# Coalition formation games for relay transmission: Stability analysis under uncertainty.

**ABSTRACT** Relay transmission or cooperative communication is an advanced technique that can improve the performance of data transmission among wireless nodes. However, while the performance (e.g., throughput) of a source node can be improved through cooperation with a number of relays, this improvement comes at the expense of a degraded performance for the relay nodes due to the resources that they dedicate for helping the source node in its transmission. In this paper, we formulate a coalitional game among the wireless nodes that seek to improve their performance by relaying each other's data. The game is classified as a coalition formation game in which the nodes can take individual and distributed decisions to join or split from a given coalition while ensuring that their individual throughput is maximized. A Markov chain model is proposed to investigate the stability of the resulting coalitional structures. Further, we consider the practical case in which the wireless nodes do not have an exact and perfect knowledge of the parameters (e.g., channel quality) in coalition formation. For this scenario, we analyze the stability of the partitions resulting from the proposed coalition formation game under uncertainty. We also define the conditions needed for obtaining the stable and unstable coalitional structures among the nodes that are performing cooperative transmission.

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**ABSTRACT:**Cooperative transmission in vehicular networks is studied by using coalitional game and pricing in this paper. There are several vehicles and roadside units (RSUs) in the networks. Each vehicle has a desire to transmit with a certain probability, which represents its data burtiness. The RSUs can enhance the vehicles' transmissions by cooperatively relaying the vehicles' data. We consider two kinds of cooperations: cooperation among the vehicles and cooperation between the vehicle and RSU. First, vehicles cooperate to avoid interfering transmissions by scheduling the transmissions of the vehicles in each coalition. Second, a RSU can join some coalition to cooperate the transmissions of the vehicles in that coalition. Moreover, due to the mobility of the vehicles, we introduce the notion of encounter between the vehicle and RSU to indicate the availability of the relay in space. To stimulate the RSU's cooperative relaying for the vehicles, the pricing mechanism is applied. A non-transferable utility (NTU) game is developed to analyze the behaviors of the vehicles and RSUs. The stability of the formulated game is studied. Finally, we present and discuss the numerical results for the 2-vehicle and 2-RSU scenario, and the numerical results verify the theoretical analysis.02/2013;

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Coalition Formation Games for Relay Transmission:

Stability Analysis under Uncertainty

Dusit Niyato1, Ping Wang1, Walid Saad2, Zhu Han3, and Are Hjørungnes4

1School of Computer Engineering, Nanyang Technological University (NTU), Singapore

2Electrical Engineering Department, Princeton University, USA

3Electrical and Computer Engineering, University of Houston, USA

4UNIK - University Graduate Center, University of Oslo, Kjeller, Norway

Abstract—Relay transmission or cooperative communication

is an advanced technique that can improve the performance

of data transmission among wireless nodes. However, while the

performance (e.g., throughput) of a source node can be improved

through cooperation with a number of relays, this improvement

comes at the expense of a degraded performance for the relay

nodes due to the resources that they dedicate for helping the

source node in its transmission. In this paper, we formulate a

coalitional game among the wireless nodes that seek to improve

their performance by relaying each other’s data. The game is

classified as a coalition formation game in which the nodes can

take individual and distributed decisions to join or split from a

given coalition while ensuring that their individual throughput

is maximized. A Markov chain model is proposed to investigate

the stability of the resulting coalitional structures. Further, we

consider the practical case in which the wireless nodes do not have

an exact and perfect knowledge of the parameters (e.g., channel

quality) in coalition formation. For this scenario, we analyze the

stability of the partitions resulting from the proposed coalition

formation game under uncertainty. We also define the conditions

needed for obtaining the stable and unstable coalitional structures

among the nodes that are performing cooperative transmission.

Keywords – Cooperative communications and networking,

coalitional game theory, Markov chain.

I. INTRODUCTION

Cooperative communication has recently emerged as a

novel communication paradigm that can significantly improve

the performance of wireless communication systems [1]. It

is proposed as a potential solution to deal with the diffi-

culty in the implementation of Multiple-Input-Multiple-Output

(MIMO) systems, where multiple antennas are required at

both transmitter and receiver. In particular, cooperative com-

munication attempts to benefit from the broadcast nature of

wireless channels to improve the performance of wireless data

transmission. In this context, the signals transmitted from a

source to a destination can also be overheard by other nodes

in the vicinity of the source, which may act as relays (virtual

antennas) to process and re-transmit the received signals. In

a cooperative network, the nodes can act as both source and

relay. That is, whenever a node wishes to transmit data, the

other nodes who decide to cooperate will perform cooperative

transmission for relaying the data of the node. In turn, when

another node in the network wants to transmit its data, the node

which previously acted as a source, can now act as a relay and

perform cooperative transmission. A key design challenge in

cooperative communication is to decide which node should be

cooperative and in which cooperative group, especially when

the network’s nodes act in a rational and selfish manner, i.e., in

a way to maximize their own performance (e.g., throughput).

Thus, it is of interest to design cooperative strategies to model

how the nodes can act strategically for cooperating (or not

cooperating) to relay each other’s data.

In this paper, the problem of coalition (i.e., group) formation

for rational nodes with relay transmission in a cooperative

network is formulated using the framework of coalitional game

theory. In the considered game, each node can autonomously

decide to join or split from a coalition while aiming to

maximize its own throughput. The decision of each node is

contingent on whether, by joining a particular coalition, a gain

in performance can be witnessed due to relay transmission by

the other nodes in the same coalition. The gain in performance

due to cooperative transmission comes at a cost that is a

function of the resources (e.g., time slots) that a given node

needs to allocate for relaying the data of the other nodes in

the same coalition. We analyze the stability of the network

partitions resulting from the proposed coalition formation

game using a Markov chain model. The stationary probability

of the defined Markov chain is used to determine the stable

coalitional structures, i.e., the stable coalitions that the nodes

can form. Further, we consider the uncertainty in the coalition

formation process due to the fact that the nodes may be

unaware of some parameters needed for cooperation, e.g., the

channel quality. The conditions for the stability of the coalition

formation process are defined based on the upper and lower

bounds of the stationary probability of the derived Markov

chain.

The rest of this paper is organized as follows: Related works

are reviewed in Section II. Section III describes the system

model and assumptions. Section IV presents the coalitional

game formulation for relay transmission. The stability analysis

under uncertainty is also presented. Section V presents the

numerical results. The summary is given in Section VI.

II. RELATED WORKS

Game theory has been applied to solve various issues in co-

operative communications [2]. In [3], a non-cooperative game

model was presented to investigate the cooperation among

Page 2

nodes using decode-and-forward (DF) cooperative transmis-

sion. The Nash equilibrium is identified under Rayleigh fading

channels as a two-state Markov model. In [4], a Nash bargain-

ing game was formulated to study the bandwidth allocation

problem between a source node and a number of relay nodes.

Also, the authors studied the conditions under which the

source and relay nodes will cooperate. The relay selection and

power control problems were formulated in [5] as a two-level

Stackelberg game. In this game, the source node, considered

as a buyer, pays to the relay nodes to provide them with an

incentive to cooperate. In [6], coalitional game theory was

used, in combination with cooperative transmission, to solve

an important problem related to the boundary nodes in an

ad hoc packet forwarding network. A grouping algorithm for

relay selection is proposed in [7] to minimize transmit power.

Also, an optimal rate allocation scheme among the relay nodes

was studied.

Although existing literature tackled several aspects of coop-

erative transmission using game theory, none of these works

considered the problem of performing coalition formation

among a number of rational nodes that seek to cooperate

for performing cooperative transmission and relaying. One

somehow related work is done in [8] where a coalitional

game framework was proposed for cooperative communica-

tions. However, no analysis on the stability of the resulting

coalitional structure was considered. Also, the uncertainty

accompanying the coalition formation process for performing

relay transmission was ignored.

III. SYSTEM MODEL AND ASSUMPTIONS

Coalition

1

Relay

transmissions

2

3

4

Destination

Fig. 1.

{4}.

System model for relay transmission with coalitions {1,2,3} and

A. Network Model

We consider a group of nodes denoted by N = {1,...,N}

where N is the total number of nodes. Each node needs to

transmit data to a given destination (e.g., a base station as

shown in Fig. 1)1. To improve their performance, the nodes

can decide to cooperate and form a group, i.e., a coalition

S ⊆ N, in which the nodes relay each other’s data and perform

cooperative transmission.

In this paper, we consider that the nodes in a given coalition

perform relaying and benefit from cooperative diversity based

1The destination of the nodes is not necessarily the same.

on a decode-and-forward strategy as in [9]. However, the

approach that we propose in the rest of this paper can easily

accommodate other strategies (e.g., amplify-and-forward). In

the first phase of the cooperative diversity scheme, the source

node i transmits using a particular adaptive modulation and

coding (AMC) mode while the relay nodes j ∈ S for j ̸= i

and the destination receive the signals. In the second phase,

the relay nodes repeat the transmission with the same AMC

mode while the source node i remains silent. At the end of

the second phase, the destination of node i achieves a gain

in SNR by combining the space-time decoded signals with

those received in the first phase. Let γi and γj denote the

instantaneous SNR from source node i ∈ S to its destination

and from relay node j to the destination of node i for all j ∈ S

and j ̸= i, respectively. For the case of multiple relays with the

perfect channel between source and relay, the post-processing

SNR at the destination of node i can be expressed as follows:

γpost

i

= γi+

∑

j∈S\{i}

j̸=i

γj.

(1)

When considering Rayleigh fading channels, the cumulative

distribution function (CDF) of the post-processing SNR for

single relay node j (i.e., S = {i,j}) is given by

γi

γi− γj

where γiand γjare the corresponding average SNRs from

source node i to its destination and from relay node j to the

destination of node i. For the multiple-relay case, (2) can be

extended as follows [9]:

j∈S\{i}

∑

where |S| ≥ 3, I = 1 − e−γ

for DF based relay transmission, the choice of an AMC mode

only depends on the post-processing SNR at the corresponding

destination. Given the available AMC modes as well as the

minimum required SNR threshold Γr for each mode, the

probability of using mode r for source node i in a given

coalition S can be calculated as αi,r(S) = F(Γr+1)−F(Γr).

The transmission rate of a source node i ∈ S can be obtained

from

Ri(S) =

r∈R

where R is a set of AMC modes, and ρris the transmission

rate of AMC mode r in packets/time slot.

F(γ) =

(1 − e−γ

γi) +

γj

γj− γi

(1 − e−γ

γj),

(2)

F(γ)=

+

∏

γi

γi− γj

γj− γi

I

(3)

j∈S\{i}

γj

∏

j′∈S\{i,j}

γj

γj− γj′

Bj,

γi, and Bj= 1 − e−γ

γj. Note that

∑

ρrαi,r(S),

(4)

B. Relay Transmission

Time division multiple access (TDMA) is considered in

which a frame is divided into multiple time slots and each node

Page 3

transmits in an allocated time slot. When node i ∈ S transmits

data, the other nodes in the same coalition (i.e., j ∈ S where

j ̸= i) receive data from node i. Then, nodes j relay the data

to the destination of node i by transmitting in their allocated

time slots.

Due to this relay transmission, each node i ∈ S will be

able to transmit for every |S| frame, where | · | denotes the

cardinality. Therefore, the throughput of node i ∈ S can be

calculated from

ϕi(S) =Ri(S)

|S|

.

(5)

By clearly inspecting the defined model, we can see that

there exists an interesting tradeoff that governs the decision

of the nodes on whether or not it is beneficial to establish

a coalition. As more nodes join a coalition, the rate per

transmission (i.e., Ri(·)) becomes higher due to the better

quality of the received signal (more diversity). However, the

nodes would have less time to transmit since the allocated

time slots have to be used for relay transmission of the other

nodes in the same coalition. As a result, the throughput ϕi(·)

may decrease. To efficiently analyze this tradeoff and devise

adequate cooperative strategies, a coalitional game formulation

will be presented in the next section.

IV. RELAY TRANSMISSION AS A COALITIONAL GAME

In this section, we present a cooperative model based on

coalitional game theory to analyze the coalition formation

process among the relay nodes and its stability given the selfish

nature of each node, i.e., its objective to maximize its own

throughput.

A. Players, Payoffs, and Actions

The problem of cooperative relaying mentioned in Sec-

tion III can be formulated as a coalition formation game with

the players being the relay network’s nodes. The set of all

players, i.e., nodes, is denoted as N. The payoff received

by each node due to relay transmission is captured using the

throughput as expressed in (5). The action of each player is

to form coalition. A coalition is denoted by Sx, where x is an

index of coalition. Given a set of current coalitions (i.e., state),

the players can decide to join or split from any coalition in a

way that can maximize their individual payoffs as follows:

• Joining: Let Mjndenote a set of candidate coalitions that

can join together to form a new single coalition Sx′. If all

nodes j ∈ Sx∈ Mjncan gain higher individual payoffs,

i.e.,

ϕj(Sx′) > ϕj(Sx)

where Sx′ =∪

• Splitting: Given a coalition Sx, the players in this coali-

tion can split (i.e., be partitioned) into multiple new

∀j ∈ Sx,

(6)

Sx∈MjnSx, then the coalitions can decide

to join together.

coalitions Sx′, if all the players j ∈ Sxcan gain higher

individual payoffs, i.e.,

ϕj(Sx′) > ϕj(Sx)

∪

∀j ∈ Sx,

(7)

where Sx =

coalitions.

We note that this cooperative game formulation has a non-

transferable utility (NTU) since the value (i.e., payoff which is

here proportional to the throughput of the node) of a coalition

cannot be transferred arbitrarily among players in the same

coalition as each individual node has its own payoff. The

distributed algorithm using which the nodes can join or split

can be implemented as in [10].

S′

x∈MspSx′ and Msp is a set of new

B. A Markov Model for Coalition Formation

To analyze the stability of the considered coalition forma-

tion game, a Markov chain model can be used [11]. The state

of the Markov chain is a partition or a coalitional structure

which can be defined as follows: A partition or a coalitional

structure is a group of coalitions that span all the players

in N and is defined as ω = {S1,...,Sx,...,SX} where

Sx∩Sx′ = ∅, for x ̸= x′, and∪X

the state space of Markov chain is defined as follows:

x=1Sx= N, and X is the total

number of coalitions in structure ω, i.e., X = |ω|. Therefore,

Ω = {ωy|y = {1,...,DN}},

(8)

where ωyrepresents a coalitional structure (spanning all play-

ers). DN is the Bell number obtained from

(

The transition probability matrix of this Markov chain is

denoted by P whose elements are pω,ω′. Each element pω,ω′

represents the probability that the coalitional structure (i.e.,

state) of all players changes from ω to ω′. Let Cω,ω′ denote

the set of candidate players who are bound to make a coalition

formation decision which will result in the change of the

coalitional structure from ω to ω′. This transition probability

can be obtained from

coalitional structure ω′is reachable from ω given the decision

of all players, then the condition ω ? ω′is true. Otherwise,

the players make a decision (e.g., δ = 0.5). βi(ω′|ω) is the

best-reply rule of player i. That is, βi(ω′|ω) is the probability

that the player i changes decision, and, hence, the coalitional

structure changes from ω to ω′. This best-reply rule is defined

as follows:

{

Di=

i−1

∑

j=0

i − 1

j

)

Dj, for i ≥ 1, and D0= 1.

(9)

pω,ω′ =

∏

i∈Cω,ω′

0,

δβi(ω′|ω),ω ? ω′,

otherwise,

(10)

where ω ? ω′is a feasibility condition. In particular, if a

condition ω ? ω′becomes false. δ is the probability that

βi(ω′|ω) =

ˆβ,

ϵ,

if ϕi(S(i)

otherwise,

x ⊂ ω′) > ϕi(S(i)

x ⊂ ω),

(11)

Page 4

where 0 <ˆβ ≤ 1 is a constant, ϵ is a small number that

corresponds to the probability that a player makes an irrational

decision. Further, we consider that a player can make an

irrational coalition formation decision due to either: (i)- a lack

of information or, (ii)- a need for “exploration” in the learning

process. In this case, the state transition probability pω,ω′ is

determined from the product of the transition probabilities of

players who do and do not make decisions.

C. Stable Coalition Formation

The solution of the coalition formation game is the coali-

tional structure, i.e., ω∗, which can exhibit internal and ex-

ternal stability notions [11]. Internal stability implies that,

given a coalition, no player in this coalition has an incentive

to leave this coalition and act alone (non-cooperatively as a

singleton), since the payoff any player receives in the coalition

is higher than that received when acting non-cooperatively.

External stability implies that, in a given partition, no player

can improve its payoff by switching its current coalition and

join another one. In particular, a coalitional structure ω∗is said

to be stable, if the conditions for internal and external stability

are verified for all the coalitions in ω∗. A stable coalitional

structure can be identified from the stationary probability of

the Markov chain defined with state space in (8) and transition

probability in (10). If the transition probability pω,ω′ is exactly

known, the stationary probability of the Markov chain can be

obtained by solving

⃗ πTP = ⃗ πT,

and

⃗ πT⃗1 = 1,

(12)

where ⃗ π =

of stationary probabilities and πω is the probability that the

coalitional structure ω will be reached.

If the probability of irrational decisions approaches zero

(i.e., ϵ → 0+), there could be an ergodic set E ⊆ Ω of states

ω in the Markov chain defined by the state space in (8) and

the transition probability in (10). This ergodic set E exists if

pω,ω′ = 0 for ω ∈ E and ω′/ ∈ E, and no nonempty proper

subset of E has this property. In this regard, the singleton

ergodic set is the set of absorbing states.

Once all players reach the state (i.e., coalitional structure) in

an ergodic set, they will remain in this ergodic set forever. In

particular, players will stop making any new decisions for join-

ing or splitting from any coalition. Therefore, the absorbing

state is referred to as the stable coalitional structure ω∗. With

this stable coalitional structure, no player has an incentive to

change the decision given the prevailing coalitional structure.

[

πω1

···

πωy

···

πωDN

]T

is a vector

D. Stable Coalition Formation under Uncertainty

In practice, there is an uncertainty in the coalition formation

process due to the fact that some system parameters have

to be estimated for relay transmission (e.g., channel quality

and best-reply rule). In particular, the transition probability

pω,ω′ of the Markov chain can be inaccurate. To analyze the

stability of coalition formation under parameter uncertainty,

we will adopt an analysis based on a Markov chain with

uncertainty [12]. To obtain the stationary probability of a

Markov chain with uncertainty, the following optimization

problem can be defined:

min

⃗ π,A

⃗ cT⃗ π,

(13)

subject to

A ∈ U,

⃗1TA =⃗0T,

⃗1T⃗ π = 1,

(14)

(15)

(16)

A⃗ π =⃗0,

⃗ π ≥⃗0,

where U is the uncertainty set of matrix A and ⃗ cTis a

vector of objective coefficients. A = I − PTand I is the

identity matrix. Let aω,ω′ denote the element of matrix A

corresponding to coalitional states ω and ω′. The off-diagonal

element of matrix A must be non-positive, i.e., aω,ω′ ≤ 0,

for ω ̸= ω′and ω,ω′∈ Ω. The diagonal elements must be

bounded by one, i.e., aω,ω≤ 1 for ω ∈ Ω.

The interval-value uncertainty set U is examined. In partic-

ular, let p−

on the transition probability of the Markov chain used for

coalition formation, respectively. The corresponding lower

and upper bounds of the elements of the matrix A can be

obtained from a−

a−

the conditions for the upper and lower bounds of the elements

of the matrix A are as follows:

• For off-diagonal element: −1 ≤ a−

ω ̸= ω′and ω,ω′∈ Ω.

• For diagonal element: 0 < a−

• For all elements with ω′∈ Ω:∑

With the interval uncertainty set U, the constraint in (14)

becomes

A−≤ A ≤ A+,

where A−and A+are matrices of lower bound a−

upper bounds a+

programming problem with the objective defined in (13) and

the constraints defined in (15), (16), and (17) can be written

as follows:

ω,ω′ and p+

ω,ω′ denote the lower and upper bounds

ω,ω′ = −p+

ω,ω′ for ω ̸= ω′and ω,ω′∈ Ω. Therefore,

ω′,ωand a+

ω,ω′ = −p−

ω′,ω, where

ω,ω′ ≤ aω,ω′ ≤ a+

ω,ω′ ≤ a+

ω,ω′ ≤ 0 for

ω,ω≤ a+

ω,ω≤ 1 for ω ∈ Ω.

ω∈Ωa−

ω,ω′ ≤ 0 and

∑

ω∈Ωa+

ω,ω′ ≥ 0.

(17)

ω,ω′ and

ω′,ω′, respectively. Then, the equivalent linear

min

⃗ π,Ξ

⃗ cT⃗ π,

(18)

subject to

πω′a−

⃗1TΞ =⃗0T,

⃗1T⃗ π = 1,

ω,ω′ ≤ ξω,ω′ ≤ πω′a+

Ξ⃗1 =⃗0,

⃗ π ≥⃗0,

ω,ω′, ω,ω′∈ Ω,

where ξω,ω′ is the element of matrix Ξ. Let π−

denote the smallest and largest possible values (i.e., lower and

upper bounds) for the stationary probability of the coalitional

structure ω. π−

problem defined in (18) with the objective coefficient vector

⃗ c = ⃗ eωwhere ⃗ eωis the unit vector (i.e., all elements are zeros

except row corresponding to coalitional state ω to be one).

Further, for π+

ωand π+

ω

ωcan be obtained by solving the optimization

ω, the objective coefficient vector is ⃗ c =⃗1−⃗ eω.

Page 5

The lower and upper bounds of the stationary probability

can be used to determine the stability conditions of the coali-

tional structure ω given that the irrational decisions approach

zero (i.e., ϵ → 0). Also, the transition probability of the

Markov chain used for coalition formation takes a value from

[p−

ω,ω′,p+

DEFINITION 1 (Stable Coalitional Structure). If π−

the coalitional structure ω will be always stable. If π+

then there exists a case in which the coalitional structure ω is

stable.

ω,ω′] for ω,ω′∈ Ω.

ω > 0,

ω> 0,

DEFINITION 2 (Unstable Coalitional Structure). If π+

the coalitional structure ω will never be stable. If π−

there exists a case in which the coalitional structure ω will

not be stable.

ω= 0,

ω= 0,

Since π−

given that the transition probability pω,ω′ is in [p−

if π−

However, if π−

case in which the coalitional structure ω is unstable. Similarly,

since π+

it only indicates that there is at least one case in which the

coalitional structure ω is stable. However, if π+

coalitional structure ω will never be stable.

ωcorresponds the minimum stationary probability

ω,ω′,p+

ω,ω′],

ω> 0, the coalitional structure ω will be always stable.

ω= 0, it just indicates that there is at least one

ωis the maximum stationary probability, if π+

ω> 0,

ω = 0, the

V. PERFORMANCE EVALUATION

246810 12

0

0.5

1

1.5

2

Average SNR from node to destination (dB)

Throughput of node 1 (packets/frame)

{1}

{1,2}

{1,2,3}

Fig. 2.

different coalitions.

Throughput under different SNR from nodes to destination with

Consider a network consisting of three nodes and a common

destination as shown in Fig. 1. Fig. 2 shows the throughput

of node 1 given variations in the average SNR from nodes to

its destination. In Fig. 2, we assume that the average SNR of

all nodes are identical for simplicity of presenting the result.

We consider three different coalitions for node 1 (i.e., {1},

{1,2}, and {1,2,3}) in order to investigate the impact of

relay transmission from the other nodes. It is observed that,

at different SNRs, node 1 obtains its highest throughput by

being a member of different coalitions. For instance, when

the average SNR is small, the throughput without cooperation

is the lowest due to poor channel quality. Therefore, node 1

finds it beneficial to cooperate by forming coalition with other

nodes. In contrast, when the average SNR is high, node 1 gains

a higher throughput without relay transmission. In this case,

forming coalitions will degrade the performance since node 1

has to dedicate some resources (e.g., time slots) to perform

relay transmission for the other nodes in the same coalition.

Based on this observation, we can clearly see that coalition

formation is a central issue for deciding on whether to perform

cooperative transmission or not. In addition, this result shows

how, in many scenarios, coalition formation can significantly

improve the performance of the network nodes, in terms of

throughput.

0.51

Difference of SNR between nodes (dB)

(a)

1.522.5

1

2

3

4

5

6

7

{1,2,3,4}

{1},{2,3,4}

{1,2,3},{4}

{1,2},{3,4}

{1},{2,3},{4}

{1},{2},{3,4}{1},{2},{3},{4}

Stable coalitional structure

|S|=4

|S|=3

|S|=2

|S|=1

0.51

Difference of SNR between nodes (dB)

(b)

1.52 2.5

0

0.5

1

1.5

2

Average throughput of nodes in

stable coalitions (packets/frame)

Node 1

Node 2

Node 3

Node 4

Fig. 3. (a) Stable coalitions and (b) average throughput of nodes under varied

SNR difference.

As the SNR difference between the nodes varies, Fig. 3(a)

shows the stable coalitional structure resulting from the pro-

posed coalition formation algorithm with a total of 4 nodes in

the network. The SNR difference used in Fig. 3(a) is defined

as ∆ = γ1−γ2= γ2−γ3= γ3−γ4where the average SNR

of node 4 is fixed at γ4= 4 dB. That is, the average SNRs

of nodes 1, 2, and 3 are varied. It is observed that, when the

SNR difference is small, all nodes can gain a higher payoff

(i.e., throughput) by forming a grand coalition (i.e., coalition

of all nodes). Therefore, the grand coalition is stable. When

the SNR difference becomes larger, the nodes can achieve a

higher throughput by being forming smaller coalitions (i.e.,

being more non-cooperative). Therefore, the stable coalitions

in this case have a smaller number of members (e.g., {2,3,4}

and {1,2,3}). When the SNR difference is very large (e.g., 2.5

dB), the majority of the nodes would have an incentive to not

form any coalition since the throughput of direct transmission

is high. Moreover, there could be multiple stable coalitional

Page 6

structures (e.g., when the SNR difference is 1 dB). In this case,

when the nodes reach any of these stable coalitional structures,

they will remain using these structures forever (assuming a

static environment).

Fig. 3(b) shows the average throughput of the nodes in a

stable coalitional structure. As the SNR difference increases,

node 4 will be the first to split from its coalition as it has the

lowest SNR, since the other nodes would have no incentive

to cooperate with node 4. As a result, the throughput of node

4 decreases first. Then, due to similar reasons, node 3 will

be split from the coalition of nodes 1 and 2. As a result,

throughput of node 3 decreases. Finally, nodes 2 will be split.

Note that the split nodes can form their own coalition (e.g.,

{3,4}) to improve their throughput. In addition, we can see

from Fig. 3(b) that node 1 has the highest throughput even

without belonging to a coalition. In particular, node 1 direct

transmission throughput can be higher than its throughput

when using relay transmission. This result is due to the fact

that, when forming coalitions, node 1 has to use its allocated

resource (i.e., time slot) to perform relay transmission for the

other nodes. As a result, we can see that, under different

conditions (e.g., channel quality), forming a coalition may or

may not lead to the best performance for the nodes. This result

motivates further the stability analysis of coalition formation

that is performed in this paper.

TABLE I

UPPER AND LOWER BOUND PROBABILITIES OF COALITIONAL STRUCTURE

Scenario 1

Lower

bound

{{1},{2},{3}}

{{1,2},{3}}

{{1,3},{2}}

{{1},{2,3}}

{{1,2,3}}

Coalitions

Scenario 2

Lower

bound

0.0

0.0

0.0

0.0

0.0

Upper

bound

0.0

0.0

0.0

0.0

1.0

Upper

bound

0.0

1.0

0.0

0.0

1.0

0.0

0.0

0.0

0.0

1.0

Table I shows the upper and lower bound probabilities

of stable coalitional formation under an interval uncertainty

for three nodes with relay transmission. The average SNR

of nodes 1 and 2 are γ1= γ2= 5 dB. We consider two

scenarios in which the range for the average SNR of node 3

is varied, i.e., γ−

and γ−

in Table I, for Scenario 1, only the grand coalition {1,2,3}

is always stable (i.e., upper and lower bound probabilities are

one) if the average SNR of node 3 falls into the range of

[4,6] dB. In contrast, for Scenario 2, if the average SNR

of node 3 falls into the range of [3,6] dB, only partitions

{{1,2,3}} and {{1,2},{3}} can be stable in some cases due

to the uncertainty of the average SNR. However, coalitional

structures {{1},{2},{3}}, {{1,3},{2}}, and {{1},{2,3}}

will never be stable.

3= 4 dB and γ+

3= 6 dB for Scenario 2. As shown

3= 6 dB for Scenario 1

3= 2 dB and γ+

VI. SUMMARY

In a cooperative network, the wireless nodes can cooperate

by forming coalitions in order to improve their throughput

through relay transmission. In this paper, we have presented

a coalitional game formulation for relay transmission when

the nodes are rational and seek to maximize their individual

throughput. In this game, the nodes can take a decision to join

or split from a coalition, while taking into account the perfor-

mance improvement, in terms of throughput, resulting from

the coalition formation decision. For analyzing the stability of

the proposed coalition formation game, we have proposed and

studied a Markov chain-based model. In addition, this paper

tackles the case in which the wireless nodes are uncertain

about the parameters that will be used to perform coalition

formation. Under this certainty, we have analyzed the stability

of the coalition formation process using the lower and upper

bounds of the stationary probability of the corresponding

Markov chain.

ACKNOWLEDGMENT

This work was done in the Centre for Multimedia and

Network Technology (CeMNet) of the School of Computer

Engineering, Nanyang Technological University, Singapore.

This work was supported by the Research Council of Norway

through projects 183311/S10 and 176773/S10, and NSF CNS-

0910461, CNS-0905556, CNS-0953377, and ECCS-1028782.

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