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1
Game Theoretic Approaches for Multiple Access in
Wireless Networks: A Survey
Khajonpong Akkarajitsakul∗, Ekram Hossain∗, Dusit Niyato†, and Dong In Kim+
∗Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada
†School of Computer Engineering, Nanyang Technological University (NTU), Singapore
+School of Information and Communication Engineering, Sungkyunkwan University (SKKU), Suwon, South
Korea
Abstract—Multiple access methods in a wireless network allow
multiple nodes to share a set of available channels for data
transmission. The nodes can either compete or cooperate with
each other to access the channel(s) so that either an individual
or a group objective can be achieved. Game theory, which is
a mathematical tool developed to understand the interaction
among rational entities, can be applied to model and to analyze
individual or group behaviour of nodes for multiple access in
wireless networks. Game theory also enables us to model the
selfish/malicious behaviour of nodes, and subsequently design the
punishment or defense mechanisms for robust multiple access
in wireless networks. In addition, game models can provide
distributed solutions to the multiple access problems, which
are based on solid theoretical foundations. In this survey, we
provide a comprehensive review of the game models (e.g., non-
cooperative/cooperative, static/dynamic, and complete/incomplete
information) developed for different multiple access schemes (i.e.,
contention-free and contention-based random channel access)
in wireless networks. We consider time-division multiple access
(TDMA), frequency-division multiple access (FDMA), and code-
division multiple access (CDMA), ALOHA, and carrier sense
multiple access (CSMA)-based wireless networks. In addition,
game models for multiple access in dynamic spectrum access-
based cognitive radio networks are reviewed. The major findings
from the game models used for these different access schemes
are highlighted. To this end, several of the key open research
directions are outlined.
Index Terms: Wireless networks, game theory, multiple
access, random access game, power and rate control game.
I. INTRODUCTION
Game theory is a branch of applied mathematics which is
concerned with how rational entities make decisions in a situ-
ation of conflict. It provides a rich set of mathematical tools to
model and analyze interactions among the rational entities, and
the rationality is based on gains or payoff perceived by these
entities. Game theory has been primarily used in Economics.
It has also been used in other disciplines such as Biology,
Political science, Engineering, and Philosophy. One of the
major areas in Engineering where game theory has been used
is data communication networking. In particular, it has been
used to model and analyze routing and resource allocation
problems in a competitive environment, and more recently to
This work was supported in part by the AUTO21 NCE research grant for the
project F303-FVT and in part by the MKE (Ministry of Knowledge Economy),
Korea, under the ITRC (Information Technology Research Center) support
program supervised by the NIPA (National IT Industry Promotion Agency)
(NIPA-2010-(C1090-1011-0005)).
security problems in wireless networks. Applicability of game
theory tools to analyze power control, waveform adaptation,
medium access, routing, and node participation was discussed
in [1] from a layered perspective.
In a multi-user wireless communication network, the trans-
mitting nodes share the limited radio resources (e.g., wireless
channels and transmission power). Therefore, one critical issue
is how the nodes share these resources to transmit data so that
the optimal network performance can be achieved. Multiple ac-
cess methods developed for wireless networks can be divided
into two main groups, namely, contention-free channel access
and contention-based random access methods. In a multiple
access scheme, nodes can either cooperate or compete to
achieve their objectives (e.g., optimal throughput and quality-
of-service (QoS)). Consequently, game theory has become a
very useful mathematical tool to model and analyze multiple
access schemes in wireless networks, and to obtain solutions
for resource allocation, channel assignment, power control,
and cooperation enforcement among the nodes. The notion
of multiple access game can be illustrated by the following
example [2], [3]. Suppose that there are two mobile nodes tx1
and tx2who want to access a shared wireless channel to send
information to the corresponding receivers rcvr1and rcvr2.
Both the receivers are within the transmission range of both
the transmitters. Each transmitter has one packet to transmit
in each time step and it can either choose to transmit during a
time step or wait. If tx1transmits, the packet is successfully
transmitted if tx2chooses not to transmit during that time
step (and hence there is no collision). For successful packet
transmission, tx1obtains a benefit at the cost of transmit
power. It is of interest to analyze the interactions between the
transmitters under different network settings and performance
objectives.
Different game models (e.g., noncooperative/cooperative,
static/dynamic, and complete/incomplete information games)
have been developed to study the behavior of transmitting
nodes to access the wireless channel(s) and obtain the multiple
access solution (or equilibrium) [2], [4]. Various game models
are considered under different scenarios, perspectives, or as-
sumptions on transmitting nodes’ behavior. Nevertheless, the
common aim of these models is to improve network perfor-
mance (e.g., throughput maximization, resource consumption
minimization, and QoS guarantee) given self-interest or group-
rationality of transmitting nodes.
2
The motivations of using game models for design, analysis,
and optimization of multiple access in wireless networks are
as follows:
•Theoretical foundation for multiple access schemes:
Game theory, which is most notably used in Economics,
usually considers a multiplayer decision problem. A
success or benefit of an individual in making decisions
depends on the decisions of others. Game theory provides
a theoretical basis to analyze interactions in multiplayer
systems including human as well as non-human players
(e.g., computers, animals, and plants) [5]. Therefore, it
can be applied to a wireless communication network in
the context of resource sharing where the players are
the nodes (e.g., mobile stations, base stations, access
points) in the network. Cooperation or competition among
mobile nodes for channel access in a wireless network is
a multiplayer decision problem, which can be modeled
as a game. The benefit of a node as a result of its chosen
action (i.e., strategy or move) can be measured in terms
of performance metrics such as throughput and delay.
An equilibrium solution of the game model defines the
actions of the different nodes (e.g., transmission power)
for which the chosen performance objective is optimized.
•Modeling selfish/malicious behavior of nodes: The trans-
mitting nodes in a wireless network may behave selfishly
in order to reap performance advantage over other nodes,
as a result of which the overall network performance may
degrade. To make the network robust against the selfish
behaviors (or attacks) by these malicious nodes, efficient
defense mechanisms need to be built into the system.
Game theory can be used to model and analyze the selfish
behavior of nodes and design the defense mechanisms for
robust multiple access in wireless networks.
•Distributed protocols: In many scenarios, wireless nodes
make their decisions in an individual (or distributed)
manner rather than in a centralized manner. Game theory
is a suitable tool to optimize wireless access distributively
[6]. In a centralized scheme, solving the problem of mul-
tiple access may become computationally expensive when
the network size increases. Also, the network control
overhead could be prohibitive. In contrast, efficient dis-
tributed algorithms can be designed based on game theory
which can reduce the communication and computation
overhead significantly. Therefore, game theory is a useful
tool to develop efficient distributed protocols for wireless
networks. With an appropriate game formulation, cross-
layer optimization can be also performed in a distributed
way.
•Mechanism design: The parameters of a game can be
designed (or varied) such that it leads the independent
and self-interested wireless nodes toward a system-wide
desirable outcome. Pricing is one technique that can be
used for such mechanism design (or incentive scheme)
to regulate the usage of radio resources by the wireless
nodes by adjusting their costs.
This article comprehensively surveys the existing researches
on game theoretic approaches for channel access in a multi-
user wireless network. The aim of this article is to familiarize
the readers with the state-of-the-art research on this topic and
the different techniques for game theoretic modeling of the
multiple access problem in wireless systems. Different types
of game models are reviewed for both contention-free and
random channel access schemes. For contention-free channel
access, time-division multiple access (TDMA), frequency-
division multiple access (FDMA), and code-division multiple
access (CDMA)-based wireless networks are considered. For
contention-based channel access, game models for ALOHA
and carrier sense multiple access (CSMA)-based channel ac-
cess methods are reviewed.
The rest of the paper is structured as follows: Section II
presents an overview of multiple access methods for wireless
networks. Then, an overview of game theory and its appli-
cations to multiple access design is presented in Section III.
Next, we review the game models for contention-free channel
access and random channel access in Sections IV and V,
respectively. Section VI provides a summary of the survey and
discusses several open research issues. Section VII concludes
the paper.
II. OVE RVIEW OF MULTI PL E ACC ES S MET HO DS
In this section, the general concepts of channel access
and performance issues related to multiple access design in
wireless networks are discussed.
A. General Concepts
Channel access methods in wireless networks can be divided
into two main groups, namely, contention-free channel access
and contention-based random channel access schemes. In
contention-free schemes, multiple nodes are allocated with
the radio resources (e.g., time slot, channel, and code) by a
central entity and the nodes use the allocated resources for
data transmission [7]. Contention-free channel access can be
used in time-division, frequency-division, and code-division
multiple access networks.
•Time-division multiple access (TDMA): In TDMA, time is
divided into fixed-length frames and each frame is divided
into multiple time slots. Time slots are allocated to the
nodes for data transmission. In TDMA, synchronization
among the nodes is required to avoid interference [8].
•Frequency-division multiple access (FDMA): In FDMA,
radio frequency band is divided into multiple channels.
The channels are allocated to the nodes for data trans-
mission. Orthogonal frequency-division multiple access
(OFDMA) is an improved version of FDMA which is
based on the orthogonal frequency-division multiplexing
(OFDM) modulation in the physical layer. In OFDMA,
frequency band is divided into multiple subcarriers which
are shared among the nodes. OFDMA is used in the IEEE
802.16-based WiMAX networks.
•Code-division multiple access (CDMA): In CDMA, mul-
tiple nodes can transmit data on the same channel simul-
taneously. The transmitted data by each node is encoded
by using a unique spreading code. The spreading codes
for the different users are orthogonal/near-orthogonal to
3
each other. The receiver of each node can decode the
original data correctly if the signal-to-interference-plus-
noise ratio (SINR) is maintained above a threshold.
For contention-based random access scheme, a node has to
compete with other nodes to transmit data over the wireless
channel. A packet transmitted by a node will be received
successfully if there is no collision. A collision occurs when
multiple nodes transmit data simultaneously and the SINR
at the receiver is lower than the minimum SINR required
to decode the original packet correctly. If collision occurs, a
node may attempt to retransmit the packet, and the specifics of
the retransmission method depend on the protocol used. The
most common contention-based channel access schemes are
as follows [9]:
•ALOHA: In ALOHA, if a node has a packet to send, it
will attempt to transmit the packet immediately. If the
packet collides with packets from other nodes, the node
will retransmit the packet later. The ALOHA protocol can
be operated in a slotted fashion, in which case, time is
divided into slots, and packet transmissions are aligned
with the time slots.
•Carrier sense multiple access (CSMA): CSMA is a prob-
abilistic medium access method in which a node senses
the status of the channel before attempting transmission.
If the channel is idle, the node initiates a transmission
attempt. If the transmission is unsuccessful due to a col-
lision, the node waits for a packet retransmission interval
and transmits again. Two of the improved variants of
CSMA are CSMA with collision detection (CSMA/CD)
and CSMA with collision avoidance (CSMA/CA). In
CSMA/CD, assuming that a node is able to detect a colli-
sion, a transmission is terminated as soon as a collision is
detected. The collision can be avoided by expanding the
retransmission interval (i.e., backoff period) for the node
to wait before a new transmission. In CSMA/CA, if the
channel is sensed busy before transmission, to decrease
the probability of collisions on the channel, transmission
is postponed for a random period of time.
B. Performance Issues in Multiple Access for Wireless Net-
works
The key requirements for the design and optimization of
multiple channel access schemes for wireless networks are as
follows [10]:
•Maximize network throughput: Throughput refers to the
amount of data successfully transmitted by the nodes over
a time period. Maximizing the overall system throughput
is a key objective of most of the multiple access schemes.
This is turn improves the spectrum efficiency in wireless
networks.
•Minimize delay: Delay refers to the time required for a
packet to be transmitted successfully since it has been
received at the transmission buffer from the upper layer.
Delay is a key performance metric for real-time traffic
(e.g., voice and video). Multiple channel access schemes
for such traffic have to minimize delay.
•Guarantee fairness: Fairness is a measure of whether the
nodes are receiving an equal (or fair) share of radio re-
sources. Multiple channel access schemes must guarantee
a certain level of fairness to all nodes in the network.
•Improve power efficiency: Power efficiency is an im-
portant performance metric for battery-powered wireless
nodes. There is a tradeoff between power efficiency and
network performance. To reduce power consumption, a
node can be put in standby mode during which the node
cannot transmit and/or receive packets. Consequently, the
throughput reduces.
In a wireless network, the nodes sharing the limited radio
resources may have different behaviors. On one hand, all nodes
can cooperate to meet the above requirements and achieve
optimal network performance. This is referred to as group ra-
tionality. On the other hand, the nodes can be noncooperative
to compete with each other for the radio resources. This is
referred to as self-interest behavior. To analyze these behaviors
and investigate their impact on network performance, game
theory can be applied through which the equilibrium solution
(i.e., behavior of the nodes at steady state) can be obtained.
III. OVERVIEW OF GAME THEORY AND ITS APPLICATIONS
TO MULTIP LE AC CE SS
In this section, the basic concepts used in game theory are
discussed and different game models are briefly introduced.
The issues pertinent to using game theory to analyze multiple
access schemes in wireless networks are also discussed.
A. General Concepts
A game is defined by a set of players, a set of actions
for each player, and the payoffs for the players. A player
chooses an action and the complete plan of action is referred
to as the strategy. When the action is chosen deterministi-
cally, it is called a pure strategy. On the other hand, when
the action is chosen probabilistically according to a certain
probability distribution, it is called a mixed strategy. Based
on the strategies of the players, their payoffs are determined.
Depending on the nature of the game, there are different
solution concepts. However, almost all of them rely on the
equilibrium concept which ensures that a player will gain a
fair or optimal payoff given the strategies of other players
in the game. Pareto optimality or Pareto efficiency is another
well-known concept in a game. A strategy is called Pareto
optimal if it is impossible to make one player better off without
necessarily making other players worse off.
B. Game Theoretic Models
Two major game-theoretic approaches which can be used
to model multiple access schemes are the noncooperative and
the cooperative game approaches. In a noncooperative game,
the players make rational decisions considering only their
individual payoff. In a cooperative game, players are grouped
together and establish an enforceable agreement in their group.
4
1) Noncooperative games: Self-interested players in a non-
cooperative game make decisions independently. The players
are unable to make enforceable contracts but it does not mean
that players do not cooperate. Any cooperation in the games
must be self-enforcing. Noncooperative game theory has been
used extensively to study various issues in wireless networks
(e.g., medium access control game, time slot competition,
and power control in CDMA). The goal of a noncooperative
game model is to find the equilibrium solution for networks
with self-interested nodes. A well-known solution concept for
a noncooperative game is Nash equilibrium [11]. A Nash
equilibrium is a set of strategies for the players such that no
player has any intention to change his/her strategy to gain a
higher payoff given that none of the other players changes
his/her strategy.
Let ibe an index of a player, i∈I={1, . . . , M}where
Iis a set of players and Mis the total number of players.
Let Sidenote a set of strategy of player i.si∈Siis any
possible strategy of player i. The Nash equilibrium satisfies
the following condition [11]:
ui(s∗
i,s∗
−i)≥ui(si,s∗
−i),∀i∈I,∀si∈Si(1)
where ui(·)is the payoff function of player i,s∗
iis a Nash
equilibrium strategy of player i, and s∗
−iis a Nash equilibrium
strategy vector of all players except player i. However, a Nash
equilibrium may not exist in a game. Also, even if a Nash
equilibrium exists, it may not be unique.
Another solution concept which is more general than the
Nash equilibrium is known as correlated equilibrium [12]. In
this concept, a strategy profile is chosen according to the joint
distribution instead of the marginal distribution of players’
strategies as in the Nash equilibrium solution. The definition
of correlated equilibrium is given below. Let Sidenote a set
of strategies of player i. A probability distribution πover
S1× · · · × SMis a correlated equilibrium if for every strategy
s∗
i∈Sisuch that π(s∗
i,s−i)>0, and every alternative strategy
si∈Si, it holds that,
X
s−i∈S−i
π(s∗
i,s−i)[ui(s∗
i,s−i)−ui(si,s−i)] ≥0,
∀i∈I,∀si∈Si.(2)
To interpret this definition, given a recommendation (i.e., a rec-
ommended strategy according to the distribution π) to player
i, a distribution πis defined to be a correlated equilibrium
if no player ican choose a strategy siinstead of s∗
iwhich
results in a higher expected payoff.
A noncooperative game can be classified as either a com-
plete or an incomplete information game. In a complete infor-
mation game, information such as the payoffs and strategies
of the players are observable to all the players. On the other
hand, in an incomplete information game, the information is
unknown by other players. An incomplete information game
can be modeled as a Bayesian game [11] in which Bayesian
analysis is used to predict the outcome of the game. The
equilibrium solution of such a game is called Bayesian Nash
equilibrium. Similar to the Nash equilibrium in a complete
information game, a Bayesian Nash equilibrium can be ob-
tained in which each player seeks for a strategy profile that
maximizes its expected payoff given its beliefs about the types
and strategies of other players.
Moreover, a game can be classified as either a static game or
a dynamic game. A static game is a one-shot game where all
players make decisions without knowledge of the strategies
that are being chosen by other players. The one-shot game
ends when actions of all players are chosen and payoffs are
received. In contrast, in a dynamic game, a player chooses
an action in the current stage based on the knowledge of the
actions chosen by the other players in the current or previous
stages. This dynamic game can be called a sequential game
since players play a static game repeatedly. The common
equilibrium solution in dynamic games is a subgame perfect
Nash equilibrium [13]. A subgame perfect Nash equilibrium
represents a Nash equilibrium of every subgame of the original
game. A common method to obtain subgame perfect equilibria
is backward induction.
A dynamic game with incomplete information can be de-
scribed as a multi-stage game when information is unknown
to other players [11]. It is similar to a dynamic game with
complete information in that the players take turns sequentially
rather than simultaneously but information is incompletely
known to others. The players follow their beliefs and dy-
namically update their beliefs by using the Bayes’ rule. In a
dynamic game with incomplete information, perfect Bayesian
equilibrium is the solution concept which can be considered as
a combination of the Bayesian Nash equilibrium and subgame
perfect equilibrium concepts.
Repeated game [11] is a special kind of dynamic game in
which the same set of players plays the same stage game or
one-shot game repeatedly over a long time period. Repeated
games can be divided into two key types, namely, finite and
infinite repeated games, depending on whether the period of
time during which the game is played is finite or infinite. Most
repeated games are typically infinite repeated games and a
player takes into account the effect of his/her current action
on the future actions of other players.
Markovian game (i.e., Markovian dynamic game or Markov
game) is an extension of game theory to Markov Decision
Process-like environments. A Markovian game can be defined
as a type of stochastic game [14]-[15] which can be regarded
as a multiagent extension of Markov decision process [16].
The key difference between a Markov game and a Markov
decision process is that a transition depends on the current state
and the action profile of the players. Also, each player may
receive different reward as a result of the action profile. Each
player has a reward function (i.e., payoff function) and tries
to maximize its expected sum of discounted reward. A more
specific type of Markovian game is a switching controlled
Markovian game where the transition probability in any given
state depends on the action of only one player. The Nash
equilibrium for such a game can be computed by solving a
sequence of Markov decision processes.
Auction game [17] is a game theoretic approach in which an
object or service is exchanged on the basis of bids submitted
by the bidders to an auctioneer. There are two main auction
mechanisms, namely, the first and second price auctions. In
first price auction, an object or service is given to a bidder
5
who submitted the highest bid and pays a price equal to the
amount of bid. In second price auction, an object or service
is given to a bidder who submitted the highest bid and pays
a price equal to the second highest amount of bid.
Stackelberg game or leader-follower game [11] is a strategic
game in which the player acting as a leader moves first and
then the rest acting as followers move afterward. Then, the
problem is to find an optimal strategy for the leader, assuming
that the followers react in such a rational way that they
optimize their objective functions given the leader’s actions.
The Stackelberg game model can be solved by subgame
perfect Nash equilibrium.
Evolutionarily stable strategy (ESS) [18] is a solution
concept in the evolutionary game theory. In this game, the
evolution of social behaviour of animals in a population is
considered. In a wireless network, a population can be a group
of mobile nodes sharing the channels. A strategy is called an
ESS if in a fixed population, the entire population using ESS
cannot be invaded by mutant strategies of a small group.
2) Cooperative games: In a cooperative game, players are
able to make enforceable contracts. The players in a coalition
cooperate to maximize a common objective of a coalition. In
this case, players can coordinate strategies and agree on how
the total payoff is to be divided among players in a coalition.
Nash bargaining game is one type of cooperative games where
the players maximize the product of their gains given what
each player would receive without cooperation (i.e., threat
point). This is referred to as the Nash bargaining solution
which can be defined as follows:
s∗= arg max
sY
i∈I
(ui(si)−ud
i)(3)
where ui(·)is the payoff function of player i,siis a strategy
of player i, and s∗is a Nash bargaining solution strategy vector
of all players, and ud
iis the threat point (i.e., the utility gained
if player idecides not to cooperate and bargain with the other
players).
Coalition formation game is a cooperative game involving
a set of players who are looking for cooperative groups (i.e.,
coalitions). A coalition S, which represents an agreement
among the players to act as a single entity, can be formed
by players in a set Nto gain a higher payoff, and the
worth of this coalition, denoted by vis called the coalitional
value. Two common forms of coalitional games are strategic
form and partition form. In the former case, the value of a
coalition Sdepends on the members of that coalition only (i.e.,
independent of how the players in N\Sare structured). In the
latter case, the value of a coalition Sstrongly depends on how
the players in N\Sare structured. Coalitional game models
can be developed with either transferable payoff or non-
transferable payoff. In a transferable payoff coalitional game,
there is no restriction on how the total payoff will be divided
among the members of a coalition. In a non-transferable payoff
coalitional game, the payoff that each player in a coalition
obtains depends on the joint actions that the players of a
coalition select [19]. A stable solution for a coalition formation
game ensures that the outcome is immune to deviations by
groups of players (i.e., no player has an incentive to move
from its current coalition to another coalition).
C. Issues in Game Theoretic Design of Multiple Access
Schemes
Game theory can be used to model and analyze cooperative
and noncooperative behaviors of nodes and their interactions
during channel access in wireless networks. There are a few
considerations when game theory is applied to model and
analyze multiple access schemes in wireless networks.
•Self-interest and group-rationality: Most of the game
theoretic models developed for multiple access in wireless
networks have the assumption of self-interest and group-
rationality for the noncooperative and cooperative game
models, respectively. A node with self-interest has the
objective to maximize only its own payoff. However, this
behavior may not provide a socially optimal solution.
The corresponding depreciation is called price of anarchy.
On the other hand, group-rational nodes can cooperate
to achieve a socially optimal solution, and for this, a
cooperative game (e.g., bargaining game) can be applied
[20]-[21]. However, in many cases, the group-rationality
condition may not hold for all nodes, and some nodes
may deviate from cooperation. Therefore, a penalization
(or punishment) mechanism is required to enforce cooper-
ation among the nodes so that a socially optimal solution
can always be achieved.
•Penalization mechanisms: These mechanisms [22] are
proposed to promote cooperation in multiple access
among nodes sharing the channel. The punishment is
commonly applied to the nodes deviating from coop-
eration since such deviation can degrade the network
performance. In this respect, two important issues are the
detection of deviating nodes and designing the punish-
ment mechanism to be applied.
•Practical implementation: Although game theory pro-
vides solutions for a situation of conflict during multiple
access, it is still difficult to implement these solutions
in a practical environment. In many cases, distributed
implementations are desirable. In some distributed imple-
mentations, the mobile stations may require information
such as SINR, power, price from the base station(s) in
order to converge to the equilibrium solution. However, in
a realistic scenario, these information cannot be observed
perfectly (e.g., channel gain of other nodes). Therefore,
the mobile stations may need to have the ability to
learn from the radio environment which may increase the
complexity of implementation and reduce the rate of con-
vergence of the solution of the game to the equilibrium
solution.
•Payoff function: A payoff function represents the benefit
or reward to a player in the game when an action is
chosen by this player. Defining a suitable payoff function
is an important issue. When the payoff function is defined
differently, the solution of the same game model applied
to the same multiple access scheme can be dramatically
different. The payoff function should be defined consid-
6
TABLE I
SUMMARY OF GAME MODE LS
Game model Key objective Solution concept Type
Noncooperative game Individual players act to maximize their own payoff. Nash equilibrium, Static game vs. dynamic game
Correlated equilibrium, (e.g., repeated game),
Bayesian Nash equilibrium, Complete information game vs.
Evolutionary stable strategy, incomplete information game,
Stackelberg equilibrium Evolutionary game,
Markovian game,
Stackelberg game,
Auction game
Cooperative game Coalitions of players are formed and Nash bargaining solution Coalitional game
players have joint actions to gain Bargaining game
mutual benefits.
ering the physical performance measures of the nodes
and/or networks.
In the following sections, we review various game theoretic
models proposed in the literature to study multiple access
schemes. This review is categorized based on different types of
multiple access methods and types of the game models used.
IV. GAME MOD EL S FO R CONTENTION-F RE E CHA NN EL
ACCESS
In this section, game models for contention-free channel
access based on TDMA, FDMA, and CDMA are reviewed.
A. TDMA-Based Channel Access Games
Since the nodes have to transmit data during their allocated
time slots, in TDMA-based channel access games, the nodes
compete for time slots to achieve their performance objectives
(i.e., QoS requirements). Four different game models, namely,
noncooperative static game,auction game,dynamic game, and
repeated game models are discussed. A summary of the key
features of these game models is provided in Table II. The
details of these models are discussed next.
1) Noncooperative static game-theoretic approach: In [23],
a noncooperative static game model for QoS-aware resource
competition is proposed for a single-hop wireless network
with multiple wireless links. The wireless nodes need to
compete for the transmission time within a time slot. The
power consumption of all nodes is assumed to be constant
and the channel fading is assumed to follow a stationary flat-
fading process. Two nodes are considered as the players in the
game. The time slot length is denoted as tiTwhere ti∈[0,1]
is the time slot share of node i(i.e., fraction of time in a slot
to be accessed by node i) and Tis the size of frame. Time slot
share tiis the strategy of nodes. Two nodes i= 1,2transmit
their data without collision if t1+t2≤1and collision occurs
if t1+t2>1. Time slot allocation to the nodes is shown
in Fig. 1(a). Since the nodes have self-interest, they minimize
the time slot usage given the channel state information (i.e.,
channel power gain denoted as hi) subject to the effective
capacity [24]. The corresponding optimization formulation for
node 1 is as follows:
min
t1
Eh{t1}(4)
subject to
Eh{e−β1t1R1} − A1≤0(5)
and
t1≤1−t2(6)
where Ehdenotes expectation over h,β1is the normalized
QoS exponent, R1is the normalized transmission rate which is
a function of channel state information. A1is equal to e−θ1C1
where C1denotes the target effective capacity exponent and
θ1denotes the QoS exponent for node 1. This effective
capacity is also considered as an objective of the system. Nash
equilibrium is the solution of this time slot competition game.
It is found that the Nash equilibrium may not be unique.
However, since the game can be formulated as a convex
optimization problem, a unique solution can be obtained by
minimizing the Lagrangian function of the objective function
if the available time slot length is enough to guarantee QoS
for both the nodes.
Fig. 1. The diagram represents different time slot allocations: (a) time slot
allocation based on [23], (b) time slot allocation based on [25], and (c) time
slot allocation based on [26].
2) Auction game-theoretic approach: In [25], a second
price and sealed bid auction for time slot competition in a
dynamic spectrum access scenario is proposed. In dynamic
spectrum access, each node i∈I={1, . . . , M }(i.e., the
bidder/player in a game) submits its bid to the base station.
The value of the submitted bid is the portion of the time slot
(i.e., between 0 and 1) that will be used to help the base station
relaying data to another distant node. The bidding value biof
node iis a non-decreasing function of the channel condition
xi. The base station (i.e., the auctioneer) allocates the down-
link channel to a node offering the highest bid (as shown in
Fig. 1(b)). The price that this winning node pays is equal to the
second highest bid. The amount of transmitted data of winning
node jis denoted as dj=xj(1 −max
i∈I,i6=jbi(xi)), where xjis
the channel condition of the winning node which is assumed to
7
TABLE II
SUMMARY OF TDMA-BASED CH ANN EL AC CES S GAM ES
Game model Key concept Method of solution Reference
Noncooperative static game Nodes minimize the time slot usage given the channel power
gain.
Convex optimization [23]
Auction game Time slot allocation with a second-price auction scheme when
a fraction of time to relay data is used as bid
Distributed bid update algorithm [25]
Time slot allocation with a second-price auction scheme when
money is used as bid
Fair and efficient centralized op-
portunistic scheduler
[26]
Dynamic game Channel competition and rate control in TDMA cognitive
radio formulated using Markov game theory
Value iteration algorithm [27]
Iteration correlated algorithm and
correlated Q-learning algorithm
[28]
Repeated game Nodes choose their power allocation under power constraint
and channel condition. To enforce node cooperation, a pun-
ishment and truth-telling mechanism is used.
Self-enforcing truth-telling mecha-
nism
[29]
be the amount of data received per unit time, and bi(xi)is the
bid submitted by a node. A node chooses a value of bid which
maximizes its expected amount of transmitted data under its
budget constraint given the probability distributions of the
channel conditions of all the nodes. The budget constraint
of node irepresents the amount of time that the node is
able to provide to the base station for data relaying. Nash
equilibrium is considered as the solution. It is found that, for
pure strategy, a Nash equilibrium exists in the two-node case,
but in a general multiple-node case, a Nash equilibrium may
not exist. A distributed algorithm is proposed for updating
the bids which converges to the Nash equilibrium. The results
show that to avoid zero throughput (i.e., maximum bid), the
budget constraint has to be smaller than one. Also, the higher
the budget constraint, the lower is the throughput for each
node.
A system model almost similar to that in [25] is considered
in [26]. The second-price auction mechanism is used for the
nodes competing for time slots in a downlink transmission
scenario. Similar to [25], the utility function of a node is the
expected amount of transmitted data and the bidding value
biof node iis a function in the channel condition xi. A
node submits a bid which maximizes its expected amount
of transmitted data under the budget constraints given the
probability distributions of the channel conditions of all the
nodes. The time slot allocation process for this auction game
can be illustrated as in Fig. 1(c). The Nash equilibrium is
considered as a solution of the auction game. It is shown that
the Nash equilibrium leads to a unique allocation which is
also Pareto optimal. For uniformly distributed channel state,
the aggregated throughput that nodes achieve at the Nash
equilibrium is at least 3/4 of the optimal aggregated throughput
achieved using an optimal centralized allocation without fair-
ness consideration. Also, a centralized opportunistic scheduler
is proposed to achieve proportional fairness. The a priori
knowledge of channel distribution is not required by this
scheduler. The centralized scheduler will assign time slots
according to the Nash equilibrium strategy.
3) Dynamic game-theoretic approach: In [27], a Markovian
dynamic game is formulated to solve the transmission rate
adaptation problem in a dynamic spectrum access-based cog-
nitive radio network. In such a network, the secondary users
(or cognitive radio users) opportunistically access the radio
spectrum, which is licensed to the primary (or licensed) users,
without causing harmful interference to the primary users. The
players of the game are secondary nodes competing for the
channel or time slot in a TDMA scenario (e.g., in the IEEE
802.16-based network). In a TDMA cognitive radio system,
the system has a predefined decentralized access rule that
allows only one secondary node to access the channel at a
time. The access rule is defined as a function of channel quality
and transmission delay. This transmission rate control problem
is formulated as a general-sum switching control Markovian
dynamic game.
In this dynamic game, the system state transition probability
at each time slot depends only on the active secondary node.
Node i(i.e., secondary node i) follows a decentralized access
rule to try to occupy a time slot at time nafter a period of time
tn
i=γi
qn
ihn
i
where γiis the QoS parameter of node i,qn
iis the
buffer occupancy state of user i, and hn
iis the channel state
of node i. The composite variable xn
i= [qn
i, hn
i]denotes the
state of user iat time n. If there are more than one node
having the same waiting period, a node will be randomly
picked with equal probability. After node jis selected to
transmit data, this node chooses action an
j(i.e., transmission
rate in bits/symbol) assuming an M-ary quadrature amplitude
modulation. The transmission cost of the selected node j,
cj(xn, an
j), is defined as its transmission bit error rate (BER),
and the cost of node i,di(xn, an
j), is defined as its delay
constraint (i.e., QoS constraint) which is a function of the
buffer state qn
i. The transition probabilities depend only on the
action of active node; hence, a Markovian dynamic game can
be formulated. The strategy of node idenotes the transmission
policy si. The Nash equilibrium policy s∗
iis computed by
minimizing the expected total discounted cost function subject
to the expected total discounted delay constraint as follows:
s∗(n)
i={sn
i: min
si
Cn
i(si)}subject to Dn
i(si)≤ˆ
Di(7)
where Cn
i(si)is the infinite expected total discounted trans-
mission cost calculated from cj(xn, an
j).Dn
i(si)is the infinite
expected total discounted delay which is calculated from
di(xn, an
j)and cannot be greater than threshold ˆ
Di.
The value iteration algorithm is used to obtain a Nash
equilibrium policy. The Nash equilibrium policy of any node i
is observed to be a randomized mixture of pure policies and the
pure policies are non-decreasing on the buffer occupancy state.
A stochastic approximation algorithm exploiting this structure
is presented to efficiently estimate the Nash equilibrium policy
by computing parameters such as buffer state thresholds and
randomization factors.
8
In [28], a system model similar to that presented in [27] is
considered; however, correlated equilibrium is studied as the
solution of the game. Two distributed correlated equilibrium
algorithms (i.e., iterative correlated equilibrium algorithm and
correlated Q-learning algorithm) are proposed to obtain the
correlated equilibrium in the Markovian game. The stationary
policy s∗is a correlated equilibrium for the Markovian game,
i.e.,
X
a−i∈A−i
s∗
x(a−i, ai)Qs∗
i(x,{a−i, ai})≥
X
a−i∈A−i
s∗
x(a−i, ai)Qs∗
i(x,{a−i, a0
i})(8)
where Qs∗
i(·)is the Q-function of user iwhich is the total
discounted reward of taking action ain state x. The Q-
function is a function of the user i’s utility plus the infinite
expected total discounted utility. The utility is the difference
between a function of achievable transmission rate and the
transmission delay. When an iterative algorithm is used to
find the correlated equilibrium, each user does not need the
information of the Q-functions of other users, and this can be
used to develop distributed algorithms. However, the proba-
bility transition matrix is required to update the Q-function
values. The correlated Q-learning algorithm can remove the
requirement of a system state transition probability matrix for
the iterative correlated equilibrium. Hence, it is more practical
than the iterative algorithm.
4) Repeated game-theoretic approach: In [29], a repeated
game model for spectrum sharing in a cognitive radio network
is presented. The game enforces the nodes to tell their true
channel conditions and to cooperate with each other. Data
transmission over a long time period is considered. Therefore,
spectrum sharing can be formulated as a repeated game where
the nodes are concerned about their payoffs (e.g., throughputs)
in the future. The actions of the nodes are the power allocation
according to the power constraint and channel condition. In
this game, the power constraint is assumed to be identical for
all nodes. If all the nodes make an agreement and share the
spectrum in an orderly fashion, every node gains benefit from
the cooperation. However, some nodes may violate the agreed
upon rule and deviate from cooperation. Then, the game model
provides a punishment mechanism which will be triggered and
applied to the deviating node for a certain period of time. The
period of time for punishment is chosen such that the expected
payoff from cooperation is greater than the expected payoff
from deviation.
To design a cooperation rule, an opportunistic time slot
allocation method is developed which maximizes the total
throughput. The node informing the best channel gain will
be allocated time slots for transmission. However, in the
incomplete information case, the channel gain of one node
may not be known to other nodes, and some node may falsely
inform its channel gain information. Therefore, a Bayesian
mechanism is introduced to enforce all the nodes to tell the
true values of their channel gains.
B. Channel Access Games in FDMA
In FDMA, the nodes compete for available channels in the
network and the solutions of the game models (i.e., equilibria)
can be obtained in the complete and incomplete information
cases. We consider three different game models, namely,
noncooperative static game,auction game, and cooperative
game models. A summary of the key features of these game
models for FDMA is provided in Table III. The details of these
models are discussed next.
1) Noncooperative static game-theoretic approach: In [30],
the optimal FDMA channel assignment problem for noncoop-
erative wireless networks is studied assuming that the nodes
can be equipped with either single or multiple radio inter-
faces. The available frequency band is divided into orthogonal
channels. The authors introduce a payment formula to ensure
the existence of a strongly dominant strategy equilibrium
(SDSE) [31], which is a stronger solution concept than the
Nash equilibrium. This payment is used to obtain the globally
optimal solution in terms of effective system-wide throughput.
The strategy of node i(si) is the channel assignment vector
which is the number of radio interfaces allocated to each
channel. The solution in terms of SDSE can be described as
follows:
∀s−i∈Si,∀si6=s∗
i, ui(s∗
i,s−i)≥ui(si,s−i)(9)
∃s−i∈Si,∀si6=s∗
i, ui(s∗
i,s−i)> ui(si,s−i)(10)
where Siis the set of all possible strategies and ui(·)is the
payoff function of node i. The payoff function is defined as
the difference between the throughput and the payment to
the system administrator. The payment is a function of the
node’s throughput plus a penalty (if the node deviates from
the globally optimal solution) or a bonus (if the node does not
deviate). An algorithm to obtain the SDSE is proposed. It is
proved that the algorithm converges to the SDSE.
Multiple channel access in multi-cell and multi-user
OFDMA networks is considered in [32]. In multi-cell net-
works, changes of resource allocation in a cell affect the
performances of other nearby cells. A noncooperative game
model for sub-channel assignment, rate adaptation, and power
control is introduced. Node iwith self-interest maximizes its
own payoff ui(·)(e.g., minimizes its transmission power) by
allocating its transmission rates rion different sub-channels
k∈ {1, . . . , K}under the required rate Riand power
constraint Pi
max as follows:
min
ri
ui(·) =
K
X
k=1
Pk
i,s.t.
K
X
k=1
rk
i=Ri(11)
where Pk
iis the transmission power of user iin subchannel
k.
Nash equilibrium is considered as a solution. This Nash
equilibrium (NE) can be obtained by using the water-filling
algorithm. However, in some cases of high channel interfer-
ence, there might be multiple Nash equilibria or the solution
might also be undesirable (i.e., the overall power for the users
is larger than the power constraint: PK
k=1 Pk
i> P i
max). A dual
noncooperative game is used if the desired NE solution cannot
9
TABLE III
SUMMARY OF FDMA-BASED CH ANN EL AC CES S GAM ES
Game model Key concept Method of solution Reference
Noncooperative
game
Channel allocation among nodes equipped with multiple radio
interfaces to obtain system-wide throughput optimality
An algorithm to obtain a strongly dominant strategy
equilibrium (SDSE) [31]
[30]
Sub-channel allocation by allocating transmission rates on
different channels subject to power constraint and required
rate
A mechanism called virtual referee is used to reduce
the complexity of the game.
[32]
Channel allocation among nodes equipped with multiple radio
interfaces when rate on each channel is allocated equally by
time-division schedule
A centralized algorithm with perfect information, a
distributed algorithm with perfect information, and a
distributed algorithm with imperfect information
[33]
Sub-channel assignment and power allocation when multiple
base stations are players instead of mobile nodes
A distributed algorithm based on a greedy approach for
sub-channel assignment problem and base on the best-
response update for power allocation problem
[34]
Channel allocation of secondary nodes when transmission
power is constrained. The game is also extended to a Stack-
elberg game.
N/A [35]
Auction game Power is allocated based on water-filling allocation according
to the result from the second-price auction.
An iterative update algorithm based on bidding effi-
ciency and the subgradient algorithm
[36]
Cooperative
game
Coalition formations among secondary base stations to im-
prove knowledge of available channels to serve secondary
nodes
A distributed algorithm to find a Nash-stable set of
coalitions
[38]
Transmission power allocation and subchannel assignment
using coalitional game when a player is a pair of one
subchannel and one node
An iterative algo. to update best-responses based on
Markov modeling
[40]
be obtained. The definition of the dual game is as follows:
max
ri
K
X
k=1
rk
i,s.t.
K
X
k=1
Pk
i=Pi
max.(12)
After the dual noncooperative game converges, if the desired
NE solution is still not reached (i.e., if any node has to play
the dual noncooperative game, the obtained NE is considered
as an undesired NE), a mechanism called virtual referee is
introduced to improve the performance of this noncooperative
game.
The key idea of the virtual referee mechanism is that a
referee monitors the nodes in the network. If the Nash equilib-
rium is reached, the referee does nothing; however, if it is not,
the referee will modify the game rule in order to remove some
nodes from accessing subchannels so that a better performance
can be achieved. The flow of this noncooperative game with
a referee is shown as Fig. 2.
A channel allocation game is studied in [33] as a static game
when nodes have multiple radio interfaces. The players are
the nodes with self-interest which aim to maximize their own
profits defined as the total rates or channel utilization. In this
game, there are Korthogonal channels. If the same channel
is used by multiple nodes, they can hear the transmissions of
each other. Moreover, a node can use multiple channels at the
same time. The strategy of each node is the channel allocation
vector or the number of radio interfaces on each channel. The
payoff of each node is the achieved bit rate. It is assumed
that the rate on each channel is allocated equally by using a
reservation-based time-division schedule among the interfaces.
The total available rate on a channel is assumed to be a non-
increasing function of the number of radio interfaces accessing
this channel. The set of channels used by node iis denoted as
Ki. The payoff function of node i, defined as ui, is the sum
of achieved bit rate ron each occupied channel k∈Kias
follows:
ui=Xli,k
lk
R(lk)(13)
where li,k is the number of radio interfaces of node icurrently
Fig. 2. The diagram illustrates the steps of the algorithm for noncooperative
game with virtual referee in OFDMA networks.
using channel k,lkis the number of radios using channel k,
and R(lk)is the total rate which is a decreasing function of
the number of radios using channel k.
Nash equilibrium is considered as a solution. If the total
number of radio interfaces is smaller than or equal to the
number of channels, then a flat channel allocation (not more
than one radio interface per channel) is the Nash equilibrium.
To find a Nash equilibrium, three algorithms are introduced.
The first one is a centralized algorithm with perfect infor-
mation. It requires sequential action of players and global
coordination. This global coordination can be achieved with
an extra radio interface per device for scanning the channels.
The second algorithm is a distributed algorithm with perfect
information. This algorithm is a round-based algorithm in
which a random radio interface assignment to the nodes over
10
the channels is used. It is assumed that there is no node that
can assign more than one radio interface to any channel. After
the initial assignment, each node evaluates the number of radio
interfaces on each channel and tries to improve its total rate
by reorganizing the allocation. However, an unstable allocation
can occur. To avoid this problem, a backoff technique is used.
Each node chooses a random initial value and then decreases
this counter value periodically. The reallocation is performed
when the counter is zero.
The third algorithm is a distributed algorithm with imperfect
information. This algorithm also uses the backoff technique. In
each round, a node calculates the average number of devices
on the channels. Then, the node can obtain a probability to
choose channel. The algorithm can reach a stable state but
it may not be the Nash equilibrium since the available local
information is incompletely known. Then, a mechanism is
introduced to resolve inefficient stable states.
The main difference of [34] from the previous game model
is that the base stations (rather than the mobile nodes) are the
players in this game. A noncooperative distributed resource
allocation game in a multi-cell OFDMA system is proposed for
Mbase stations serving Lnodes. All of the base stations share
the same frequency band with the total bandwidth Bdivided
into Ksub-channels. The players are the base stations and a
strategy is the sub-channel assignment and power allocation.
The transmission power at a sub-channel kof base station i
is denoted as pi
k. That is, pi= (pi
1, . . . , pi
k, . . . , pi
K)is the
transmission power vector of all the sub-channels of base sta-
tion i. The constraint on the transmission power of each base
station is PK
k=1 pi
k≤Pmax. The payoff function is defined
as the difference between the weighted sum of the data rates
(Pj∈UiβjRj(P,Ai)) and the cost of total power (PK
k=1 pi
k)
where Uiis the set of nodes in cell i,βiis a weighting factor,
P= [p1,...,pi,...,pM]and Ai= [ai
kj ]K×L,ai
kj is 1 if
sub-channel kis assigned to node j; otherwise, it is 0. The
payoff function is defined as follows:
ui(P,Ai) = X
j∈Ui
βjRj(P,Ai)−c
K
X
k=1
pi
k(14)
where cdenotes the price per unit power, having the unit
bps/W.
Each base station maximizes its payoff function. To find the
optimal sub-channel assignment given a network power vector
P0, the sub-channel assignment game can be represented as
maxAiPj∈UiβjRj(P0,Ai). Using a greedy approach, the
solution A∗i(P)can be found when Pis determined. Hence,
we can obtain the optimal power allocation, which is the Nash
equilibrium, by solving maxpiPj∈UiβjRj(P0,A∗i(P)).
The existence and uniqueness of Nash equilibrium of the
power allocation game can be proved. Moreover, a distributed
resource allocation algorithm is proposed to obtain both the
sub-channel assignment and power allocation. The algorithm
iteratively converges to an equilibrium point. The key concept
of the algorithm is that each base station updates the sub-
channel assignment according to a greedy approach and the
power allocation according to the best-response update using
local information from nodes (i.e., SINR in each sub-channel).
The work of [35] presents a noncooperative game model
for spectrum access in distributed cognitive radio networks. In
such a network, Msecondary nodes opportunistically transmit
data on the channel allocated to the primary node. Let pi
be transmission power of secondary node i(i.e., transmitter
and receiver). Secondary node ihas a maximum transmission
power constraint (pi≤ˆpi) in each channel. It is assumed that
the total power in each channel must not exceed the maximum
total power of all users using the channel k∈ {1, . . . , K},
which is Pi∈Ukpi≤ˆ
Pwhere Ukdenotes the set of all nodes
using channel kand ˆ
Pis the maximum total power of all users.
The strategies of secondary nodes are the choices among
Kavailable channels. The objective of secondary node iis
to maximize its payoff, which is a function of SINR γk
ion
each channel k(i.e., max ui(γk
i)) subject to the maximum
transmission power constraint ˆp(i.e., pi≤ˆp). Using an N-
channel bi-matrix game, the existence of pure strategy Nash
equilibrium is proved. Next, this noncooperative game model
is extended to the Stackelberg game since the channel access
of a disconnecting secondary node depends on the other
secondary nodes’ strategies. There are events which cause
disruption in channel access. Arrival of a primary node is the
main cause of channel access disruption since the secondary
node has to leave the channel immediately. Interference from
multiple secondary nodes accessing the channel and channel
fading may also cause disruption in channel access.
In an unexpected event, a secondary node who has any strat-
egy that can help uninterrupted channel access is considered
to be a leader (i.e., a leader has information on channel access
by primary users). The rest of the secondary nodes who do
not have any information on accessing channel are followers.
Then, the payoff of the leader is the summation of ui(γk
i)
and the cost that the followers pay for switching the channel.
The payoff for a follower is the difference between ui(γk
i)
and the price that the leader sets for switching the channel.
Note that this leader-follower scenario is temporary. A node
finds a channel and broadcasts the new channel information.
Only one node can be a leader in each channel. The numerical
results show the existence of an equilibrium solution.
2) Auction game-theoretic approach: In [36], a distributed
resource control scheme is presented to achieve fairness in
OFDMA systems. Specifically, an auction game-theoretic re-
source allocation scheme based on iterative multi-unit second
price auction is applied. A base station (BS) controls trans-
mission power and bidding to maximize system capacity and
node fairness. From an information-theoretic point of view,
the medium access control (MAC)-layer throughput capacity
region is achievable by successive decoding [37] when at each
subchannel k, the first node’s decoded signal is subtracted
from the sum signal, then the next node’s signal is decoded,
and so on.
In this auction, first each node isubmits bid biwhich
includes power control variable and bid value. Each node cal-
culates its bid by maximizing the expected Shannon capacity,
and each node submits its bid and waits to be assigned the
decoding priority for each sub-channel from the base station.
After the bids are received by the base station, the decoding
priority is assigned to each node following the weighted sum-
11
rate capacity maximization of the base station. The cost that
each node ipays is the cost for winning the lth decoding
priority at subchannel k. Then, transmission power will be
allocated based on the optimal and fair water-filling allocation
according to the result of the decoding order. Also, the cost
that the nodes have to pay will be announced.
To obtain the Nash equilibrium for bidding in this auction,
an iterative update algorithm is proposed. The key concept is
to update the bid value based on the difference between the
current bidding efficiency and the target bidding efficiency
at each time slot t. Bidding efficiency is computed by a
node’s achievable transmission rate divided by the cost of the
node. Also, the bidding control variable is updated using the
subgradient algorithm as follows:
x(t+1) =x(t)+αtg(t)(15)
where x(t)is the bidding control variable at time t,αtis
a constant step size, and g(t)is a subgradient which is a
function of the total cost that node has to pay for and the
total bid money that node can use during the game. The
analytical and simulation results show that this iterative update
algorithm can converge to the stable and optimal equilibrium
which can achieve fairness among users when the channel
conditions of the subchannels for the different nodes are
uniformly distributed.
3) Cooperative game-theoretic approach: In [38], a coop-
erative game theoretic model is proposed for secondary base
stations (SBSs) in a cognitive radio network. The main concept
of this work is to form cooperative groups among the SBSs in
a multi-channel cognitive radio network by using a game the-
oretic approach called coalition formation game. To improve
the quality of information about availability of primary nodes
(PNs) to serve secondary nodes (SNs), the SBSs share their
information through control channels such as a cognitive pilot
channel (CPC) to other SNs. SBS i∈I={1, . . . , M }detects
the presence of any PU k∈K={1, . . . , K}by using a
channel and serves LiSNs. Each SBS can gather information
on the availability of channels as a subset Ki⊆K. The false
alarm probability obtained by SBS iover PN channel kis
denoted as Pi
fal,k. Then, the total potential utility of SBS iin
a noncooperative approach is given as follows:
u({i}) = X
k∈Ki
Li
X
j=1
[(1 −Pi
fal,k)θkρj i −αk(1 −Pi
det,k)
(1 −θk)(ρkrk−ρj
krk)] (16)
where ζkis the probability that channel kis available and
αkis a penalty factor defined by PN kfor any SN causing
interference. The term (1 −Pi
det,k)is the probability of mis-
detection. ρij is the probability of successful transmission
of SN jto its serving SBS iat the time when channel k
is available. The term (ρkrk−ρj
krk)is the reduction in the
probability of successful transmission of PN kat its receiver
rkwhenever SN jtransmits over channel kat a time of the
presence of PN kdue to the mis-detection of PN k.
To improve the utility, the SBSs can share the available
knowledge of the presence of PNs; however, there is a trade-
off between the utility gained from learning new channels
(through information sharing) and the cost to obtain coop-
erative information. In a coalition S, the set of known PNs by
any SBS iin the coalition is defined as KS=∪i∈SKi. Hence,
the payoff of any SBS i∈Sis defined as follows:
ui(S) = X
k∈KS
Li
X
j=1
[(1 −Pik
fal,k)θk
iikρji −αk(1 −Pik
det,k)
(1 −θk
iik)(ρkrk−ρj
krk)] (17)
where θk
iikis the probability that SBS ican obtain the
knowledge of channel kfrom another SBS ik∈S.ik=i
if SBS ihas its own information on channel k. SBS ikgiving
the maximum utility will be selected by SBS i. Since the
payoff of SBS idepends only on the identity of the SBSs in
the coalition which SBS iis a member of, this game can be
considered as a hedonic coalition game [39]. The formulation
of the game is described next.
Given two coalitions S1and S2, and i∈S1and i∈S2,
S1S2means SBS iprefers to be a member of coalition
S1over being a member of coalition S2, and S1S2means
SBS istrictly prefers to be a member of coalition S1over
being a member of coalition S2. Then, the proposed coalition
formation game can be defined as follows:
S1S2⇔wi(S1)≥wi(S2)(18)
where wi(S)is a preference function for SBS iand coalition
S. SBS imakes a decision to leave its current coalition Sx
and then to join another coalition Sywhen Sx6=Syif
and only if Sy∪ {i} iSx. This can be interpreted as: an
SBS will switch to a new coalition if it can strictly gain
more payoff without decreasing other members’ payoffs in
the new coalition. A partition or a set of all coalitions is
Nash-stable if no SBS has an incentive to move from its
current coalition to another coalition or to deviate and act
alone. A distributed algorithm to find a Nash-stable partition
is proposed and the simulation results show that the average
payoff per SBS of the coalition formation scheme outperforms
one of the noncooperative schemes when the number of SBS
increases.
Fig. 3. Coalitions of players are formed following the game model in [40]
when there are 3 mobile nodes and 3 subchannels.
Another coalitional game for transmission power alloca-
tion and subchannel assignment in the uplink channel of
an OFDMA system is presented in [40]. In the considered
system model, there are Mnodes located in the coverage
12
area of a same base station. The base station provides K
subchannels to node i∈I={1, . . . , M }to guarantee
the target rate requirement. Let kdenote each subchannel
k∈K={1, . . . , K}. Let Ribe the target rate requirement of
node i. Suppose that the total bandwidth is B, then the carrier
spacing of every subchannel is 4f=B/K. A player defined
in this game is a pair of one subchannel and one node. Hence,
MK players are considered in this game. The strategy of each
player is the transmission power assigned to subchannel pik.
Then, there are Mcoalitions ζ= [S1,...,Si,...,SM]to be
assigned to the Mnodes and each coalition Sicontains K
players (e.g., shown in Fig. 3).
In this game, the members in each coalition do not change
during the game. Consequently, the coalition Siachieves its
rate Ci=Pk∈KCik where Cik =4flog2(1 + γik )is the
Shannon capacity achieved by node ion subchannel k.γik is
the SINR at the base station. The payoff that each coalition
will obtain is defined as follows:
u(Si) = 1
Ck/Rk−1−α.t(1 −Ck/Rk)(19)
where t(·)is the step function with t(y)=1if y≥0and
t(y) = 0 if y < 0, and αis a positive constant. A coalition
will achieve the highest payoff (i.e., positive infinite) when
Ck=Rk. An iterative algorithm based on Markov modeling
of the TU coalitional game is proposed to update the best-
responses. The analytical and numerical results show that the
algorithm can be considered as a Markov process. The process
can quickly converge to an absorbing state which is also a
Nash equilibrium solution with probability of one.
C. Channel Access Games in CDMA
CDMA systems use spread-spectrum technology in which
each node is assigned with a different code to allow multiple
users to be multiplexed over the same channel at the same
time. Power control for multiple access is crucial for CDMA
to ensure that the received signal can be decoded correctly. In
a CDMA system with self-interested nodes, the transmission
power control problem can be modeled as both the complete
and incomplete information noncooperative games. Also, co-
operative game models can be used for group-rational nodes
in a CDMA system to achieve a Pareto optimal power control
strategy. A summary of the key features of these game models
for CDMA is provided in Table IV. The details of these models
are discussed next.
1) Noncooperative static game-theoretic approach: In [41],
a noncooperative game model is presented for power control.
Each node has an objective to maximize its own utility.
The game considers a multi-carrier direct-sequence CDMA
system in which the data stream for each node is divided into
multiple parallel streams. Each stream is first spread using
a spreading sequence and then transmitted on a carrier. The
strategy of each node is to choose its transmission power.
A high transmission power may yield high SINR and high
transmission rate. However, it may also cause high interference
to the other nodes in the network. The utility of a node
is defined as the ratio of the total throughput and the total
transmission power for all Kcarriers.
Assuming that all the nodes use equal transmission rates, the
utility function of a node can be expressed as the ratio of the
summation of the efficiency functions and the summation of
transmission powers for all Kcarriers. The efficiency function
(f(γ)) represents packet success probability. The utility is a
non quasi-concave function of the transmission power of the
node. Nash equilibrium is considered as a solution. At the
Nash equilibrium, each node transmits only on the carrier
with the best effective channel. This best effective channel
is the channel that requires the least amount of transmission
power to achieve optimal SINR γ∗at the output of the uplink
receiver. Optimal SINR γ∗=γf 0(γ)is the solution to the
efficiency function. A unique Nash equilibrium in this game
can be achieved under a certain set of conditions.
Also, an iterative and distributed algorithm based on best-
response update is proposed to obtain the Nash equilibrium.
The results show that, at the Nash equilibrium, the total
network utility of this multicarrier system is higher than that
of a single carrier system. Also, it is higher than that of a
multicarrier system with the nodes choosing their transmission
powers to maximize their utilities over each carrier indepen-
dently.
In [42], a noncooperative static Bayesian game is presented
for uplink power control in a CDMA network. Each node
chooses its transmission power. The payoff is a function of
the difference between throughput and power consumption.
The throughput part in the payoff function is composed of the
gain from achievable bit rate and a ‘success function’. The
‘success function’ is a Sigmoid function of SINR. Since the
path loss information for the other nodes is not completely
known, each node uses path loss probability density functions
to estimate the SINR (and hence payoff) of the other nodes.
The solution of this incomplete information game is the
Bayesian Nash equilibrium (BNE), which can be obtained
from the best-response dynamics. This dynamics represents the
strategy update rules based on the expected utility when path
loss information is not completely known to the other nodes.
The existence of the Bayesian Nash equilibrium is proved and
it can be obtained in a distributed way.
In [43], a noncooperative power control game for multirate
CDMA networks is studied. All nodes in this multirate CDMA
system use the same chip rate. However, they are able to
adjust their processing gains to increase their data rates. The
objective of the game is similar to that of [41], [42]. However,
the payoff of each node is defined as the difference between
the throughput in bits per second and the cost of transmission
power. The cost that each node has to pay is a function of its
received power divided by the total received power of all nodes
plus noise at the base station. The existence and uniqueness of
the Nash equilibrium are proved for two channel models, i.e., a
binary-input Gaussian output channel and a binary symmetric
channel. Also, the spectral efficiency is derived for both the
channel models.
A joint rate and power control game model is presented
in [44] for uplink CDMA communications. The system model
and the concept of power control and rate updates in this game
are similar to those in [46]. In particular, each node can adjust
both transmission rate and transmission power to maximize its
13
TABLE IV
SUMMARY OF CDMA-BASED CH ANN EL AC CES S GAM ES
Game model Key concept Method of solution Reference
Noncooperative
game
Nodes choose transmission power for all carriers when their
optimal SINRs are taken into consideration.
An iterative best-response update algorithm [41],[42]
Power control game when nodes can adjust the processing
gain in a multirate CDMA system
N/A [43]
The game considers both rate and power control for the uplink
CDMA systems.
An NRPG algorithm when the information of interfer-
ence plus noise is required
[44]
Nodes choose transmission power based on their objective to
minimize cost.
A distributed SIR-based power update algorithm [47]
Cooperative
game
Nodes minimize power consumption while satisfying the
SINR requirements.
N/A [48]
payoff (i.e., utility). The payoff of each node is in bits/J, which
can be calculated from packet length, transmission power,
transmission rate, and an efficiency function which is related
to the SINR. The strategy of each node is to choose rate
and power for transmission. The sets of rate and power are
shown to be convex. The existence of Nash equilibrium for
joint rate and power control is proved by using the Nikaido-
Isoda theorem [45]. Moreover, an algorithm to find the Nash
equilibrium in this noncooperative joint transmission rate
and power control game, shortly called NRPG, is proposed.
The requirement of this algorithm is that each node has to
obtain the SINR of the other nodes. The algorithm is proved
to converge to the same Nash equilibrium when the nodes
are assigned with different initial powers. Also, NRPG can
converge to the solution faster than the algorithm proposed by
Zhao and Lu [46].
Another noncooperative game model for power control is
proposed in [47]. The strategy of each node is to choose
transmission power pi. Each node in this system model has
an objective to minimize its cost (instead of maximizing its
utility). The cost function should be convex and non-negative
and in [47] it is chosen to be a weighted sum of power (βipi)
and square of SIR error (δi(γtar
i−γi)) which is the difference
between the actual SIR and the target SIR. βiand δiare
weighting constants. A distributed algorithm is proposed to
obtain the Nash equilibrium. The power update algorithm is
expressed as follows:
pt+1
i=γtar
ipt
i
γt
i−βi
2δipt
i
γt
i2
(20)
where tdenotes the tth iteration of the algorithm. However,
the information of interference power and SIR is still required
by the node from the base station in order to calculate γt
iin
each iteration. The convergence of the algorithm is proved.
The proof shows that the algorithm converges to a unique
fixed solution under a set of conditions. Also, the proposed
algorithm outperforms the traditional power balance algorithm
(e.g., smaller number of iterations are required for convergence
to a solution, higher efficiency in power saving, and more
nodes can be handled).
2) Cooperative static game-theoretic approach: In [48], a
cooperative game is applied to obtain the optimal power allo-
cation in a CDMA system. A multiuser CDMA system with
perfectly known channel information and fixed signature and
linear sequences is considered. The objective is to minimize
power consumption given minimum SINR of each node. It
is shown that the power region (i.e., a feasible set of power
allocation such that the SINR requirement of each node is
met) is convex and log-convex. If the power region is not
empty, then there is a unique power allocation that satisfies
the SINR requirements of all nodes. To obtain the unique,
Pareto optimal, and proportional fair solution, a bargaining
game similar to that in (3) is formulated and solved. In this
case, a node’s strategy is its transmission power. The results
show that the utility should be appropriately selected as a
function of transmission power. The payoff function can be
chosen to be ui(si) = −esi, where si, node i’s strategy is the
choice of transmission power.
V. GAME MO DE LS O F RANDOM CHA NN EL ACCESS
In this section, the game models for random channel access
are reviewed. In particular, channel access based on ALOHA
and CSMA/CA protocols are considered.
A. Channel access games in ALOHA-like protocols
In the literature, different game models, namely, noncooper-
ative game, cooperative game, evolutionary game, and Stackel-
berg game models have been used for analyzing ALOHA-like
channel access schemes with (and without) power control and
rate adaptation. A summary of these game models is provided
in Table V. The details of these models are described below.
1) Noncooperative game-theoretic approach: In [49], a
noncooperative static game analysis is applied to the slotted
ALOHA protocol with Mselfish nodes. Actions of nodes
are “To transmit” and “Not to transmit”. A node has the
objective to maximize its expected payoff given other nodes’
transmission probabilities. The payoff is zero when a node
chooses not to transmit, one when a node chooses to transmit
and it is successful, and −ciwhen a node chooses to transmit
but it is unsuccessful (here ciis the cost of unsuccessful
transmission for node i). Mixed strategy Nash equilibria are
considered as the solutions which can be described as the
transmission probability (i.e., the probability to perform action
“To transmit” and “Not to transmit”) of the nodes.
In [50], a noncooperative ALOHA game model is presented.
The actions of the nodes are similar to those in [49]. However,
the payoff is the utility defined as the difference between
a logarithmic function of a node’s SINR and the cost of
transmission. Note that the transmission power is assumed to
be identical for all nodes. The channel gains of other nodes
are unknown, and a node’s objective is to maximize its own
14
TABLE V
SUMMARY OF ALOHA-LIK E CHAN NEL ACCESS GA ME S
Game model Key concept Transmission power Information of other nodes Reference
Noncooperative
game
Nodes choose to transmit data or not to transmit data;
however the payoff function of each game model is
different.
Fixed
N/A [49]
Unknown channel states [50],[51],[52]
Nodes choose their access probabilities independently. Indirectly known payoff and
outcome
[53]
Nodes play both random access game and power-controlled
MAC game Not fixed Unknown types of nodes (self-
ish and malicious)
[54]
Known state information
(backlogged nodes)
[55]
Incompletely-
cooperative
game
High and low priority nodes access the channel in both
contention phase and contention-free phase
N/A Both untruthful and truthful
types reported to the access
point
[57]
Pricing mechanism used to enforce nodes to cooperate with
others
Fixed N/A [58]
Evolutionary
game
Nodes are classified into two populations: Transmit and Not
to transmit
Fixed N/A [59]
Stackelberg
game
Nodes are classified into a leader and followers. A leader node
chooses its transmission strategy according to best response
strategies of the followers
Fixed N/A [60]
expected utility. Note that only one node with the highest
channel gain can capture the channel. Each node will gain
zero payoff if it does not transmit. A node will transmit
if its expected payoff is greater than zero. Bayesian Nash
equilibrium is considered as a solution of the game.
In a noncooperative ALOHA game, the Bayesian Nash
equilibrium is always the threshold strategy of a channel
gain. That is, a node will transmit if its channel gain is not
lower than the threshold. The threshold strategy enables the
system to exploit multiuser diversity by giving more chance
of transmission to the node with better channel gain. To find
the optimal strategy, the optimal threshold strategy has to
be obtained first. In this model, only a symmetric case is
considered where the cumulative distribution function (CDF)
of channel gains and weights of the payoff function are
identical for all nodes. The existence of a unique symmetric
Bayesian Nash equilibrium is proved.
Noncooperative Bayesian static ALOHA games are also
presented in [51] and [52]. Both the game models consider
interference. As in [49], a fixed power is assumed in both
the MAC games. The nodes do not know others’ channel
states (i.e., signal-to-noise ratio (SNR)). Each node decides to
transmit or not to transmit the data (i.e., strategies) based on
the SNR. In [51], each node will then obtain its payoff which
is the difference between the utility function of SNR and the
cost function if its transmission is successful. The node will
pay the cost if its transmission is unsuccessful and will gain
nothing if the node makes a decision not to transmit data.
Also, in [52], the payoff is the network throughput expressed
as the difference between a logarithmic function of SNR and
the cost of transmission power. Each node has an objective
to maximize its expected payoff given a belief about other
nodes’ channel states (i.e., probability density functions of
other nodes’ SNRs or channel gains) and the transmission
probabilities. As in [50], only symmetric case is considered,
and Bayesian Nash equilibrium is obtained as the solution of
both of these games. It is found that a node will transmit if
its channel gain is not lower than the SNR threshold. The
existence of a unique symmetric Bayesian Nash equilibrium
is proved. It is mentioned in [52] that in the static game with
symmetric Bayesian Nash equilibrium, a threshold exists such
that the expected payoff of “To transmit” action is equal or
greater than the expected payoff of “Not to transmit” action
if and only if its channel gain is greater than the equilibrium
threshold and the expected payoff is equal to or greater than
zero. The best-response dynamics is used to obtain a pure
Bayesian Nash equilibrium strategy. The convergence time of
the best-response dynamics is of the order of a polynomial of
number of nodes.
In [53], nodes can observe multiple contention signals which
are functions of nodes’ channel access probabilities. The action
of each node is to select the channel access probability. The
payoff of each node is the utility which is the difference
between a function of its channel access probability and the
cost. The cost is defined as a function of contention mea-
sure signals (e.g., collision probability and idle time between
channel access). Nash equilibrium is considered as a solution.
The conditions under which the Nash equilibrium becomes
efficient are established. The utility functions can be defined
by using reverse engineering from existing protocol and by
using forward engineering from desired operating points and
based on heuristics. Since a node can observe the outcome of
others’ actions and payoffs indirectly, the node can use these
observed information to update their distributed algorithms to
converge to the Nash equilibrium.
The dynamics of the random access game is studied. Three
basic dynamic algorithms (i.e., best-response based, gradient-
play based, and Jacobi-play based algorithms) are presented.
Also, a variant of the basic best response-based dynamic
algorithm is proposed when the propagation delay is taken
into account. Moreover, a dynamic algorithm under estimation
error is considered. It is proved that the stochastic gradient-
play algorithm converges to the equilibrium point without
error.
A power-controlled MAC game and a random access game
with incomplete information are presented in [54]. A node
can be either selfish or malicious (i.e., type of node). In the
power-controlled MAC game, the payoff of a selfish node
is the expected value of the difference between a function
of SINR and the energy cost. The payoff of a malicious
node can be defined as two different functions depending
on its opponent nodes. Two utility functions are considered,
15
i.e., based on SINR and Shannon rate. Nash equilibrium
is considered as the solution when the types of the nodes
are known. On the other hand, Bayesian Nash equilibrium
is considered as the solution when the types of the nodes
are unknown. Each node maximizes its expected payoff by
varying its transmission power. A Bayesian random access
game, in which the nodes transmit with probabilities such
that their payoffs are maximized, yields the same result as
that of the power-controlled game. The payoff of a selfish
node is the expected value from successful and unsuccessful
transmissions. Also, each node dynamically updates its belief
about the opponent’s type using Bayes’ rule.
Similar to [54], in [55], both random access game and
power control game are studied. First, a random access game
is presented. The payoff of node iis calculated based on
the expected payoff when it transmits data, and the expected
utility when it waits for transmission. A dynamic random
access game is considered. At each stage of the game, nodes
follow a mixed strategy (i.e., transmit or wait). Each node
maximizes its payoff by choosing the transmission probability
appropriately. The decision at each stage can be described as a
state (i.e., the number of backlogged nodes) which is a general
property of Markov games. Given the state information, the
Markov perfect equilibrium at each stage of the game can be
computed as the Nash equilibrium of the mixed strategy game.
Cooperation among the nodes yields a higher payoff than that
at the Nash equilibrium.
Next, a power control game is presented. Unlike the mixed
strategy of transmitting or waiting in the random access game,
power control is the mixed strategy of each node in the power
control game. Each node selects its transmission probability
with different power levels from a feasible set of power levels
to maximize its expected payoff. This payoff depends on
the probability that the captured power level at the receiver
belongs to any node i.
Similar to the power control game, in a rate adaptation
game, each node selects the probabilities of employing dif-
ferent modulation schemes (i.e., using different transmission
rates) that maximize its payoff. This payoff depends on the
probability that the captured rate belongs to any node i. Zero
power level or zero transmission rate can be considered as
the action of waiting for a transmission. To this end, a joint
power control and rate adaptation game is formulated. Nodes
determine their equilibrium strategies which are both the
power level and transmission rate maximizing their expected
payoffs. Numerical results show that power and rate control
game improves the expected utilities compared to the random
access game discussed in the same paper. However, the joint
power and rate control game incurs a higher computational
complexity.
In all the above works, it is assumed that the network has
the single packet reception capability only. In contrast, in
[56], a noncooperative game model is developed for optimal
decentralized transmission control in a slotted ALOHA-like
protocol for a finite-size random wireless network having the
multipacket reception capability. The objective of each node
in the network is to optimize its transmission probability such
that its own utility is maximized. It is proved that, for a
node, when the probability of successful transmission is a non-
decreasing function of the corresponding SINR, there exists
a threshold transmission policy which maximizes its utility.
Subsequently, it is shown that there exists a Nash equilibrium
at which every node adopts a threshold policy.
2) Incompletely-cooperative game-theoretic approach: In
[57], a game-theoretic model of a slotted ALOHA-like MAC
is presented. The model considers nodes with traffic of ei-
ther high-priority (HP) or low-priority (LP). Since the nodes
transmitting low priority traffic can experience an unfair
channel access (i.e., HP packets have higher probability to be
transmitted than that of LP packets), selfish nodes can cheat
by classifying the low priority traffic as high priority traffic
to gain performance improvement. To solve this problem, an
access point (AP) can decide the size of contention phase
(CP) and contention free phase (CFP). In the contention phase,
LP queues contend for channel with probability qwhile HP
queues contend with probability p > q. The access point can
switch to a contention-free phase for a fraction of the time
αto poll the nLP nodes. Then, the throughput that each LP
node receives during CFP is α/n as shown in Fig. 4.
Average delay and throughput are used to compute the
utility of HP and LP nodes, respectively. Nash equilibrium,
which is a solution of this game, is any fraction of time αin
which the throughput of nodes with LP traffic pretending to
be the HP traffic is lower than the throughput of truthful nodes
with LP traffic. Therefore, Nash equilibrium is the point where
none of the nodes has an incentive to lie about its traffic type.
The access point can choose a value of αfrom an admissible
range to ensure the truthful Nash equilibrium. This point can
be chosen as the Nash bargaining solution from cooperative
game theory.
Fig. 4. High and low priority queues access the channel with different
probabilities during contention phase and low priority nodes are polled equally
during contention-free phase.
In [58], a pricing-based noncooperative slotted ALOHA
MAC game is presented. The key idea of this game is to
motivate the nodes to cooperate with each other by using a
pricing mechanism in the payoff function so that the multiuser
diversity gain can be achieved. A static game is proposed in
16
which the actions of each player i∈I={1, . . . , M }are
“To transmit” and “Not to transmit”. If a player successfully
transmits its packet(s), the payoff is 1−ci−µi, where ciis the
cost of transmission and µiis the price charged per successful
packet transmission. If the transmission is unsuccessful, the
payoff is −ci−vi. If a player chooses not to transmit and it
waits, the payoff is −vi, where viis the waiting cost which
is defined as 1−ci−µi.
In this game, each node maximizes its payoff given the
medium access probabilities of all nodes. The probabilities
of medium access are identical for all nodes since a fair
game is considered. To maximize the expected payoff, a
node will choose “To transmit” when the expected utility
of “To transmit” action is not lower than that of “Not to
transmit” action. Nash equilibrium, which is considered as a
solution, can be found to be of threshold type. The equilibrium
threshold is the cost of the corresponding action. Therefore,
the transmission is successful only if there is exactly one
transmitting node and transmission cost is smaller than the
equilibrium threshold.
3) Evolutionary game-theoretic approach: In [59], an evo-
lutionary game-theoretic model is formulated for ALOHA
protocol. An evolutionary game is a dynamic game where
players interact with other players and adapt their strategies
based on payoff (fitness). The dynamics (i.e., stability) of
the population adopting different strategies is studied. Also,
an evolutionary stable strategy (ESS) is considered. In the
evolutionary game model, if an ESS is reached, the proportions
of population adopting different strategies do not change over
time. In particular, the population with ESS is immune from
being invaded by a population with non-ESS strategy. The
effect of time delay on the dynamics of the evolutionary game
model is studied. Similar to the other ALOHA games, each
player has two possible strategies (i.e., “To transmit” and “Not
to transmit”).
For the two-player case, if a player transmits a packet, it
incurs a transmission cost (c∈(0,1)) irrespective of whether
the transmission is successful or not. The payoffs are 1−
c,0, and −cif the player has a successful transmission, no
transmission, and collision, respectively. It is found that this
game has two pure Nash equilibria (i.e., (Player I - Transmit,
Player II - Not to transmit) and (Player I - Not to transmit,
Player II - Transmit)) and one mixed Nash equilibrium (1−c,
c) where 1−cand crepresent proportions of individuals which
transmit and do not transmit, respectively. The strategy (1−c,
c) can also be an ESS since this strategy is a unique symmetric
Nash equilibrium.
4) Stackelberg game-theoretic approach: In [60], slotted
ALOHA protocols are analyzed using game theory. The model
considers throughput of the system when nodes are of self-
interest and compete for bandwidth using a generalized version
of slotted-ALOHA protocols. First, an analysis based on a two-
state Markov model is presented when the nodes cooperate
to equally share the bandwidth and maximize the system
throughput. The states are “Free state” when the most recent
transmission of node is successful, and “Backlogged state”
when the most recent transmission is unsuccessful due to
collision. The results show that the lower bound of aggregated
throughput is one half and this bound is independent of the
number of nodes. Next, an analysis is presented for the case
when the nodes are selfish to maximize their own throughputs.
Since in this case all nodes transmit with probability one, the
system throughput will be zero.
Next, a Stackelberg game model is presented. A leader
is any node that takes the selfish nodes (i.e., the followers)
into account. The follower and leader nodes choose their
best strategies (i.e., transmission probabilities in both states)
by maximizing their throughputs (i.e., payoffs) subject to
constraints on the budgets of the nodes. The budget should be
higher than the cost of transmission. The followers maximize
their throughputs based on the leader’s strategy while the
leader maximizes its own throughput according to the best
response strategies of followers. Backward induction is used to
find the Stackelberg equilibrium. The leader achieves a higher
throughput than that of the followers when the budget is large.
B. Channel Access Games in CSMA/CA Systems
In this section, the game models formulated for analyzing
CSMA/CA-based channel access are reviewed. The solution
of a CSMA/CA game describes how the nodes in the network
should choose their backoff windows so that the equilibrium
point can be reached. Noncooperative static game-theoretic ap-
proach, noncooperative dynamic game approach, and repeated
game approach can be used to model and analyze CSMA/CA
systems. Since the nodes are selfish, to maximize their payoffs,
the nodes may set the backoff windows to the smallest value.
However, if all the nodes do so, the network throughput will
be zero due to collision. To avoid this problem, incompletely-
cooperative game models are used in which the nodes are
enforced to cooperate in the system by using a penalizing
mechanism. A summary of these approaches is provided in
Table VI. The details are described below.
1) Noncooperative static game-theoretic approach: In [51],
a medium access contention game model is formulated for
the CSMA protocol. Similar to the slotted ALOHA protocol,
the possible results from transmission attempts of each node
are successful transmission, collision, and no-transmission.
Transmission after kbackoff slots is added to the action space
of each node. The action set is A={1, . . . , K, K + 1}, where
k∈K=A\ {K+ 1}={1, . . . , K}denotes transmitting at
slot kand index K+1 denotes the action of not-transmitting a
packet. The payoff function of node iis the difference between
the utility and the cost of transmission if this node selects a
backoff slot number which is less than the backoff slot number
of each of the other nodes. Node iincus a cost of transmission
if a collision occurs, that is, when the earliest backoff slot
chosen by one (or more) of the other nodes is the same as
that of node i. Node igains nothing if it selects a backoff slot
number greater than the lowest backoff slot number among
the other nodes. The nodes play the game by maximizing
their expected payoffs (similar to the ALOHA game models
discussed in Section V-A1 before) given the type spaces (i.e.,
channel SNR, h) and beliefs (i.e., probabilities of channel
states of other nodes, P−i(h−i)). A symmetric mixed strategy
(in terms of the probability that the node will not transmit at
17
TABLE VI
SUMMARY OF CSMA/CA CHAN NEL AC CES S GAM ES
Game model Key concept Symmetric strategy Method of Solution Reference
Noncooperative
game
Nodes choose to transmit or not to backoff when channel
states of nodes are unknown or known to other nodes. Yes N/A [51], [61]
Best-response and gradient update
algorithms
[62]
Dynamic game
Kbackoff slots represent Kstages in a dynamic game. Nodes
choose to transmit or not to transmit. The game ends when
there is at least one transmission at any stage.
Yes N/A [51]
Each time slot represents a stage. Their probabilities of
transmission depend on the number of contending nodes.
Yes A dynamic Bayesian-learning
mechanism
[61]
Each time slot represents a stage. Their probabilities of
transmission depend on the observations of other nodes’
actions
Yes N/A [63]
Incompletely-
cooperative
game
Nodes choose to transmit or to backoff. A penalizing mecha-
nism is used to punish nodes deviating from Nash bargaining
solution.
No A round-based distributed algo-
rithm and a coordination algorithm
[64]
Player 1 is node iand Player 2 corresponds to all the
opponents. Both the players help each other by minimizing
their transmission probabilities.
No A distributed approach used to up-
date nodes’ information
[66]
Repeated game Nodes choose backoff slots to maximize their long-term
throughput. An enforcement mechanism is used to prevent
backoff attacks from misbehaving nodes.
No CRISP cooperative mechanism [67]
the first k∈Kslots) Bayesian Nash equilibrium is found for
this single-stage (static) Bayesian game.
In [61], CSMA/CA is first modeled as a static game and
then as a dynamic game (e.g., Bayesian learning game with
incomplete information). In the static game, the action can be
“To transmit” or “Not to transmit” (i.e., wait). After node i
selects its action, the utility of node iis calculated as a function
of status of the packet transmission and actions of all nodes.
The status can be “idle” (i.e., no transmission), “successful”,
or “fail” (i.e., collision). If the nodes “decide to transmit at the
beginning of a given slot with probability pi” or “stay quiet
with probability 1−pi”, for the same transmission probability
by all nodes (i.e., symmetric behaviour), there is a unique
solution of this static game (s∗
1=· · · =s∗
i=· · · =s∗
M=p∗)
which is a symmetric mixed strategy Nash equilibrium given
as follows:
s∗
i= 1 −uw−uf
us−uf
1
M−1
(21)
where there are Mnodes in the system and us,uf, and uw
are the payoffs that a node obtains if its status is successful,
failed, and idle, respectively. The dynamic game model for
CSMA/CA will be described in Section V-B2.
In [62], a noncooperative game theoretic model is presented
for contention control in a point-to-multipoint network (e.g.,
WiMAX network). Multiple subscriber stations (SSs) are
connected to a base station. A time-division duplex mode
for wireless access is used for best-effort traffic. To provide
multiple access services, a node has a limited number of time
slots to transmit request messages (REQs). The node enters a
contention resolution process when it has packets to send. The
node sets its backoff counter (i.e., the number of slots that the
node needs to wait before it transmits an REQ). If an REQ
is successfully received and there is enough bandwidth, then
the node can transmit data without collision in the scheduled
time slots. The transmission is unsuccessful if a permission
is not received by the node from the base station within a
defined period of time, in which case, a new exponential
backoff process is started. The objective here is to obtain high
throughput by avoiding collision in the system. This can be
achieved by gradually adjusting the contention windows of all
contending nodes to the optimal values.
The game in [62] considers a saturated system (i.e., nodes
always have packets to transmit). The channel access proba-
bility of node ican be found to be pi= 2/(CWi+ 1) where
CWiis a constant contention window of node i. The utility
function has to be continuously differentiable, strictly concave,
and with finite curvatures bounded away from zero. The utility
is chosen to be a function of channel access probability which
is the strategy of a node. The cost of transmission is the
probability of collision (piqi(p)) where qi(i)is the conditional
collision probability of node i. In the game, the utility function
is defined as follows:
µ(pi) = ln(pi)−pi/wi
1/vi−1/wi
.(22)
µ(pi)is an increasing function in the strategy space of node i
(i.e., channel access probability (pi)) where pi∈[vi, wi]and
0< vi< wi<1. The payoff function is then u(pi) = µ(pi)−
piqi(p). Nash equilibrium (defined as p∗) is the solution of
the game. The proof of the existence of unique non-trivial
Nash equilibrium (i.e., ui(p∗
i) = qi(p∗), where qi(i)is the
conditional collision probability of node i), is provided. The
best-response play and gradient play algorithms are presented
to obtain the Nash equilibrium solution. The results show that,
with this algorithm, a higher throughput is achieved with fewer
transmissions than that of standard binary exponential backofff
protocol.
2) Noncooperative dynamic game-theoretic approach: The
single-stage CSMA Bayesian game in [51] described before is
extended to a dynamic game where the static one-stage game
is played repeatedly. The action of node ican be either to
transmit a packet or not to transmit a packet based on node
i’s channel gain hiand node i’s type. Kstages associated with
Kbackoff slots are considered in this Bayesian dynamic game.
At stage k∈ {1, . . . , K}, if node isuccessfully transmits its
packet, it will obtain the payoff function, µi(hi)−ci(hi)where
µiis the utility function and ci(hi)is the cost function. If
node iunsuccessfully transmits its packet, it will pay −ci(hi)
as a cost of transmission; otherwise, node igains nothing
18
(i.e., zero payoff). If there is no transmission, the stage of
the game increases from kto k+ 1. When there is at least
one transmission at any stage k, the game ends. Each node
maximizes its expected payoff from stage 1 to kto obtain the
perfect Bayesian equilibrium (PBE).
A symmetric PBE is considered since it is a proper operating
point of a distributed protocol for the following reasons.
First, it might not be possible to distinguish among nodes in
the random access network. Second, asymmetric PBE is not
sustainable since it causes unfairness problem by assigning
unequal shares of channel to the nodes. Third, it is much
simpler to operate a network with a single strategy in a
symmetric equilibrium for all nodes than to operate a network
with different strategies for different nodes. The symmetric
PBE is shown to be a threshold strategy. That is, any node i
decides to transmit at stage kwhen its SNR is greater than
SNR threshold hk
th (i.e., hi> hk
th). The numerical results
show that the proposed protocols provide better robustness
and higher multi-user diversity gain than those of conventional
random access protocols.
The static game is extended to a dynamic Bayesian-learning
game with the unknown number of contending nodes in [61]
(i.e., the static game is played repeatedly). There are maximum
of Mnodes in the system. These nodes compete for transmis-
sion at the beginning of a time slot. Fairness requirement is
considered. In order to maximize the payoffs of the nodes, only
symmetric strategy with s∗
1=· · · =s∗
i=· · · =s∗
M=p∗,
which is the probability of transmission, is considered. Each
time slot corresponds to a stage in this dynamic game. Since
the number of contending nodes is unknown, each node needs
to observe the feedbacks from its previous play. Then, the node
can build its belief about the network using Bayes’ rule.
It is assumed that all nodes can keep track of their historical
information perfectly. Three counters (i.e., the total number of
passed time slots, the total number of successfully transmitted
data packets, and the number of times that ACK control-frame
is not received in a pre-defined time space after transmitting
the data packet) are used to compute the belief. The current
packet transmission probability and the frame collision prob-
ability of each node can be computed by these three counters
under the assumption that all nodes always have packets to
transmit. Each node has to listen to the channel to receive any
possible packet from its neighbouring nodes when it is idle.
After that, each node can obtain the posterior belief of the
number of concurrently contending nodes (n) for the channel
by using both the transmission and collision probabilities (i.e.,
pand q, respectively) as follows:
n=f(p, q) = 1 + log(1 −q)
log(1 −p).(23)
It is found that the equilibrium optimal solution of this game
depends on the number of contending nodes n. The trans-
mission probability can be varied according to the following
equation in which only contention parameter CWmin (i.e.,
minimum contention window) is considered:
p=2(1 −2q)
(1 −2q)(CWmin + 1) + q.CWmin.(1 −(2q)n)(24)
where CWmin = min([n×rand(7,8)], CWmax ),rand(x, y)
returns a random value between xand y, and nis the number
of concurrently contending nodes. The game state ncan
be then updated using (23) after a node updates its beliefs.
Simulation results show that the performance of the dynamic
CSMA game-based MAC is superior to the IEEE 802.11 DCF
MAC in terms of throughput, delay, and packet-loss rate.
Another dynamic game model for CSMA/CA is proposed in
[63], where the probabilities of transmission are the strategies
of the nodes. Each node estimates its conditional collision
probability and adjusts the persistence probability. The payoff
of each node is the difference between the utility when the
node accesses the channel with probability piand the cost (i.e.,
probability of collision). For a network with homogeneous
nodes, the game model has a unique nontrivial Nash equilib-
rium which is a symmetric equilibrium. This guarantees fair
sharing of wireless channel among the same class of nodes.
Next, the dynamics of the game is studied. Although one node
can observe the outcome of other nodes, it does not have
complete knowledge of actions and payoffs of other nodes.
Every node adjusts its current channel access probability
gradually in the gradient direction based on the observations
of other nodes’ actions. Then, the Nash equilibrium can be
reached.
Based on the dynamics of this random access game, a
new MAC protocol based on CSMA/CA is proposed. Instead
of executing exponential backoff upon collisions, each node
estimates its collision probability and contention window
according to gradient play. Each node can estimate its condi-
tional collision probability by observing the average number
of consecutive idle slots. Therefore, the size of contention
window can be adjusted accordingly. Throughput and short-
term fairness of the proposed MAC protocol are better than
those of the IEEE 802.11 DCF protocol. Service differentiation
is also considered. When there is more than one class of nodes,
the throughput ratio can converge to a constant when the total
number of nodes increases.
3) Incompletely-cooperative game-theoretic approach: In
[64], a CSMA/CA-based MAC game model is presented for
dynamic spectrum access in a cognitive radio network. This
game model can be divided into two sub-games. The first sub-
game is a channel allocation game in which the nodes compete
to allocate radio interfaces to the channels. The second sub-
game is a multiple access game among the nodes contending to
transmit packets in the same channel. The available frequency
band is divided into Kchannels of the same bandwidth. Each
node is equipped with lradio interfaces (for l < K). Each
node can hear other nodes’ transmissions if the same channel
is used. Each node determines the number of interfaces to
be used in each channel. This is the action of nodes in the
first sub-game of channel allocation. Each node maximizes
its utility function which is the sum of throughputs achieved
by the node in all allocated channels. Each node can observe
other nodes’ information perfectly. The solution of the channel
allocation game is the Nash equilibrium if the difference
between the number of interfaces in any channel xand that
in any other channel yis lower than or equal to 1. Also, the
number of interfaces allocated to any channel xby node iis
19
lower than or equal to 1 for any channel y. It is found that if
the rate function of each channel is independent of the number
of interfaces in any channel, then any Nash equilibrium of
channel allocation is Pareto optimal. The existence of Nash
equilibrium is shown and its efficiency (i.e., price of anarchy)
is studied. It is found that the price of anarchy is close to
one (i.e., Nash equilibrium yields a payoff close to that of the
socially optimal solution).
Next, the second sub-game for CSMA/CA channel con-
tention is formulated. This sub-game aims not only to optimize
the network performance, but also to provide incentives to
the nodes to behave optimally. The actions of the nodes are
“To transmit”, “Not to transmit”, and “To backoff” in which a
contention window value between one and the maximum value
is chosen by a node. Node iselects the value of contention
window on each channel cto maximize its throughput (i.e.,
payoff). The static CSMA/CA game shows that the Nash
equilibrium (i.e., contention window is chosen to be one) is
inefficient and unfair.
A desirable solution for the CSMA/CA game should have
three properties: uniqueness, per-radio fairness, and Pareto
optimality. Using the Nash bargaining framework from the co-
operative game theory, these three properties can be achieved.
However, in the noncooperative regime, the Nash bargaining
solution is not a Nash equilibrium and might not be stable.
Therefore, a penalizing mechanism is introduced by which
the node deviating from Nash bargaining solution will be
punished. A jamming mechanism is presented to penalize
the deviating node. The deviating node is selectively jammed
for a short duration by other nodes using the same channel
when the deviating node is detected doing selfishly for its
transmission. Using the penalty function and the jamming
mechanism, the game can reach a Nash equilibrium unilateral
deviation from which is not profitable. A distributed algorithm
is proposed to obtain the Pareto-optimal Nash equilibria.
The algorithm can converge to the equilibrium point even in
case of imperfect information. The algorithm is based on a
round-based distributed algorithm [65]. Also, a coordination
algorithm is proposed for CSMA/CA in which one node acts as
a coordinator for the observed channel by inflicting penalties
to the other nodes which receive a higher throughput.
Fig. 5. Strategies of two players when Player 1 is node iand Player 2 refers
to all opponent nodes [66].
In [66], an incompletely cooperative game model is pre-
sented for wireless mesh networks. The routers and clients
communicate wirelessly with each other in a mesh architec-
ture. CSMA/CA is used as a channel access mechanism. Since
a packet can be retransmitted only for a certain number of
times, the game model can be formulated as a finite repeated
game. The number of opponents can be estimated by using
conditional collision probability and transmission probability.
These probabilities are computed by the node using two local
counters (the total number of successfully transmitted data
frames and the total number of transmitted data frames which
are unsuccessful). However, these estimations are accurate
only under saturated conditions (i.e., nodes always have pack-
ets to transmit).
A virtual CSMA/CA mechanism, V-CSMA/CA is proposed.
V-CSMA/CA follows the CSMA/CA scheme but it handles
virtual frames. V-CSMA/CA will send a virtual frame and then
the probability of collision is estimated. If the channel is idle,
the node will observe that the virtual frame is successfully
transmitted. A collision of virtual frame will be detected
whenever any other node chooses the same time slot for
transmission of their real data frame. If the node has no packet
to transmit, the game state is estimated by using V-CSMA/CA.
If the node has packets to transmit, the game state is estimated
by using CSMA/CA.
In the analysis of the game, a player is not always fixed.
That is, Player 1 stands for node iand Player 2 stands
for all other opponent nodes. The actions of Player 1 are
“Transmission” and “Backoff”. The actions of Player 2 are
“Successful Transmission” (i.e., Player 1 selects “Backoff”
and none of the nodes in the group of Player 2 selects
“Transmission”), “Unsuccessful Transmission” (i.e., Player
1 selects “Backoff”, but a node in the group of Player 2
selects “Transmission”), or “Backoff” as shown in Fig. 5.
µi
b,µi
f, and µi
sare utility obtained when Player ichooses
to backoff, to transmit and transmission is successful, and
to transmit but transmission fails, respectively. The optimal
strategy of each player (s∗
i) can be found by minimizing its
payoff by varying the transmission probability pi. It can be
considered as a mixed strategy solution. Then, the cooperation
among the nodes considered. In a two-node scenario, one node
adjusts its transmission probability to help the other node to
achieve the optimal payoff (i.e., p∗
1= arg minp1u1(p2, p1)
and p∗
2= arg minp2u2(p1, p2), where u1(·)and u2(·)are
payoff functions of Players 1 and 2 which are based on each
other’s utility of transmission, respectively.
The transmission probabilities can be changed by tuning
the MAC contention parameters as in [61]. After estimating
the game state, the mesh router broadcasts its estimated
information to all nodes. Since in a dynamic network frequent
information updates are needed, this may result in large over-
head. To reduce the overhead, a distributed approach is used by
each node to detect the channel, estimate the game state, and
adjust the contention parameters. Moreover, the estimation is
performed after a packet is transmitted or a packet is discarded
(rather than in every time slot). Also, the contention parameter
is adjusted accordingly. The simulation results show that the
incompletely cooperative game can improve throughput and
decrease delay, jitter, and packet loss rate. The fairness of this
game is comparable to that of the IEEE 802.11 DCF protocol.
4) Incompletely-cooperative repeated game-theoretic ap-
proach: In [67], a game-theoretic study of CSMA/CA under
a backoff attack is presented. An enforcement mechanism
20
is introduced for the misbehaving nodes in the network.
Although this enforcement mechanism is similar to that in
[64], here it is used in the context of a mobile ad hoc network
and the game formulation is for a repeated game in which a
long-term utility is to be maximized. First, a noncooperative
game is formulated for a finite number of nodes. Each node
chooses an action which is a backoff configuration from a
feasible set. The payoff function is defined to be the bandwidth
share function depending on the backoff configuration profile
(s= (s1, . . . , si, . . . , sM)). Nash equilibrium is the solution of
this one-shot noncooperative game which might be unfair or
inefficient. Therefore, to obtain a better solution, a repeated
game is proposed in which a node takes into account the
effect of its current action on the future actions of other nodes.
The number of stages is finite and should be large enough to
approach the steady state values. Nodes can switch between
standard or non-standard backoff configuration (i.e., fair or
more-than-fair bandwidth share, respectively) to maximize
their own long-term payoffs.
To prevent the backoff attack and to obtain a fair Pareto
optimal and sub-game perfect Nash equilibrium, a strategy
profile called cooperation via randomized inclination to self-
ish/greedy play (CRISP) is introduced. This optimal solution
is a probability distribution over the selected backoff configu-
ration at stage k(sk
i). An invader node deviating from CRISP
will experience lower bandwidth than that of nodes playing
CRISP.
VI. SUMMARY OF GAME MOD EL S AN D OPE N RESEARCH
ISS UE S
A. Summary of Game Models
Table VII summarizes the game models formulated for the
key multiple access mechanisms. In the channel access games
for TDMA, nodes compete with each other to obtain time slots
for their transmissions. Time slot allocation among the nodes is
performed by using various game models. In the auction game
models, the nodes bid for time slots and they have to pay to the
base stations for the allocated time slots. Game models can be
formulated in which the nodes are able to choose transmission
power in their allocated time slots. To enforce cooperation
among the nodes, a punishment and truth-telling mechanism
can be used.
In the channel access games for FDMA, most of the models
consider how nodes (with single or multiple radio interfaces)
choose channels for transmission. In these game formulations,
the number of radios, transmit rate, and power rate assigned
to each channel correspond to nodes’ actions. In the channel
access games for CDMA, power control is the key objective of
all the proposed games. Both cooperative and noncooperative
games can be formulated. Nodes select their transmission
powers to meet their requirements in terms of SINR and
transmission cost.
In most of the ALOHA-like game models, the nodes can
choose either “To transmit” or “Not to transmit” as their
possible actions and the transmission powers of the nodes are
assumed to be fixed. Then, the games have mixed strategy
solutions. Some of the games can be shown to have solutions
which are threshold strategies. In some of the CSMA/CA
game models, the actions are “To transmit” and “To wait
for kbackoff time slots”. The solutions of these game mod-
els are mixed strategies (i.e., the transmitting probability of
nodes at the first ktime slots). In some CSMA/CA-like
MAC game models, the action set of nodes is defined as
transmission probabilities. In addition, most of the CSMA/CA-
like MAC game models consider only the symmetric strategy
case by assuming that all nodes are identical and throughput
maximization is the key objective. Since in random access
schemes, nodes access the channel(s) in a distributed manner,
some nodes may misbehave. A penalizing mechanisms is
required to address this problem. A summary of game models
for contention-free channel access approaches is shown in
Table VIII and a summary of game models for random channel
access approaches is shown in Table IX.
B. Open Research Issues
Based on the summary of the game models presented in
the previous section, several open research issues on the
application of game theoretic models for design, analysis, and
optimization of multiple access schemes for wireless networks
can be identified as follows:
•Investigation of different equilibrium concepts: Many
works in channel access games only study Nash equilib-
rium as the solution concept. However, Nash equilibrium
does not always provide the best network performance.
Therefore, other equilibrium concepts need to be inves-
tigated. For example, Nash bargaining solution, which
is Pareto optimal, can be considered for efficiency and
fairness reasons. Another solution is correlated equilib-
rium which is a more general equilibrium concept than
the Nash equilibrium and also incurs less computational
complexity. It can be computed in polynomial time for
essentially all kinds of multi-player games.
•Utility and cost function design: In the application of
game theory to multiple channel access, utility and cost
indicate, respectively, the preference of players to be
maximized and to be minimized for channel access. There
are many different functions used to represent payoff of
players in channel access games. Comparisons among the
utility functions in terms of performance and effective-
ness may be required. Also, the utility functions should
consider system parameters such as the priorities of pack-
ets, amount of battery power available, and application-
layer QoS parameters. To obtain suitable utility functions
that can converge to desirable equilibria, reverse engineer-
ing from existing protocols, desired operating points, or
forward engineering from heuristics [68] can be used.
•Cooperation enforcement mechanism: In random channel
access games, many works propose cooperation enforce-
ment mechanisms in noncooperative scenarios. Penalizing
and pricing mechanisms are introduced. However, the
complexity and implementation issues of these mecha-
nisms (e.g., jamming and detection mechanisms) need to
be investigated while developing practical protocols. In
this case, a pricing mechanism may be preferred over a
21
TABLE VII
SUMMARY OF CHAN NEL AC CES S GAM ES
Access Scheme Summary
TDMA In TDMA access games nodes compete for time slots to achieve their objectives and meet QoS requirements. Noncooperative static
game, auction game, dynamic game, and repeated game models can be applied for TDMA.
FDMA In FDMA, nodes compete for the available channels in the network (e.g., through an auction mechanism). The solution in terms
of equilibrium can be achieved for the complete and incomplete information cases. Noncooperative static game, auction game, and
cooperative game models can be used for FDMA.
CDMA In a CDMA system, each node is assigned with a different code to allow multiple users to be multiplexed over the same channel at
the same time. Power control is crucial for CDMA to ensure that the received signal can be decoded correctly. In a CDMA system
with self-interested nodes, the transmission power control can be modeled as complete and incomplete information noncooperative
games. Also, cooperative game model for group-rational nodes can be used to achieve a Pareto optimal power control strategy.
ALOHA Noncooperative game, cooperative game, evolutionary game, and Stackelberg game models can be used for ALOHA-like channel
access. For the majority of the models, the solution is a threshold strategy. Along with channel access, power control and rate
adaptation are also considered in the models.
CSMA/CA In CSMA/CA games, the nodes in the network choose their backoff windows so that the equilibrium point can be achieved.
Noncooperative static game, noncooperative dynamic game, and repeated game models can be applied for CSMA/CA. Since the
nodes can be selfish (i.e., to maximize their payoffs, they may set the backoff windows to be the smallest value), a penalizing
mechanism is required for the misbehaving nodes.
TABLE VIII
GAM E THE ORY M ODE LS FO R CO NTE NT ION -FR EE CH AN NEL A CCE SS
Issues Game type Player Strategy Payoff Solution
Time slot competition [23] Noncooperative game Mobile nodes Fraction of time in the slot Time slot usage Nash equilibrium
Time slot competition [25] Auction game Mobile nodes and a base station Bid value between 0 and 1 Amount of data transmitted Nash equilibrium
Time slot competition [26] Auction game Mobile nodes and a base station Monetary units Amount of data transmitted Nash equilibrium
Time slot competition and Markov game Secondary nodes in Transmission rate Discounted transmission cost Nash equilibrium
transmission rate control [27] TDMA cognitive radio
Time slot competition and Markov game Secondary nodes in Transmission rate Discounted transmission cost Correlated equilibrium
transmission rate control [28] TDMA cognitive radio minus a function of transmission delay
Time slot allocation [29] Repeated game Mobile nodes Power allocated Long-term throughput Nash equilibrium
to each time slot
Time slot allocation [29] Repeated game with Mobile nodes Channel gain Transmission rate plus Nash equilibrium
incomplete information to each time slot transfer function
Channel allocation [30] Noncooperative game Mobile nodes with The number of radios assigned Throughput minus cost Strongly dominant -
multiple radios to each channel strategy equilibrium
Sub-channel allocation [32] Noncooperative game Mobile nodes Rate assigned to each sub-channel Transmission power minimization Nash equilibrium
Channel allocation [33] Noncooperative game Mobile nodes with The number of radio assigned Achieve bit rate Nash equilibrium
multiple radios to each channel
Sub-channel and Noncooperative game Base stations Transmission power and sub-channel Data rate Nash equilibrium
power allocations [34] assigned to mobile nodes
Channel allocation [35] Noncooperative game Secondary nodes Channel selection Function of SINR Nah equilibrium
Channel allocation [35] Stackelberg game Secondary nodes Channel selection Function of SINR and Nash equilibrium
cost of switching channel
Transmission power allocation [36] Auction game Mobile nodes and a base station Power control variable Shannon capacity Nash equilibrium
and bid value
Cooperative group formation [38] Coalitional game Secondary base stations Coalition’s member selection Utility from learning new channels Nash equilibrium
in multi-channel network minus cost of cooperation (Nash-stable formation)
Sub-channel assignment and [40] Coalitional game Mobile nodes Transmission power assigned Function of Shannon- Nash equilibrium
power allocation to each channel capacity
Power control in CDMA [41] Noncooperative game Mobile nodes Transmission power selection Fraction of throughput Nash equilibrium
to transmission power
Power control in CDMA [42] Bayesian game Mobile nodes Transmission power selection Difference between Bayesian Nash equilibrium
throughput and power consumption
Power control in CDMA [43] Noncooperative game Mobile nodes Transmission power selection Difference between throughput Nash equilibrium
(bits/second) and cost of transmission power
Joint rate and power Noncooperative game Mobile nodes Transmission power and Utility in Nash equilibrium
control in CDMA [44],[46] rate selections bits/J
Power control in CDMA [47] Noncooperative game Mobile nodes Transmission power selection cost a function of Nash equilibrium
power and SIR
Optimal power allocation [48] Bargaining game Mobile nodes Transmission power selection Any appropriate function Nash bargaining solution
penalizing mechanism since pricing schemes are easier
to be implemented by service providers.
•Uncertainty in wireless and mobile networks and accu-
racy of available information: Some game models are
formulated as dynamic or repeated games in order to
analyze the outcomes in long-run periods. Most games
assume that the number of nodes in the network is
constant; however, in realistic scenarios, the number of
nodes changes over time (i.e., nodes leave or join the
network). In random channel access games, the number
of nodes can be estimated by using probabilities of
packet transmission and collision. It might be acceptable
if the time duration of one stage of the game is short
or the game is not played infinitely. However, it is not
completely true for all games. Moreover, most games
assume that the nodes completely know other nodes’
information. The assumption may not always hold in a
mobile wireless network. Then, unknown information has
to be estimated based on some available knowledge of
mobiles and their belief models (i.e., probability distribu-
tion). Currently, there are few works using game-theoretic
approaches with incomplete information. Bayesian game
and Bayesian-learning mechanism are useful tools to
study the multiple access problem under incomplete in-
formation. More works in this area are required to model
and analyze distributed multiple access methods in large-
scale wireless networks.
•Cost of information: The Bayesian Nash equilibrium for
incomplete information game may be inefficient due to
the lack of complete information. Nodes can implement
the information gathering mechanism so that a better
decision can be made. However, the cost of information
22
TABLE IX
GAME MODELS FOR RANDOM CHANNEL ACCESS
Issues Game type Player Strategy Payoff Solution
Slotted ALOHA channel access [49] Noncooperative game Mobile nodes “To transmit” and “Not to transmit” Reward and cost of transmission Mixed strategy Nash equilibrium
Slotted ALOHA channel access [50] Bayesian game Mobile nodes “To transmit” and “Not to transmit” Function of SINR minus Bayesian Nash equilibrium
with unknown channel gains cost of transmission (Strategy threshold)
Slotted ALOHA channel access [51] Bayesian game Mobile nodes “To transmit” and “Not to transmit” Function of SNR minus Bayesian Nash equilibrium
with unknown SNR cost function of SNR (Strategy threshold)
CSMA/CA channel access [51] Noncooperative game Mobile nodes Transmission and Backoff Reward and cost of transmission Mixed strategy Nash equilibrium
CSMA/CA channel access [51] Dynamic game Mobile nodes “To transmit” and “Not to transmit” Utility and cost- Perfect Bayesian
with incomplete information function of SNR equilibrium
Interference-aware MAC [52] Bayesian game Mobile nodes “To transmit” and “Not to transmit” Function of SNR minus Bayesian Nash equilibrium
with unknown channel gains cost of transmission power (Strategy threshold)
Multiple channel access[53] Noncooperative game Mobile nodes Channel access probabilities Function of access probability Nash equilibrium
minus function of contention signals
Power control / static game/ Bayesian game Mobile nodes Transmission power/ SINR minus energy cost/ Nash/ Bayesian Nash equilibrium
random access [54] “To transmit” and “Not to transmit” Reward and cost of transmission (Strategy threshold)
Power control, joint power and Noncooperative static/ Mobile nodes Transmission power and rate / Function of captured packets/ Mixed strategy Nash equilibrium
rate control /random access [55] Markov game “To transmit” and “Not to transmit” power level/ rate probabilities/
ALOHA-like MAC [57] Incompletely-cooperative Access point Fraction of time Delay (HP nodes) and Nash equilibrium and
with prioritized nodes game throughput (LP nodes) Nash Bargaining Solution
Slotted ALOHA channel access [58] Incompletely-cooperative Mobile nodes “To transmit” and “Not to transmit” Transmission cost, waiting cost Nash equilibrium
game and price (Strategy threshold)
Slotted ALOHA channel access [59] Evolutionary game Subpopulations of mobile nodes “To transmit” and “Not to transmit” Cost of transmission Evolutionary stable strategy
using same strategies
Slotted ALOHA channel access [60] Stackelberg game Mobile nodes Transmitting probability Throughput Stackelberg equilibrium
New CSMA/CA-like MAC [61] Noncooperative game Mobile nodes “To transmit” and “Not to transmit” Not specified Mixed strategy Nash equilibrium
New CSMA/CA-like MAC [61] Dynamic game Mobile nodes “To transmit” and “Not to transmit” Not specified Mixed strategy Nash equilibrium
Contention control in WiMAX [62] Noncooperative game Mobile nodes Transmitting probability Function of channel access probability Non-trivial Nash equilibrium
New CSMA/CA-like MAC [63] Dynamic game Mobile nodes Transmitting probability Function of transmission probability Mixed strategy Nash equilibrium
minus collision probability
CSMA/CA channel access[64] Incompletely-noncooperative Mobile nodes Transmission and Backoff Throughput Pareto-optimal
game Nash equilibrium
CSMA/CA channel access [66] Incompletely-noncooperative Mobile nodes Transmission and Backoff Not specified Mixed strategy Nash equilibrium
CSMA/CA under attack [67] Repeated game Mobile nodes Transmission and Backoff Bandwidht share Perato optmal and subgame-
perfect Nash equilibrium
collection needs to be considered and a cost-benefit
analysis needs to be performed.
•Develop defense mechanisms against multiple access
attack: Defending attacks from untrusted nodes is an
important research issue in wireless networks. Malicious
nodes prevent other nodes from accessing the network
(i.e., denial-of-service (DoS) attack). For example, jam-
ming attack is a form of DoS attack. There are some
channel access games which take security issues into
consideration. However, most of these game-theoretic
models analyze the interactions between pairs of jammers
and communicating nodes or between pairs of jammers
and detecting nodes when an intrusion detection system
(IDS) is deployed [69]-[70]. A few researches such as
those in [71]-[73] formulate games between jammers and
communicating nodes in order to analyze equilibrium
points as defense strategies against jamming attacks.
More researches in this area are required. For example,
we can formulate a noncooperative game with incom-
plete information when types of nodes such as selfish,
malicious, or defending (i.e., when IDS is deployed) are
incompletely known to other nodes.
•Multiple access in heterogeneous wireless networks:
Channel allocation and multiple access problem in a het-
erogeneous wireless access network can be modeled by
using game theory. In a heterogeneous network, the ser-
vice providers for the different wireless access networks
as well as the mobile users (who are able to access the
different networks using multiple radio interfaces) may
cooperate or compete with each other. Game theoretic
modeling and analysis of the interactions among the mo-
bile nodes and service providers is an interesting research
topic. Again, in a heterogeneous network where a bi-
level hierarchy exists (e.g., in a cellular network where
macrocells are underlaid with femtocells), transmission
power and rate control problem (e.g., by the macro base
stations and the femto access points) for multiple access
can be modeled by using game theory, and an optimal
multiple access method can be designed.
•Develop application-centric game models for multiple
access: Traditional multiple access schemes may not be
efficient in some specific wireless access scenarios such
as in vehicular networks and wireless sensor networks.
There are two steps in designing MAC protocols for a
specific application [74]. First, the application require-
ments and the resource constraints are specified. Next, a
protocol that satisfies all these constraints is designed. For
example, in vehicular networks, high mobility that causes
the topology of the network to vary rapidly is one of the
application specifications, and transmission delay (i.e.,
emergency messaging delay) is an application require-
ment. Limited bandwidth due to high vehicle mobility and
vehicle density is one of the resource constraints. These
specifications and requirements need to be considered
when designing a multiple access method for vehicular
networks. For a game theoretic multiple access scheme,
the utility functions for the players should take the
related parameters into account. Due to the applications’
specific requirements and specifications, developing game
models for application-centric multiple access is more
challenging.
VII. CONCLUSION
Game theory has been widely used to model and analyze the
noncooperative and cooperative behaviours of mobile nodes
in the context of multiple access in wireless networks. The
game models are useful for designing distributed channel
access mechanisms in wireless networks to achieve stable
and efficient solutions. This article has presented a compre-
hensive survey of the game models developed for multiple
access in wireless networks. These game models have been
23
categorized based on the types of protocols (e.g., contention-
free or contention-based) and types of games (e.g., complete
and incomplete information, and static and dynamic games).
The major findings from these game models have been dis-
cussed. Unavailability of complete information, misbehaviour
of nodes, consideration of system (i.e., implementation) as-
pects give rise to major challenges in designing game theoretic
design of multiple access schemes in wireless networks. From
these perspectives, several open research directions have been
outlined.
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Khajonpong Akkarajitsakul received the B.E. and
M.E. degrees in computer engineering from King
Mongkut’s University of Technology Thuonburi,
Bangkok, Thailand, in 2005 and 2007, respectively.
He is currently working toward a Ph.D. degree in
Electrical and Computer Engineering at the Uni-
versity of Manitoba, Winnipeg, MB, Canada. His
research interests are in vehicular networks and
applied game theory.
Ekram Hossain (S’98-M’01-SM’06) is currently
a Professor in the Department of Electrical and
Computer Engineering at University of Manitoba,
Winnipeg, Canada. He obtained his Ph.D. in Elec-
trical Engineering from the University of Victoria,
Canada, in 2001. Dr. Hossain’s primary research
interest is in the area of resource management and
multiple access in wireless and mobile communica-
tions networks, and cognitive radio systems. He is
an author/editor of the books “Cooperative Cellular
Wireless Networks” (Cambridge University Press,
2011), “Dynamic Spectrum Access and Management in Cognitive Radio
Networks” (Cambridge University Press, 2009), “Heterogeneous Wireless
Access Networks” (Springer, 2008), “Introduction to Network Simulator NS2”
(Springer, 2008), “Cognitive Wireless Communication Networks” (Springer,
2007), and “Wireless Mesh Networks: Architectures and Protocols” (Springer,
2007). Dr. Hossain serves as the Area Editor for the IEEE Transactions on
Wireless Communications in the area of “Resource Management and Multiple
Access”, an Editor for the IEEE Transactions on Mobile Computing, the IEEE
Communications Surveys and Tutorials, and IEEE Wireless Communications.
He is a registered Professional Engineer in the Province of Manitoba, Canada.
25
Dusit Niyato (M’09) is currently an Assistant Pro-
fessor in the School of Computer Engineering, at
the Nanyang Technological University, Singapore.
He received B.E. from King Mongkut’s Institute
of Technology Ladkrabang (KMITL) in 1999. He
obtained his Ph.D. in Electrical and Computer Engi-
neering from the University of Manitoba, Canada in
2008. His research interests are in the area of radio
resource management in cognitive radio networks
and broadband wireless access networks.
Dong In Kim (S’89-M’91-SM’02) received the B.S.
and M.S. degrees in Electronics Engineering from
Seoul National University, Seoul, Korea, in 1980
and 1984, respectively, and the M.S. and Ph.D.
degrees in Electrical Engineering from University of
Southern California (USC), Los Angeles, in 1987
and 1990, respectively. From 1984 to 1985, he
was a Researcher with Korea Telecom Research
Center, Seoul. From 1986 to 1988, he was a Korean
Government Graduate Fellow in the Department of
Electrical Engineering, USC. From 1991 to 2002, he
was with the University of Seoul, Seoul, leading the Wireless Communications
Research Group. From 2002 to 2007, he was a tenured Full Professor in
the School of Engineering Science, Simon Fraser University, Burnaby, BC,
Canada. From 1999 to 2000, he was a Visiting Professor at the University
of Victoria, Victoria, BC. Since 2007, he has been with Sungkyunkwan
University (SKKU), Suwon, Korea, where he is a Professor and SKKU Fellow
in the School of Information and Communication Engineering. Since 1988,
he is engaged in the research activities in the areas of wideband wireless
transmission and access. His current research interests include cooperative
relaying and base station cooperation, multiuser cognitive radio networks,
and cross-layer design for interference management in wireless networks.
Dr. Kim was an Editor for the IEEE Journal on Selected Areas in Com-
munications: Wireless Communications Series and also a Division Editor for
the Journal of Communications and Networks. He is currently an Editor for
Spread Spectrum Transmission and Access for the IEEE Transactions on
Communications and an Area Editor for Cross-Layer Design and Optimization
for the IEEE Transactions on Wireless Communications. He also serves as co-
Editor-in-Chief for the Journal of Communications and Networks.
