Content uploaded by Panagiotis Promponas

Author content

All content in this area was uploaded by Panagiotis Promponas on May 13, 2021

Content may be subject to copyright.

IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. XX, NO. XX, XXXX 2020 1

Rethinking Power Control in Wireless Networks:

The Perspective of Satisfaction Equilibrium

Panagiotis Promponas, Student Member, IEEE, Eirini Eleni Tsiropoulou, Member, IEEE, and Symeon

Papavassiliou, Senior Member, IEEE

Abstract—In this paper we propose a holistic framework

that aims at a paradigm shift in the treatment of the uplink

power control problem in wireless networks, under the per-

spective of games in satisfaction form. Novel satisfaction

equilibrium points of special interest within the considered

problem - such as the Minimum Efﬁcient Satisfaction Equi-

librium (MESE) and the Minimum Satisfaction Equilibrium

(MSE) - are introduced, while their beneﬁts, existence and

uniqueness are investigated, considering a realistic and

generic user utility function being quasiconcave with re-

spect to its transmission power. It is proven that at the

MESE and MSE points the system achieves the lowest

possible cumulative cost, while each user individually is

penalized with the minimum cost compared to the cor-

responding cost of any Efﬁcient Satisfaction Equilibrium

(ESE) and of any Satisfaction Equilibrium (SE), respectively.

A decentralized low complexity algorithm, based on the

Best Response Dynamics, is proposed that converges to

the MSE equilibrium, while it can efﬁciently handle the

dynamic behaviors of the users in the network. Numerical

results are provided that validate and evaluate the beneﬁts

of the proposed novel power control framework, underlin-

ing the superiority of the MSE against other equilibrium

points.

Index Terms—Satisfaction equilibrium, energy efﬁ-

ciency, game theory, power control, resource management

I. INTRODUCTION

THE emergence and evolution of 5G and Internet of

Things (IoT), has pushed researchers and industries to be

looking at the technological transformation to move towards an

environment, where multiple devices will be able to connect,

share information, interpret, and deliver a seamless experience

for users. Despite the fact that signiﬁcant advances have been

realized through the use of enhanced network architectures

and technologies, large amounts of spectrum – being a scarce

resource - are still required to deliver massive increases in

The research project was supported by the Hellenic Foundation for

Research and Innovation (H.F.R.I.) under the “1st Call for H.F.R.I. Re-

search Projects to support Faculty Members & Researchers and the the

procurement of high-cost research equipment grant” (Project Number:

HFRI-FM17-2436). The research of Dr. Tsiropoulou was conducted as

part of the NSF CRII Award #1849739.

P. Promponas and S. Papavassiliou are with the the School of

Electrical and Computer Engineering, National Technical University of

Athens, Zografou, Greece, 15780 (e-mail: ppromponas@mail.ntua.gr;

papavass@mail.ntua.gr).

E.E. Tsiropoulou is with the Department of Electrical and Computer

Engineering, University of New Mexico, New Mexico, USA, 87131 (e-

mail: eirini@unm.edu).

capacity and achieve high throughput. Unless a paradigm shift

occurs in the resource allocation decision making process

the problem of spectral efﬁciency will still remain a barrier

towards the realization of 5G’s full potential [1], [2].

Traditionally, towards devising intelligent resource alloca-

tion approaches in such resource constrained environments,

the Expected Utility Theory (EUT) has been adopted targeting

at the maximization of the users’ beneﬁts from allocating the

available resources. Following the principles of EUT, each user

aims at maximizing its personal utility in a selﬁsh manner

targeting at the highest possible performance [3], [4], [5].

Moreover, to enable the users’ distributed intelligent decision

making in a computationally efﬁcient manner, while at the

same time capturing the users’ competitive behavioral patterns,

Game Theory has arisen as a theoretical and practical powerful

tool [6], [7]. The solution of the corresponding resource

orchestration problems is the Nash Equilibrium (NE) point,

where the users maximize their own utility, while they cannot

achieve a better outcome by unilaterally changing their own

strategies given the strategies of the rest of the users [8].

However, is the NE point really the best solution that it

can be achieved by the users [9], in communications and

computing systems where users’ decision are interdependent?

Even more, is the goal of maximizing each user’s utility a rea-

sonable and meaningful goal within such resource constrained

systems? Those are the fundamental questions that this work

aims to address, while introducing a novel efﬁcient resource

control framework based on the theory of Satisfaction Games.

A. Related Work & Motivation

Various resource management problems in wireless net-

works have been considered in the recent literature, based

on the concept of EUT and non-cooperative Game Theory

(e.g., [10]–[12]). However, the NE points stemming from

users’ selﬁsh decision-making are generally inefﬁcient. To-

wards guiding the selﬁsh users to a more efﬁcient operating

point, various pricing mechanisms that penalize the users with

respect to their resources’ consumption, were introduced (e.g.

[13]). These approaches constituted a ﬁrst step towards treating

the aforementioned inefﬁciency, without however offering a

holistic treatment to the main disadvantage of the NE points.

The latter is due to the fact that customized heuristic pricing

mechanisms are required each time to treat different resource

types and networking environments. Furthermore, even when

pricing is considered, each user still aims at maximizing its

own perceived Quality of Service (QoS).

2 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. XX, NO. XX, XXXX 2020

Towards treating the above issue in a formal and universal

manner, a new concept of equilibrium is introduced, namely

Satisfaction Equilibrium (SE), where the users aim to satisfy

their minimum QoS prerequisites instead of targeting at QoS

maximization [1], [14]. In particular, in [15] the deﬁnition of

the SE and the general conditions for examining its existence

have been discussed in detail. Furthermore, the concept of

users’ effort investment to achieve the SE has been introduced,

leading to a reﬁnement of the SE, namely the Efﬁcient SE

(ESE). At the ESE point, all the users conclude to a resource

allocation strategy, which requires the lowest effort to satisfy

their minimum QoS prerequisites. In [16] and [17], the con-

cepts of SE and ESE are applied in a simpliﬁed uplink power

control problem considering interference channels in a single-

cell environment. In [18] machine learning has been adopted

to determine the SEs and ESEs under different conditions and

uncertainties, while in [19] a distributed learning algorithm

that converges to speciﬁc correlated equilibria is provided.

Nevertheless, many interesting properties that emerge when

the holistic satisfaction equilibrium framework is applied have

not been revealed yet [14], while several critical challenge

remain unexploited.

B. Contributions and Outline

Our work aims at ﬁlling this gap, while focusing on the

transformation and treatment of the uplink power control

problem in next generation wireless networks under the per-

spective of satisfaction games, while proposing new concepts

in the ﬁeld of satisfaction games. A key differentiating aspect

of our work, is the relaxation of the common assumption

of using strictly increasing user utilities with respect to the

user’s uplink transmission power, when adopting the concept

of satisfaction equilibrium in the current literature [15], [20].

Such assumption is quite restrictive, thus considerably limit-

ing the exploitability and applicability of the corresponding

approaches. Instead, in our work we consider generic enough

and realistic users’ utility functions, which are assumed to be

quasiconcave with respect to the user’s uplink transmission

power. Two representative examples of such utility functions

widely used in the literature regarding wireless networks, is

the Shannon capacity and the energy efﬁciency function [21].

Accordingly, the novel concepts of Minimum Efﬁcient

Satisfaction Equilibrium (MESE) and Minimum Satisfaction

Equilibrium (MSE) are introduced building on the existing

concepts of SE and ESE, and their special interest and

properties are underlined (Section II). Based on this new

introduced framework, the corresponding uplink power control

problem is formulated and studied as a game in its satisfaction

form (Section III). In particular, assuming that each user is

associated with a usage-based cost function that is increasing

with respect to its transmission power, we prove that the MESE

point is unique. The intuition behind and the physical notion

of the MESE point is, that at this novel equilibrium point,

the system achieves the lowest cumulative cost from every

other ESE of the system, while at the same time each user is

penalized with the minimum cost (i.e., transmission power)

that could experience in every other ESE. Capitalizing on

this observation subsequently we prove that the unique MESE

point is also the unique MSE point of the game, which is

the ultimate targeted and desired operation point. That is, in

the considered uplink power control game, the MSE and the

MESE points coincide, and from every other power allocation

that satisﬁes the users’ QoS prerequisites, this point allocates

to each user the minimum possible transmission cost.

A decentralized algorithm based on the Best Response

Dynamics is proposed, that enables the system to efﬁciently

converge to its MESE/MSE point, or alternatively determine

the non-existence of an SE (Section IV). Furthermore, capi-

talizing on the aforementioned theoretical foundations and al-

gorithm, a holistic operationally efﬁcient framework is offered

to accommodate the users’ dynamic behavior in the examined

system (i.e. user decrease/increase of QoS demands, or user

entrance/departure from the system), which typically occur in

5G networks (Section VI). A series of simulation experiments

are performed that provide a proof of concept of the validity

of the introduced theoretical framework, by: (i) comparing the

MSE with other existing equilibria in the literature (SE, ESE,

NE) while underlining its properties and superiority, and (ii)

studying the behavior and the convergence of the proposed

novel holistic framework based on games in satisfaction, under

different scenarios (Section VI). Finally, Section VII concludes

the paper.

II. GAMES IN SATI SFACT ION FORM

In this section, we provide some deﬁnitions and the basic

notation that will be used in the rest of the paper. A game in

satisfaction form is deﬁned as ˆ

G= (K, {Ak}k∈K,{fk}k∈K),

where K={1, . . . |K|} represents the set of players, Akis the

strategy set of player k∈K,uk(ak,a−k)represents player’s

kpayoff (i.e., utility function), and fk(a−k) = {ak∈Ak:

uk(ak,a−k)≥uthr}determines the set of actions of player

kthat allows its satisfaction, that is its payoff to be above a

threshold value uthr, given the actions a−kplayed by all the

other players [17]. A strategy proﬁle is denoted by a vector

a= (a1, . . . , a|K|)∈A,A=A1× · · · × Ak× · · · × A|K|.

Deﬁnition 1: An action proﬁle a+is an SE point for the

game ˆ

G= (K, {Ak}k∈K,{fk}k∈K)if

a+

k∈fk(a+

−k),∀k∈K(1)

From this deﬁnition it is evident that at the SE point,

each player satisﬁes its QoS prerequisites. It should be

noted that there could exist multiple strategy vectors a+=

(a+

1, . . . , a+

|K|)satisfying player’s minimum QoS prerequisites,

some of which are of particular interest. A representative

example is the Efﬁcient Satisfaction Equilibrium (ESE) where

each player of the system achieves its minimum QoS prereq-

uisites via being simultaneously penalized with the minimum

cost. To capture the notion of the players’ penalty and effort

associated with a given action choice, the concept of the cost

function for each player is introduced. For all k∈K, the

cost function ck:Ak→[0,1] satisﬁes the following condition:

ck(ak)< ck(a0

k),∀(ak, a0

k)∈A2

k, if and only if, akrequires

a lower effort by player kthan action a0

k.

Deﬁnition 2: An action proﬁle a∗is an ESE point for the

game ˆ

G, with cost functions {ck}k∈K,if

PROMPONAS et al.: RETHINKING POWER CONTROL IN WIRELESS NETWORKS: THE PERSPECTIVE OF SATISFACTION EQUILIBRIUM 3

a∗

k∈fk(a∗

−k),∀k∈K(2a)

ck(ak)≥ck(a∗

k),∀k∈K, ∀ak∈fk(a∗

−k)(2b)

At the ESE point, each player satisﬁes its personal QoS pre-

requisites with its minimum possible personal cost. It is noted

that an ESE point is also an SE point. Another equilibrium

point of special interest is the Minimum Efﬁcient Satisfaction

Equilibrium (MESE). At the MESE point, all players satisfy

their QoS prerequisites (Eq. 3a), with the minimum cost for

themselves (Eq. 3b) and the minimum total cost from the

system’s perspective (Eq. 3c).

Deﬁnition 3: An action proﬁle a†is a Minimum Ef-

ﬁcient Satisfaction Equilibrium (MESE) for the game

ˆ

G= (K, {Ak}k∈K,{fk}k∈K),with cost functions {ck}k∈K,

and set of action proﬁles that are ESEs {E}if

a†

k∈fk(a†

−k),∀k∈K(3a)

ck(ak)≥ck(a†

k),∀k∈K, ∀ak∈fk(a†

−k)(3b)

X

k∈K

ck(ek)≥X

k∈K

ck(a†

k),∀e∈E(3c)

From this deﬁnition it is implied that an MESE point

is also an ESE point. Last, but not least, the concept of

Minimum Satisfaction Equilibrium (MSE) is introduced, where

all players satisfy their QoS prerequisites (Eq. 4a) and the

system achieves its minimum possible cost (Eq. 4b).

Deﬁnition 4: An action proﬁle aopt is a Minimum

Satisfaction Equilibrium (MSE) for the game

ˆ

G= (K, {Ak}k∈K,{fk}k∈K),with cost functions {ck}k∈K,

and set of action proﬁles that are SEs {S}if

aopt

k∈fk(aopt

−k),∀k∈K(4a)

X

k∈K

ck(sk)≥X

k∈K

ck(aopt

k),∀s∈S(4b)

III. RETHINKING UPLINK POWE R CONT RO L

A. System Model and Assumptions

Let us consider Ktransmitter/receiver pairs denoted by

index k∈K. For all k∈K, transmitter kuses power level

pk∈Ak, with Akgenerally deﬁned as a compact sublattice.

We denote pmin

kand pmax

kthe minimum and maximum power

levels in Ak, respectively, while gij is the channel gain coef-

ﬁcient between transmitter iand receiver j. We study uplink

power control games in which each user has a utility function

that is quasiconcave with respect to its own transmission power

and decreasing with respect to the total summation over the

powers of the rest of users, as the latter quantity acts as

interference to the examined user’s transmission. A general

example of such utility function that satisﬁes this realistic

assumption, is the commonly adopted in the literature energy

efﬁciency function in typical interference limited communica-

tion environment, as presented in the seminal paper [21] and

[17], and presented below :

uk(pk,p−k) = f(γk)

pk

, γk=W

R

hkpk

Pj6=khjpj+σ2

k

(5)

where σ2

kdenotes the Additive White Gaussian Noise variance

at receiver k,Ris the requested user service data rate and

Wdenotes the system bandwidth. f(γk)is an efﬁciency

function representing the probability of a successful packet

transmission for user kand is an increasing and sigmoidal

function with respect to pk. An indicative form of this function

that has been used in the existing literature, and is also adopted

in this paper for evaluation purposes, is f(γk) = (1−e−aγk)M,

where parameters a, M > 0control the shape of this function.

[13], [21]. It has been shown that if every user adopts the utility

function of Eq. 5, then the corresponding non-cooperative

power control game possesses at least one NE [21].

In the following we consider discrete power levels, that is

the user’s strategy set is discrete, which in turn is translated to

taking a sample from the energy efﬁciency function’s possible

values. This means that one interval of the possible energy

efﬁciency values of each user is increasing with respect to

power. This interval is from pmin

kto a power that maximizes

uk(), that is [pmin

k, pM

k]. We refer to that interval as the left

interval. Note that for a ﬁxed value of p−k,uk(pM

k,p−k)

is the maximum possible value of the sampled powers. That

means that pM

k(p−k)depends on the strategies of the others.

In the other interval, i.e., the right interval, (pM

k, pmax

k],uk()

is decreasing with respect to pk. Note that the sampling we

have over the utility function is following a quasiconcave

function. Nevertheless, uk(pM

k,p−k)could be less or equal

to the maximum value of the utility in the corresponding

continuous interval of user’s transmission power.

Deﬁnition 5: Given a strategy proﬁle p∈A, we deﬁne the

set of users R(p)as the users k∈K:pk> pM

k(p−k).

The following proposition states that for every possible SE

of the game that there are users that transmit with power in

the interval (pM

k, pmax

k], there exists another one that all of the

users transmit in the interval [pmin

k, pM

k(p−k)].

Proposition 1: Let an uplink power control game in the

satisfaction form ˆ

Gwith utility functions {uk}k∈K(Eq. 5)

and the set of all possible SEs {S}of the game. Then

∀p+∈S:R(p+)6=∅,∃e+∈S:R(e+) = ∅.

Proof: See Appendix I.

Proposition 1 shows that transmitting in (pM

k, pmax

k], given

the strategies of the others, is inefﬁcient as there would also

be a lower pkthat could yield higher utility. For the rest of

the paper, we assume increasing cost functions, {ck}k∈K, with

respect to the users’ transmission powers.

B. Best Response in Uplink Power Control

In the following we initially assume that each user al-

ways possesses a strategy that satisﬁes its QoS prerequisites.

Nevertheless, as argued later in Section IV, this assumption

is not restrictive and can be relaxed. Thus, we can easily

conclude that given p−k, there is a pk∈[pmin

k, pM

k(p−k)]

which satisﬁes the QoS prerequisites of the examined user k,

and if a lower transmission power is used, this will leave the

user unsatisﬁed. Contrary, if the user transmits with a greater

power in that interval, then the user will remain satisﬁed. We

will refer to that power as the Best Response BRk(p−k) =

{pk∈Ak:pk= arg minpk∈fk(p−k)c(pk)}of user kgiven

p−k. With the following propositions we study users’ best

responses in the context described above.

Proposition 2: Given a strategy proﬁle p−k, the user’s k

best response is in the user’s left interval of transmission

4 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. XX, NO. XX, XXXX 2020

powers: BRk(p−k)∈[pmin

k, pM

k].

Proof: See Appendix II.

Based on proposition 2, if {E}is the set of all possible

ESEs of the game, it holds that ∀p∗∈ {E}, R(p∗) = ∅.

The proposition below states that if some users increase their

transmitting powers, their best responses, if they exist, will

also be increased or remain the same.

Proposition 3: Let a user k∈K, and two strategy proﬁles

p1,p2∈A. Then: p1

−kp2

−k⇒BRk(p1

−k)≤BRk(p2

−k).

Proof: See Appendix III.

C. Existence of ESE and MESE

To prove the existence of at least one ESE point in the

uplink power control game ˆ

Gin our setting we ﬁrst mention

the Tarski and Knaster’s ﬁxed point theorem [22].

Theorem 1 (Tarski and Knaster’s ﬁxed point theorem):

Let Lbe a complete lattice and let f:L→Lbe an

order-preserving function. Then, the set of ﬁxed points of f

in Lis also a complete lattice.

Let Abe the set of the strategy space of the game ˆ

Gas

deﬁned above. Let us also deﬁne the lattice L=hA, i,

where is the component-wise less or equal. Note that L

is a complete lattice as all its subsets have both a supremum

and an inﬁmum. The next step is to construct an appropriate

function g:L→L. Thus we deﬁne g:L→Las follows:

g(p) = (BR1(p−1), . . . , B R|K|(p−|K|)) ∀p∈A

Note that if fk(·)6=∅for every user k, then BRk(p−k)∈

Ak,∀p−k∈A−k,∀k∈K. Following those deﬁnitions we

conclude to the following proposition.

Proposition 4: If an uplink power control game in satisfac-

tion form ˆ

Gwith cost function {ck}k∈Kand utility function

{uk}k∈K(Eq. 6), has fk(·)6=∅,∀k∈Kfor every input then

it possesses at least one ESE.

Proof: See Appendix IV.

Given the existence of at least on ESE, we can readily

conclude to the existence of at least one MESE as well.

D. Uniqueness and Beneﬁts of MESE

In the following, for simplicity in the discussion and without

loss of generality, we assume that the fkfunctions are non

empty for every input and every user k, thus ensuring the

possession of at least one ESE (Proposition 4). However, in

Section IV it is argued that this assumption can be relaxed

through the use of an additional auxiliary power stage.

Proposition 5: If there exists an action proﬁle p+that is

SE of the game ˆ

Gthere also exists one action proﬁle p∗that

is an ESE and it holds true that ck(p+

k)≥ck(p∗

k),∀k∈K.

Proof: See Appendix V.

Subsequently we prove the uniqueness of the MESE point.

Proposition 6: The MESE point p†of the game ˆ

Gis

unique.

Proof: Let {E}be the set of action proﬁles that are

ESEs. Let us consider two MESEs p†(1) and p†(2) such that

for one user kit holds that ck(p†(1)

k)6=ck(p†(2)

k). In order for

them to be MESEs, it should hold true that:

∀p∗∈E, X

k∈K

ck(p∗

k)≥X

k∈K

ck(p†(1)

k) = X

k∈K

ck(p†(2)

k)(6)

There is one user kthat ck(p†(1)

k)6=ck(p†(2)

k), thus, p†(1)

k6=

p†(2)

k. Without loss of generality, we assume p†(1)

k< p†(2)

k.

Thus, the total summation over the costs of all users in p†(1)

would be lower than the one of p†(2) if they do not differentiate

in any other strategy. This means that there should be one other

user j6=kthat cj(p†(1)

j)> cj(p†(2)

j), so p†(1)

j> p†(2)

j.

Let p+be an action proﬁle with p+

k=p†(1)

kand p+

j=p†(2)

j.

Note that p+has lower summation over the costs of users k,

jfrom both p†(1) and p†(2). Continuing in that fashion, p+

strategy proﬁle selects for every user kthe lower power from

p†(1)

kand p†(2)

kand thus the lower cost. Note that p+is an SE

as each user kwas satisﬁed by playing p+

keither at p†(1) or

at p†(2) while all the other users have played greater or equal

transmission powers. So, at p+it holds true that:

X

k∈K

ck(p+

k)<X

k∈K

ck(p†(1)

k) = X

k∈K

ck(p†(2)

k)(7)

Applying proposition 5 on p+gives us an ESE p†with

X

k∈K

ck(p+

k)≥X

k∈K

ck(p†

k)(8)

Combining Eq. 7 and Eq. 8, we conclude that:

X

k∈K

ck(p†

k)≤X

k∈K

ck(p+

k)<X

k∈K

ck(p†(1)

k) = X

k∈K

ck(p†(2)

k)

which leads to contradiction with Eq. 6, as p†is an ESE. So,

ck(p†(1)

k) = ck(p†(2)

k),∀k∈Kand p†(1)

k=p†(2)

k,∀k∈K.

Thus, the MESE point p†is unique.

The following proposition shows that each user achieves the

minimum cost at a MESE point compared to the experienced

cost at any other ESE point.

Proposition 7: In the considered uplink power control

game, let p†be a MESE of the game and {E}the set of

ESEs, it holds true that ck(p†

k)≤ck(p∗

k),∀k∈K, ∀p∗∈E.

Proof: Let us study the strategy proﬁle pthat:

∀k∈K, ∀p∗∈E, pk= arg min

p∗

k

ck(p∗

k)(9)

Thus, the strategy proﬁle ppicks for each user the power

that gives the lowest cost for the user over all its strategies that

belong to the set of ESEs, i.e., ∀k∈K, ∀p∗∈E, pk≤p∗

k.

Let us focus on a random user k. Let p∗be one ESE such

that pk=p∗

k. So, from all the ESEs, p∗gives the lowest cost

to user k,ck(p∗

k). As proved, ∀i∈K, pi≤p∗

i. Owing to the

above, user kwill certainly be satisﬁed in strategy proﬁle p

because it was satisﬁed at the ESE p∗in which the other users

have played greater or equal transmission powers. The above

analysis holds for every user k, thus every user in strategy

proﬁle pis satisﬁed, thus pis an SE. Now, we can apply

Proposition 5 that gives us an ESE p†that:

∀k∈K, ck(pk)≥ck(p†

k)(10a)

X

k∈K

ck(pk)≥X

k∈K

ck(p†

k)(10b)

Taking into consideration Eq. 9, we can note that only

the equality can hold in inequalities (12a), (12b) so: ∀k∈

K, ck(pk) = ck(p†

k)and Pk∈Kck(pk) = Pk∈Kck(p†

k). Note

that we cannot ﬁnd an ESE that has lower total cost than p.

Thus, p†is the MESE. That means that every MESE allocates

to each user the minimum cost that it could possibly have in

an ESE, as exactly pdoes.

PROMPONAS et al.: RETHINKING POWER CONTROL IN WIRELESS NETWORKS: THE PERSPECTIVE OF SATISFACTION EQUILIBRIUM 5

Below we can harness the monotonicity of the assumed cost

functions to prove that the MESE point is the best strategy

proﬁle that the system could possibly converge to, while when

it does not exist, the system does not possess any SE at all.

Proposition 8: In the considered uplink power control

game, the MESE point, p†, is also the MSE point, popt.

Proof: Let us apply proposition 5 in the MSE point, popt

which gives an ESE point p∗that ck(popt

k)≥ck(p∗

k),∀k∈K.

Let a user k∈Khave popt

k6=p∗

k. That would imply that

X

k∈K

ck(popt

k)>X

k∈K

ck(p∗

k)(11)

The above inequality leads to a contradiction because of the

MSE’s deﬁnition 4. Thereby, p∗=p†=popt .

Following the above proposition and discussion it is noted

that the MESE point is also an ESE point, and accordingly an

SE point, as discussed in detail in Section II. Following the

previous pattern, we can prove that the MSE is component-

wise lower than any SE.

Proposition 9: In the considered uplink power control

game, let popt be the MSE of the game and {S}the set of

SEs, it holds that ck(popt

k)≤ck(p+

k),∀k∈K, ∀p+∈S.

Proof: Because of proposition 8, the MSE is also the

MESE point of the game. Because of that and based on

proposition 7, we have that

ck(popt

k)≤ck(p∗

k),∀k∈K, ∀p∗∈E(12)

Let a random strategy proﬁle that is an SE, p+. Applying

proposition 5 in p+, we have an ESE p∗with

ck(p∗

k)≤ck(p+

k),∀k∈K(13)

Because of Eq. 12:

ck(popt

k)≤ck(p∗

k),∀k∈K(14)

Thus, because of Eq. 13, 14 we have that:

ck(popt

k)≤ck(p∗

k)≤ck(p+

k),∀k∈K(15)

Because p+was a random SE of the game it holds that:

ck(popt

k)≤ck(p+

k),∀k∈K, ∀p+∈S

IV. ALGORITHM & CONVERGENCE

In this section, we present a decentralized algorithm that

converges at a Minimum Satisfaction Equilibrium (MSE) of

the game ˆ

G= (K, {Ak}k∈K,{fk}k∈K), based on the concept

of Best Response Dynamics (BRD), properly applied in the

context of a game in satisfaction form. In particular, Best

Response Dynamics (BRD) is deﬁned as the behavioral rule

in which each user always chooses its strategy (i.e., its uplink

transmission power) to be its best response (BR) to the

strategies of the rest of the users. In the context of this paper,

the dynamics should not be sequential but rather asynchronous.

As it has been shown in [17], when all the users adopt

utility functions given by the Shannon capacity and the BRD

starts from an SE as an initial strategy proﬁle, they converge

monotonically to an ESE.

Algorithm: SDA Turn Phase

1: if pk∈fk(p−k)then {If user kis still satisﬁed with its

previous power}

2: play pk;{transmit with the same power}

3: else

4: pM

k(p−k)←ModiﬁedBinarySearch(

Pk[],1,|Ak|, uk(),p−k);{Finds the power that max-

imizes k’s utility in that turn}

5: BRk(p−k)←BinarySearch(

Pk[],|Ak|, uk(),p−k, pk, pM

k(p−k));{Finds new BR

(as the vector p−khas changed) using binary search in

Pk[] from previous power (pk) to pM

kusing the utility

function of the user}

6: play BRk(p−k);{play the lowest power that satisﬁes}

7: end if

ModiﬁedBinarySearch(Pk[], low, high, uk(),p−k)

1: mid ←(low +high)/2;

2: if low =high then

3: returnPk[low];

4: else if uk(Pk[mid],p−k)> uk(Pk[mid + 1],p−k)then

5: result ←ModiﬁedBinarySearch(

Pk[], low, mid, uk(),p−k);

6: else if uk(Pk[mid],p−k)< uk(Pk[mid + 1],p−k)then

7: result ←ModiﬁedBinarySearch(

Pk[], mid + 1, high, uk(),p−k);

8: end if

A. Satisfaction Dynamics Algorithm (SDA)

Initially, each user transmits with its minimum power Pk[0]

having sorted its possible transmission powers in a vector

Pk[]. This power could be considered as their best response

in the initialization of the game. Note, that because of the

monotonicity of the cost functions, by minimizing the power

of a user we also minimize its cost in a turn. After the initial

transmission, pstart = (P1[0], . . . , P|K|[0]) each user chooses

the power that minimizes its cost function. That said, each user

who is in turn to play executes the Turn Phase of the SDA

algorithm (summarized in the pseudocode above) in order to

ﬁnd its BR and transmits with it. Note that each user k, ﬁrst,

has to ﬁnd its pM

kwith a modiﬁed binary search, given the total

interference, i.e., P∀j∈Khjpj. The total interference may be

simply broadcasted by the base station to all the users, in

order each user to determine its own sensed interference, i.e.,

P∀j∈Khjpj−hkpM

k. It is noted here that the base station

does not make any decision with respect to the power control

problem, a process that is fully executed at each user side.

Then, because of Proposition 2, with a second binary search

in the interval [pmin

k, pM

k(p−k)] it can ﬁnd its BR in that turn.

Due to the the fact that each user either does not change or

increases its transmission power at each turn (as we will prove

in the section IV.B), user kshould only do binary search from

the BR of its previous turn to its new pM

k(p−k)to ﬁnd its

new BR. The algorithm stops when no user has a new best

response strategy to play.

6 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. XX, NO. XX, XXXX 2020

B. Convergence & Complexity Analysis

In this section we initially prove that the SDA algorithm

converges to an MESE, which is also the MSE point of the

uplink power control game, under ﬁnite number of steps.

Subsequently the complexity of the algorithm is analyzed.

Proposition 10: When an SE exists in a game, the SDA

algorithm monotonically converges to a strategy proﬁle popt ∈

Athat is the MSE of the game.

Proof: See Appendix VI.

It is noted that in practice the convergence condition of the

SDA algorithm is that the best responses of all the users within

the examined network have not changed in two consecutive

cycles of turns (i.e. iterations ite) of the algorithm, i.e.,

|BRite

k(p−k)−BRite−1

k(p−k)|= 0,∀k∈K. Note, that in

line 1 of the SDA turn phase, each player, k, should check

whether the previous BR coincides with the BR of the next

turn, i.e., |BRite

k(p−k) = BRite−1

k(p−k)|. In the case that the

latter holds true, then BRite

k(p−k) = BRite−1

k(p−k) = pk,

which in turn means that the convergence criterion is met.

Furthermore, it should be clariﬁed that so far we proved

that SDA algorithm converges to the MSE point, under the

assumption that fk(·),∀k∈Kis not empty. In principle

this assumption is not required and it can be easily relaxed

by adding for each user kone auxiliary (virtually maximum)

transmission power, pV

k, in its strategy space such that ∀p−k∈

A−k, pV

k∈fk(p−k)and ck(pV

k)=+∞. If SDA converges

to the strategy proﬁle p†= (pV

1, . . . , pV

|K|)then the game does

not possess any SE.

Below, the complexity of the algorithm is studied in the

case of the users are playing sequentially in a given order.

Let us concentrate on one user kin order to specify its CPU

time complexity excluding the time that other users take in

order to make their decisions as the proposed framework

is implemented and executed in a decentralized manner. In

every cycle of turns, someone should always increase its

power, or else the algorithm converged to popt. The worst

case is bounded by the case where the game would have

C=|A1|+· · · +|A|K||cycles of turns. So, in C − |Ak|

cycles, user kwill ﬁnd out, in constant time, that it is satisﬁed.

On the other hand, in |Ak|cycles of turns the user runs

the modiﬁed binary search to ﬁnd its current maximum, pM

k,

and then one binary search in Pk[] in order to ﬁnd out its

next strategy. Therefore, for all of the cycles it will need

O((C − |Ak|)+2· |Ak| · log2(|Ak|)). Thus, the total time

complexity is O((C − |Ak|) + |Ak| · log2(|Ak|)). Note, that

if each user has the same cardinality in its strategy space, N,

the total complexity will be O(|K| · N+N·log2(N)).

V. DYNAMIC SYSTEM CHANGES - ENHANCEMENTS

In this section, we discuss how the proposed framework

can efﬁciently handle possible system changes that commonly

arise in 5G networks, without having to re-initialize the SDA

algorithm, in terms of: a) Increase/Decrease of the QoS thresh-

olds, and b) Entrance/Departure of users from the system. We

prove that we can harness the knowledge from the algorithm’s

previous run, to speed up the next run and accordingly ﬁnd the

MSE of the new game, in an evolutionary manner. Note, that

to prove the following propositions we focus on the MESE

points, however the same holds true for the MSE due to

Proposition 9.

A. Study of MESE properties with system changes

In this section initially we study the properties (Proposition

11) of the MESE of the game, when the QoS requirements

of all the users become stricter. Subsequently, Propositions 12

and 13, argue how the SDA framework can capitalize on these

properties, in order to efﬁciently handle relevant changes in the

system. Then we demonstrate how the obtained observations

are used in order to treat different types of dynamic behaviors

of the users (Sections V.B and V.C).

Proposition 11: Let two games be ˆ

G1=

(K, {Ak}k∈K,{f1

k}k∈K)and ˆ

G2= (K, {Ak}k∈K,{f2

k}k∈K)

with f1

k(p−k)⊇f2

k(p−k),∀k∈K, ∀p−k∈A−k. Then, for

the MESEs of ˆ

G1,ˆ

G2,p†(1),p†(2) it holds that p†(1) p†(2) .

Proof: First, note that it cannot be p†(1) p†(2) as in

this situation p†(1) would not be the MESE of ˆ

G1. Let now

a set of users J∈K, J 6=∅:∀j∈J, p†(1)

j> p†(2)

j. Let

also pbe a strategy proﬁle that for each user k, it picks the

lower transmission power among p†(1)

kand p†(2)

j. Speciﬁcally

pk=p†(1)

k∀k∈K\Jand pj=p†(2)

j∀j∈J. Summarizing

the above discussion we have:

p≺p†(1) and pp†(2).(16)

Let us now focus on the strategy proﬁle pin game ˆ

G1:

-∀k∈K\J, pk∈f1

k(p−k)as they were satisﬁed while the

others, i.e., j∈J, had played p†(1)

j> p†(2)

j=pjin p†(1). So,

the interference was decreased for them.

-∀j∈J, pj=p†(2)

j∈f2

j(p†(2)

−j)so from our assumptions

∀j∈J, pj=p†(2)

j∈f1

j(p†(2)

−j)and because of Eq. 16, it holds

true that ∀j∈J, pj=p†(2)

j∈f1

j(p−j).

Thus, we have: ∀k∈K, pk∈f1

k(p−k)which means

that pis an SE for the game ˆ

G1. Thus, from proposition

5, we have an ESE p∗:∀k∈K, p∗

k≤pk. The afore-

mentioned fact combined with Eq.16 gives: Pk∈Kp∗

k≤

Pk∈Kpk<Pk∈Kp†(1)

k, which gives: Pk∈Kck(p∗

k)≤

Pk∈Kck(pk)<Pk∈Kck(p†(1)

k), which is a contradiction,

so J=∅.

Proposition 12: Let the MESE of the game be p†and a

strategy proﬁle p:pk≤p†

k,∀k∈K. If SDA algorithm is

initiated with p, it will also converge to p†.

Proof: If each user k∈Kexcludes the powers pd

k:

pd

k< pkand executes the SDA algorithm, it will converge to

the MESE p†.

Proposition 13: Let the MESE of the game be p†and a

strategy proﬁle p:pk≥p†

k∀k∈K. If each user initiates the

SDA algorithm with pmax

k=pk, it will also converge to p†.

Proof: Once again, if each user k∈Kexcludes the

powers pd

k:pd

k> pkand executes the SDA algorithm it will

converge to the MESE p†.

In both cases, although the functions {fk}k∈Kmay change,

we can exploit the information of the previous convergence

point of the SDA algorithm to make the next run more

efﬁcient. In the ﬁrst case, i.e. stricter QoS prerequisites, we

run the SDA algorithm from the point that it stopped, while

PROMPONAS et al.: RETHINKING POWER CONTROL IN WIRELESS NETWORKS: THE PERSPECTIVE OF SATISFACTION EQUILIBRIUM 7

in the latter we exclude the powers that are greater from the

previous convergence point.

B. Change in the utility thresholds

Proposition 14: In an uplink power control game ˆ

G=

(K, {Ak}k∈K,{fk}k∈K)in case a user j∈Kincreases

its threshold utrh

jto utrh(2)

j, the system can be modeled by

a game ˆ

G2= (K, {Ak}k∈K,{f2

k}k∈K)with fk(p−k)⊇

f2

k(p−k)∀p−k∈A−k.

Proof: For the users k∈K:k6=j, fk(p−k) =

f2

k(p−k)as their QoS requirements remained the same. For

user j,fj(p−j)⊇f2

j(p−j)∀p−j∈A−j. That holds true as

∀pj∈Aj,∀p−j∈A−j, we have: uj(pj,p−j)≥uthr(2)

j⇒

uj(pj,p−j)≥uthr

jand uj(pj,p−j)< uthr

j⇒uj(pj,p−j)<

uthr(2)

j. Thus: fk(p−k)⊇f2

k(p−k)∀p−k∈A−k.

Inversely and with similar arguments if a user j∈

Kdecreases its threshold utrh

jto utrh(2)

j, the system can

be modeled by an uplink power control game ˆ

G2=

(K, {Ak}k∈K,{f2

k}k∈K),fk(p−k)⊆f2

k(p−k)∀p−k∈A−k.

C. Entrance and departure of users

In this section, we treat a possible change in the set Kof

the users in the system (i.e. entrance or departure).

Proposition 15: Let an uplink power control game ˆ

G1=

(K, {Ak}k∈K,{fk}k∈K)with MESE p†(1). Let also the

same game with an extra user j:ˆ

G2= (K+

{j},{Ak}k∈K+{Aj},{fk}k∈K+{fj})with MESE p†(2).

Then (p†(1), pmin

j)p†(2).

Proof: Let us consider the game ˆ

G0= (K+

{j},{Ak}k∈K+{A0

j},{fk}k∈K+{f0

j})with MESE p†(0).

Let also A0

j=Aj+{Ø}and f0

j(p−j) = fj(p−j) +

{Ø},∀p−j∈A−j. Note that we added a virtual power to

user’s jstrategy space that corresponds to zero transmission

and we allowed user jto be satisﬁed by not transmitting at all.

Let us also consider the game ˆ

G00 = (K+{j},{Ak}k∈K+

{A0

j},{fk}k∈K+{fj})with MESE p†(00). Note that this

game has the same strategy space for user jbut it is not

satisﬁed by not transmitting at all. By deﬁnition, f0

j(p−j)⊇

fj(p−j),∀p−j∈A−j. Thus, because of proposition 11 we

have that p†(0)p†(00). Given that in ˆ

G00 , the {Ø}power will

be useless to user jin every satisfaction equilibrium it also

means that: p†(2) =p†(00)Therefore we conclude that:

p†(2) =p†(00)p†(0)(17)

On the other hand, in game ˆ

G0, we know for sure that at the

MESE p†(0), user jwill transmit with {Ø}power thus adding

zero interference to the other users. That would mean that:

p†(1)

k=p†(0)

k,∀k∈K, k 6=j(18)

Finally because of Eq. 17 and Eq. 18 we get that: ∀k∈K, k 6=

j, p†(2)

k≥p†(1)

kand p†(2)

j≥pmin

j.

From proposition 12, we can conclude that when a user

enters the system, we could execute SDA algorithm starting

from the previous MESE for the existing users and from the

minimum power for the entering user. Similarly, for the case

of a user departure the following proposition holds true (the

proof follows similar steps with the case of a user entering the

system and is omitted due to space limitation).

Proposition 16: Let an uplink power control game ˆ

G1=

(K, {Ak}k∈K,{fk}k∈K)with MESE p†(1). Let also the

same game without the user jbe ˆ

G2= (K\

{j},{Ak}k∈K\ {Aj},{fk}k∈K\ {fj})with MESE p†(2).

Then (p†(1)

1, . . . , p†(1)

j−1, p†(1)

j+1, . . . , p†(1)

|K|)p†(2).

In a nutshell, Fig. 1 below provides a ﬂow diagram of the

operations of the aforementioned holistic framework. Speciﬁ-

cally, based on Proposition 14-16 the required arguments for

the efﬁcient operation of the SDA algorithm at a given instance

- that is the starting points and corresponding users’ maximum

powers - are deﬁned. Note also that the evolutionary operation

of the framework enables the users to harness the resources

of the network as much as possible, ensuring on one hand

the satisfaction of the QoS requirements of the existing users,

while on the other hand allowing the dynamic increase in the

system capacity in terms of satisﬁed users, if this is feasible.

D. Discussion and application in 5G systems

The proposed holistic framework offers and supports the

realization of user-centric operating models, as the ones

emerging in 5G wireless systems. Such approach, owing to

its decentralized nature, is a promising alternative to network-

centric solutions that are more complex while also bearing

signiﬁcantly higher overhead and signaling for implementa-

tion purposes. The adoption and realization of our proposed

satisfaction equilibrium-oriented game theoretic power control

framework, supports the proliferation of 5G networks, due to

its ﬂexibility, dynamicity and adaptability.

There exist several key characteristics of the emerging

wireless communication environment that call for the use of

approaches like the ones proposed in our framework based on

the adoption of satisfaction equilibrium (increasing the system

capacity in terms of satisﬁed users) and game theory for

decentralized operation. Indicatively, we outline the following:

(i) the densiﬁcation of the wireless communication systems

with heterogeneous types of cells [23], (ii) the increasing

number of nodes in 5G networks along with the variety of

the communication types [24], (iii) the heterogeneity of the

available communication and multiple access techniques in 5G

networks [25], [26], and (iv) the need of supporting a large

number of devices with diverse and dynamically changing

QoS requirements and behavior [23]. Indeed, Game Theory

is largely considered as a building block of the artiﬁcial intel-

ligent solutions envisioned in next generation communications

and computing systems [27].

The observations drawn form the analysis and discussion

in previous sections, supports the claim that an approach

adopting the principles introduced in our framework, arises

as a powerful tool providing intelligence to the end-user to

make optimal decisions about itself, considering the available

feedback from the heterogeneous communications environ-

ment. The latter is well aligned with the new advances in the

intelligence and processing capability of the next generation

end-user smart devices. Additionally, the security and privacy

concerns can be explicitly and/or implicitly mitigated, given

that control information is not exchanged among the end-users

8 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. XX, NO. XX, XXXX 2020

Initialization

p=SDA((pmin

1, . . . , pmin

|K|),(pmax

1, . . . , pmax

|K|))

InE quilibrium(p)

Abort & Notify p=ptemp

Is ptemp Equilibrium?

ptemp =SDA((p,pmin

j),(pmax

1, . . . , pmax

|K|, pmax

j))

ptemp =SDA((pmin

1, . . . , pmin

j−1, pmin

j+1 , . . . , pmin

|K|),(p1, . . . , pj−1, pj+1,...p|K|))

ptemp =SDA(p,(pmax

1, . . . , pmax

|K|))

ptemp =SDA((pmin

1, . . . , pmin

|K|),p)

Equilibrium Interrupt:

jIncreases Threshold

Equilibrium Interrupt:

jDecreases Threshold

Equilibrium Interrupt:

Departure of User j

Equilibrium Interrupt:

Entrance of User j

No

Yes

Figure 1 A ﬂow diagram of the holistic framework

and a central entity in the case of the game-theoretic power

control, making the users less susceptible to intrusions. Last,

but not least, it has been concluded that the proposed approach

can effectively embed, as needed, the opportunistic behavior

and rationale to the end-users, while it can efﬁciently handle

dynamic system and user requirement changes, events that

often occur in next generation wireless networks. The latter, in

our framework is realized in an incremental and evolutionary

manner thus facilitating the real time processing required in

5G systems. Along these lines, indicative numerical results

presented later in Section VI, show that the proposed SDA

algorithm converges very fast (in approximately 10msec) to

the MSE point for the overall examined system, which is well

aligned with the requirements in 5G systems [28].

VI. NUMERICAL RE SULTS

In this section, we provide indicative numerical results to

evaluate the performance of the SDA algorithm and illustrate

the key beneﬁts of the MSE point as well as the operation

of the framework as a whole. In particular, the focus is

placed on the evaluation of the validity and superiority of the

introduced theoretical framework by comparing the MSE with

other existing equilibria (SE, ESE, NE) (Sections VI.A and

VI.B), and on the study of the behavior and convergence of the

proposed novel holistic framework, under different scenarios

(Section VI.C). Finally, in Section VI.D a comparative study

demonstrating its beneﬁts against approaches targeting directly

utility maximization outcomes is provided. The user distance

dk,∀k∈Kfrom the base station is randomly and uniformly

distributed within the range of 20 to 150 m. The gain gk

of each user kis inversely proportional to the square of its

distance dk, i.e., gk=1

d2

k

. Each user is assumed to have 150

discrete achievable power levels, randomly chosen within the

interval of [0.1,1.7] Watts. The utility function of each user,

unless otherwise explicitly stated, follows the form of Eq. 5.

Finally, for demonstration only purposes and without loss of

generality, we have assumed R= 64Kbps and W= 106H z.

A. Satisfaction Equilibria and Convergence of SDA

Fig. 2 presents the set of all possible Satisfaction Equilibria

as well as the convergence of the SDA algorithm and the

unique NE of a 2-user (Fig. 2a) or a 3-user game (Fig.

2b). Speciﬁcally, the colored region represents all the strategy

proﬁles that are SEs and each point’s color depends on the

cumulative transmission power of the users, where the light

and dark color represent high and low summation, respectively.

It is noted that the SDA algorithm monotonically converges to

the unique MSE, which is also the SE that charges each user

with the lowest power.

In this experiment, the value of uthr

kfor user kis set to be

the utility it scores if all of the users are transmitting with the

powers that the NE point indicates. Thereby, all the strategy

proﬁles in the colored region represent points where the users

enjoy the same or greater energy efﬁciency compared to the

unique NE point. That said, at any middle point, whereas users

are transmitting with lower power than in the NE, they also

enjoy greater or equal energy efﬁciency. Thus, the framework

of satisfaction games and speciﬁcally the MSE point along

with the SDA algorithm propose more power efﬁcient strategy

proﬁles. It is also observed that the unique NE - commonly

adopted in literature - leads to ultimately arbitrary solutions.

Although there are seemingly plenty of strategy proﬁles with

the same or greater energy efﬁciency, which are basically a

combination of transmission power and the achieved channel

capacity, the NE is utterly arbitrary and depends merely on the

conﬁguration of the network, thus, ignoring the user needs.

B. Comparison of the NE with the MSE

The previous result arises the question of whether there

are strategy proﬁles that the system can converge to, where

all the users achieve strictly greater energy efﬁciency score

than the one of the NE. Indeed, Fig. 3 presents what happens

if all of the users gradually increase their QoS prerequisites

from the NE outcome. Surprisingly, in the 3-user game (Fig.

PROMPONAS et al.: RETHINKING POWER CONTROL IN WIRELESS NETWORKS: THE PERSPECTIVE OF SATISFACTION EQUILIBRIUM 9

Transmission Power

0.1 0.2 0.3 0.4

Transmission Power

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Nash Equilibrium

SDA

Cumulative Cost

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(a) 2-user game

Transmission Power

1

0.5

0

1

Transmission Power

0.8

0.6

0.4

0.2

0

0.5

1

1.5

2

0

Transmission Power

Nash Equilibrium

SDA

Cumulative Cost

0.5

1

1.5

2

2.5

3

(b) 3-user game

Figure 2 Satisfaction Equilibria and Convergence of the

SDA algorithm in a 2 and 3-user power control game

3a) the three users can augment their QoS prerequisites and

simultaneously achieve, when in the MSE, 860% of their

achieved energy efﬁciency score of the NE with just 10.5% of

the cumulative transmission power for the system. Similarly,

when considering 20 users (Fig. 3b), it was rather possible to

achieve 220% of the energy efﬁciency score of the NE with

just 10.8% of the cumulative power.

Thus, directly maximizing the energy efﬁciency utility turns

to be not a good incentive for a user, as by stating its

prerequisites, strategy proﬁles with greater energy efﬁciency

score can be obtained. With the framework of satisfaction

games, we could alternatively conclude to energy beneﬁcial

solutions for the system, by simply targeting channel capacity

instead of energy efﬁciency. For instance, assuming that the

users’ utility is the Shannon capacity, then the users can

achieve their quality prerequisites, but with the minimum

power consumption. That is, if the system converges to the

MSE of the game, there would not exist any other strategy

proﬁle where everyone meets his threshold, while someone

transmits with lower power.

Percentage Increase of the NE thresholds

0 200 400 600 800

Power Consumption

Percentage

7.5

8

8.5

9

9.5

10

10.5

(a) 3-user game

Percentage Increase of the NE thresholds

0 20 40 60 80 100 120

Power Consumption

Percentage

9

9.5

10

10.5

11

(b) 20-user game

Figure 3 Satisfaction Equilibria that lead to strictly greater

energy efﬁciency score with lower power consumption

C. Holistic framework dynamic operation

Following the above argument and in order to show the

holistic nature of our framework, below we adopt the Shannon

capacity as the considered user utility function. To demonstrate

the efﬁcient dynamic operation of the proposed framework,

we assume six different events (system stages) taking place

sequentially, as follows: a) The system starts with 200 users;

b) 10 users enter; c) 1user enters; d) 3users double their QoS

prerequisites; e) 3users set their thresholds to 0.8times the

previous one; and f) 21 users depart.

Fig. 4a represents the transmission powers of three different

users in the MSEs of the system after each of the six different

events took place. In particular, the user with id 1doubled its

threshold (event 4), the one with id 2decreased its threshold

(event 5), while the user with id 3has stayed in the system after

the event 6without changing its threshold. As it is expected,

after the entrance of the 10 users (event 2), the users had to

increase their transmission power to achieve the same QoS

levels. After the 3users increased their thresholds (event 4),

we observe that all users had to increase their powers as well,

in the new MSE. While the user with id 1was one of the users

that had to achieve greater QoS prerequisites, the others had to

also increase their powers to achieve their previous thresholds

due to the increased interference in the system. Similarly, when

the 3 users decreased their thresholds (event 5), they decreased

their transmission powers to be in the MSE. Finally, when the

network was left with 190 users (event 6), they all were able to

decrease their transmission powers to meet their prerequisites.

Fig. 4b presents the time required for convergence during

the occurrence of those events (horizontal axis), under the

scenario (static) where the users had to completely re-run

the SDA algorithm (orange line) and the scenario (dynamic)

where the dynamic proposed holistic framework (blue line)

was applied. It is noted that using the dynamic framework

signiﬁcant execution time savings are obtained, particularly

for minor changes in the system. From the obtained results

we notice that the convergence time of our proposed frame-

work is approximately in the range of 10 msecs, which is

within the requirements of 5G for real time communications.

Furthermore, it should be noted that in a realistic 5G network,

the channel gain conditions do not change that fast and often,

i.e., in the order of magnitude of msec, thus in a realistic

implementation the outcome of the proposed algorithm can be

used for a consecutive number of time slots, reducing further

the corresponding overhead.

Event ID

123456

Transmission Power

0.1

0.2

0.3

0.4

0.5

0.6

0.7

User 1

User 2

User 3

(a) Transmission powers of 3

users at the MSE point.

Event ID

123456

Time (microseconds)

103

104

105

Dynamic

Static

(b) Convergence to MSE under

static and dynamic operation

Figure 4 SDA Static & Dynamic Operation

D. Comparative Results of Different Strategy Proﬁles

In this section, we compare the MSE point with the corre-

sponding NE points achieved when either Energy Efﬁciency

Maximization or Shannon Maximization is targeted. The latter

is selected for comparison and benchmarking purposes, as the

Shannon capacity has been commonly and widely used in the

relevant literature to capture the users’ achievable data rate

[17], [21]. In this scenario, six users are considered that are

located at decreasing distances from the base station, with

users with lower ID having the highest distances from the

base station, thus worse channel conditions. In particular, Fig.

5a suggests that for the ﬁrst 3 users (the 3 users that are the

farthest from the base station) the energy efﬁciency maximiza-

tion approach, achieves as expected higher scores in the energy

efﬁciency metric. It is noted here that the presented energy

10 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. XX, NO. XX, XXXX 2020

efﬁciency is measured in [bits/Joule], and it is calculated per

one unit of available bandwidth measured in [Hz]. However, as

shown in Fig. 5b this happens at the cost that each of the three

users transmits with a very high transmission power compared

to the MSE, hence gaining higher bit rate than required

from their QoS prerequisites (Fig. 5c) while, dramatically

increasing the interference in the system. The latter wasteful

energy consumption and corresponding negative impact, is

observed by the power allocations of the last 2users, where

although they transmit with higher powers under the energy

efﬁciency maximization strategy proﬁle - than the respective

ones in the MSE - they ultimately are assigned lower Shannon

capacity scores than their requirements (QoS thresholds). On

the other hand, the MSE strategy proﬁle converges to quite low

transmission powers, while assigning to each user transmission

rate close to its threshold (green line in Fig. 5c), therefore

satisfying each user’s requirement.

Users ID

123456

Energy Efficiency Score

10-60

10-40

10-20

100

1020

SDA

Energy Effic. Maxim.

Shannon Maximization

(a) Energy Efﬁciency Utility

User ID

123456

Power Allocation

0

0.2

0.4

0.6

0.8

1

SDA

Energy Maximization

Shannon Maximization

(b) Power Allocation

User ID

123456

Shannon Score

0

1

2

3

4SDA

Energy Efficiency Maxim.

Shannon Maximization

Thresholds

(c) Shannon Capacity

Figure 5 Comparison of strategies for SDA, Energy-

Efﬁciency Maximization & Shannon Maximization

VII. CONCLUSION

In this paper we adopted the concept of games in satisfaction

form in order to treat the uplink power control problem

in wireless networks. First, we deﬁned different types of

equilibrium points (SE, ESE, MESE, MSE) that are of special

interest within this framework, while highlighting the beneﬁts,

existence and uniqueness of the MSE equilibrium point. In

particular, we proved that in this strategy proﬁle the users

of the network meet their QoS prerequisites, while being

penalized with the lowest possible cost/power. Underlining

the need of the system to transmit in this speciﬁc point, we

proposed a decentralized and low complexity algorithm (SDA)

that is shown to converge to this point. Capitalizing on the key

properties of the MSE operation point and the SDA algorithm,

a holistic framework was proposed to efﬁciently deal with the

dynamic behavior of the users in the network. Finally, detailed

numerical results were presented to reveal the properties

and superiority of the MSE equilibrium, especially compared

to other equilibrium points that have been proposed in the

literature with respect to the resource allocation problems in

wireless networks.

APPENDIX I

PROOF O F PROPOSITION 1

Let p+∈S:R(p+)6=∅and a user k∈R(p+). Then,

the strategy proﬁle p= (p+

1, p+

2, . . . , pM

k(p+

−k), . . . , p+

|K|)will

be an SE of the game as user kreceived greater utility than

in p+while it lowered its power, something that proves that

the others will still be satisﬁed. Repeating this process for

every user in R(p)will eliminate this set (each user in R(p+)

decreases its power) and conclude to the strategy proﬁle e+.

APPENDIX II

PROOF O F PROPOSITION 2

Let us assume that for a user k,BRk(p−k) = p,p∈

(pM

k, pmax

k], when the others have played p−k. Due to the fact

that pis a best response it should hold true that uk(p, p−k)≥

uthr

kand uk(pM

k,p−k)< uthr

k. However this cannot hold true

as by deﬁnition uk(pM

k)is the maximum possible value for a

ﬁxed p−k.

APPENDIX III

PROOF O F PROPOSITION 3

Let p1be a random strategy proﬁle, and p=BRk(p1

−k)

be user k’s best response. Let p2

−kbe a strategy proﬁle that

is acquired when a set of users in K− {k}increase their

powers from p1

−k. That is p2

−k>p1

−k, thus, Pj6=khjp2

j≥

Pj6=khjp1

j. Given that the user’s kutility function is decreas-

ing with respect to the interference for a ﬁxed transmitting

power of k, we have that: uk(p, p1

−k)> uk(p, p2

−k). Given

that pis the BR(p1

−k), then based on proposition 2, we have

that p∈[pmin

k, pM

k], when the others are playing p1

−k. We

can distinguish the following two cases.

I) uk(p, p1

−k)> uk(p, p2

−k)≥uthr

k: In that case, p∈

fk(p2

−k)as uk(p, p2

−k)≥uthr

k. Moreover, p=BR(p2

−k)as

it was the best response in p1

−kwhere the others had lower or

equal transmission powers. Thus, BRk(p1

−k) = BRk(p2

−k).

Again, because of proposition 2, p∈[pmin

k, pM

k]when the

others have played p2

−k.

II) uk(p, p1

−k)≥utrh

k> uk(p, p2

−k): Because of the fact

that p∈[pmin

k, pM

k]when the others are playing p1

−kand is

equal to BRk(p1

−k)there cannot be any satisfying power that

is less than p. That will also hold true when the others change

strategies to p2

−kbecause for sure the corresponding utilities

will be further decreased. If p∈(pM

k, pmax

k]when the others

have played p2

−k, then user kdoes not have a best response

nor a satisfying power when the others have played p2

−kor

a strategy proﬁle that is component wise greater. In the other

case, where p∈[pmin

k, pM

k], when the others have played p2

−k,

again if there is no satisfying power there will also not exist

any satisfying power for strategy proﬁles that are component

wise greater than p2

−k. Nevertheless, if there exists a satisfying

power that is also the best response we have proven that it

will be in the interval [pmin

k, pM

k]and consequently it will be

greater than p. Because the above cases represent the only

two possible orderings of those quantities, we have proven

that p=BRk(p1

−k)≤BRk(p2

−k).

PROMPONAS et al.: RETHINKING POWER CONTROL IN WIRELESS NETWORKS: THE PERSPECTIVE OF SATISFACTION EQUILIBRIUM 11

APPENDIX IV

PROOF O F PROPOSITION 4

The proof comes from the Theorem 1. As mentioned,

Lis a complete lattice. Thus, ∀p,p

0∈A:pp

0

it holds true that: (BR1(p−1), . . . , B R|K|(p−|K|))

(BR1(p

0

−1), . . . , BR|K|(p

0

−|K|)), or equivalently

g(p)g(p

0), based on proposition 3. Therefore, we

also proved that gis an order-preserving function. Following

the previous analysis, Tarski-Kraskel’s theorem ensures the

existence of a ﬁxed point of function g. That is, ∃p∈A:p=

g(p)⇔(p1, . . . , p|K|)=(BR1(p−1), . . . , B R|K|(p−|K|).

That would mean that for the strategy proﬁle p, every user

has played its best response strategy given the strategies of

the rest of the users. So, pis an ESE for the game ˆ

G.

APPENDIX V

PROOF O F PROPOSITION 5

For the proof, we exclude the powers pd:pd> p+

k,∀k∈K,

as they do not conclude to an ESE. Thus, the modiﬁed strategy

space is denoted by A0

k, and the corresponding game is ˆ

G0.

In the game ˆ

G0, we know that the strategy p+

kwill satisfy the

user k,∀k∈K, regardless the strategies of the rest of the

users as the interference can only be decreased. By applying

the proposition 4, we prove the existence of an action proﬁle

p∗that is an ESE for ˆ

G0, i.e., ∀k∈K, ∀pk∈A0

k:pk∈

fk(p∗

−k), ck(pk)≥ck(p∗

k). Because by default p+

kis the

maximum transmission power of the set A0

kof the kth user in

ˆ

G0, it means that p+

k≥p∗

kand consequently ck(p+

k)≥ck(p∗

k).

So, because the excluded powers (i.e., pd> p+

k) were greater

than p+

k, we can conclude to the following statement regarding

the initial game ˆ

G:

∀k∈K, ∀p∈Ak:p∈fk(p∗

−k), ck(p)≥ck(p∗

k)

Due to the above statement and given that p∗is an SE in ˆ

G,

we conclude that p∗is also an ESE in ˆ

G. Thus we have also

proven that Pk∈Kck(p+

k)≥Pk∈Kck(p∗

k).

APPENDIX VI

PROOF O F PROPOSITION 10

From proposition 5, because of the existence of an SE we also

have the existence of at least one ESE. Therefore, the MESE

p†also exists. Because of the fact that there exists a strategy

proﬁle that is ESE (and the MESE in that case), in every

strategy proﬁle that is component wise less than p†, each

user will have a satisfying power. The starting strategy proﬁle

of SDA is the pstart = (pmin

1,...pmin

|K|). As proved above, if

each user kdoes not exceed p†

k(and no one else does also),

it will always possess a BRkthat is increasing with respect

to the powers of the others (Proposition 3). Thereby, in each

turn a user either keeps its transmission power (if satisﬁed)

or increases it by playing its new BR. Let us assume that

user jwas the ﬁrst one that exceeded its p†

jduring one of

its turns. Let us also denote the strategy proﬁle of the SDA

before jexceeded its p†

jas p. That would mean that for every

user k∈K,pk≤p†

k, thus, p−jp†

−j. Therefore, from

proposition 3 we have that BRj(p−j)≤BRj(p†

−j) = p†

j.

Given that in every turn each user responds with its BR, we

note that as long as each user kis below its p†

k, user jwill

not exceed its p†

j(contradiction). Because of that, no one will

exceed its p†

kwhen the dynamics start from pstart. Given

that everyone increases its power by playing their BR when

they are not satisﬁed and they do not exceed p†

j, SDA will

converge at the MESE p†which is also the MSE popt.

REFERENCES

[1] H. Elhammouti, E. Sabir, M. Benjillali, L. Echabbi, and H. Tembine,

“Self-organized connected objects: Rethinking qos provisioning for iot

services,” IEEE Comm. Mag., vol. 55, no. 9, pp. 41–47, 2017.

[2] A. Zappone, L. Sanguinetti, G. Bacci, E. Jorswieck, and M. Debbah,

“Energy-efﬁcient power control: A look at 5g wireless technologies,”

IEEE Trans. on Signal Processing, vol. 64, no. 7, pp. 1668–1683, 2015.

[3] E. E. Tsiropoulou, P. Vamvakas, and S. Papavassiliou, “Supermodular

game-based distributed joint uplink power and rate allocation in two-tier

femtocell networks,” IEEE Transactions on Mobile Computing, vol. 16,

no. 9, pp. 2656–2667, 2017.

[4] H. Iiduka, “Distributed optimization for network resource allocation with

nonsmooth utility functions,” IEEE Transactions on Control of Network

Systems, vol. 6, no. 4, pp. 1354–1365, 2018.

[5] A. Sinha and A. Anastasopoulos, “Mechanism design for resource allo-

cation in networks with intergroup competition and intragroup sharing,”

IEEE Trans. on Control of Net. Syst., vol. 5, no. 3, pp. 1098–1109, 2017.

[6] Y. Zhang and M. Guizani, Game theory for wireless communications

and networking. CRC press, 2011.

[7] T. Alpcan, T. Bas¸ar, R. Srikant, and E. Altman, “Cdma uplink power

control as a noncooperative game,” Wireless Networks, vol. 8, no. 6, pp.

659–670, 2002.

[8] T. Lin, T. Alpcan, and K. Hinton, “A game-theoretic analysis of energy

efﬁciency and performance for cloud computing in communication

networks,” IEEE Systems Journal, vol. 11, no. 2, pp. 649–660, 2015.

[9] J. R. Marden and J. S. Shamma, “Game theory and distributed control,”

in Handbook of game theory with economic applications. Elsevier,

2015, vol. 4, pp. 861–899.

[10] P. Xu, X. Fang, M. Chen, and Y. Xu, “A Stackelberg game-based

spectrum allocation scheme in macro/femtocell hierarchical networks,”

Computer Commun., vol. 36, no. 14, pp. 1552–1558, 2013.

[11] A. H. Salati, M. Nasiri-Kenari, and P. Sadeghi, “Distributed subband,

rate and power allocation in OFDMA based two-tier femtocell networks

using Fractional Frequency Reuse,” in IEEE Wireless Communications

and Networking Conf., 2012, pp. 2626–2630.

[12] J. Zhang, Z. Zhang, K. Wu, and A. Huang, “Optimal distributed

subchannel, rate and power allocation algorithm in OFDM-based two-

tier femtocell networks,” in IEEE Vehicular Technology Conference

(VTC 2010-Spring), 2010, pp. 1–5.

[13] E. E. Tsiropoulou, G. K. Katsinis, and S. Papavassiliou, “Distributed

uplink power control in multiservice wireless networks via a game

theoretic approach with convex pricing,” IEEE Trans. on Parallel and

Distr. Systems, vol. 23, no. 1, pp. 61–68, 2012.

[14] M. Fasoulakis, E. E. Tsiropoulou, and S. Papavassiliou, “Satisfy instead

of maximize: Improving operation efﬁciency in wireless communication

networks,” Comp. Net., vol. 159, pp. 135–146, 2019.

[15] S. M. Perlaza, H. Tembine, S. Lasaulce, and M. Debbah, “Quality-of-

service provisioning in decentralized networks: A satisfaction equilib-

rium approach,” IEEE Journal of Selected Topics in Signal Processing,

vol. 6, no. 2, pp. 104–116, 2012.

[16] M. Goonewardena, S. M. Perlaza, A. Yadav, and W. Ajib, “Generalized

satisfaction equilibrium: A model for service-level provisioning in

networks,” in 22th Eur. Wireless Conf., 2016, pp. 1–5.

[17] F. M´

eriaux, S. Perlaza, S. Lasaulce, Z. Han, and V. Poor, “Achievability

of efﬁcient satisfaction equilibria in self-conﬁguring networks,” in Int.

Conf. on Game Th. for Net. Springer, 2012, pp. 1–15.

[18] L. Rose, S. Lasaulce, S. M. Perlaza, and M. Debbah, “Learning

equilibria with partial information in decentralized wireless networks,”

IEEE Commun. Magazine, vol. 49, no. 8, pp. 136–142, 2011.

[19] H. P. Borowski, J. R. Marden, and J. S. Shamma, “Learning to play

efﬁcient coarse correlated equilibria,” Dynamic Games and Applications,

vol. 9, no. 1, pp. 24–46, 2019.

[20] P. Promponas, P. A. Apostolopoulos, E. E. Tsiropoulou, and S. Pa-

pavassiliou, “Redesigning resource management in wireless networks

based on games in satisfaction form,” in 12th IFIP Wireless and Mobile

Networking Conf. (WMNC). IEEE, 2019, pp. 24–31.

12 IEEE TRANSACTIONS ON CONTROL OF NETWORK SYSTEMS, VOL. XX, NO. XX, XXXX 2020

[21] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, “Efﬁcient power

control via pricing in wireless data networks,” IEEE transactions on

Communications, vol. 50, no. 2, pp. 291–303, 2002.

[22] B. Knaster and A. Tarski, “Un theoreme sur les fonctions d’ensembles,”

Ann. Soc. Polon. Math., vol. 6, pp. 133–134, 1928.

[23] I. Ahmad, Z. Kaleem, R. Narmeen, L. D. Nguyen, and D.-B. Ha,

“Quality-of-service aware game theory-based uplink power control for

5g heterogeneous networks,” Mobile Networks and Applications, vol. 24,

no. 2, pp. 556–563, 2019.

[24] C. Yang, J. Li, M. Sheng, A. Anpalagan, and J. Xiao, “Mean ﬁeld

game-theoretic framework for interference and energy-aware control in

5g ultra-dense networks,” IEEE Wir. Com., vol. 25, no. 1, pp. 114–121,

2017.

[25] H. Munir, S. A. Hassan, H. Pervaiz, Q. Ni, and L. Musavian, “Energy

efﬁcient resource allocation in 5g hybrid heterogeneous networks: A

game theoretic approach,” in 2016 IEEE 84th VTC-Fall, 2016, pp. 1–5.

[26] A. Benamor, O. Habachi, I. Kammoun, and J. Cances, “Game theoretical

framework for joint channel selection and power control in hybrid

noma,” in ICC 2020, 2020, pp. 1–6.

[27] R. Li, Z. Zhao, X. Zhou, G. Ding, Y. Chen, Z. Wang, and H. Zhang,

“Intelligent 5g: When cellular networks meet artiﬁcial intelligence,”

IEEE Wireless Communications, vol. 24, no. 5, pp. 175–183, 2017.

[28] I. Parvez, A. Rahmati, I. Guvenc, A. I. Sarwat, and H. Dai, “A survey

on low latency towards 5g: Ran, core network and caching solutions,”

IEEE Comm. Surv. & Tut., vol. 20, no. 4, pp. 3098–3130, 2018.

Panagiotis Promponas received his Diploma

in Electrical and Computer Engineering (ECE)

from the National Technical University of Athens

(NTUA), Greece, in 2019. His main scientiﬁc

interests lie in the area of resource allocation

in constrained interdependent systems and op-

timization of algorithms. Currently, he is a re-

search associate in the Network Management

and Optimal Design Laboratory at NTUA. He is

also the recipient (co-author) of the Best Paper

Award at the 12th IFIP WMNC 2019.

Eirini Eleni Tsiropoulou is currently an As-

sistant Professor at the Department of Elec-

trical and Computer Engineering, University of

New Mexico. Her main research interests lie in

the area of cyber-physical social systems and

wireless heterogeneous networks, with empha-

sis on network modeling and optimization, re-

source orchestration in interdependent systems,

reinforcement learning, game theory, network

economics, and Internet of Things. Five of her

papers received the Best Paper Award at IEEE

WCNC in 2012, ADHOCNETS in 2015, IEEE IFIP WMNC 2019, IN-

FOCOM 2019 by the IEEE Communications Systems Integration and

Modeling Technical Committee, and IEEE/ACM BRAINS 2020. She was

selected by the IEEE Communication Society - N2Women - as one of

the top ten Rising Stars of 2017 in the communications and networking

ﬁeld. She has received the Early Career Award by the IEEE ComSoc

Internet Technical Committee 2019.

Symeon Papavassiliou is currently a Profes-

sor in the School of ECE at National Technical

University of Athens. From 1995 to 1999, he

was a senior technical staff member at AT&T

Laboratories, New Jersey. In August 1999 he

joined the ECE Department at the New Jersey

Institute of Technology, USA, where he was an

Associate Professor until 2004. He has an es-

tablished record of publications in his ﬁeld of ex-

pertise, with more than 300 technical journal and

conference published papers. His main research

interests lie in the area of computer communication networks, with

emphasis on the analysis, optimization, and performance evaluation

of mobile and distributed systems, wireless networks, and complex

systems. He received the Best Paper Award in IEEE INFOCOM 94,

the AT&T Division Recognition and Achievement Award in 1997, the

US National Science Foundation Career Award in 2003, the Best Paper

Award in IEEE WCNC 2012, the Excellence in Research Grant in

Greece in 2012 and the Best Paper Awards in ADHOCNETS 2015, ICT

2016 and IFIP WMNC 2019. He also served on the board of the Greek

National Regulatory Authority on Telecommunications and Posts from

2006 to 2009.