Forty-Eighth Annual Allerton Conference
Allerton House, UIUC, Illinois, USA
September 29 - October 1, 2010
Nash equilibria for spectrum sharing of two bands
among two players
Ilaria Malanchini
Steven Weber
Matteo Cesana
Drexel University, Dept. of ECE
[email protected]
Drexel University, Dept. of ECE
[email protected]
Politecnico di Milano, DEI
[email protected]
Abstract—The spectrum sharing game and the quality of its
equilibria have been widely studied in a variety of contexts.
In this paper we consider two pairs of communicating users
that share two bands of spectrum. Through the analysis of
the Nash equilibria, we provide the conditions, with respect to
the normalized signal and interference strengths, for the set of
equilibria power allocations to coincide with the set of optimal
allocations. In contrast, when these sets do not coincide, we
characterize the quality of the equilibria using the price of
stability and the price of anarchy measures. In the more general
case of N pairs of transmit receive pairs in an ad hoc network,
we provide simulation results of a simple distributed player
power allocation update heuristic that improves the sum rate
utility above that achieved by the equilibrium of splitting the
power evenly between the two bands.
I. I NTRODUCTION
Growth in demand for wireless spectrum in the last few
years has shown that the current policies for use of available
wireless spectrum are increasingly inadequate. New, more
flexible spectrum management between users and operators
is necessary to improve the efficiency of spectrum usage.
This problem of how to share spectrum has been widely addressed in the technical literature, especially in the context of
heterogeneous and cognitive networks. Many of the adopted
models are based on game theory since, in general, the quality
of service perceived by a system (e.g., user, network) strictly
depends on the behavior of the other entities. Therefore, the
interaction and the competition among multiple systems can
be analyzed using game theoretic models. In particular, noncooperative game theory is suitable in distributed networks,
where control and management are inherently decentralized.
In this work we consider a Gaussian interference game
(GIG) [1], where two transmitter and receiver pairs (each
pair being a player) spread power across two bands of
spectrum, with each player subject to a sum power constraint.
Each player’s payoff function is the sum of the Shannon
rate achieved on each band, where the Shannon rate is
calculated assuming random Gaussian codebooks and where
interference is treated as noise. Each pair of users makes
selfish decisions regarding which portion of the spectrum
to use. This corresponds to deciding how to split the total
power budget for transmission among the two sub-channels.
We model the problem as a non-cooperative game in which
selfish users aim at maximizing the achieved throughput.
Related work. The spectrum sharing game has been
widely addressed in the literature. In [1], the authors study
978-1-4244-8216-0/10/$26.00 ©2010 IEEE
the spectrum sharing game for multiple players capable of
spreading their power budget across a shared portion of
spectrum. The authors investigate and analyze the equilibria
of the game, focusing on the issues of fairness and efficiency.
They also propose punishment strategies that allow competing entities to reach a fair and efficient operating point. The
game between users is analyzed also in [2]. The authors
consider the power control problem with SINR as objective
function, in both the selfish and the cooperative scenario.
They characterize the equilibria and identify the conditions
where the Pareto and the Nash equilibrium coincide.
The spectrum sharing problem has also been analyzed
from the network’s point of view. In [3], the authors analyze
the spectrum competition between two contending networks.
They characterize the Nash equilibria of the game and
discuss the different behavior varying the pathloss exponent.
In [4], a similar model is proposed to analyze the spectrum
sharing problem between two networks operators sharing two
carriers. Based on the water-filling algorithm proposed in [5],
the authors prove the existence of the Nash equilibria of
the game, and characterize the different behavior varying
the channel gains between the two contending communicating pairs. In [6], two system pairs sharing two frequency
bands are considered. In particular, the model comprises an
interference channel in parallel with an interference relay
channel. For two different relaying strategies, the existence
and the uniqueness of the equilibrium are analyzed. A similar
scenario is discussed also in [7], where the authors consider
a network with both shared and protected bands, and power
allocation is similarly based on the water-filling solution. A
formal proof that the game is supermodular is derived as well
as conditions for the existence and uniqueness of the Nash
equilbria.
Spectrum sharing problem has been discussed also in the
context of Cognitive Radio Networks [8]. Cognitive networks are emerging as a solution for the spectrum efficiency
problem, allowing unlicensed users to access vacant portions
of the spectrum. In [9] and [10], the authors consider the
problem of spectrum sharing among secondary users. In
contrast with other work, the power control is analyzed
with a constraint on the total interference perceived by the
primary system. Namely, the interference is measured using
the interference temperature. This is a parameter that measures the power and bandwidth occupied by the secondary
interference. In both the papers, the authors identify the Nash
783
equilibria and analyze their properties. Also in [11], the
authors analyze the spectrum sharing problem with multichannels, considering the co-channel interference among
secondary users and the interference temperature regulation
imposed by primary systems. Existence and properties of
the Nash equilibria are investigated. In [12], the authors
model the spectrum competition among networks that have to
decide both the channel and the power allocation. The Nash
equilibria of the non-cooperative game are characterized and
a cooperative technique is proposed in order to improve the
opportunistic solution. In [13], the channel competition is
played among selfish users, instead of networks. The authors
characterized the equilibria and discuss the efficiency with
respect to the optimal channel allocation. In [14], a spectrum
selection game is proposed. Different from previous work,
competitive users take decisions on which channels to use,
not only depending on the number of other users that are
sharing the same channel, but also taking into account the
different parameters that characterize the available spectrum
opportunity. Furthermore, a multi-stage game is proposed for
the dynamic spectrum management.
Contributions. We consider first a symmetric scenario
where the received normalized interference powers (X) are
equal and the received normalized signal powers (D) are
equal, so that the problem has only these two parameters. By
this parameter reduction we are able to explicitly characterize
allocations that are socially optimal as well as all Nash equilibria, and thereby explicitly compute the Price of Stability
(PoS) and Price of Anarchy (PoA) in terms of (X, D). We
then consider an asymmetric scenario where the interference
powers are unequal (X1 6= X2 ), and we characterize the
socially optimal allocations and the Nash equilibria. Again,
we are able to study the PoS and PoA, now as a function
of the three key parameters (X1 , X2 , D). In the case of N players, we use our observation on the sum-rate optimal
allocations in the 2-player asymmetric game to motivate
a distributed player power allocation update heuristic that
improves the sum rate utility above that achieved by the
full-spread equilibrium. Our simulation results demonstrate
that for networks with low spatial density a fixed full-spread
equilibrium is socially optimal, but the gap between the fullspread equilibrium and the social optimum increases in the
spatial density. This heuristic manages to achieve a sum
utility above that of the full-spread equilibrium but is at a
significant gap from the maximum sum utility.
The paper is organized as follows. In §II, we describe
the scenario and the game model for the 2-player case.
§III and §IV analyze the 2-player and the N -player game,
respectively. §V concludes the paper, discussing some open
issues.
II. S YSTEM M ODEL
In this section, we describe the system and game theoretic
model for the 2-player case that is analyzed in §III. The
spectrum band is assumed to be divided into two orthogonal
channels with equal bandwidths B. Players are indexed as
1, 2 and we let i ∈ {1, 2} indicate a player and j 6= i as the
R1
D
X1
Player 1
(P1 , P̄1 ) T1
U1 (P1 , P2 )
T2
D
X2
Player 2
(P2 , P̄2 )
U2 (P1 , P2 )
R2
Fig. 1. Two transmitter and receiver pairs (players) split their power over
two orthogonal bands.
other player. The distance between the generic receiver i and
the contending transmitter j is denoted by xi , whereas di is
the distance between transmitter i and receiver i. To simplify
the problem and reduce the number of model parameters we
assume throughout this paper that d1 = d2 = d. Each pair has
to decide how to split its total transmission power P between
the two available bands, taking into account the behavior of
the contending pair. Moreover, given a certain power budget
P available for each one of the two pairs, the strategy space
of the generic transmitter i is Pi ∈ [0, 1]. Namely, Pi is the
fraction of P that pair i uses in the left band and P̄i = 1−Pi
is the fraction of power in the right band. The utility (payoff)
function of player i is defined as the sum achievable Shannon
rate over the two bands when the interference from player
j is treated as noise. We assume a channel model with pure
pathloss attenuation so that the received power from a unit
power transmission over a distance r is r−α , for α > 2 the
pathloss exponent. Assuming a noise power of η̃ on each
band, the player i utility function for transmission powers
(P̃i , P̃j ) ∈ [0, P ]2 is
!
!
P̃i d−α
P̃¯i d−α
Ũi = B log2 1 +
+B log2 1 +
η̃ + P̃j x−α
η̃ + P̃¯j x−α
i
i
(1)
in units of bits per second. To summarize, under the above
model assumptions, the spectrum sharing problem can be
modeled as a non-cooperative game, where the players are
the two transmitter and receiver pairs, the actions are the
possible power splits over the two bands, and the payoffs are
the sum of the achievable Shannon rates over the two bands.
We remove nuissance parameters B, P, η, α from the
model as follows. First note the payoff is linear in B and
hence we define the normalized utility Ui = Ũi /B with units
of spectral efficiency (bits per second per Hertz); this removes
B. Multiply the numerator and denominator in the SINR in
each band by 1/P and define the normalized transmission
power as Pi = P̃i /P ∈ [0, 1], and the normalized noise
power as η = η̃/P ; this removes P . Define the received signal to noise ratio (SNR) under maximum transmission power
on a band as D = d−α /η, and the received interference to
noise ratio (INR) under maximum transmission power on a
band as Xi = x−α
i /η; this removes η, α. The utility function
becomes:
Pi Di
P̄i Di
Ui = log2 1 +
+ log2 1 +
. (2)
1 + Pj Xi
1 + P̄j Xi
784
Each one of the two players is assumed to be selfish.
This means that each transmitter allocates power between
the two bands trying to maximize the player’s achieved
throughput (i.e., without taking into account the global optimum). Therefore, the stable operating points for the two
players are the Nash equilibria. A Nash equilibrium is a pair
(P1 , P2 ) ∈ [0, 1]2 from which neither player has incentive to
unilaterally deviate.
Since, in general, the Nash equilibrium reflects the selfish
behavior of the players, it is often inefficient from the system
point of view. Therefore, it is useful to compare the Nash
equilibria of a game with the “globally” optimal solution,
i.e., the one that could be achieved with a centralized control.
Usually, this comparison is done in terms of social utility,
e.g., the sum of the utility of all the players, UT = U1 + U2 .
In particular, the social/global optimal sum utility is
UT∗ =
max
(P1 ,P2 )∈[0,1]2
UT (P1 , P2 ).
(3)
This quantity exists by virtue of the fact that UT is continuous
and bounded on [0, 1]2 . The “quality” of an equilibrium can
be assessed using the concepts of Price of Stability (PoS) [15]
and Price of Anarchy (PoA) [16]. They are, respectively, the
ratio between the optimal solution that could be achieved
by players in a centralized system and the best/worst Nash
equilibrium. Namely, for NE ⊆ [0, 1]2 the set of Nash
equilibria, we have:
PoS
=
PoA
=
UT∗
max(P1 ,P2 )∈NE UT (P1 , P2 )
UT∗
.
min(P1 ,P2 )∈NE UT (P1 , P2 )
(4)
Clearly 1 ≤ PoS ≤ PoA where PoS = 1 when the social
optimal allocation is a Nash equilibrium, and PoS = PoA
when the Nash equilibrium is unique (or at least multiple
equilibria all have common sum utility). Combining these
observations, we have PoS = PoA = 1 when the unique
Nash equilibrium coincides with the social optimal allocation.
The aim of this analysis is to characterize the Nash
equilibria of the game and assess their quality with respect
to the optimal solution using the PoS and PoA. In particular,
we characterize the PoS and PoA as a function of the three
key parameters (D, X1 , X2 ).
a game has selected the best response (or one of the best
responses) to the other players’ strategies.
To evaluate the best response function of the generic player
i, we consider the partial derivative with respect to Pi (5) of
the utility function (2). Therefore, the best response of player
i is, by definition, the maximum of the utility function, over
all possible strategies of player j. In other words, the best
response function is given by the points in which the partial
derivative is equal to zero. The above function is equal to
zero when the numerator is null, that is:
1 X 1
Pi = +
− Pj .
(6)
2 D 2
For the sake of brevity, we do not report here the analysis
that shows that these points are maxima of the function (and
not minima).
Since the domain of the function is [0,1], i.e., (Pi , Pj ) ∈
[0, 1]2 , the best response function cannot be outside this
range. Therefore, we consider the following cases:
1 X 1
1
D
Pi = +
− Pj < 0 ⇐⇒ Pj >
1+
(7)
2 D 2
2
X
1 X 1
1
D
Pi = +
− Pj > 1 ⇐⇒ Pj <
1−
(8)
2 D 2
2
X
This implies that whenever Pi is less than 0 (or greater
than 1) the best response is fixed to 0 (or 1).
Finally, the best response for player i, Pi∗ (Pj ), is the
following function:

D
 1 if X > D and 0 ≤ Pj <12 1 − X
1
D
Pi∗ (Pj ) =
0 if X > D and
2 1 + X < Pj ≤ 1
 1 X 1
else
2 + D 2 − Pj
(9)
These best response functions are illustrated in Fig. 2. Their
intersection give the Nash equilibria, as stated in the following theorem.
P1�P2�
P1�P2�
X<D
1.0
P2∗ (P1 )
0.8
�
0.6
1 1
,
2 2
A. The symmetric case
For this subsection we assume X1 = X2 = X. Generally
speaking, the Nash equilibria of a game can be found using
the best response functions. The best response is the strategy
(or strategies) which produces the most favorable outcome for
a player, taking other players’ strategies as given. Therefore,
the Nash equilibrium is the point at which each player in
�
2
1
2
0.2
�
1−
0.4
�
X
D
0.6� 0.8� 1.0
1
2
1+
1+
X
D
�
P1∗ (P2 )
0.8
1
2
0.6
P1∗ (P2 )
�
1
0.2
This section analzes the quality of the equilibria in the
2-player game. In the first part we consider the symmetric
case, i.e., when X1 = X2 = X, and in the second part we
extend the analysis to the more general asymmetric case with
X1 6= X2 .
1
2
�
0.4
III. T HE 2-P LAYER G AME
X>D
1.0
X
D
�
0.4
X
1−
D
�
0.2
P2�P1�
1
2
1 1
,
2 2
�0.2
�0.4
D
1−
X
�
P2∗ (P1 )
1
2
0.6�
1
2
1+
0.8�
1.0
�
1+
D
X
�
�
1−
D
X
�
P2�P1�
D
X
Fig. 2. The intersection of the best response functions P1∗ (P2 ), P2∗ (P1 )
give the Nash equilibria. The set of equilibria depends upon X ≶ D.
Theorem 3.1: The 2-player symmetric game admits the
following Nash equilibria:

if X < D
 {(0.5, 0.5)}
{(Pi , Pj ) : Pi + Pj = 1} if X = D
NE =
(10)

{(0, 1), (0.5, 0.5), (1, 0)} if X > D
Proof: A pure strategy Nash equilibrium is a point at
which each player has selected the best response to the other
785
∂Ui
[(1 − 2Pj )Xi + (1 − 2Pi )D]D log2 e
=
∂Pi
(1 + Pj Xi + Pi D)(1 + (1 − Pj )Xi + (1 − Pi )D)
players’ strategies. In this game, the point in which the two
best responses cross is the solution of the following system:
(1 − 2Pj )X + (1 − 2Pi )D = 0
(11)
(1 − 2Pi )X + (1 − 2Pj )D = 0
that is Pi = Pj = 0.5. This is the unique equilibrium when
X < D. However, when X > D the best responses cross
in other two points, i.e., (0, 1) and (1, 0). When X = D the
two best responses coincide with the line Pi + Pj = 1, i.e.,
there is an infinite number of equilibria.
We assess the quality of the Nash equilibria via the PoS
and PoA given in (4). Therefore, we consider the sum utility
function UT (P1 , P2 ) in (13) with X1 = X2 = X. The
globally optimal allocations are the maximizers of UT :
Opt =
arg max UT (P1 , P2 )
(12)
(P1 ,P2 )∈[0,1]2
Claim 3.1: The set of optimal power allocations for the
2-player symmetric game is:

if X < D̃
 {(0.5, 0.5)}
Opt =
(14)
{(Pi , Pj ) : Pi + Pj = 1} if X = D̃

{(0, 1), (1, 0)}
if X > D̃
√
where D̃ = 1 + D − 1. Moreover, the function UT is
concave for X > D̃, and has a saddle point at P = (0.5, 0.5)
for X < D̃.
Rationale: The function we want to maximize is a
continuous twice differentiable function defined over the set
C = [0, 1]2 . The necessary condition for a point (Pi , Pj )
to be a stationary point is the gradient to be zero. The
two components of the gradient are given by (18) with
Xi = Xj = X. We are looking for a pair (Pi , Pj ) that
satisfy ∂UT (P1 , P2 )/∂P1 = 0 and ∂UT (P1 , P2 )/∂P2 = 0.
For any point P = (P1 , P2 ) ∈ C define the projection
Q(P) onto the line L = {(P1 , P2 ) : P1 + P2 = 1} as
Q1 (P) = 0.5(1 + P1 − P2 ) and Q2 (P) = 0.5(1 + P2 − P1 ).
We claim without proof that UT (Q(P)) ≥ UT (P) for all
P ∈ C. With this claim it follows that
max UT (P) = max UT (P) = max UT (P, P̄ ).
P∈C
P∈L
P ∈[0,1]
7.38
7.38
7.375
7.379
7.37
7.365
7.378
7.36
7.377
7.355
7.376
7.35
7.375
7.345
7.374
1
7.34
1
0.5
0
P2
0
0.2
0.6
0.8
0.5
1
0
P2
P1
0
0.2
0.4
0.6
0.8
1
P1
The two previous theorems highlight how the equilibria
for the two players depend upon (X, D). In particular,
the optimal allocation is for both players to spread their
power equally over the two bands when the interference is
small relative to the signal (X < D̃), while the optimal
allocation is to perform frequency division and put all power
in (complementary) bands when the interference is large
relative to the signal (X > D̃). Fig. 3 illustrates the function
UT for X > D̃ (left) and X < D̃ (right).
Finally, we can assess the quality of the Nash equilibria
with respect to the optimal solution. To do this, we provide
the analytical expression of the PoS and the PoA.
Theorem 3.2: The Price of Stability (PoS) for the symmetric game is:
(
1
if X ≤ D̃ or X > D
log2 (1+D)
PoS =
(19)
if D̃ < X ≤ D
0.5D
2 log2 (1+ 1+0.5X )
and it has a maximum that is unbounded in D, i.e.,
limD→∞ maxX≥0 PoS(X, D) = ∞.
Proof: The price of stability is given by (4). It is easy
to see that D̃ ≤ D. Therefore there are two cases. When
X ≤ D̃ or X > D the best equilibrium coincides with the
global optimum, i.e., PoS = 1. Otherwise, PoS > 1. Namely,
it is possible to see that the limX→D̃+ PoS = 1. Since:
log2 (1 + D)
=1
0.5D
2 log2 1 + 1+0.5
D̃
(20)
it follows that the PoS is continuous at X = D̃. In contrast,
the function is discontinuous at X = D. This point is also
the maximum of the PoS, since the function is monotonically
increasing in the interval (D̃, D). The upper bound of the
PoS (as X → D− ) is:
lim PoS =
X→D −
and it is easy to see that:
lim
log2 (1 + D)
2 log2 2(1+D)
2+D
lim PoS = 1, lim
D→0 X→D −
(17)
0.4
Fig. 3. The sum utility function UT (P1 , P2 ) has a saddle at (0.5, 0.5)
(left) for X > D̃ but is concave for X < D̃ (right).
It follows that
UT (0.5, 0.5) ≷ UT (0, 1) ⇔ X ≷ D̃.
7.381
7.385
(15)
i.e., all global maximizers are in L. We now have a onedimensional optimization problem over P ∈ [0, 1]. We leave
the fact that UT is concave for X < D̃ and has a saddle
at (0.5, 0.5) for X > D̃ unproven. To find the threshold for
which UT changes its shape and its set of maximizers, we
evaluate UT at the key points P = (0.5, 0.5) and P = (0, 1)
(equivalently, P = (1, 0)):
0.5D
UT (0.5, 0.5) = 4 log2 1 +
1 + 0.5X
UT (0, 1) = UT (1, 0) = 2 log2 (1 + D)
(16)
(5)
lim PoS = ∞
D→∞ X→D +
Therefore, the PoS is unbounded when D → ∞.
786
(21)
(22)
UT = log2 1 +
Pi D
1 + Pj Xi
+ log2 1 +
P̄i D
1 + P̄j Xi
∂UT
(1 − 2Pj )Xi + (1 − 2Pi )D
= D log2 e
+
∂Pi
(1 + Pj Xi + Pi D)(1 + P̄j Xi + P̄i D)
+
+ log2 1 +
P1�P2�
1.0
+ log2 1 +
P1�P2�
X1 ≥ D, X2 ≥ D
1.0
(1, 0)
1.3
log2 (1 + D)
�
�
1
D
2 log 1 + 1+2 1 X
1.2
1.0
√
0.5
1+D−1
{(0.5, 0.5)}
P1 (P2 )
0.8
P oA
0.6
2
1.1
P oS
1.0
0.4
1.5
X
P2 (P1 )
{(0.5, 0.5)}
0.8
(0, 1)
(2, 4, 1)
0.2
{(0, 1), (1, 0)}
P1�P2�
{(0.5, 0.5), (0, 1), (1, 0)}
0.4
0.6
0.8
X1 X2 = D
P1 (P2 )
1.0
lim
log2 (1 + D)
=∞
0.5D
2 log2 1 + 1+0.5X
(24)
Therefore, the limX→∞ PoA = ∞.
Fig. 4 shows PoS√and the PoA as a function of X for D =
1 (for which D̃ = 2 − 1 ≈ 0.414). Both curves are plotted
to highlight the interval in which they coincide. Note that the
PoS is in general bounded, and goes to infinity only when D
goes to infinity. In contrast the PoA goes to infinity whenever
X goes to infinity, i.e., for every D. The conclusion is that in
general the PoS is very close to one, i.e., the best equilibrium
is very close to the optimum, but at the same time, when X
is large, e.g., in dense networks, the worst equilibrium could
be “infinitely” worse than the optimal solution. A similar
result is also given in [1], where the authors show that the
inefficiency resulting from choosing to spread power equally
among available bands can be arbitrarily large.
B. The asymmetric case
In this section, we consider the case in which Xi 6= Xj .
In order to characterize the Nash equilibria of the game,
we consider the partial derivative of the utility function of
player i, reported in (5). The best response of each player is
similar to the previous case, since the function of player i
depends only on Xi . Namely, the best response for player i,
(β2+ , 0)
(2, 3/4, 1)
0.2
P1�P2�
0.4
0.6
0.8
X1 X2 ≤ D
1.0
P2�P1�
2
1.0
P2 (P1 )
0.8
0.6
P2 (P1 ) (0.5, 0.5)
(0.5, 0.5)
0.4
0.2
and it is unbounded, i.e., limX→∞ PoA = ∞.
Proof: The PoA is given by (4). It is easy to see that
(18)
P2 (P1 )
2
0.4
P1 (P2 )
0.2
(2, 1/4, 1)
(2, 1/2, 1)
Theorem 3.3: The Price of Anarchy (PoA) for the symmetric game is:
(
1
if X ≤ D̃
log2 (1+D)
PoA =
(23)
if X > D̃
0.5D
2 log2 (1+ 1+0.5X
)
(0.5, 0.5)
P2�P1�
0.8
0.6
(13)
X1 ≤ D, X1 X2 ≥ D2
(β2− , 1)
P1 (P2 )
0.4
0.2
D
PoS and PoA for the 2-player symmetric case for D = 1.
X→∞
P̄j D
1 + P̄i Xj
0.6
(0.5, 0.5)
0.2
1.0
P̂
Fig. 4.
−Xj Pj
Xj P̄j
+
(1 + Pi Xj )(1 + Pi Xj + Pj D) (1 + P̄i Xj )(1 + P̄i Xj + P̄j D)
PoS,PoA
1.4
P∗
Pj D
1 + Pi Xj
0.2
Fig. 5.
0.4
0.6
0.8
1.0
P2�P1�
0.2
0.4
0.6
0.8
1.0
P2�P1�
Best response functions for the 2-player asymmetric game.
i.e., Pi∗ (Pj ), is the following function:

D

 1 if Xi > D and 0 ≤ Pj < 12 1 − X

i
∗
1
D
Pi (Pj ) =
0
if
X
>
D
and
1
+
<
P
≤1
i
j
Xi


2
 1 Xi 1
+
−
P
else
j
2
D 2
(25)
The Nash equilibria are the intersections of the two best
response functions P1∗ (P2 ), P2∗ (P1 ), illustrated in Fig. 5.
The following theorem gives the Nash equilibria of the
asymmetric game.
Theorem 3.4: The 2-player asymmetric game has Nash
equilibria given by (26).
Proof: As in the previous case, the pure strategy Nash
equilibrium is the point at which each player has selected
the best response to the other players’ strategies. The two
best responses always cross at (0.5, 0.5). This is the unique
equilibrium when Xi Xj < D2 . However, when Xi Xj > D2 ,
the best responses cross in other two points. Namely, when
both Xi and Xj are greater than D, these two points are (0, 1)
and (1, 0). In contrast, when Xi ≤ D (and, consequently,
i
Xj > D), these points are Pi = 0.5 1 ± X
D , with Pj =
0 and Pj = 1, respectively. When Xi Xj = D2 the two
best responses coincide and there is an infinite number of
equilibria.
To assess the quality of the Nash equilibria, we characterize the optimal solution of the asymmetric game.
Claim 3.2: The optimal solution for the asymmetric game
is given by (27). Moreover, the function UT is concave for
787

{(0.5, 0.5)}


 {(P P ) : P = 0.5 + D (0.5 − P )}
i
j
Xi
i, j
NE =
i

(0.5, 0.5), 0.5 1 + X
, 0 , 0.5 1 −

D

{(0.5, 0.5), (0, 1), (1, 0)}
Xi + Xj + Xi Xj < D, and has a saddle point at P =
(0.5, 0.5) for Xi + Xj + Xi Xj > D.
Rationale: The function we want to maximize and its
partial derivative are reported in (13) and (18), respectively.
As in the previous case, it is possible to show that the only
point in which the gradient is zero is (0.5, 0.5). Therefore
the maximum of the function can be attained at (0.5, 0.5)
or at (0, 1) and (1, 0). We do not provide a formal proof of
this claim. The threshold for which the total utility function
changes its maximum (varying X1 and X2 ) can be found
comparing its value at these two points. We obtain that:
,1
Xi Xj < D2
Xi Xj = D2
Xi ≤ D, Xi Xj > D2
Xi > D, Xj > D
(26)
X2 2
1.5
NE: (0.5,0.5), (0,1), (1,0)
OPT: (0,1), (1,0)
PoS=1 and PoA≥ 1
NE:
(0.5,0.5),
(β+2,0),(β−2,1)
OPT: (0,1), (1,0)
PoA ≥ PoS ≥ 1
(D,D)
1
X1X2=D2
NE: (0.5,0.5)
OPT: (0,1), (1,0)
PoS = PoA ≥ 1
0.5
UT (0.5, 0.5) ≷ UT (0, 1) ⇔ Xi + Xj + Xi Xj ≶ D. (28)
The behavior of the Nash equilibria and the optimal
solution (varying X1and X2 ) is highlighted in Fig. 6, where
i
βi± = 0.5 1 ± X
D . Note that there exist four regions.
2
When Xi Xj > D , the game admits three equilibria and
the optimum is in (0, 1) and (1, 0). In particular, when
both Xi and Xj are greater than D, the PoS is one, since
the best equilibrium and the optimum coincide. In contrast,
when Xi (or Xj ) is less than D, the best equilibrium is
worse than the optimum, then the PoS is greater than one.
The PoA is in both the two cases greater than one. We
do not report the expressions, that can be easily derived
as done for the symmetric case. When Xi Xj < D2 and
Xi Xj +Xi +Xj > D, the game admits a unique equilibrium,
that does not coincide with the optimum, then PoS and PoA
coincide and are greater than one. In contrast, below the
curve given by (28), the optimal solution and the unique
equilibrium coincide. Similar conditions on the uniqueness
of the equilibrium are discussed also in [5] and [7]. Finally,
along the curve Xi Xj = D2 the two best responses coincide
and there exists an infinite number of equilibria. Note that
the corresponding regions of the symmetric case, i.e., [0, D̃],
[D̃, D], [D, ∞), can be identified along the line X1 = X2 .
Xi
D
if
if
if
if
~ ~
(D,D)
NE:+(0.5,0.5),
(0,β1),(1, β−1,)
X1+X2+X1X2=D
NE: (0.5,0.5)
OPT: (0.5,0.5)
PoS=PoA=1
0
0
0.5
OPT: (0,1), (1,0)
PoA ≥ PoS ≥ 1
1
1.5
2
X1
D
Fig. 6. Nash equilibria and optimal allocations as a function of X1 , X2
(for D = 1).
Theorem 4.1: The strategy profile Pi = 0.5, ∀ i ∈ N is
always a Nash equilibrium of the N -player game.
Proof: The best response function Pi∗ (P−i ) of player
i is obtained by solving ∂Ui (P)/∂Pi = 0 for Pi , where
∂Ui (P)/∂Pi is given by (32):
X
(1 − 2Pj )Xij + (1 − 2Pi )D = 0.
(30)
j6=i
Projecting Pi∗ onto [0, 1] gives:


1
X Xij 1
1
Pi∗ (P−i ) =  +
− Pj  .
2
D
2
j6=i
(31)
0
Therefore, we have a system with N equalities and in N
variables. It is straightforward to see that Pi = 0.5, ∀i ∈ N
is always a solution of the previous system. This proves the
IV. T HE N -P LAYER G AME
theorem.
We show why this solution is not guaranteed to be the only
In this section, we consider the N -player game, i.e., there
equilibrium
of the game. First, we have to consider the case
are N pairs of users competing for the same spectrum that
is divided into two orthogonal sub-channels. Let N be the in which the N equalities are linearly dependent. In this case
set of players. The utility function of the generic player i is we have a hyperplane of dimension N −1 on which there is an
infinite number of solutions (Nash equilibria). Note that this
the following:
!
! is consistent with the 2-player case. When X1 X2 = D2 the
PD
P̄ D
system (30) is composed by two linearly dependent equalities
P i
P i
+log2 1+
Ui = log2 1+
1+ j6=i Pj Xij
1+ j6=i P̄j Xij and there is an infinite number of Nash equilibria. Therefore,
(29) (31) is always a solution but it is not guaranteed to be the
where Xij = x−α
/η
is
the
interference
to
noise
ratio
from
only one. In [1], the authors provide a sufficient condition for
ij
transmitter j to receiver i when full power is used on a band, this solution to be the unique equilibrium. Furthermore, the
and xij is the distance between transmitter j and receiver i. characterization of the best response for the N -player game
We prove the following theorem, which is found in [1].
is discussed also in [4], [7].
788

if Xi + Xj + Xi Xj > D
 {(0, 1), (1, 0)}
{(Pi , Pj ) : Pi + Pj = 1} if Xi + Xj + Xi Xj = D
Opt =

{(0.5, 0.5)}
if Xi + Xj + Xi Xj < D
hP
i
j6=i (1 − 2Pj )Xij + (1 − 2Pi )D D log2 e
∂Ui
P
P
=
∂Pi
(1 + j6=i Pj Xij + Pi D)(1 + j6=i (1 − Pj )Xij + (1 − Pi )D)
(27)
(32)
Since we are unable to characterize either all possible Nash
equilibria or the optimal solution, we provide a numerical
evaluation. The goal is to compare, through simulations, the
optimal solution with the Nash equilibrium in which every
player plays the strategy Pi = 0.5.
Finally, in order to improve the Nash equilibria of the N player game, we propose a heuristic algorithm that starts from
the Nash equilibrium in which all users play the strategy Pi =
0.5 and aims at improving the efficiency of the allocation.
Namely, from the results of the 2-player game, we have found
that whenever two pairs of users are such that Xij and Xji
satisfy:
Xij + Xji + Xij Xji > D
(33)
they should use different halves of the spectrum, instead of
the whole band. Therefore, we can motivate the users to
implement local moves towards the optimum, reducing the
interference perceived from their neighbors. The goal of the
heuristic is to approximate the global optimal solution only
with local changes. Future work will investigate distributed
protocols and mechanisms based on local moves that could
be adopted by the users in order to improve their throughput.
The heuristic is illustrated in Algorithm 1.
Algorithm 1 Heuristic algorithm
1: Initialize Pi = 0.5
∀ i = 1...N
2: while ∃(i, j)|Xij + Xji + Xij Xji > D and Pi = Pj =
0.5 do
(0, 1), w.p. 0.5
3:
Set (Pi , Pj ) =
(1, 0), w.p. 0.5
4: end while
We have implemented an instance generator able to create
instances representing the network scenario. The generator
takes as input the following parameters: the number of
pairs N , the distance d between each transmitter and receiver
and the edge L of the square arena where the N pairs are
randomly deployed. Once all the pairs are deployed over
the arena, the INRs Xij between each transmitter j and
each receiver i are evaluated by the generator. The optimal
solution has been obtained writing the optimization problem,
i.e., (12)), in AMPL [17] and solving it with the non-linear
solver MINOS [18]. Numerical results are discussed in the
following. Namely, we consider an arena with L = 200
and d = 25. The parameters for the evaluation of the total
utility are η = 10−3 , α = 4 and B = 1. An example
of the considered topology is reported in Fig. 7. Hereafter,
all the reported results are averaged values on 30 randomly
generated instances.
Fig. 7.
Sample of the considered topology when N = 10
In Fig. 8, we report the total utility achieved by the users
as a function of increasing N . When the number of users is
small (say 2, . . . , 10) the optimum and the Nash equilibrium
are very close. This is consistent with the theoretical results
obtained for the 2-player game. In fact, when the number
of pairs is small, they are far enough apart (i.e., the average
Xij is small enough) and the optimal solution coincides with
the equilibrium, i.e., each player splits power equally over the
two bands. As the spatial density of the network increases, the
average interference increases, and the equal split equilibrium
achieves an increasingly poor sum utility relative to the
optimal utility. The heuristic is able to improve the sum utility
above that achieved by the equal split equilibrium, but there
is still a significant gap between the heuristic sum utility and
the optimal.
V. O PEN I SSUES
In this paper, we have considered a scenario composed by
multiple pairs of users sharing two bands of the spectrum.
We have modeled the problem as a non-cooperative game
and we have analytically characterized the Nash equilibria.
Furthermore we have provided measures of their quality with
respect to the optimal solution, both for the 2-player and the
N -player case. We conclude this paper briefly discussing two
open issues that are currently part of the ongoing work.
A. The optimal solution of the asymmetric 2-player game
As reported in §III-B, the optimal solution of the 2-player
game is based only on a conjecture, and we do not provide a
formal proof. However, we aim at providing a formal proof,
789
Fig. 8. Top: total utility (optimal, full-spread equilibrium, heuristic) for
the N -player game as a function of N . Bottom: price of anarchy of the
full-spread equilibrium.
based on the following idea. As done for the asymmetric
case, we want to show that the point (0.5, 0.5) is the only
stationary point of the interior of the domain set, therefore
the maximum should be in (0.5, 0.5) or at the boundary of
the domain. Our result is supported also by the Karush-KuhnTucker conditions, that show that the points (0, 1) and (1, 0)
are candidate to be maxima of the utility function only when
Xi + Xj + Xi Xj > D.
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and shared bands: uniqueness and efficiency of Nash equilibria,” in
GAMECOMM, Pisa, Italy, 2009, pp. 1–10.
[8] S. Haykin, “Cognitive radio: brain-empowered wireless communications,” IEEE Journal on Selected Areas in Communications, vol. 23,
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[9] J. Jia and Q. Zhang, “A non-cooperative power control game for
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B. The equilibria of the N -player game
Concerning the N -player scenario, our analysis aims at
fully characterize the Nash equilibria of the game. We have
shown that the equilibrium in which all players choose Pi =
0.5 is not the only one. We would like to prove conditions
under which this equilibrium is the only, the best, or the worst
among all the equilibria.
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