INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL
ISSN 1841-9836, 10(4):520-538, August, 2015.

Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels

M. Hayajneh, C. Abdallah

Mohammad Hayajneh*
United Arab Emirates University
College of IT , AlAin, P.O.Box 17551
*Corresponding author: Mhayajneh@uaeu.ac.ae

Chaouki Abdallah
University of New Mexico
Department of Electrical & Computer Engineering
MSC01 1100, 1, Albuquerque, NM 87131-0001, USA
chaouki@ece.unm.edu

Abstract: In this paper we present a game-theoretic power control algorithms
for wireless data in CDMA cellular systems under two realistic channels: (a1) Fast
flat fading channel and (a2) Slow flat fading channel. The fading coefficients under
both (a1) and (a2) are studied for three appropriate small scale channel models
that are used in the CDMA cellular systems: Rayleigh channel, Rician channel and
Nakagami channel. This work is inspired by the results presented by [1] under non-
fading channels. In other words, we study the impact of the realistic channel models on
the findings in [1] through the followings: we evaluate the average utility function, the
average number of bits received correctly at the receiver per one Joule expended, for
each channel model. Then, using the average utility function we study the existence,
uniqueness of Nash equilibrium (NE) if it exists, and the social desirability of NE
in the Pareto sense. Results show that in a non-cooperative game (NPG) the best
policy for all users in the cell is to target a fixed signal-to-interference and noise
ratio (SINR) similar to what was shown in [1] for non-fading channel. The difference
however is that the target SINR in fading channels is much higher than that in a
non-fading channel. Also, for spreading gain less than or equal to 100, both NPG
and non-cooperative power control game with pricing (NPGP) perform poorly, where
all the terminals except the nearest one were not able to attain their corresponding
minimum SINR even if sending at the maximum powers in their strategy spaces.
Keywords: Code-division-multiple-access (CDMA), utility function, power control,
game theory, non-cooperative game (NPG), wireless data.

1 Introduction

The mathematical theory of games was introduced by Von Neumann and Morgenstern in
1944 [18], and by the late 1970’s became an important tool whenever a player’s decision depends
on what the other players did or will do. A core idea of game theory is how strategic interactions
between rational agents (players) generate outcomes according to the players’ utilities [10], [19].
Game theory thus forms a suitable framework to obtain more insight into the interactions of self-
interested rational agents with potentially conflicting interests. A player in a non-cooperative
game responds to the actions of other players by choosing a strategy (from his strategy space)
in an attempt to maximize a utility function that quantifies its level of satisfaction.

In a cellular system each user desires to have a high SINR at the base station (BS) coupled
with the lowest possible transmit power. It is important in such systems to have high SINR, as
this will reflect a low error rate, a more reliable system, and high channel capacity, so that more
users can be served per cell [11]. It is also important to decrease the transmit power to lengthen

Copyright © 2006-2015 by CCC Publications



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 521

battery life and to alleviate the near-far problem [17]. In power control algorithms exploiting
game theory however, the tendency of each user to maximize his/her utility function in response
to other users’ actions, leads to a sequence of power vectors that converges to a point where
no user has incentive to individually increase his power. This operating point is called a Nash
equilibrium. Due to the lack of cooperation between the users this point may not be efficient,
in the sense that it may not be the most desirable social point [3]. In Pareto sense, the most
desirable social point is actually the power vector that Pareto dominates all other power vectors.

It should be noted that extensive work has been done on non-game theoretic power control
algorithms for wireless data and multimedia CDMA cellular networks, e.g. [9, 10, 15, 16]. The
power control problem for wireless data CDMA systems was first addressed in the game theoretic
framework in [1]- [8]. In this paper the work in [1], which only dealt with deterministic (non-
fading) channels, is extended to a realistic wireless CDMA channels by considering the following
cases of fading models: A Rayleigh fast/slow flat fading channel model, a Rician1 fast/slow flat
fading channel model and a Nakagamifast/slow flat fading channel model. Where we use the
same utility function and evaluate its average in the fading channels mentioned above, then we
use these averaged utility functions to study the existence, uniqueness and social desirability of
NE operating point under each channel model.

The remaining of this paper is organized as follows: In section 2 we present the utility function
and the system model studied in this paper. In section 3 we evaluate the performance of the
system for the channel models mentioned above. Non-cooperative power control game (NPG)
and Non-cooperative power control game with pricing (NPGP) are discussed briefly in sections
4 and 5, respectively. We then point out the constraints on the new modified strategy spaces to
guarantee the existence and uniqueness of Nash equilibrium points for NPG and NPGP under
the assumed channel models in section 6. Simulation results are outlined in section 7, and our
conclusions are given in section 8.

2 Utility Function and System Model

In general utility functions are used to quantify the satisfaction level a player achieves by
choosing an action from its strategy profile, given the other players’ actions. A utility function
thus maps the player’s preferences onto the real line. A formal definition of a utility function
may be found in [10].

In a CDMA cellular system, a number of users sharing the spectrum and air interface. Hence-
forth, each user’s transmission adds to the interference of all users at the BS. Each user desires
to achieve a high quality of reception at the BS, i.e., a high SINR, while using the minimum
possible amount of power in order to extend the battery’s life. The conflicting goal of each user
to have a high SINR at the BS makes the game theoretic framework suitable for studying and
solving the problem.

In this paper we consider the same system model and the same utility function of [1]: Uplink
single-cell direct sequence code division multiple access (DS-CDMA) system with N users, where
each user transmits frames (packets) of M bits with L information bits. The rate of transmission
is R bits/sec for all users. Let Pc represent the average probability of correct reception of a
frame at the BS, and let p represent the average transmit power level. The utility function for a
CDMA system is given by:

u =
LR

M p
f(γ) (1)

1For space limitation, we omit the findings related to Rician channel model. The reader can find these findings
in the supplementary document [22].



522 M. Hayajneh, C. Abdallah

where f(γ) is an efficiency function that approximates Pc, u thus represents the number of infor-
mation bits successfully received at the BS per joule of consumed energy. With the assumption
of no error correction, and random packet correct reception rate P̃c, i.e., Pc = E{P̃c}, is then
given by

∏M
l=1(1−P̃e(l)), where P̃e(l) is the random bit error rate (BER) of the lth bit at a given

SINR γi. Pe is the average BER, that is Pe = E{P̃e} (c. f. (9) and (23)). We are assuming that
all users in a cell are using non-coherent binary frequency shift Keying (BFSK) modulation, and
are transmitting at the same rate R. It should be noted that the efficiency function f(γ) has the
same expression of Pc in terms of P̃e, except that P̃e is replaced by 2P̃e [1].

3 Evaluation of The Performance

In this section we find closed-form formulas of the average utility functions under the six
assumed channel models. We then use these formulas to study the existence and uniqueness of
Nash equilibrium point in section 6. The SINR γi at the receiver for the ith user is assumed to
be large (γi ≫ 1) to combat the fading effect, it is given by [13]:

γi =
W

R

pi hi α
2
i∑N

k ̸=i pk hk α
2
k + σ

2
(2)

Where αi is the path fading coefficient between ith user and the BS and is constant for each
bit in a fast flat fading channels (a1), while it is constant for each packet in a slow flat fading
channels (a2). W is the spread spectrum bandwidth, pk is the transmitted power of the kth
user, hk is the path gain between the BS and the kth user, and σ2 is the variance of the AWGN
(additive-white-gaussian-noise) representing the background thermal noise in the receiver. For
simplicity we express the interference from all other users as xi, i.e.

xi =

N∑
k ̸=i

pk hk α
2
k (3)

therefore (2) can be written as:

γi = γi(αi,xi) =
W

R

pi hi
xi + σ2

α2i = γ
′
iα

2
i (4)

For a given αi and xi, the BER, P̃(e|αi,xi), of the ith user using BFSK is given by [13]:

P̃(e|αi,xi) =
1

2
e−

γi(αi,xi)

2 (5)

The average BER and average utility functions for this modulation scheme is evaluated next
under the previously mentioned channel models.

3.1 Rayleigh Flat Fading Channel

In this case αi is modelled as a Rayleigh random variable with a probability distribution
given by:

fαi(ω) =
ω

σ2r
e−(1/2σ

2
r)ω

2
, i = 1,2, · · · ,N (6)

Where σ2r = E{α2i}/2 is the measure of the spread of the distribution. In all following cal-
culations, and as a consequence of the multiplicative effect of small and large scale models, it
is assumed that σ2r = 1/2. Using (4) and (6) the distribution of γi for a given xi becomes:

fγi|xi(ω) = 1
γ′i
e
−( 1

γ′
i
)ω



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 523

Rayleigh Fast Flat Fading Channel

For the lth bit in the frame, we can rewrite the SINR (4) and the interference (3) for the
ith user as follows: γi(l) = WR

pi hi α
2
i (l)

xi(l)+σ2
, and xi(l) =

∑N
k ̸=i pk hk α

2
k(l). In this paper, we assume

that the fading channel is fast enough to have the fading coefficients {αi(l)}Ml=1 statistically
independent, that is iid (identical independent distributed) random variables. Henceforth, the
averaged correct reception Pc is given as (1 −Pe)M , while the efficiency function f(γi) is given
as (1−2Pe)M , where Pe is averaged BER for each bit in the frame, that is Pe = E{P̃e}. We will
calculate the averaged Pe next. We can find the conditioned error probability P̃(e|xi) by taking
the average of (5) with respect to fγi|xi(ω):

P̃(e|xi) = E
{
P̃(e|γi,xi)

}
=

∫ ∞
0

P̃(e|ω,xi)fγi|xi(ω)dω

=
1

2γ′i

∫ ∞
0

e
−(

2+γ′i
2γ′

i
)ω
dω =

1

2 + γ′i
(7)

Notice that we dropped the bit index l because the average BER does not depend on l. For large
SINR, (7) behaves like:

P̃(e|xi) ≈
1

γ′i
=
xi + σ

2

W
R
pi hi

(8)

Now, we can find the averaged BER Pe by taking the expectation of (8):

Pe = E
{
P̃(e|xi)

}
=
E{xi}+ σ2

W
R
pi hi

=
1

γi
(9)

where γi is the ratio of the mean of the received power from user i to the mean of the interference
at the receiver and given by:

γi =
W

R

pi hi∑N
k ̸=i pk hk + σ

2
(10)

Therefore, the average utility function of the ith user is given by:

ui =
L R

M pi
(1−

2

γi
)M (11)

Rayleigh Slow Flat Fading Channel

In a slow flat fading channel model, αi is assumed to be constant for each packet/frame. The
averaged efficiency function f(γi) is therefore given as the expectation of (1 − 2P̃(e|αi,xi))M
with respect to the random variables αi and xi. One can evaluate ui(p|xi) as follows:

ui(p|xi) =
∫ ∞
0

ui(p|ω,xi)fγi|xi(ω)dω =
∫ ∞
0

L R

M pi
(1−e−ω/2)M

1

γ′i
e
−( 1

γ′
i
)ω
dω

=
L R

M pi γ
′
i

M∑
k=0

(−1)k
(
M

k

)
×
∫ ∞
0

e
−( k

2
+ 1

γ′
i
) ω
dω =

L R

M pi

M∑
k=0

(
M

k

)
2 (−1)k

k γ′i + 2
(12)

For large SINR (γ′i ≫ 1), (12) can be approximated by:

u(p|xi)≈
L R

M pi

(
1 +

1

γ′i

M∑
k=1

(
M

k

)
2 (−1)k

k

)
(13)



524 M. Hayajneh, C. Abdallah

Averaging (13) with resect to xi we obtain the average utility function for high SINR:

ui = E{ui(p|xi)}≈
L R

M pi

(
1 +

E{xi}+ σ2
W
R
pi hi

M∑
k=1

(
M

k

)
2 (−1)k

k

)

=
L R

M pi

(
1 +

1

γi

M∑
k=1

(
M

k

)
2 (−1)k

k

)
(14)

ui ≈
L R

M pi

(
1−

β

γi

)

where β = −
∑M

k=1

(
M
k

) 2 (−1)k
k

> 0.

3.2 Nakagami Flat Fading Channel

Here, the fading coefficient αi is modelled as a Nakagami random variable with a probability
distribution given by [13]: fαi(ω) = 2m

m

Γ(m)Ωm
ω2m−1 e(−

m
Ω
)ω2; i = 1,2, · · · ,N where Ω = E{α2i}

controls the spread of the distribution. The fading figure m = Ω
2

E{(α2i −Ω)
2} is a measure of

the severity of the fading channel, where m = ∞ corresponds to a nonfading channel. In
the following it is assumed that Ω = 1. Then the distribution of γi for fixed xi is given as:

fγi|xi(ω) = 1
Γ(m)

(
m
γ′i

)m
ωm−1 e

−( m
γ′
i
) ω

Nakagami Fast Flat Fading Channel

We find the conditioned error probability P̃(e|xi) as:

P̃(e|xi) =
∫ ∞
0

P̃(e|ω,xi)fγi|xi(ω)dω =
1

2Γ(m)

(
m

γ′i

)m ∫ ∞
0

ωm−1 e
−(

γ′i+2m
2γ′

i
) ω
dω

=
1

2

(
2m

2m + γ′i

)m
(15)

For fixed m and γ′i ≫ 1, (15) can be rewritten as:

P̃(e|xi) ≈
1

2

(
2m

γ′i

)m
(16)

To find the average Pe, we need to find the mean of (xi + σ2)m. Here, xi is a summation of
independent random variables each distributed according to a Gamma density function. This
makes the evaluation of (xi + σ2)m tedious and it may be easier to find an approximate density
function of xi. To do this, let us recall Esseen’s inequality which estimates the deviation of the
exact distribution of a sum of independent variables from the normal distribution [21].

Theorem 1. let Y1, · · · ,YN be independent random variables with EYj = 0, E|Yj|3 < ∞ (j =
1, · · · ,N). Let σ2j = EY

2
j , BN =

∑N
j=1 σ

2
j , LN = B

−3/2
N

∑N
j=1 E|Yj|

3. Let ψN(z) be the c.f.

(cumulative distribution ) of the random variable B−1/2N
∑N

j=1 Yj. Then

|ψN(z)−e−z
2/2| ≤ 16LN |z|3 e−z

2/3 (17)



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 525

Define Ỹk = pkhkα2k and Yk = Ỹk −pkhk. By simple calculations we can find that Ỹk,(k =
1, · · · ,N) are Gamma distributed random variables, such that fỸk(ω) = (m/pkhk)

m

Γ(m)
ωm−1 e−(m/pkhk)ω

and EỸk = pkhk, which means that Yk,(k = 1, · · · ,N) are zero mean random variables. Note
that σ2k = EY

2
k = (pkhk)

2/m,∀k = 1, · · · ,N, and therefore, BN = 1m
∑N

k=1(pkhk)
2. It is

fairly simple to find out that the third moment E|Yk|3 = EY 3k =
2(pkhk)

3

m2
(Yk ≥ 0), and

LN =
2
∑N

k=1(pkhk)
3

√
m (

∑N
k=1(pkhk)

2)3/2
. For large N, LN has a very small value, i.e., LN << 1. Exam-

ining (17) for small values of z, LN takes care of the bound and making it very small, while
for large values of z, the exponential will decrease the bound and make it approach zero. In
conclusion, we can approximate xi as a Gaussian random variable with mean ζxi and variance
σ2xi given by:

ζxi = E{xi} = E




N∑
k ̸=i

α2kpk hk


 =

N∑
k ̸=i

pk hk E{α2k} =
N∑
k ̸=i

pk hk (18)

and

σ2xi = E{x
2
i}− ζ

2
xi

= E




N∑
l ̸=i

N∑
k ̸=i

plhlpk hk α
2
l α

2
k


− ζ2xi = 1m

N∑
k ̸=i

(pk hk)
2 (19)

where (19) was obtained using the fact that αk and αl are statistically independent for all k ̸= l.

So, we can write fxi, the PDF of xi, as follows: fxi(w) =
δi√

2πσxi
e
−

(w−ζxi )
2

2σ2xi , where w ≥ 0 and

δi = 2/(1 + Erf[ζxi/
√
2σxi]) is a scaling factor such that f

xi(w) is a valid PDF. Erf[.] is the
error function. By examining equations (18) and (19), one can see that ζxi ≫ σxi , therefore
δi ≈ 1. Averaging (16) over fxi(ω) we obtain the average error probability Pe for high SINR
below:

Pe ≈
1

2

(
2m

W
R
pi hi

)m ∫ ∞
0

(
xi + σ

2
)m × 1√

2πσxi
e
−

(xi−ζxi )
2

2σ2xi dxi

=
1

2

(
2m

W
R
pi hi

)m ∫ ∞
σ2

ym ×
1

√
2πσxi

e
−

(y−(ζxi +σ
2))2

2σ2xi dy

≈
1

2

(
2m

W
R
pi hi

)m ∫ ∞
0

ym ×
1

√
2πσxi

e
−

(y−(ζxi +σ
2))2

2σ2xi dy (20)

where we used the change of variable y = xi + σ2 and the last approximation in (20) used the
fact that σ2 ≪ 1. By examining (20) one can see that it is the mth moment of a random variable
y normally distributed with mean ζy = ζxi + σ

2 and variance σ2y = σ
2
xi

. Therefore, the average
Pe is given by:

Pe =
1

2

(
2m

W
R
pi hi

)m
E{ym} =

1

2

(
2m

W
R
pi hi

)m
E{((y − ζy) + ζy)m}

=
1

2

(
2m

W
R
pi hi

)m m∑
k=0

(
m

k

)
ζm−ky Ck =

1

2

(
2mζy
W
R
pi hi

)m m∑
k=0

(
m

k

)
Ck
µky

= 2m−1
(
m

γi

)m m∑
k=0

(
m

k

)
Ck
ζky

(21)



526 M. Hayajneh, C. Abdallah

where γi is given in (10), and Ck is the kth central moment and it is given by [13]:

Ck =

{
1.3 · · ·(k −1)σkxi k even
0 k odd

By splitting up the summation in (21), we obtain:
m∑
l=0

(
m

l

)
Cl
ζly

=1 +

(
m

2

)
σ2xi

(σ2 +
∑N

k ̸=i pk hk)
2
+ · · ·+

(
m

m′

)
1.3 · · ·(m′ −1)σm

′−1
xi

(σ2 +
∑N

k ̸=i pk hk)
m′

(22)

where m′ = m if m is even and m′ = m − 1 if m is odd. Since σ2x (see (19)) is very small
compared to ζxi (see (18)), we can approximate the summation by its leading term which is 1.
Therefore the average Pe at high SINR behaves like:

Pe ≈ 2m−1
(
m

γi

)m
(23)

And the average utility function of the ith user is given by:

ui =
L R

M pi

(
1−2m

(
m

γi

)m)M
(24)

Notice that if we set m = 1, we obtain the same performance as in the Rayleigh fast flat fading
case.

Nakagami Slow Flat Fading Channel

As done earlier, ui(p|xi) can be determined as follows:

ui(p|xi) =
∫ ∞
0

ui(p|ω,xi)fγi|xi(ω)dω

=

∫ ∞
0

L R

M pi
(1−e−ω/2)M

1

Γ(m)

(
m

γ′i

)m
ωm−1 e

−( m
γ′
i
) ω

dω (25)

By factorizing (1−e−γi/2)M and using the identity
∫ ∞
0
yne−a y dy =

Γ(n+1)
an+1

we obtain:

ui(p|xi) =
LR

M pi

M∑
k=0

(−1)k
(
M

k

)(
2m

k γ′i + 2m

)m
(26)

For fixed m and high SINR, γ′i ≫ 1 (26) can be approximated as:

ui(p|xi) ≈
LR

M pi

[
1 + (

1

γ′i
)m

M∑
k=1

(−1)k
(
M

k

)(
2m

k

)m]
(27)

Averaging (27) with respect to the distribution of xi and using the same argument as in (20),
(21) and (22) we end up with the final approximate averaged utility function given by:

ui ≈
LR

M pi

[
1 + (

1

γi
)m

M∑
k=1

(−1)k
(
M

k

)(
2m

k

)m]

ui ≈
LR

M pi

[
1− ξ (

1

γi
)m
]

(28)

where ξ = −
∑M

k=1(−1)
k
(
M
k

)(
2 m
k

)m
> 0.

In the following two sections, we introduce briefly both NPG and NPGP games.



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 527

4 Non-Cooperative Power Control Game (NPG)

Let N = {1,2, · · · ,N} be the index set of the users currently served in the cell and {Pj}j∈N
represent the set of strategy spaces of all users in the cell. Let G = [N ,{Pj},{uj(.)}] denote a
noncooperative game, where each user, based on local information, chooses a power level from
a convex set Pj = [pj−min,pj−max] and where pj−min and pj−max are the minimum and the
maximum power levels in the jth user strategy space, respectively. Assuming that the power
vector p = [p1,p2, · · · ,pN] is the result of NPG, the utility of user j is given by [1]:

uj(p) = uj(pj, p−j) (29)

where pj is the power of user j, and p−j is the vector of powers transmitted by all other users.
The right side of (29) emphasizes the fact that user j can only control his own power. We rewrite
(1) for user j as:

uj(pj, p−j) =
LR

M pj
f(γj) (30)

The formal expression for the NPG is given in [1] as:

G : max
pj∈Pj

uj(pj, p−j), for all j ∈N (31)

This game will produce a sequence of power vectors until it converges to a point where all users
are satisfied with their utility level. This operating point is called a Nash equilibrium operating
point of NPG. In the next subsection, we define the Nash equilibrium point and describe its
physical interpretation.

4.1 Nash Equilibrium in NPG

Definition 2. [1] A power vector p = [p1,p2, · · · ,pN] is a Nash equilibrium of the NPG defined
above if for every j ∈N ,uj(pj, p−j) ≥ uj(p′j, p−j) for all p

′
j ∈ Pj.

One interpretation of Nash equilibrium is that no user can increase its utility by changing its
power level unilaterally. If we multiply the power vector p by a constant 0 < λ < 1 we may get
higher utilities for all users, as was the case in nonfading channels. This means that the Nash
equilibrium is not efficient, that is, the resulting p is not the most desired social operating point.
This results from the lack of cooperation between the users currently in the system. To impose
a level of cooperation between users in order to reach a Pareto dominant Nash point, a pricing
technique was introduced in [1]. We discus this modified NPG game next.

5 Non-Cooperative Power Control Game with Pricing (NPGP)

In NPGP each user maximizes the difference between his/her utility function and a pricing
function. This aims to allow more efficient use of the system resources within the cell, as each
user is made aware of the cost of aggressive resources use, and of the harm done to other users
in the cell. We use here a linear pricing function, i.e., a pricing factor multiplied by the transmit
power. The base station broadcasts the pricing factor to help the users currently in the cell reach
a Nash equilibrium that improves the aggregate utilities of all users at power levels lower than
those of the pure NPG. In other words, the resulting power vector of NPGP is Pareto dominant
compared to the resulting power vector of NPG, but is still not Pareto optimal in the sense that
we may multiply the resulting power vector of NPGP by a constant 0 < λ < 1 to get higher



528 M. Hayajneh, C. Abdallah

utilities for all users. Let Gc = [N ,{Pj},{ucj(.)}] represent an N-player noncooperative power
control game with pricing (NPGP), where the utilities are [1]:

ucj(p) = uj(p)− cpj for all j ∈N (32)

where c is a positive number chosen to get the best possible improvement in the performance.
Therefore, NPGP with a linear pricing function can be expressed as:

Gc : max
pj∈Pj

{uj(p)− cpj} for all j ∈N (33)

6 Existence and Uniqueness of Nash Equilibrium Point

In this section we show that NPG and NPGP introduced by [1] admit a unique Nash equi-
librium points under the assumed channel models. However, to guarantee the existence and
uniqueness of NE point in both games, the terminals’ strategy spaces defined in [1] should be
constrained more. That is, some transmit power values which were allowed in a non-fading chan-
nel, may not be allowed under a fading channel. In the following, we refer to the unconstrained
maximizing transmit power level of user i by pmaxi . Pi refers to the convex strategy space of user
i.

Lemma 3. In NPG under Rayleigh fast flat fading channel with the average utility function
ui given in (11), the existence of a Nash equilibrium point is guaranteed if and only if the
strategy space is modified to Pi = {pi : γi ∈

(
γi−min,γi−max

)
}, where γi−min = 2(M + 1) −√

2M(M + 1) and γi−max = 2(M + 1) +
√

2M(M + 1). The best response vector of all users
r1(p) =(r11(p),r

1
2(p), · · · ,r

1
N(p)), where r

1
i (p) = min(p

max
i ,pi−max), and

pmaxi = 2(M + 1)Ii, Ii =
R (
∑N

k ̸=i hkpk + σ
2)

W hi
(34)

is a standard interference function, therefore by [12], the Nash equilibrium point is unique. Ii is
the effective interference for user i.

Proof: In all following proofs we make use of the classical results of game theory, where the
existence of a Nash equilibrium point is guaranteed if the utility function is quasiconcave and
optimized on a convex strategy space. Thus, to prove the existence of Nash equilibrium point,
it is enough to prove that the utility function ui is concave (a concave function on some set is
also a quasiconcave function on the same set) in pi given p−i on the convex set Pi = {pi : γi ∈(
γi−min,γi−max

)
}. Let us find the first and second order derivatives with respect to pi as follows:

∂ui
∂pi

=
LR

M p2i
(
2(M + 1)

γi
−1) (1−

2

γi
)M−1, (35)

then ∂
2ui
∂p2i

=
2L R (1− 2

γi
)M [2(M+1)(2+M)−4(1+M) γi+γi2]

M p3i (−2+γi)
2 . Therefore,

∂2ui
∂p2i

< 0,∀ γi ∈
(
γi−min,γi−max

)
,

where γi−min = 2(M +1)−
√

2M(M + 1) and γi−max = 2(M +1)+
√

2M(M + 1). This implies
that the strategy space should be modified to the convex set Pi = {pi : γi ∈

(
γi−min,γi−max

)
}

to guarantee the concavity of the utility function, and then to guarantee the existence of Nash
equilibrium point.

To prove the uniqueness of Nash equilibrium point we need to prove that r1(p) is a standard
function. By setting (35) to zero we find the maximizing transmit power level that lies in the
convex strategy space Pi is given as in (34). Before proving that r1(p) is a standard interference
function we introduce the definition of an arbitrary standard interference function ϕ(p) as follows:



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 529

Definition 4. [12] An interference function ϕ(p) is called a standard interference function if it
satisfies the following: 1) Positivity: ϕ(p) > 0, i.e., each element is positive, 2) Monotonicity: if
p > p̂ then ϕ(p) ≥ ϕ(p̂) (component wise), and 3) Scalability: ∀δ > 1, δϕ(p) > ϕ(δ p) (component
wise).

To prove that r1(p) is a standard interference function we proceed as follows: The proof of
positivity is trivial, since Pi ⊂ R+ and r1i (p−i) ∈ Pi, ∀i ∈ N , where r

1
i (p−i) = r

1
i (p). Also, it is

obvious that pmaxi (p) > p
max
i (p̂) for all i if p > p̂ by looking at (34), henceforth the monotonicity

of r1(p) is satisfied. To prove the scalability, it is enough to prove that pmaxi (p−i) is a scalable
function and then the scalability of r1(p) comes through. Let us rewrite equation (34) as follows:

pmaxi (p−i) =
2R (M+1) (

∑N
k ̸=i hkpk+σ

2)

W hi
then

pmaxi (δp−i) =
2R (M + 1) (δ

∑N
k ̸=i hkpk + σ

2)

W hi
, (36)

while

δpmaxi (p−i) =
2δ R (M + 1) (

∑N
k ̸=i hkpk + σ

2)

W hi
(37)

It is clear that δpmaxi (p−i) > p
max
i (δ p−i), therefore r

1(p) is a standard interference function, and
the Nash equilibrium point is unique. 2

In the following lemmas we omit the proof of existence and/or uniqueness as they are similar
to those of lemma 3.

Lemma 5. In NPG under Rayleigh slow flat fading channel with the average utility function
(14), the existence of a Nash equilibrium point is guaranteed if and only if the strategy space
is modified to the convex set Pi = {pi : γi ∈ (1,3β)}. The best response vector of all users
r2(p) =(r21(p),r

2
2(p), · · · ,r

2
N(p)), where r

2
i (p) = min(p

max
i ,pi−max), and p

max
i = 2β Ii, is a stan-

dard interference function, therefore by [12] Nash equilibrium point is unique.

Proof: Similarly, we need to find the first and second order derivatives of ui with respect to pi:
∂ui
∂pi

= L R
M p2i

(
2 β
γi
−1
)
, then ∂

2ui
∂p2i

= 2 L R
M p3i

(
1− 3 β

γi

)
, which means that ∂

2ui
∂p2i

< 0,∀ γi ∈ (1,3β) so
the convex strategy space should have the following form: Pi = {pi : γi ∈ (1,3β)} to guarantee
the concavity of ui and then to guarantee the existence of Nash point. 2

Lemma 6. In NPG under Nakagami fast flat fading channel with the average utility function ui
given in (24) with m = 2, the existence of a Nash equilibrium point is guaranteed if the strategy
space is modified to the following convex set Pi = {pi : γi ∈

(
γi−min,γi−max

)
}, where γi−min =

√
8
√

2 + 5M −
√
M (8 + 17M) and γi−max =

√
8
√

2 + 5M +
√
M (8 + 17M). The best response vector of all users r5(p) =(r51(p), r

5
2(p), · · · ,

r5N(p)), where r
5
i (p) = min(p

max
i ,pi−max), and p

max
i = 4

√
1 + 2M Ii, is a standard interference

function, therefore by [12] Nash equilibrium point is unique.

Proof: We find the first and second order derivatives of ui in (24) after setting m = 2 with
respect to pi as follows:

∂ui
∂pi

=
LR

M p2i
(
16(2M + 1)

γ2i
−1) (1−

16

γ2i
)M−1, (38)

then

∂2ui
∂p2i

=
1

M p3i (−16 + γ
2
i )

2

(
2LR (1−

16

γ2i
)M ×

[
256(1 + M)(2M + 1)−16(2 + 5M)γ2i + γ

4
i

])
(39)



530 M. Hayajneh, C. Abdallah

and this implies that ∂
2ui
∂p2i

< 0,∀ γi ∈
(
γi−min,γi−max

)
,

where γi−min =
√

8(2 + 5M)−8
√
M (8 + 17M) and γi−max =

√
8(2 + 5M) + 8

√
M (8 + 17M).

Henceforth, the strategy space should have the following convex set: Pi = {pi : γi ∈
(
γi−min,γi−max

)
}

to guarantee that ui is strict concave on Pi, then a Nash equilibrium exists. 2

Lemma 7. In NPG under Nakagami slow flat fading channel with the average utility function
ui given in (28), a Nash equilibrium point is guaranteed if and only if the strategy space is the
following convex set Pi = {pi : γi ∈

(
1,
√
6ξ
)
}. The best response vector of all users r6(p) =

(r61(p), r
6
2(p), · · · , r

6
N(p)), where r

6
i (p) = min(p

max
i ,pi−max), and p

max
i =

√
3ξ Ii, is a standard

interference function, therefore by [12] Nash equilibrium point is unique.

Proof: The first derivative and second order derivatives of ui after setting m = 2 with respect
to pi are given by:

∂ui
∂pi

= L R
M p2i

(
3 ξ
γ2i
−1
)
, and ∂

2ui
∂p2i

= 2 L R
M p3i

(
1− 6 ξ

γ2i

)
, therefore ∂

2ui
∂p2i

< 0,∀ γi ∈(
1,
√
6ξ
)
. As a result, the convex strategy space should be Pi = {pi : γi ∈

(
1,
√
6ξ
)
} to guarantee

the strict concavity of ui and then the existence of a Nash equilibrium point is guaranteed. 2

Now, we turn to the existence and uniqueness of Nash equilibrium point of NPGP under the
assumed channel models discussed above.

Lemma 8. In NPGP under Rayleigh fast flat fading channel model with utility function uci =
ui −c pi, where ui is given in (11), a Nash equilibrium point existence is guaranteed if and only
if the strategy space is the following convex set: Pi = {pi : γi ∈

(
γi−min,γi−max

)
}, where γi−min

= 2 (M + 1) −
√

2M(M + 1) and γi−max = 2 (M + 1). The best response vector of all users
r7(p) =(r71(p),r

7
2(p),· · · ,r

7
N(p)), where r

7
i (p) = min(p

max
i ,pi−max), and

pmaxi ≈
−6 21/3 a + 22/3(27bi +

√
108a3 + 729b2i )

2/3

6 (27bi +
√

108a3 + 729b2i )
1/3

; a =
LR

M c
, bi =

2 (M + 1)LRIi
M c

(40)

is a standard interference function, therefore by [12] Nash equilibrium point is unique.

Proof: Let us find the maximizing power pmaxi in terms of the SINR γi as follows:

∂uci
∂pi

=
LR

M p2i
(
2 (M + 1)

γi
−1) (1−

2

γi
)M−1 − c = 0, (41)

then

pmaxi =

√
LR

M c
(
2 (M + 1)

γi
−1) (1−

2

γi
)M−1 (42)

For pmaxi to be feasible, i.e., real and positive we need to satisfy the following condition on the
strategy space: Pi = {pi : γi ∈ (1,M + 1)}. But, to guarantee the concavity of the utility
function uci , we have to have Pi = {pi : γi ∈

(
γi−min,γi−max

)
}, where γi−min = 2 (M + 1) −√

2M(M + 1) and γi−max = 2 (M + 1) +
√

2M(M + 1). Therefore, to fulfill the two conditions
the convex strategy space should be the intersection of the two sets, that is Pi = {pi : γi ∈(
γi−min,γi−max

)
}, where γi−min = 2 (M + 1) −

√
2M(M + 1) and γi−max = 2 (M + 1). Since

γi >> 1 on the convex strategy space Pi given above, one can approximate pmaxi , the solution
of (41), as the feasible solution of the following equation:

p3i +
LR

M c
pi −

2 (M + 1)LRIi
M c

= 0 (43)



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 531

This equation has only one real and positive solution which is given in (40). It is fairly easy to
prove that r7(p) with pmaxi as given in (40) is a standard interference function. Therefore, Nash
equilibrium point is unique. 2

In the following lemmas we may omit the proof of existence and/or uniqueness if it can be
argued the same way as in lemma 8.

Lemma 9. In NPGP under Rayleigh slow flat fading channel model with utility function uci =
ui−c pi, where ui is given in (14), a Nash equilibrium point existence is guaranteed if and only if
the strategy space is the following convex set: Pi = {pi : γi ∈ (1,2 β)}. The best response vector
of all users r8(p) =(r81(p),r

8
2(p),· · · ,r

8
N(p)), where r

8
i (p) = min(p

max
i ,pi−max), and

pmaxi =
−6 21/3 a + 22/3(27bi +

√
108a3 + 729b2i )

2/3

6 (27bi +
√

108a3 + 729b2i )
1/3

; a =
LR

M c
, bi =

2LRβ Ii
M c

(44)

is a standard interference function, therefore by [12] Nash equilibrium point is unique. We have
equality in (44) because there was no approximation in getting pmaxi .

Lemma 10. In NPGP under Nakagami fast flat fading channel model with utility function
uci = ui −c pi, where ui is given in (24), a Nash equilibrium point existence is guaranteed if and
only if the strategy space is the convex set: Pi = {pi : γi ∈

(
γi−min,γi−max

)
}, where γi−min =

√
8
√

2 + 5M −
√
M (8 + 17M) and γi−max = 4

√
1 + 2M. The best response vector of all users

r11(p) = (r111 (p), r
11
2 (p), · · · , r

11
N (p)), where r

11
i (p) = min(p

max
i ,pi−max), and

pmaxi ≈
√

LR

2M c

√
−1 +

√
1 +

64 (1 + 2M)I2i M c

LR
(45)

is a standard interference function, therefore by [12] Nash equilibrium point is unique.

Proof: The maximizer transmit power pmaxi is the feasible solution of
∂ ui
∂ pi
− c = 0, where ∂ ui

∂ pi
is given in (38), and results in a polynomial of degree 2M + 4. It is a tedious and may be
impossible to find a closed-form for the feasible solution of this polynomial. Recall that γi > 4
to guarantee ui(p) > 0, so the maximizer transmit power level pmaxi can be approximated by the
feasible solution of the following equation.

p4i +
LR

M c
p2i −

16 (1 + 2M)LRI2i
M c

= 0 (46)

The only feasible solution of the equation above is given by (45). 2

Lemma 11. In NPGP under Nakagami slow flat fading channel model with utility function
uci = ui −c pi, where ui is given in (28), a Nash equilibrium point existence is guaranteed if and
only if the strategy space is the following convex set: Pi = {pi : γi ∈

(
1,
√
3ξ
)
}. The best response

vector of all users r12(p) = (r121 (p), r
12
2 (p), · · · , r

12
N (p)), where r

12
i (p) = min(p

max
i ,pi−max), and

pmaxi =

√
LR

2M c

√
−1 +

√
1 +

12ξ I2i M c

LR
(47)

is a standard interference function, therefore by [12] Nash equilibrium point is unique.



532 M. Hayajneh, C. Abdallah

Proof: The maximizer transmit power level pmaxi is the feasible solution of the following equa-
tion.

p4i +
LR

M c
p2i −

3ξ LRI2i
M c

= 0 (48)

The only feasible solution of the equation above is as given by (47). It is simple to check that
r12(p) with the maximizer power in (47) satisfies all the conditions of a standard interference
function. Henceforth, the Nash equilibrium point is unique. 2

Observing lemmas 3-7, we see that the maximizing SINR γmaxi for all users are the same:
γmaxi = 2(M + 1), ∀i ∈ N under fast Rayleigh and fast Rician fading channels. On the other
hand γmaxi = 2β, ∀i ∈ N under slow Rayleigh and slow Rician fading channels. γ

max
i =

4
√
1 + 2M ∀i ∈ N under fast Nakagami fading channels, and γmaxi =

√
3ξ∀i ∈ N under slow

Nakagami fading channels. For nonfading channels it was shown in [1] that γmaxi = 12.4, ∀i ∈N .
This implies, as expected, that in order to overcome the fading effect, users in fading channels
have to target higher SINR values.

Next, we introduce an algorithm that converges to Nash equilibrium point of NPG and NPGP.
We need to keep in mind that the strategy space denoted by Pi in the algorithm differs according
to the channel model. The algorithm is the same as in [1] except that the strategy spaces are
modified to the forms given in lemmas 3-7 to guarantee the existence of Nash equilibrium point
under the studied channel models.

Assume user j updates its power level at time instances that belong to a set Tj, where
Tj = {tj1, tj2, · · ·}, with tjk < tjk+1 and tj0 = 0 for all j ∈ N . Let T = {t1, t2, · · ·} where
T = T1

∪
T2
∪
· · ·
∪
TN with tk < tk+1 and define p to be the smallest power vector in the

modified strategy space P = P1
∪

P2
∪
· · ·
∪

PN .

Algorithm 12. Consider non-cooperative game G as given in (31) and generate a sequence of
power vectors as follows:
Initialize
k = 0;
N = total numberof active users;
p[0] = p;
for j = 1 to N do
set Tj = set of times user j updates its power;
end for
set T =

∪
j∈N Tj = {t1, t2, t3, ...};

k ← k + 1;
STEP1 : for j = 1 to N do
if tk ∈ Tj;
set pmaxj (tk) = arg max

pj∈ Pj
uj(pj, p−j(tk−1));

else pmaxj (tk) = p
max
j (tk−1);

end if;
end for
if p(tk) = p(tk−1)
stop and declare p(tk) as the NE Point;
else
k ← k + 1
Goto STEP1

The next algorithm finds the best pricing factor cBest for NPGP, keeping in mind that the
strategy space should be according to lemmas 8-11.



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 533

Algorithm 13. Initialize
∆c = real positive number;
c = 0;
use algorithm 12 to obtain uc=0j for all users j = 1 to j = N ;
STEP2:for j = 1 to N do
set cj ← c + ∆c; {Broadcast the pricing factor to all users} ;
end for
use algorithm 12 to obtain uc+∆cj for all users j = 1 to j = N ;
if uc+∆cj is Pareto dominates u

c
j for all users;

goto STEP2
else
stop and declare cBest = c−∆c

7 Simulation Results

We show the effects of time-varying, fast and slow fading wireless channels on the equilib-
rium utilities and powers, that are the outcomes of the extended NPG and NPGP algorithms
(algorithms12 and 13) which were originally studied for non-fading wireless channels in [1].

The system studied is a single-cell DS-CDMA cellular mobile system with 9 stationary users,
all are using the same data rate R and the same modulation scheme, non-coherent BFSK. The
system parameters used in this study are given in Table 1. The distances between the 9 users
and the BS are d = [310,460, 570,660, 740,810, 880,940,1000] in meters. The path attenuation
between user j and the BS using the simple path loss model [17] is hj = 0.097/d4j , where 0.097
approximates the shadowing effect. Results of simulations show that under Rayleigh, Rician, and
Nakagami fast flat fading channels with spreading gain W/R = 102, users do not reach a Nash
equilibrium point where all users except the nearest user to the BS are using the highest power
level in the strategy space without achieving the maximizing SINRs (γmaxi = 2(M +1), ∀i ∈N).

In Fig.1 we demonstrate the equilibrium utilities and the equilibrium powers of NPG under
a fast fading channels (a1) with the three small scale fading models with spreading gain W/R =
103. All users were able to achieve their maximizing SINR under two small scale fading models,
namely, Nakagami and Rician channels. Under the Rayleigh channel model however some users
failed to achieve the maximizing SINR. One can see in Fig.1 that the farthest 4 users in the
Rayleigh channel were forced to send at their maximum allowable power to achieve their minimum
SINRs. The equilibrium utilities and equilibrium powers of the NPGP under (a1) are shown in
the left and right graphs of Fig.2, respectively. Results show that a Pareto improvement over
NPG for Rayleigh, Rician, and Nakagami channels was obtained such that all users succeeded
to attain SINRs more than their corresponding minimum SINRs (γi > γi−min, ∀i ∈N).

Then we present the effect of a slow flat fading channels (a2) on the equilibrium utilities
and powers which are the outcomes of NPG algorithm 12 as shown in Fig. 3. This figure
shows that, unlike fast fading channels, all users succeeded to achieve the maximizing SINR
(γmaxi = 2β = 19.8619 under Rician and Rayleigh channels and γ

max
i =

√
3ξ = 25.1182 under

Nakagami channels). Left graph of Fig. 3 shows that under both Rayleigh and Nakagami channel
models users were equally satisfied, i.e., the equilibrium utilities are the same for both models.
However, under Rayleigh channels, the equilibrium powers are less than those under Nakagami
channels. This could be due to the fact that users under Nakagami channels target a higher
maximizing SINR as we just mentioned above.

As for the effect of slow fading channels on the outcomes of NPGP, equilibrium utilities and
equilibrium powers, our simulations showed that Pareto improvement (dominance) over NPG



534 M. Hayajneh, C. Abdallah

was not possible under the three small scale models. At c = cBest, simulations showed that the
best policy is that all users to target a fixed SINR, that is γmaxi = 19.8619 under Rician and
Rayleigh channels and γmaxi = 25.1182 under Nakagami channels, which is exactly the same
situation as in NPG. To demonstrate this result for the three small scale models more clearly, we
present Fig. 4 for Rician and Rayleigh channel models and Fig. 5 for Nakagami channel model.
Fig. 4 shows that with c = cBest the maximizing transmit power pmaxi given in (44) behaves with
respect to the effective interference Ii (feasible values of Ii ) the same as pmaxi = 2β Ii given in
Lemmas 2 and 4. While Fig. 5 shows that pmaxi given in (47) behaves with feasible values of
Ii the same as pmaxi =

√
3ξ Ii given in Lemma 6. Surprisingly, both figures suggest that NPGP

with linear pricing does not admit a Pareto dominance over the NE operating point of NPG in
a slow flat fading channels under the three small scale fading models.

10
2

10
3

10
−6

10
−5

10
−4

10
−3

10
−2

10
−1

10
0

d (meters)

E
q

u
ili

b
ri
u

m
 P

o
w

e
rs

 (
W

a
tt
s)

10
2

10
3

10
3

10
4

10
5

10
6

10
7

10
8

10
9

10
10

d (meters)

E
q

u
ili

b
ri
u

m
 U

til
iti

e
s 

(b
its

/J
o

u
l)

Deterministic
Rayleigh
Rician
Nakagami

Figure 1: Equilibrium powers and equilibrium utilities of NPG for Rician fast flat fading channel
gain (+), Rayleigh fast flat fading channel gain (o), Nakagami fast flat fading (△) and determin-
istic channel gain (*) versus the distance of a user from the BS in meters with W/R = 103.

Table 1: The values of parameters used in the simulations.

L, number of information bits 64
M length of the codeword 80
W , spread spectrum bandwidth 106, 107 Hz
R, data rate 104 bits/sec
σ2, AWGN power at the BS 5×10−15
N, number of users in the cell 9
s2, specular component 1
W/R, spreading gain 102, 103

m, fading figure 2
pi−max, maximum power in ith user’s strategy space 1 Watts



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 535

10
2

10
3

10
4

10
5

10
6

10
7

10
8

10
9

10
10

d (meters)

E
q

u
ili

b
ri
u

m
 U

til
iti

e
s 

(b
its

/J
o

u
l)

Deterministic c
Best

=6e8
Nakagami c

Best
=6e3

Rician c
Best

=1e9
Rayleigh c

Best
=1e8

10
2

10
3

10
−6

10
−5

10
−4

10
−3

10
−2

10
−1

d (meters)

E
q

u
ili

b
ri
u

m
 P

o
w

e
rs

 (
W

a
tt

s)

Figure 2: Equilibrium utilities and equilibrium powers of NPGP for Rician fast flat fading
channel gain (+), Rayleigh fast flat fading channel gain (o), Nakagami fast flat fading (△)
and deterministic channel gain (*) versus the distance of a user from the BS in meters with
W/R = 103.

10
2

10
3

10
6

10
7

10
8

10
9

10
10

d (meters)

E
q

u
ili

b
ri
u

m
 U

til
iti

e
s 

(b
its

/J
o

u
l)

10
2

10
3

10
−6

10
−5

10
−4

10
−3

10
−2

d (meters)

E
q

u
ili

b
ri
u

m
 P

o
w

e
rs

 (
W

a
tt
s)

Rician
Nakagami
Rayleigh

Figure 3: Equilibrium utilities and equilibrium powers of NPG for Rician slow flat fading channel
gain (+), and Rayleigh slow flat fading channel gain (o), Nakagami slow flat fading (△) versus
the distance of a user from the BS in meters with W/R = 103.



536 M. Hayajneh, C. Abdallah

10
−6

10
−5

10
−4

10
−3

10
−2

10
−5

10
−4

10
−3

10
−2

10
−1

10
0

Interference plus noise, I
 i
 (Watts)

U
se

r 
i t

ra
n

sm
it 

p
o

w
e

r 
le

ve
l, 

p  
im
a

x  
(W

a
tt

s)

p
i
max in Eq.(54)

p
i
max (Lemma 2 or 4)

Figure 4: pmaxi as a function of Ii as in Eq. (44) (o), and the linear expression p
max
i = 2β Ii

given in Lemmas 2 and 4 (solid line) with c = cBest.

10
−6

10
−5

10
−4

10
−3

10
−2

10
−5

10
−4

10
−3

10
−2

10
−1

10
0

Interference plus noise, I
 i
 (Watts)

U
se

r 
i t

ra
n

sm
it 

p
o

w
e

r 
le

ve
l, 

p  
im
a

x  
(W

a
tt

s)

p
i
max in Eq.(32)

p
i
max (Lemma 2)

Figure 5: pmaxi as a function of Ii as in Eq. (47) (o), and the expression of p
max
i =

√
3ξ Ii given

in Lemma 6 (solid line) with c = cBest .



Game Theoretic Distributed Power Control Algorithms for
Uplink Wireless Data in Flat Fading Channels 537

8 Conclusions

We studied a noncooperative power control game (NPG) and noncooperative power control
game with pricing (NPGP) introduced in [1] for realistic channel models, where we studied the
impact of power statistical variation in Rayleigh, Rician and Nakagami fast/slow flat fading
channels on the powers and utilities vectors at equilibrium. The results showed that an equi-
librium with an equal maximizing SINR is not attainable in both games with spreading gain
(W/R = 102). In fast fading with spreading gain W/R = 103, fixed target SINR NPG admitted
NE point only under both Rician and Nakagami small scale models. And unlike fast fading,
NPG admitted NE point under all small scale fading models in slow fading channels. Results
demonstrated that in slow flat fading channels, NPGP with linear pricing does not exhibit a
Pareto dominance over NPG outcomes at equilibrium.

In order to overcome the fading effects the SINRs targeted at equilibrium are higher for all
users at equilibrium in the Rician, Rayleigh and Nakagami flat fading cases than SINRs under
deterministic (nonfading) channels.

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