’s customers. Let denote by maxi , and by ni respectively, the maximum and the current number of customers in the i-th access network. In the following we denote by ri (ni ) the revenue rate for the i-th provider when ni customers and by ei (ni ) the energy cost per unit time payed by the i-th provider when there are ni customers in its network (for i = 1, . . . , n). These costs can be expressed in terms of kWh or a corresponding monetary cost.
In the following we assume two different kinds of energy functions: ei (ni ) = ki′ ei (ni ) = hi · ni + ki′′ ,
(1a) (1b)
where ki′ , ki′′ and hi are non negative constants, that may be provider-dependent. The first function models access networks where the power consumption does not depend on the number of active customers in the network, while the latter function accounts for access networks where the power consumption is function of the number of active customers plus a fixed cost. We assume that the traffic profile exhibits a periodic behavior. In particular, a daily pattern spanning over 24 h is rather common, see, for instance, Figure 1.
Traffic Load (%)
100% 80%
coalitions, in order to improve their positions in the game. Any coalition S ⊆ N represents an agreement among the players in S to act as a single entity. In the system under study a coalition S among the providers implies that only one access network is powered on while the other |S| − 1 are powered off and the unique active network serves all the customers of the providers in S. The second fundamental concept to define a coalition game is the coalition value v that quantifies the worth of a coalition in a game. The value is a function over real numbers v : 2N −→ IR. This function associates with every coalition S ⊆ N a real number that quantifies the gain of S. In this manner a coalition game can be defined by the pair (N , v). In our study we focus the attention on the games with transferable utility (TU) and in this case the utility can be divided in any manner among the coalition members. In particular, the value (or utility) can be represented by monetary value that the members of the coalition can distributed among themselves by using an appropriate (fairness) rule. For the system under study we define the coalition value as = ri (ni ) − ei (ni ), and ∑ ∑ nj , ri (ni ) − min ej v(S) = j∈S
v({i})
60% 40% 20%
i∈S
3h
6h
9h
12h 15h Time
18h
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24h
Fig. 1: Typical daily traffic profile l(t) The simple observation of traffic patterns similar to the one depicted in Figure 1 allows us to say that when the traffic load decreases due to the normal variations, networks are overdimensioned. In general service areas are normally served by several competing providers, when the traffic is low, resources become redundant, and a few access networks can carry the traffic within the service are under study. In this manner, a subset of the access networks can be switched off when the overall traffic load is such that the networks that remain active can carry all the traffic without service deteriorations. Note that we are assuming that the customers can be served by any network. In particular, when a network is switched off, the customers of the corresponding provider roam to the networks that remain active. Furthermore, in this paper we assume that the overall traffic load, and the network configurations are such that in principle, all the traffic can be carried by only a single network while all the other one can be switched off. In particular, the focus of the paper is on the framework that the providers can use to reach such kind of agreement. B. Coalition Games Coalitional games involve a set of players N = {1, . . . , n} (e.g., the providers) who seek to form cooperative groups, i.e.,
(2)
j∈S
where ni is the number of customers of the i-th provider. The members of the coalition S decide that a particular network remains active while all the networks of the coalition S are switched off. The active { network one with minimum (∑ is the)} n . energy cost (i.e., minj∈S ej j j∈S The amount of utility that a player i ∈ S receives from the division of v(S) is the player’s payoff and is denoted by xi while the vector x ∈ IR|N | , with each component xi being the payoff of player i ∈ S is the payoff allocation. The system under study can be modeled by using the most common form of coalition games, i.e., the characteristic form [8] where the value of a coalition S depends solely on the members of that coalition, with no dependence on how the players in N \S are structured. From the payoff allocation vector x we can easily derive the side-payment for each provider. To this aim we denote by i the index of the access network that remains active. That is, the access network with minimum energy cost (see Equation (2)). The side-payment in charge to the i-th provider can be defined as ∑ nj 1I(i, i) , (3) yi = xi − ri (ni ) − min ej j∈S j∈S
where 1I(i, i) = 1 if i = i and 0 otherwise. A positive value of yi means that the i-th must receive yi from the coalition, while a negative value means that i must transfer yi to the coalition. Since we are assuming that the overall load and the network capacities are such that all the customers can be served by a single network and the remaining ones can be switched off
in terms of cooperative games this means that the formation of larger coalitions is never detrimental to all the providers. In canonical games no group of players can di worse by cooperating than by acting non-cooperatively. This is the mathematical property called superadditivity, v(S ∪ T ) ≥ v(S) + v(T ), ∀ S ⊂ N , T ⊂ N , such that S ∩ T = ∅.
(4)
Furthermore, a game is convex if v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T ).
(5)
means that some players or groups of players are better off when acting alone than when cooperating with other players (the grand coalition N ). The class of convex games was introduced by Shapley [10] and has attracted a lot of attention because the games in this class have very nice properties: for example, the core of a convex game is non-empty. In particular, we can easily prove that if the energy functions do not depend on the player, i.e., ei (ni ) = k ′
(7a)
ei (ni ) = h · ni + k ′′ ,
(7b)
One of the goals of the game theory model we develop is the study of the properties and stability of the grand coalition, i.e., the coalition of all the providers, and the distribution of the gains in a fair manner among the providers.
the game is convex (see Appendix). In general we can check whether the core is not-empty by solving the linear equations defined by (6).
III. C OOPERATIVE G AME T HEORY
IV. A LLOCATION RULES AND P RINCIPLES
A. The Core The core [7] is the most used solution concept to analyze stability issues for coalition games. In particular, the core is the set of payoff allocations which guarantees that no group of players has an incentive to leave the grand coalition N to form another coalition S ⊂ N . To specify the core, we first introduce the following definitions. A payoff vector x ∈ IR|N | is said to be group rational or ∑ efficient if i∈N xi = v(N ). A payoff vector x is said to be individual rational if every player can obtain a benefit (i.e., a payoff) no less than acting alone. In other words, for any player i, xi ≥ v({i}). An inputation is a payoff vector that satisfies these definitions. An imputation ∑ x is said to be unstable through a coalition S if v(S) > i∈S xi . In other words, the players have incentive for coalition S and upset the proposed payoff allocation x. The set C of stable imputations is called the core, i.e., { ∑ C = x such that xi = v(N ), and i∈N
∑
xi ≥ v(S), ∀S ⊂ N
} (6)
i∈S
If x ∈ C, then no coalition S has incentive to split off if x is ∑ the proposed payoff allocation in N , because the payoff i∈S xi allocate to S is not smaller than the value v(S) which the players can obtain by forming the alternative (sub) coalition S. The cores of cooperative games can be empty and hence in these cases the grand coalition cannot be stabilized. There are several techniques to prove that the core of a cooperative game is not empty. The core is a solution concept that defines a set of allocations in which no player or subgroup of players can on its own attain a better allocation. A game with a non-empty core contains allocations that can be voluntarily agreed by all players and is thereby stable. An empty core on the other hand
In this section we address the methods for the payoff allocation that could be applied to the problem under study. In the literature there are plenty of allocation methods1 , that (in principle) could applied to our problem. What are the essential principles and properties that characterize different methods ? In [11] there is a list of some basic properties that the methods for the payoff allocation should have. These are: (i) additivity, (ii) monotonicity, and (iii) consistency (or stability). Additivity is a decomposition property. That is, if an allocation problem can be decomposed, can their solution be added ? Monotonicity: as the data of the problem change, do solutions change in parallel fashion ? For example, if the value of the game increases (e.g., by using a more energy efficient access network hardware) then no player should be allocated less than what they had before. Similarly, if costs increases (e.g., for increase of energy costs) no one should receive more than what they had before. Consistency (or Stability): are the solution invariants when restricted to subgroups of players? This means that an overall allocation for v(N ) should be viewed as fair by all possible subgroups of N , and hence no subset of players should find incentive to change the allocation. Moreover, there are other basic properties that an allocation method should have, they are the following: (1) an payoff allocation ∑ vector x is said to be an efficient cost allocation method if i∈N xi = v(N ). (2) A symmetry assumption requires that the value function v(·) contains all the data relevant for the allocation problem. In other words, this assumption rules out biased methods that allocate by using some outside information. (3) A player i is a dummy if it does not contribute to any coalition. The dummy axiom states that a dummy player will not receive any allocation of either costs or benefits. As pointed out in [11], these properties/axioms can not all be satisfied in the same payoff allocation method but they can be helpful in determining the methods that are appropriate for 1 In the literature these methods are called cost or surplus sharing, see [6] for details and references.
the problem under study. In the following we review a number of well known payoff allocation methods.
particular, an interesting result from game theory is that for convex games the Shapley value lies [10].
A. Equal Share
C. The Nucleolus
The simplest allocation method is the Equal Share mechanism where the coalition value v(N ) is equally shared among the players, i.e.,
If the stability or consistency of the allocation are of primary importance the core plays a fundamental role. The nucleolus is an allocation that minimizes the dissatisfaction of the players from the allocation they can receive in a given (N , v) game. For a coalition S, the measure of dissatisfaction from an allocation x is defined as the excess ∑ s(x, S) = v(S) − xj . (11)
xi = v(N )/n, for i = 1, . . . , n.
(8)
This mechanism has many undesirable properties. In particular, it ignores the contribution of a player to the coalition, so that a participant who supplies a large contribution to the coalition value receives exactly the same share as one who supplies no input (note that the method does not possess the dummy player property). A simple generalization of this method is the Proportional Share that allocates the payoff in proportion to some criterion such as the population (i.e., for the system under study this is the number of users in each access network) or usage. The advantages of the Equal Share method (and its generalizations) is its simplicity and the fact that is monotonic, but we must remark that it does not possess the the dummy player property, and does not use game theory concepts such as individual rationality and marginal contribution. B. The Shapley Value This is a solution method, introduced by Shapley in [9], based on the concept of marginal contribution. The method assumes that players join a colation in some random order. The marginal contribution that a player makes to the overall value of the coalition can depend on at which point in coalition formation process that the player joins. This process has a parallel interpretation in case of modelling costs or benefits (see [11]). The basic idea of the Shapley value is that a player’s expected marginal contribution (or cost) can be found as the average marginal contribution out of all possible orders of joining. This average calculated for each player is the Shapley value allocation. If we define the marginal contribution of player i to coalition S as { v(S) − v(S − {i}) if i ∈ S m(i, S) = (9) v(S ∪ {i}) − v(S) if i ̸∈ S The Shapley value of player i can be defined as ∑ |S − {i}|! |N − S|! m(i, S), ϕi (v) = n!
(10)
S⊆N i∈S
where the sum is over all subsets S containing i. The Shapley value fulfils several important properties such as efficiency, symmetry, additivity, monotonicity, and dummy property (see [11] for details and proofs). In general, the Shapley value is unrelated to the core. However, in some applications, one can show that the Shapley value lies in the core. In these cases the Shapley value also combines the previous properties with consistency (or stability). In
j∈S
The excess of a coalition S measures the amount (a quantification of dissatisfaction) by which coalition S falls short of its value v(S) in the allocation x. ∑ Since the core is defined as the set of imputations such that ∈S xi ≥ v(S), this implies that an imputation x is in the core iff all its excesses are negative or zero. Let denote as O(x) the vector of excesses arranged in decreasing order. In particular, we use the lexicographically order. That is, we say that a vector y = (y1 , . . . , yk ) is lexicographically less than a vector z = (z1 , . . . , zk ), and write y
