IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 6, JUNE 2006 1115
A Game-Theoretic Approach to Energy-Efficient
Power Control in Multicarrier CDMA Systems
Farhad Meshkati, Student Member, IEEE, Mung Chiang, Member, IEEE, H. Vincent Poor, Fellow, IEEE, and
Stuart C. Schwartz, Life Fellow, IEEE
Abstract—A game-theoretic model for studying power control
in multicarrier code-division multiple-access systems is proposed.
Power control is modeled as a noncooperative game in which
each user decides how much power to transmit over each carrier
to maximize its own utility. The utility function considered here
measures the number of reliable bits transmitted over all the
carriers per joule of energy consumed and is particularly suitable
for networks where energy efficiency is important. The multidi-
mensional nature of users’ strategies and the nonquasi-concavity
of the utility function make the multicarrier problem much more
challenging than the single-carrier or throughput-based-utility
case. It is shown that, for all linear receivers including the matched
filter, the decorrelator, and the minimum-mean-square-error
detector, a user’s utility is maximized when the user transmits
only on its “best” carrier. This is the carrier that requires the
least amount of power to achieve a particular target signal-to-in-
terference-plus-noise ratio at the output of the receiver. The
existence and uniqueness of Nash equilibrium for the proposed
power control game are studied. In particular, conditions are
given that must be satisfied by the channel gains for a Nash
equilibrium to exist, and the distribution of the users among the
carriers at equilibrium is characterized. In addition, an iterative
and distributed algorithm for reaching the equilibrium (when
it exists) is presented. It is shown that the proposed approach
results in significant improvements in the total utility achieved at
equilibrium compared with a single-carrier system and also to a
multicarrier system in which each user maximizes its utility over
each carrier independently.
Index Terms—Energy efficiency, game theory, multicarrier
code-division multiple-access (CDMA), multiuser detection, Nash
equilibrium, power control, utility function.
I. INTRODUCTION
P
OWER CONTROL is used for resource allocation and in-
terference management in both the uplink and downlink of
code-division multiple-access (CDMA) systems. In the uplink,
the purpose of power control is to allow each user to transmit
enough power so that it can achieve the required quality-of-
service (QoS) at the uplink receiver without causing unnec-
essary interference to other users in the system. One of the
key issues in wireless system design is energy consumption at
users’ terminals. Since in many scenarios, the users’ terminals
Manuscript received April 1, 2005; revised October 1, 2005. This research
was supported in part by the National Science Foundation under Grant ANI-03-
38807, Grant CNS-04-27677, Grant CNS-04-17607, and Grant CCF-04-48012.
This work was presented in part at the 2005 IEEE Wireless Communication and
Networking Conference, New Orleans, LA, March 14–17, 2005.
The authors are with the Department of Electrical Engineering, Princeton
University, Princeton, NJ 08544 USA (e-mail: meshkati@princeton.edu;
chiangm@princeton.edu; poor@princeton.edu; stuart@princeton.edu).
Digital Object Identifier 10.1109/JSAC.2005.864028
are battery-powered, efficient energy management schemes are
required in order to prolong the battery life. Hence, power con-
trol plays an even more crucial role in such systems. Recently,
game theory has been used to study power control in data net-
works and has been shown to be a very effective tool for ex-
amining this problem (see, for example, [1]–[9]). In [1], the au-
thors provide some motivation for using game theory to study
communication systems, and in particular power control. In [2]
and [3], power control is modeled as a noncooperative game in
which users choose their transmit powers in order to maximize
their utilities, where utility is defined as the ratio of throughput
to transmit power. In [4], pricing is introduced to obtain a more
efficient solution. Similar approaches are taken in [5]–[8] for
different utility functions. In [9], the authors extend the ap-
proach in [2] to study the cross-layer problem of joint multiuser
detection and power control.
Multicarrier CDMA, which combines the benefits of orthog-
onal frequency-division multiplexing (OFDM) with those of
CDMA, is considered to be a potential candidate for next-gen-
eration high data-rate wireless systems (see [10]). In particular,
in multicarrier direct-sequence CDMA (DS-CDMA), the data
stream for each user is divided into multiple parallel streams.
Each stream is first spread using a spreading sequence and is
then transmitted on a carrier [11]. In a single-user scenario
with a fixed total transmit power, the optimal power allo-
cation strategy for maximizing the rate is waterfilling over
the frequency channels [12]. The multiuser scenario is more
complicated. In [13]–[15], for example, several waterfilling
type approaches have been investigated for multiuser systems
to maximize the overall throughput. However, there are many
practical situations where enhancing power efficiency is more
important than maximizing throughput. For such applications,
it is more important to maximize the number of bits that can
be transmitted per joule of energy consumed rather than to
maximize the throughput.
Consider a multiple-access multicarrier DS-CDMA network
where each user wishes to locally and selfishly choose its
transmit powers over the carriers in such a way as to maxi-
mize its own utility. However, the strategy chosen by a user
affects the performance of other users in the network through
multiple-access interference. There are several questions to ask
concerning this interaction. First of all, what is a reasonable
choice of a utility function that measures energy efficiency in
a multicarrier network? Second, given such a utility function,
what strategy should a user choose in order to maximize its
utility? If every user in the network selfishly and locally picks
its utility-maximizing strategy, will there be a stable state at
0733-8716/$20.00 © 2006 IEEE
1116 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 6, JUNE 2006
which no user can unilaterally improve its utility (Nash equilib-
rium)? If such a state exists, will it be unique? What will be the
distribution of users among the carriers at such an equilibrium?
Because of the competitive nature of the users’ interaction,
game theory is the natural framework for modeling and studying
such a power control problem. This work is the first game-theo-
retic treatment of power control in multicarrier CDMA systems.
We propose a noncooperative power control game in which each
user seeks to choose its transmit power over each carrier to max-
imize its overall utility. The utility function here is defined as the
ratio of the user’s total throughput to its total transmit power
over all the carriers. This utility function, which has units of
bits/joule, measures the total number of reliable bits transmitted
per joule of energy consumed and is particularly suitable for ap-
plications where saving power is critical. Because of the nonco-
operative nature of the proposed game, no coordination among
the users is assumed. Compared with prior work on noncoop-
erative power control games, there are two difficulties to the
problem studied in this paper. One is that users’ strategies in
the multicarrier case are vectors (rather than scalars) and this
leads to an exponentially larger strategy set for each user (i.e.,
many more possibilities). Second, the energy efficiency utility
function, which is considered here, is nonquasi-concave. This
means that many of the standard theorems from game theory as
well as convex optimization cannot be used here. In this work,
we derive the Nash equilibrium [16] for the proposed power
control game and study its existence and uniqueness. We also
address the following questions. If there exists a Nash equilib-
rium for this game, can the users reach the equilibrium in a dis-
tributive manner? What kind of carrier allocations among the
competing users will occur at a Nash equilibrium? Will there be
an even spread of usage of the carriers among users? How does
the performance of this joint maximization of utility over all the
carriers compare with that of an approach where utility is maxi-
mized independently over each carrier? How does a multicarrier
system compare with a single-carrier system in terms of energy
efficiency?
The rest of this paper is organized as follows. In Section II,
we provide some background for this work by discussing the
power control game for the single-carrier case. The power con-
trol game for multicarrier systems is presented in Section III.
The Nash equilibrium and its existence for the proposed game
are discussed in Sections IV and V, respectively. In particular, in
Section IV, we derive the utility-maximizing strategy for a user
when all the other users’ transmit powers are fixed. In Section V,
we show that depending on the channel gains, the proposed
power control game may have no equilibrium, a unique equilib-
rium, or more than one equilibrium, and we derive conditions
that characterize the existence and uniqueness of Nash equilib-
rium for a matched filter (MF) receiver. The case of two-carrier
systems is studied in more detail in Section VI, where we ob-
tain explicit expressions for the probabilities corresponding to
occurrence of various possible Nash equilibria. In Section VII,
we present an iterative and distributed algorithm for reaching
the Nash equilibrium (when it exists). Numerical results are pre-
sented in Section VIII. We show that at Nash equilibrium, with a
high probability, the users are evenly distributed among the car-
riers. We also demonstrate that our proposed method of jointly
Fig. 1. A typical efficiency function (single-carrier case) representing the
packet success probability as a function of received SINR.
maximizing the utility over all the carriers provides a signifi-
cant improvement in performance compared with a single-car-
rier system, as well as the multicarrier case in which each user
simply optimizes over each carrier independently. Finally, con-
clusions are given in Section IX.
II. P
OWER CONTROL GAMES IN SINGLE-CARRIER NETWORKS
Let us first look at the power control game with a single
carrier. To pose the power control problem as a noncoopera-
tive game, we first need to define a utility function suitable for
data applications. Most data applications are sensitive to error
but tolerant to delay. It is clear that a higher signal-to-interfer-
ence-plus-noise ratio (SINR) level at the output of the receiver
will generally result in a lower bit-error rate, and hence higher
throughput. However, achieving a high SINR level requires the
user terminal to transmit at a high power, which in turn results
in low battery life. This tradeoff can be quantified (as in [2])
by defining the utility function of a user to be the ratio of its
throughput to its transmit power, i.e.,
(1)
Throughput is the net number of information bits that are trans-
mitted without error per unit time (sometimes referred to as
goodput). It can be expressed as
(2)
where
and are the number of information bits and the total
number of bits in a packet, respectively;
and are the trans-
mission rate and the SINR for the
th user, respectively; and
is the efficiency function representing the packet success
rate (PSR), i.e., the probability that a packet is received without
an error. Our assumption is that if a packet has one or more bit er-
rors, it will be retransmitted. The efficiency function,
, is as-
sumed to be increasing, continuous, and S-shaped
1
(sigmoidal)
1
An increasing function is S-shaped if there is a point above which the func-
tion is concave, and below which the function is convex.
MESHKATI et al.: A GAME-THEORETIC APPROACH TO ENERGY-EFFICIENT POWER CONTROL IN MULTICARRIER CDMA SYSTEMS 1117
Fig. 2. User’s utility as a function of transmit power for fixed interference
(single-carrier case).
with . We also require that to ensure that
when . These assumptions are valid in many
practical systems. An example of a sigmoidal efficiency func-
tion is given in Fig. 1. Using a sigmoidal efficiency function, the
shape of the utility function in (1) is shown in Fig. 2 as a func-
tion of the user’s transmit power keeping other users’ transmit
powers fixed. It should be noted that the throughput
in (2)
could be replaced with any increasing concave function as long
as we make sure that
when . A more detailed dis-
cussion of the efficiency function can be found in [9]. It can be
shown that for a sigmoidal efficiency function, the utility func-
tion in (1) is a quasi-concave
2
function of the user’s transmit
power [17]. This is also true if the throughput in (2) is replaced
with an increasing concave function of
.
Based on (1) and (2), the utility function for user
can be
written as
(3)
This utility function, which has units of bits/joule, captures very
well the tradeoff between throughput and battery life and is
particularly suitable for applications where energy efficiency is
crucial.
Power control is modeled as a noncooperative game in which
each user tries to selfishly maximize its own utility. It is shown
in [4] that, in a single-carrier system, when MFs are used as the
uplink receivers, if user terminals are allowed to choose only
their transmit powers for maximizing their utilities, then there
exists an equilibrium point at which no user can improve its
utility given the power levels of other users (Nash equilibrium).
This equilibrium is achieved when the users’ transmit powers
are SINR-balanced with the output SINR being equal to
, the
solution to
. Furthermore, this equilibrium is
unique. In [9], this analysis is extended to other linear receivers.
2
The function
f
defined on a convex set
S
is quasi-concave if every superlevel
set of
f
is convex, i.e.,
f
x
2Sj
f
(
x
)
a
g
is convex for every value of
a
.
In this work, we extend this game-theoretic approach to mul-
ticarrier systems. In the multicarrier case, each user’s strategy is
a vector (rather than a scalar). Furthermore, the utility function
is not a quasi-concave function of the user’s strategy. Hence, the
problem is much more challenging than the one in the single-
carrier scenario.
III. N
ONCOOPERATIVE POWER CONTROL GAME IN
MULTICARRIER SYSTEMS
Let us consider the uplink of a synchronous multicarrier
DS-CDMA data network with
users, carriers and pro-
cessing gain
(for each carrier). The carriers are assumed to
be sufficiently far apart so that the (spread-spectrum) signal
transmitted over each carrier does not interfere with the signals
transmitted over other carriers [11]. We also assume that the
delay spread and Doppler spread are negligible for each indi-
vidual carrier. At the transmitter, the incoming bits for user
are divided into parallel streams and each stream is spread
using the spreading code of user
. The parallel streams are
then sent over the
(orthogonal) carriers. For the th carrier,
the received signal at the uplink receiver (after chip-matched
filtering and sampling) can be represented by an
1 vector
as
(4)
where
, , are the th user’s transmitted bit, transmit
power and path gain, respectively, for the
th frequency channel
(carrier);
is the spreading sequence for user which is as-
sumed to be random with unit norm; and
is the noise vector
which is assumed to be Gaussian with mean
and covariance
. Let us express the channel gain as
(5)
where
is the distance of user from the uplink receiver and
is a Rayleigh random variable representing the small scale
channel fading. Here,
and are constants which determine the
path loss as a function of distance.
We propose a noncooperative game in which each user
chooses its transmit powers over the
carriers to maxi-
mize its overall utility. In other words, each user (selfishly)
decides how much power to transmit over each frequency
channel (carrier) to achieve the highest overall utility. Let
denote the proposed noncoopera-
tive game where
, and is
the strategy set for the
th user. Here, is the maximum
transmit power on each carrier. Each strategy in
can be
written as
. The utility function for user
is defined as the ratio of the total throughput to the total
transmit power for the
carriers, i.e.,
(6)
where
is the throughput achieved by user over the th car-
rier, and is given by
with denoting
1118 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 6, JUNE 2006
the received SINR for user on carrier . Hence, the resulting
noncooperative game can be expressed as the following maxi-
mization problem:
(7)
under the constraint of nonnegative powers (i.e.,
for all
and ). Without significant loss of
generality, if we assume equal transmission rates for all users,
(7) can be expressed as
(8)
The relationship between the
’s and the ’s is dependent
on the uplink receiver.
It should be noted that the assumption of equal transmission
rates for all users can be made less restrictive. For our anal-
ysis, it is sufficient for the users to have equal transmission rates
over different carriers but the transmission rate can be different
for different users. More generally, the proposed power con-
trol game can be extended to allow the users to pick not only
their transmit powers but also their transmission rates over the
carriers. While joint power and rate control is important, par-
ticularly for data applications, our focus throughout this work
is on power control only (see [21] for a recent result on joint
optimization of power and rate in single-carrier case). We will
briefly comment on the joint power and rate control problem at
the end of Section IV.
IV. N
ASH EQUILIBRIUM FOR THE
PROPOSED
GAME
For the noncooperative power control game proposed in the
previous section, a Nash equilibrium is a set of power vectors,
, such that no user can unilaterally improve its utility
by choosing a different power vector, i.e.,
is a Nash
equilibrium if and only if
(9)
and for
. Here, denotes the set of transmit
power vectors of all the users except for user
.
We begin by characterizing utility maximization by a single-
user when other users’ transmit powers are fixed.
Proposition 1: For all linear receivers and with all other
users’ transmit powers being fixed, user
’s utility function,
given by (6), is maximized when
for
for
(10)
where
with being the transmit power
required by user
to achieve an output SINR equal to on the
th carrier, or if cannot be achieved. Here, is the
unique (positive) solution of
.
Proof: We first show that
is maximized when is
such that
. For this, we take the derivative of with
respect to
and equate it to zero to obtain
(11)
Since for all linear receivers
[9], is max-
imized when
, the (positive) solution to .
It is shown in [17] that for an S-shaped function,
exists and
is unique. If
cannot be achieved, is maximized when
.Now,define as the transmit power required by
user
to achieve an output SINR equal to on the th carrier
(or
if is not achievable) and let .
In case of ties, we can pick any of the indices corresponding
to the minimum power. Then, based on the above argument, we
have
for any . Also, be-
cause
,wehave
for all and . Based on the above
inequalities, we can write
(12)
Adding the
inequalities given in (12) and rewriting the re-
sulting inequality, we have
(13)
This completes the proof.
Proposition 1 suggests that the utility for user is maximized
when the user transmits only over its “best” carrier such that the
achieved SINR at the output of the uplink receiver is equal to
. The “best” carrier is the one that requires the least amount of
transmit power to achieve
at the output of the uplink receiver.
Based on Proposition 1, at a Nash equilibrium each user trans-
mits only on one carrier. This significantly reduces the number
of cases that need to be considered as possible candidates for a
Nash equilibrium. A set of power vectors,
, is a Nash
equilibrium if and only if they simultaneously satisfy (10).
It should also be noted that the utility-maximizing strategy
suggested by Proposition 1 is different from the waterfilling ap-
proach that is discussed in [18] for digital subscriber line (DSL).
This is because in [18], utility is defined as the user’s throughput
and the goal there is to maximize this utility function for a fixed
amount of available power. Here, on the other hand, the amount
of available power is not fixed. In addition, utility is defined here
as the number of bits transmitted per joule of energy consumed
which is particularly suitable for wireless systems with energy
constraints.
Alternatively, user
’s utility function can be defined as
. This utility function is maximized when each of
the terms in the summation is maximized. This happens when
the user transmits on all the carriers at power levels that achieve
for every carrier. This is equivalent to the case in which
each user maximizes its utility over each carrier independently.
We show in Section VIII that our proposed joint maximization
approach, through performing a distributed interference avoid-
ance mechanism, significantly outperforms the approach of in-
dividual utility maximization. Throughout this paper, the ex-
pression in (6) is used for the user’s utility function.
Since at Nash equilibrium (when it exists), each user must
transmit on one carrier only, there are exactly
possibilities
MESHKATI et al.: A GAME-THEORETIC APPROACH TO ENERGY-EFFICIENT POWER CONTROL IN MULTICARRIER CDMA SYSTEMS 1119
for an equilibrium. For example, in the case of ,
there are four possibilities for Nash equilibrium.
• User 1 and user 2 both transmit on the first carrier.
• User 1 and user 2 both transmit on the second carrier.
• User 1 transmits on the first carrier and user 2 transmits
on the second carrier.
• User 1 transmits on the second carrier and user 2 transmits
on the first carrier.
Depending on the set of channel gains, i.e., the
’s, the pro-
posed power control game may have no equilibrium, a unique
equilibrium, or more than one equilibrium. In the following, we
investigate the existence and uniqueness of Nash equilibrium
for the conventional MF receiver and also comment on the ex-
tensions of the results to other linear multiuser receivers such
as the decorrelating and minimum-mean-square-error (MMSE)
detectors [19], [20].
For the joint power and rate control problem, it can be shown
by using a similar technique as the one used in the proof of
Proposition 1 that for each user to maximize its own utility, the
user must transmit only on its “best” carrier. Furthermore, the
combined choice of power and rate has to be such that the output
SINR is equal to
. This implies that there are infinite combi-
nations of power and rate that maximize the user’s utility given
that the powers and rates of other users are fixed.
V. E
XISTENCE AND
UNIQUENESS OF NASH EQUILIBRIUM
If we assume random spreading sequences, the output SINR
for the
th carrier of the th user with a MF receiver is given by
(14)
Let us define
(15)
as the “effective channel gain” for user
over the th carrier.
Based on (14) and (15), we have
.
Let us for now assume that the processing gain
is suffi-
ciently large so that even when all
users transmit on the same
carrier,
can be achieved by all users. This is the case when
. We later relax this assumption. The following
proposition helps identify the Nash equilibrium (when it exists)
for a given set of channel gains.
Proposition 2: For a MF receiver, a necessary condition for
user
to transmit on the th carrier at equilibrium is that
(16)
where
is the number of users transmitting on the th carrier
and
(17)
In this case,
.
Proof: Based on Proposition 1, in order for user
to
transmit on carrier
at equilibrium, we must have
(18)
Since
users (including user ) are transmitting on the th
carrier and
users are transmitting on the th carrier and all
users have an output SINR equal to
,wehave
(19)
and
(20)
where
and
are the received powers for each user on the th and th carriers,
respectively. Now, define
to get and . Substituting
and into (19) and (20) and taking advantage of the fact that
, we get
(21)
and
(22)
Consequently, (16) is obtained by substituting (21) and (22) into
(18). Furthermore, since
,wehave
, and this completes the proof.
Note that, based on (17), when ,wehave
with .
For each of the
possible equilibria, the channel gains
for each user must satisfy
inequalities similar to (16).
Furthermore, satisfying a set of
of such inequalities
by the
users is sufficient for existence of Nash equilibrium
but the uniqueness is not guaranteed. For example, for the case
of
, the four possible equilibria can be characterized
as follows.
• For both users to transmit on the first carrier at equilib-
rium, we must have
and .
• For both users to transmit on the second carrier at equi-
librium, we must have
and
.
