Cooperative Game Theory Framework for Energy
Efficient Policies in Wireless Networks
Matteo Sereno
Dipartimento di Informatica, Università 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
21h
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 <L z, if y1 < z1 , or if y1 = z1 and y2 < z2 , or if y1 = z1 ,
y2 = z2 and y3 < z3 , or . . ., if y1 = z1 , . . . , yk−1 = zk−1
and yk < zk . That is, y <L z if in the first component in
which y and z differs, that component of y is less than the
corresponding component of z. We write y ≤L z if either
y <L z or y {
= z.
}
∑n
Let X = x : j=1 xj = v(N ) be the set of efficient
allocations. The vector ν is said to be a nucleolus if for every
x ∈ X we have that O(ν) ≤L O(x)
The idea of the nucleolus is to find a solution in the core
that is ”central” in the sense of being as far away from the
boundaries as possible (see Figures 2 and 3).
The nucleolus of a canonical coalition game exists, is
unique, and satisfies several interesting properties such as
symmetry, dummy property, consistency. Moreover, if the core
of the game is non-empty, the nucleolus solution to the game
is always in the core. To find the nucleolus allocation for a
cooperative games requires linear programming.
In the following we provide two simple example that show
some of the characteristics of the coalition game we defined.
Example 1: Consider a game with three providers N =
{1, 2, 3}, with maxi = 32, ei (ni ) = ei (maxi ) = 1 (for
i = 1, . . . , 3). We assume that the revenue function is
r(ni ) = 81 ·ni . r(ni ) = b·ni , Note that according to Proposition
1, with these energy cost functions the game is convex. The
configuration in terms of number of users is n1 = 10, n2 = 8,
and n3 = 6.
With these definitions of costs, revenues, and parameters the
value function v is given by
v({1}) = 0.25
v({2}) = 0
v({3}) = −0.25
v({1, 2}) = 1.25
v({1, 3}) = 1.0
v({2, 3}) = 0.75
v({1, 2, 3}) = 2.0
The imputation are the points (x1 , x2 , x3 ) such that x1 + x2 +
x3 = 2.0 and x1 ≥ 0.25, x2 ≥ 0.0, x3 ≥ −0.25. This set is
the triangle with vertices (2.25, 0, −0.25), (0.25, 0, 1.75) and
(0.25, 2.0, −0.25).
Scenario
(a)
(b)
(c)
ϕ1 (v)
0.9167
1.0500
1.3333
Shapley value
ϕ2 (v)
ϕ3 (v)
0.6667
0.4167
0.6000
0.1750
1.0833 -0.2917
ν1
0.9167
0.9833
1.0833
Nucleolus
ν2
ν3
0.6667
0.4167
0.7333
0.1083
0.8333
0.2083
TABLE I: Shaple values and nucleolus allocations for the
three-providers game with three different scenarios
x2=0
(0.25, 0, 1.75)
x2+x3=0.75
(1.25, 0, 0.75)
unstable
{1, 2}
x1+x3=1
x1+x2=1.25
(0.25, 1, 0.75)
core
{1, 2, 3}
unstable
{2, 3}
unstable
{1, 3}
(2.25, 0, -0.25)
(0.25, 2, -0.25)
(1.25, 1, -0.25)
x3=-0.25
x1=0.25
Fig. 2: Barycentric plot of the three-providers game with
ei (ni ) = ei (maxi ) = 1, with the Shapley value (symbol )
It is useful to plot this triangle in barycentric coordinates.
This is done by requiring that the plane of the plot is the plane
x1 + x2 + x3 = 2, and giving each point on the plane three
coordinates which add to 2. We can draw the lines x1 = 0.25,
x2 = 0, x3 = −0.25, or the line x1 + x3 = 1 (which is the
same as the line x2 = 1), and so on.
Let us find the imputation that are unstable. The coalition
{2, 3} can guarantee v({2, 3}) = 0.75, so all points
(x1 , x2 , x3 ) with x2 + x3 < 0.75 are unstable through {2, 3}.
In other words the coalition composed by {2, 3} is unstable
because the coalition {1, 2, 3} guarantees a better value of
v({1, 2, 3}). The same happens for the coalition {1, 2} that
can guarantee itself v{1, 2}) = 1.25, all points above of the
line x1 + x2 = 1.25 are unstable. Finally, since {1, 3} can
guarantee itself v{1, 3}) = 1.0, all points to the right of the
line x1 + x3 = 1.0 are unstable. The core is the remaining
set of points in the set of imputations.
The Shapley Value: We can compute the marginal contributions
for each player i and all the coalition S such that i ∈ S. In
particular, we can write
m(1, {1}) = 0.25
m(1, {1,2}) = 1.25
m(1, {1,2}) = 1.25
m(1, {1,2,3}) = 1.25
m(2, {2}) = 0
m(2, {1,2}) = 1.0
m(2, {2,3}) = 1.0
m(2, {1,2,3}) = 1.0
m(3, {2}) = −0.25
m(3, {1,3}) = 0.75
m(3, {2,3}) = 0.75
m(3, {1,2,3}) = 0.75
and hence
ϕ1 (v)
=
ϕ2 (v)
=
ϕ3 (v)
=
1
1
1
1
0.25 + 1.25 + 1.25 + 1.25 = 0.9167
3
6
6
3
1
1
1
1
0 + 1.0 + 1.0 + 1.0 = 0.667
3
6
6
3
1
1
1
1
(−0.25) + 0.75 + 0.75 + 0.75 = 0.4167.
3
6
6
3
The Nucleolus: The nucleolus requires the minimization the
maximum
of the excesses s(x, S) over all S subject to
∑
xj = v(N ). This problem of minimizing the maximum of
a collection of linear functions subject to linear constraints
can be converted to a linear programming problem that can be
solved by using the simplex method, for example. For n = 3
finding the nucleolus is not too difficult. For the case with
three providers defined in Example 1 with constant energy
cost function we can write the excesses as (note that we may
omit the empty set and the grand coalition from consideration
since their excesses are always zero). We we provide the
first step of the computation. In particular, as initial guess
we try x = (1.25, 0, 0.75). In the table below, we see that
the maximum excess occur at coalitions {2}, {1, 2}, and {2, 3}
S
{1}
{2}
{3}
{1, 2}
{1, 3}
{2, 3}
v(S)
0.25
0
-0.25
1.25
1.0
0.75
s(x, S)
0.25 − x1
−x2
−0.25 − x3
1.25 − x1 − x2 = x3 − 0.75
1 − x1 − x3 = x2 − 1
0.75 − x2 − x3 = x1 = 1.25
x = (1.25, 0, 0.75)
-1
0
-1
0
-1
0
If we iterate the minimization we get the nucleolus ν =
(0.9167, 0.66667, 0.4167).
Example 2: Next example shows a variant of the three
providers game that shows other characteristics of the game
under study. We assume that maxi = 32, n1 = 10, n2 = 8,
and n3 = 6, while r(ni ) = 81 · ni . We use three different
energy functions to point out the different cases that may
happen. In particular, the scenario (a) is the one described in
the previous example, where ei (ni ) = 1, for any i = 1, 2, 3,
and ni = 1, 2, . . . , maxi , the scenario (b) is characterized
with e1 (n1 ) = e3 (n3 ) = 0.8, and e2 (n2 ) = 1.2; while the
scenario (c) is characterized with e1 (n1 ) = e2 (n2 ) = 0.5,
and e3 (n3 ) = 2. Table I reports the Shapley values and
the nucleolus for the three scenarios. Figure 2 depicts the
barycentric plot for the scenario (a) while Figure 3 shows the
barycentric plots for the scenario (b), and (c). It is interesting
to observe that the relations between the Shapley value and the
nucleolus. In particular, we have that the may be coincident
as in scenario (a) (see Figure 2). the Shapley value may be in
the core but not coincident with the nucleolus as in scenario
(b). It may also happen that the Shapley value is not in the
core. In this case the Shapley value is not a consistent/stable
allocation.
(0.45, -0.2, 1.95)
(0.75, 0.5, 1.25)
(1.25, 0.5, 0.75)
(0.75, 1, 0.75)
(1.25, 1, 0.25)
(1.25, 0.2, 0.75)
(0.45, 1, 0.75)
(2.45, -0.2, -0.05)
(0.45, 1.8, -0.05)
(3.25, 0.5, -1.25)
(0.75, 3, -1.25)
(1.25, 1, -0.05)
(b)
(c)
Fig. 3: Barycentric plots of the three-providers game for the scenario (a) (left plot), and (b) (right plot)
V. P ROPERTIES OF A LLOCATION T ECHNIQUES
In the previous sections we present several techniques to
allocate costs and/or benefits. For the problem under study,
what basis is there for choosing among them ? We must point
out that, as summarized in several papers (see for instance
[11]), there is no allocation method suitable for all the cases
and principles which seem compelling in one context may not
be so in another. In this section we analyze the axiomatic
method presented in [11] taking into account the peculiarities
of the system under investigation. This would help to narrow
down the set of possible allocation methods.
Here it is worthy mentioning implementation issues of
the provider game. We can assume that there are several
traffic thresholds that enable different players’ behaviors. That
is, when the overall traffic load is greater than the highest
threshold each provider independently serves its customers. On
the other hand, when the traffic load is low and the resources
become redundant, the providers can work in a cooperative
mode where a subset of the access networks can be switched
off such that the networks (or the network) that remain active
can carry all the traffic.
a) Dummy Player Axiom: The first issue that we address
concerns the dummy axiom. That is, a dummy player will not
receive any allocation of costs or benefits. It is clear that this
is a mandatory property otherwise malicious player behaviors
are encouraged (i.e., dummy players that join the coalition
only to receive benefits). This implies Equal Share strategies
are not suitable for our provider game.
b) Additivity: This is decomposition property. This
means that if an allocation problem can be split up into
subproblems and the solutions of the subproblems are added,
then these would add up to the total allocations of the original
problem. The Shapley value is the allocation method that
satisfies the dummy and the additivity axioms [11].
c) Monotonicity: This axiom has to do with how allocation change when the value of the game, v, changes. For
example, if the some player’s contribution to all coalitions to
which he belongs increases or stays fixed, then that player’s
allocation should not decrease. Similarly, if costs increase (and
the value thereby decrease) no one should receive more that
they had before.
It worth to note that this concept is relevant in situations
similar to the system we are studying in which an allocation
is not a one-shot affair, but is reassessed periodically as
new information emerges (e.g., changes in the numbers of
customers in the access provider networks, or changes in
reward rates, or in the energy costs).
There are several definition of monotonicity, in its most
elementary form, monotonicity says that an increases in v(N )
does not cause any player’s allocation to decrease. Let denote
by ν(·) n allocation method (e.g, the Shapley value, the nucleolus, or the equal share, or some variation of these methods).
In particular, we use the aggregate monotonicity, defined in
[5]. The allocation ν(·) satisfies aggregate monotonicity, if
for all value functions v, v ′ with v(S) = v ′ (S) for all S ⊂ N
and v(N ) < v ′ (N ) it holds that ν(v) ≤ ν(v ′ ).
The Shapley value satisfies aggregate monotonicity but the
nucleolus is non-monotonic (see [11] for details). In the
following we use an example presented in [5], to point out
the (negative) effects of non-monotonic allocation method.
Consider a three player game with N = {1, 2, 3} with two
value functions v and v ′ defined v({1}) = 4, v({2}) =
v({3}) = 6, v({1, 2}) = v({1, 3}) = 7.5, v({2, 3}) = 12,
and v({1, 2, 3}) = 13; and v ′ ({1}) = 4, v ′ ({2}) = v ′ ({3}) =
6, v ′ ({1, 2}) = v ′ ({1, 3}) = 7.5, v ′ ({2, 3}) = 12, and
v ′ ({1, 2, 3}) = 13.1. Table II shows the Shapley value and
the nucleolus for this game with the two value functions.
We can see that the nucleolus does not satisfy aggregate
monotonicity, that is, while the value has increased (from v
to v ′ ), the contribution of player 1 has decreased. We can see
that the Shapley value does not suffer of this drawback.
It is important to point out that the non-monotonicity
value
v
v′
Shapley value
[2.167, 5.417, 5.417]
[2.2, 5.45, 5.45]
Nucleolus
[1.75, 5.625, 5.625]
[1.727, 5.6865, 5.6865]
TABLE II: Shapley value and the nucleolus for the example
presented in [5]
may inspire malicious behaviors, that is, a player may cause
damage to other player (or other players) by modifying its
contribution.
In the literature there are several results (see for instance
[11], [5], and [3]) showing that allocation methods based on
the nucleolus and its variations are non-monotonic.
d) Consistency or Stability: This property means that an
overall allocation of v(N ) should be viewed as fair by all
possible subgroups of N . No subset of players should find
incentive to change the allocation. As pointed out by [11] (and
by the example depicted in Figure 3(c)) Shapley value is nonconsistent. On the other hand the nucleolus is a method that
guarantees an allocation in the core (e.g., a consistent solution)
whenever the core is non-empty.
For the provider cooperative game under study if we compare the importance of the monotonicity with respect to the
consistency we choose an allocation method based on the
Shapley value. Moreover, this method satisfies also the dummy
and the additivity axioms.
VI. C ONCLUSIONS AND FURTHER INVESTIGATIONS
In this paper we developed a cooperative game theory
framework that can be used for the design of cooperative
strategies among cellular/wireless network providers aiming
at reducing energy consumption. Our framework allows us to
illustrate the existence of stable allocation strategies which
guarantees that no group of players has an incentive to leave
the coalition to form another coalition.
The future developments of this research is following several directions. First of all, we are going to model situations
where it is not possible to set up ”the grand coalition”. In such
cases, canonical coalitional games are not suited for modeling
the cooperative behavior of the players. In this cases we can
use the so called coalition formation games [8] to study the
network coalitional structure, i.e., answering questions like
which coalitions will form, what is the optimal coalition size
and how can we assess the structures characteristics, and so
on.
A PPENDIX
Proposition 1: The coalition game (N , v), where v(·) is
defined by Eq. (2) and the energy functions by Eq. (7a) or
Eq. (7b), is convex.
Proof From the definition of v(·) we can easily see that the
superadditivity holds because for any S and T with S ∩T = ∅
we have that


(
)
(
)
∑
∑
∑
e
ni + e 
nj  ≥ e
ni .
i∈S
j∈T
i∈S∪T
In particular, if we consider the different energy function
defined by Eq. (7a) for any S and T with S ∩ T = ∅ we
can derive that
∑
v(S ∪ T ) =
ri (ni ) − k ′
=
i∈S∪T
(
∑
)
ri (ni )
(
+
i∈S
∑
)
− k′
ri (ni )
i∈T
′
= v(S) + v(T ) + k .
If the energy function is defined by Eq. (7b) we have that
∑
∑
v(S ∪ T ) =
ri (ni ) − h
ni − k ′′
=
i∈S∪T
(
∑
)
ri (ni )
i∈S
−h
(
∑
i∈S∪T
(
)
∑
+
ri (ni ) +
)
ni
−h
i∈T
(
∑
i∈S
)
ni
− k ′′
i∈T
= v(S) + v(T ) + k ′′ .
To summarize, when S ∩ T = ∅, we can write
v(S ∪ T ) = v(S) + v(T ) + k,
where k = k ′ if the energy function is defined by Equation
(7a), or k = k ′′ if it is defined by Equation (7b).
If we define a partition of S such that S = S ′ ∪ S ′′ , and
′
S ∩ S ′′ = ∅, with S ′ = S ∩ T , and S ′′ = S\T we can derive
that
v(S) + v(T ) =
=
=
v(S ′ ) + v(S ′′ ) + k + v(T )
v(S ∩ T ) + v(S ′′ ) + k + v(T )
v(S ∩ T ) + v(S ′′ ∪ T ),
since S ′′ ⊆ S and (S ′′ ∪ T ) ⊆ (S ∪ T ), by definition of
function v(·) (Eq. (2)) we have that v(S ′′ ∪ T ) ≤ v(S ∪ T ).
This allows us to write that
v(S) + v(T ) ≤ v(S ∩ T ) + v(S ∪ T ).
R EFERENCES
[1] M. Ajmone Marsan and journal = Computer Networks volume = 55
number = 2 year = 2011 pages = 386-398 Meo, M. title = Energy
efficient wireless Internet access with cooperative cellular networks.
[2] M. Ajmone Marsan, Meo. Energy Efficient Management of two Cellular
Access Networks. In ACM SIGMETRICS Performance Evaluation
Review, Special Issue on the 2009 GreenMetrics Workshop, volume 37,
pages 69–73, 2009.
[3] T. Hokary. Population Monotonic Solutions On Convex Games. International Journal Of Game Theory, 29(3):355–358, 2000.
[4] A. P. Jardosh, K. Papagiannaki, E. M. Belding, K. C. Almeroth,
G. Iannaccone, and B. Vinnakota. Green WLANs: On-Demand WLAN
Infrastructures. Mob. Netw. Appl., 14, Dec 2009.
[5] N. Megiddo. On the nonmonotonicity of the bargaining set, the kernel
and the nucleolus of a game. SIAM Journal on Applied Mathematics,
27:355–358, 1974.
[6] H. Moulin. Axiomatic cost and surplus sharing. In Handbook of Social
Choice and Welfare, pages 289–357. North-Holland, 2002.
[7] R. B. Myerson. Game Theory: Analysis of Conflict. Harvard Univ. Press,
Cambridge, MA, 1991.
[8] W. Saad, Z. Han, M. Debbah, A. Hjorungnes, and T. Basar. Coalitional
Game Theory for Communication Networks. IEEE Signal Processing
Magazine, 26(5):77–97, September 2009.
[9] L. S. Shapley. A Value for n-person Games. In H. W. Kuhn and A. W.
Tucker, editors, Contributions to the Theory of Games, volume II ,nnals
of Mathematical Studies No. 28, pages 307–317. Princeton University
Press, 1953.
[10] L. S. Shapley. Cores of Convex Games. International Journal of Game
Theory, 1(1):11–26, 1971.
[11] H. P. Young. Cost Allocation: Methods, Principles, Application. NorthHolland, 1985.