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Telecommunication Systems manuscript No.
(will be inserted by the editor)
A Game Theoretical Approach to Clustering of Ad-Hoc and Sensor
Networks
Georgios Koltsidas · Fotini-Niovi Pavlidou
Abstract Game theory has been used for decades in fields
of science such as economics and biology, but recently it
was used to model routing and packet forwarding in wireless
ad-hoc and sensor networks. However, the clustering prob-
lem, related to self-organization of nodes into large groups,
has not been studied under this framework. In this work
our objective is to provide a game theoretical modeling of
clustering for ad-hoc and sensor networks. The analysis is
based on a non-cooperative game approach where each sen-
sor behaves selfishly in order to conserve its energy and thus
maximize its lifespan. We prove the Nash Equilibria of the
game for pure and mixed strategies, the expected payoffs
and the price of anarchy corresponding to these equilibria.
Then, we use this analysis to formulate a clustering mech-
anism (which we called Clustered Routing for Selfish Sen-
sors - CROSS), that can be applied to sensor networks in
practice. Comparing this mechanism to a popular clustering
technique, we show via simulations that CROSS achieves a
performance similar to that of a very popular clustering al-
gorithm.
Keywords Game Theory · Clustering · Ad Hoc and Sensor
Networks · Nash Equilibrium
1 Introduction
Wireless sensor networks (WSNs) are constantly gaining
popularity the last few years in both the research commu-
nity and commercial applications. The reason is their unique
characteristics. A typical WSN is comprised of a large set of
wireless nodes, with sensing, monitoring and processing ca-
pabilities, deployed in an ad hoc fashion that typically coor-
dinate to perform a common task. These wireless nodes are
Georgios Koltsidas · Fotini-Niovi Pavlidou
Dept. Electrical and Computer Engineering, Aristotle University of
Thessaloniki, Panepistimioupolis, 54124, Thessaloniki, Greece
E-mail: {fractgkb, niovi}@auth.gr
usually small, low-cost, autonomous, battery-operated de-
vices, with limited energy capacity and computational pro-
cessing capability. Therefore, energy-aware mechanisms are
required, so as to ensure a long-lasting operation without the
need for battery replacement [2].
Since the number of nodes in such a network can ex-
tend to very large values, an efficient method to reduce the
expenditure of the batteries’ power is a grouping technique
known as clustering. The essential operation in clustering
consist in selecting a set of nodes to become clusterheads
(CH), and group the rest of the nodes in clusters, around
every clusterhead. The clusterhead is responsible for coor-
dination among the nodes inside its cluster, and for forward-
ing the collected data to the sink node, usually after effi-
ciently aggregating them. Clustering is particularly useful
for applications that require scalability to hundreds or thou-
sands of nodes. Scalability in this context implies the need
for load balancing and adaptability to changes in network
size, node density and topology. Moreover, self-organized
clustering algorithms constitute natural solutions for large
networks or networks lacking centralized control, such as
ad-hoc and sensor networks.
On the other hand, game theory [12] has been developed
and extensively used in the context of economy and biology.
It is a very powerful mathematical tool for analyzing and
predicting the behavior of rational and selfish entities. Due
to its interesting and sometimes unexpected results, its pop-
ularity reached the field of communications and networking
technology [3]. Recently, it has been used to model packet
forwarding in wireless ad-hoc networks with energy con-
straints ([7], [6], [13]) or as a basis to propose cooperation
enforcements mechanisms for these networks ([15], [10],
[4]). In this work we use the terminology and theorems of
game theory to form and analyze the problem of clustering
in sensor networks.
The rest of the paper is organized as following: In sec-
tion 2 an overview of the related works on ad-hoc and sen-
2
sor network clustering algorithms is provided. The follow-
ing section is dedicated to describing and analyzing the clus-
tering game, finding its equilibria, the expected payoffs and
the price of anarchy. In section 4 a new clustering technique
is described, based on the aforementioned analysis, and its
evaluation is presented in section 5, followed by reports on
the results. Finally, section 6 concludes the paper.
2 Related Work
Many sensor networks clustering algorithms have been pro-
posed so far. The most well-known is the Low-Energy Adap-
tive Clustering Hierarchy (LEACH) algorithm [9]. Accord-
ing to this protocol, the role of the clusterhead is not perma-
nently assigned to particular nodes. Instead, each sensor is
randomly self-elected as clusterhead, but a mechanism en-
sures that all nodes will play this role within a predefined
time interval. Consequently, the role of clusterhead is rotated
between the nodes in a probabilistic way, so that the energy
consuming operation is distributed among all the nodes of
the network. Later, a centralized version of LEACH, called
LEACH-C has been proposed in [8], where the decision on
which nodes will play the role of clusterheads is not dis-
tributed, but it is decided by the base station node (alter-
natively called sink node or just sink). The benefit is that
the base station has advanced computation capabilities and
practically unlimited power. Besides, by collecting all the
information, it maintains a global knowledge of the network
and so it can make much better decisions and network plan-
ning. The results show that LEACH-C increases the network
lifetime due to global knowledge and the ability to ensure
that the optimum number of clusterheads is selected in each
round.
Another centralized mechanism is described in [11], na-
mely the Base-station Controlled Dynamic Clustering Pro-
tocol - (BCDCP). This technique forms balanced clusters,
where each cluster has the same number of members to avoid
cluster overloading. In addition to this, data from distant
clusterheads is sent to the base station via other clusterheads.
BCDCP outperforms LEACH both in terms of lifetime and
of the number of delivered messages to base station. The
Base-station Centralized Simple Protocol (BCSP), on the
other hand, aims at extending the network lifetime by bas-
ing the clustering decision on the remaining energy of ev-
ery node. The base-station redistributes the role of cluster-
head among the nodes from time to time. BCSP does not
require location knowledge, like other algorithms, but each
node should send its energy level information along with the
sensing information, increasing the overhead. The drawback
of the two aforementioned techniques is that due to their
centralized implementation, they are not so appropriate for
sensor networks with a large number of nodes.
In [5] authors propose a clustering scheme that consid-
ers multiple parameters such as mobility, battery power and
maximum number of cluster members. The proposed Weigh-
ted Clustering Algorithm (WCA) uses a weighted sum of
the parameters in a combined metric that should be opti-
mized. The weights permit the protocol’s performance to
vary among several operating points, depending on the de-
sire of the operator and the particular needs of the applica-
tion. However, the additive character of its metric means that
not all its parameters could be optimized simultaneously, so
a compromise among the them needs to be made, limiting
its performance.
HEED [16] is another clustering mechanism for ad hoc
and sensor networks that makes no assumption regarding the
location knowledge and it is completely distributed. Clus-
terheads are randomly selected on the basis of their residual
energy normalized by their initial energy. In addition, it can
be combined with a metric that takes the node density into
account. HEED is proved to perform better than a modified
version of LEACH algorithm in case the nodes’ initial ener-
gies are not identical.
To the best of our knowledge, very few works used game-
theoretic terms to study clustering for sensor networks. In
[17] the authors propose a centralized algorithm where the
base station (or sink) decides on both the number of clusters
and the nodes that will become clusterheads. This decision is
based on the information of location and remaining energy
of every node. The authors show that this way of cluster-
head selection is more energy-efficient than a random one
following the LEACH algorithm. However, no theoretical
analysis is provided and the clusterhead selection is central-
ized, an operation that requires excessive overhead and thus
consumes additional energy.
In [1], the authors formulate a cooperative game between
the nodes of a sensor network. The payoffs for every node
depend on three parameters: the reputation, the cooperation
and the quality of security of every node. The first one refers
to the received signal strength, the second one to the per-
centage of the received packets that a node has forwarded
and the third one refers to the ratio of the exchanged mes-
sages between two nodes to the dropped messages between
them. The strategy of the players corresponds to a probabil-
ity of cooperation. The clusterheads of the clusters are ini-
tially selected. Nodes’ mobility results in changes in cluster
formation and new clusters may be formed or others may
be deleted. Simulations revealed that although initially the
number of clusters is large, cooperation between the nodes
increases with time and thus the network converges to a
constant value of clusterheads. Although the comparison of
the proposed methodology seems to need less message ex-
changes than a technique based only on the distance metric,
the authors do not provide any information regarding the
energy consumption, which is crucial for sensor networks.
3
Moreover, the main objective of the algorithm is trust en-
forcement and trust maintenance between sensors. The au-
thors do not mention what will happen in the case of static
nodes, where the probability of some nodes being closer
than others is different between the sensors.
In this paper we attempt to examine the problem of clus-
tering from a new perspective: We assume that the nodes
in the network behave selfishly, meaning that their primary
goal is to maximize their own benefits while minimizing
their own contribution for the benefit of other nodes. This
context applies in energy-aware ad-hoc networks where the
nodes are usually individual users acting without any prede-
termined agreement. Since the devices are mobile, energy is
a precious resource that should be consumed wisely. Apply-
ing this model to sensor networks is also meaningful, when
collocated sensors do not belong to the same authority. Fi-
nally, this effort provides another point of view to the prob-
lem of clustering specifying the limitations of selfish behav-
ior in this context and possible a new methodology of de-
signing clustering techniques.
3 The Clustering Game
3.1 Analysis and Equilibria
We begin with the definition of the clustering game (CG).
The clustering game is the game played by the nodes of a
network, when the purpose is to select a number of nodes
as clusterheads. It actually corresponds to choosing at least
one clusterhead from the population of the nodes. The re-
sponsibility of the clusterhead is to collect data from all
other nodes (the cluster members) and forward them (af-
ter efficiently aggregating them) to another node, which is
usually located far from the cluster. We formally define the
game as CG =< N, S,U >, where N is the set of players,
S = {S
i
} is the set of the available strategies and U = {U
i
} is
the set of utility functions of the nodes. The players are the
sensors, the N nodes participating in the network. In pure
strategies, the strategy space corresponds to two choices: a
sensor decides to either declare itself as CH or not. Letting
D be the strategy ”declare myself as CH” and ND the strat-
egy ”do not declare myself as CH”, the strategy space is
S = {Declare, Not Declare} = {D,ND}. Regarding payoffs,
if a player chooses not to become a clusterhead, then if no
other node becomes a clusterhead either, its payoff will be
zero, as the player will be unable to sent its data towards
the sink. If at least one other neighbor declares itself as CH,
then its payoff will be v, i.e. the gain in successfully deliv-
ering the data to sink. Finally, if the player declares itself
as CH, its payoff for successfully delivering the data v will
be reduced by an amount equal to the cost c of becoming a
clusterhead. So, in that case the final payoff will be v − c.
Table 1 The payoffs for the simple two player clustering game
Declare Not Declare
Declare (v −c, v − c) (v −c, v)
Not Declare (v,v − c) (0,0)
Let us now discuss the possible equilibria in the case of
two players, whose payoffs are summarized in Table 1. It is
clear that the game is symmetrical, since the payoff is only
dependent on the strategies of the players and not on which
player we consider for examination. The strategy (D,D) is
not a Nash Equilibrium. This is because each player is better
off to change its strategy to ND, as in this case its payoff will
be v > v − c. For a similar reason the strategy (ND,ND) is
not a Nash Equilibrium either, as any player would prefer to
deviate and declare itself as a CH, since this leads to a pos-
itive payoff. In case the first player plays D and the second
one plays ND, then none of them has any incentive to change
its choice. Hence, the strategy set (D, ND) is a Nash Equilib-
rium. For the same reason, (ND,D) is a Nash Equilibrium
too. Although these two strategies, (D,ND) and (ND, D),
are Nash Equilibria, there is no symmetrical Nash Equilib-
rium in this game, since no common strategy for all players
exists that results in an equilibrium. In the following we pro-
vide some useful propositions and theorems corresponding
to the N player clustering game.
Extending the game to be played by N players, let s =
{s
1
,s
2
,...,s
N
} be the vector profile of the strategies followed
by the players. If no player ”Declares”, then all players’ pay-
offs are zero. If at least one player, say k, plays ”D”, then the
payoff of all other players except k will be v, while player
k’s initial gain v will be reduced by the cost of declaring it-
self. Hence, the utility function U
i
(s) of an arbitrary player i
has the following form:
U
i
(s) =
0 ,if s
j
= ND, ∀ j ∈ N
v − c ,if s
i
= D
v ,if s
i
= ND and ∃ j ∈ N s.t. s
j
= D
(1)
In the following we list some helpful propositions, whose
proofs are omitted because they are straightforward.
Proposition 1 For the symmetrical clustering game, the strat-
egy S
allD
= {D,D,D,...,D} is not a Nash Equilibrium.
Proposition 2 For the symmetrical clustering game, the strat-
egy S
allND
= {ND, ND, ND,...,ND} is not a Nash Equilib-
rium.
Proposition 3 For the symmetrical clustering game, the strat-
egy where a single player plays D and all other players play
ND is a Nash Equilibrium and there are N Nash Equilibria
in the game.
4
Proposition 4 For the symmetrical clustering game, no sym-
metric pure strategies Nash Equilibria exist.
In order to permit the game to have symmetrical Nash Equi-
libria, we need to allow the players to play mixed strate-
gies. This means that the players choose their strategies ran-
domly following a probability distribution. In other words,
every player has now a probability of declaring itself as CH
and a probability not doing so. Let us denote the probabil-
ity of playing D as p and the probability of playing ND as
q = 1 − p. A interesting theorem follows.
Theorem 1 For the symmetrical clustering game, a sym-
metric mixed strategies Nash Equilibrium exists and the equi-
librium probability p that a player declares itself as cluster-
head is
p = 1 −
³
c
v
´
1/(N−1)
.
Proof We are going to search for symmetrical Nash Equi-
libria in mixed strategies, that will correspond to a particu-
lar probability p of a node declaring itself as CH, using the
methodology found in [14]. We first need to calculate the ex-
pected payoff for each choice available. The expected payoff
when playing D is U
D
= v − c, as the payoff is independent
of the other players’ strategies. The expected payoff when
playing ND is
U
ND
= Pr{no one else declares} · 0
+ Pr{at least someone else declares} · v
= v · (1 − Pr{no one else declares})
= v ·
¡
1 − q
N−1
¢
= v ·
¡
1 − (1 − p)
N−1
¢
. (2)
At the equilibrium, these two payoffs are equal, so that no
player has incentive to alter its strategy. Thus,
v − c = v ·
¡
1 − (1 − p)
N−1
¢
. (3)
Solving the above expression, we can compute the probabil-
ity p that corresponds to the equilibrium:
p = 1 −
³
c
v
´
1/(N−1)
. (4)
Letting
ω
= c/v < 1, the above equilibrium probability
never exceeds 1 and can be also written as:
p = 1 −
ω
1/(N−1)
. (5)
3.2 Asymptotic Analysis
Let us have a more detailed view of the equilibrium prob-
ability we have computed. The parameter
ω
is positive but
lower than 1. Thus, the probability we have computed re-
mains within the interval [0,1]. What is more, p falls as the
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Probability p
Number of Players (Nodes)
ù = 0.1
ù = 0.3
ù = 0.5
ù = 0.7
ù = 0.9
Fig. 1 Probability p of a player declaring itself as CH versus the total
number of players (nodes).
number of players increases, which means that players be-
come less cooperative as their number increases. Another
interesting probability is that of at least one player declaring
itself as clusterhead:
P
A
= Pr{at least one node plays D}
= 1 − Pr{no one plays D}
= 1 − (1 − p)
N
= 1 −
ω
N/(N−1)
. (6)
From the above expression we may observe that when there
is only one player in the game, both the probability p of
declaring itself as CH and the probability P
A
of at least one
node is self-elected as CH is equal to 1. For 2 players, p =
1 −
ω
and P
A
= 1 −
ω
2
. As N tends towards infinity,
lim
N→∞
p = 0 (7)
and
lim
N→∞
P
A
= 1 −
ω
. (8)
Thus, the higher the number of nodes, the less the probabil-
ity that at least one node declares itself as CH. On the other
hand, when N tends to 1 (meaning that there is only one
node left), then P
A
tends to 1, which means that if a player
is alone, then it is always self-declared as CH. Figs. 1 and 2
depict the values of the two probabilities as the number of
players increases for 5 different values of the parameter
ω
(0.1, 0.3. 0.5, 0.7 and 0.9).
These characteristics of the computed probability can be
very useful in the designation of a clustering mechanism that
will base the cooperation only on the rationality of selfish
sensors and will not attempt to force any complex algorithm
that each sensor should follow.
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ø = 0.1
ø = 0.3
ø = 0.5
ø = 0.7
ø = 0.9
Probability P
A
Number of Players (Nodes)
Fig. 2 Probability P
A
of at least one player declares itself as CH versus
the total number of players (nodes).
3.3 Costs and Payoffs
The above model can be modified to become more realis-
tic if we consider the additional cost in the case of data ex-
change between the cluster member and the clusterhead, re-
lated to the energy expenditure. When a sensor needs to send
a data packet of k bits to another sensor, it consumes energy
in both ways. Firstly, the energy is spent at the transmitter’s
electronic circuitry and it is denoted as e
elec
. Secondly, the
energy is consumed by the transmitter’s amplifier in order
to achieve the required signal level at the receiver for cor-
rect decoding and it is denoted as e
amp
. There are usually
two variations of this parameter. The first one correspond
to a square law distance attenuation (e
amp2
) and the second
one to an attenuation proportional to the forth power of the
distance (e
amp4
). The former is used for the communication
between a cluster member and a clusterhead and the latter
for the communication between a clusterhead and the sink,
since the distance in this case is much greater. Typical values
for these parameters can be found in Table 2 [16].
The energy spent when a sensor i transmits a packet of k
bits to its clusterhead CH
i
that is located at distance d
i,CH
i
is
calculated by the following equation:
E
i,CH
i
= k ·
¡
e
elec
+ e
amp2
· d
2
i,CH
i
¢
. (9)
The receiver, on the other hand, consumes energy when re-
ceiving a data packet of k bits, equal to
E
rx
= k · e
elec
. (10)
The clusterhead is usually responsible for aggregating the
data it receives and compressing them in order to minimize
the required packet length that it will transmit to the sink.
This is in accordance to the selfishness of the players in the
CG described earlier, as it saves energy in this way. The en-
ergy spent by the clusterhead for aggregating N
u
packets of
the same length k is computed using the following expres-
Table 2 Typical values for the energy expenditure parameters
e
elec
50 nJ/bit
e
amp2
10 pJ/bit/m
2
e
amp4
0.0013 pJ/bit/m
4
e
f use
5 nJ/bit/message
sion
E
aggr
= N
u
· k · e
f use
. (11)
For the case of communication between a clusterhead and
the sink, the energy expenditure of the clusterhead is derived
from the following formula:
E
CH,Sink
= k ·
¡
e
elec
+ e
amp4
· d
4
CH,Sink
¢
. (12)
Let us now discuss on these costs. When a sensor sends
data to a clusterhead, the consumed energy E
i,CH
i
depends
on their distance d
i,CH
, for a fixed packet length. We denote
this energy consumption by
δ
i
to refer to this cost. Since
the clusterheads may vary with time and due to the random
distances between the sensor and the clusterhead, we can
calculate an expected value for
δ
i
. As there is no reason to
distinguish between the players (expecting that at least in a
long-term run they all will have the same expected values),
the parameter
δ
i
can be considered identical for all players
and constant.
The other parameter that needs to be determined is the
cost c for the CH due to aggregation of the collected data
and transmitting them to the sink. There are a number of
ways that sink can be reached: The CH can send the data
directly to the sink, it could use intermediate nodes to reach
it, or another clustering hierarchy could be used. Regardless
of the method used, the parameter c will depend on the size
of data, the number of packets received and the distance be-
tween the node and the node that will receive them, either
this is the sink itself or another node in the network.
Nevertheless, as a first approximation, we may assume
that the average degree (or average number of neighbors N
u
)
is the same for all nodes, for a given number of total nodes
1
.
Hence, the expected number of packets received by a CH
can be considered constant and thus the energy spent for re-
ception E
rx
and aggregation of packets E
aggr
is the same for
all clusterheads. Furthermore, we assume that the distance
between the clusterhead and the sink is approximately the
same for all clusterheads. This is true if the sink is located
outside the region covered by the sensors and at a far dis-
tance or if the sink is mobile and it changes positions so
that in long-term its average distance from every node is the
same for all nodes. Consequently, the distance between an
arbitrary clusterhead and the sink d
CH,Sink
is assumed con-
stant and thus the energy expenditure E
CH,Sink
can be con-
1
We will remove this assumption later in the paper
6
sidered constant too. This means that
c = N
u
· E
rx
+ E
aggr
+ E
CH,Sink
> E
i,CH
i
=
δ
. (13)
Based on the previous assumptions, we may recalculate the
probability of a node declaring itself as clusterhead in the
clustering game with N nodes, taking into account that the
benefit when a node plays ND while at least one other plays
D is v −
δ
instead of v:
p = 1 −
µ
c −
δ
v −
δ
¶
1/(N−1)
=⇒
p = 1 −
ω
1/(N−1)
, (14)
where we use the letter
ω
to represent the ratio (c−
δ
)/(v −
δ
) and 0 <
ω
< 1. Thus, we have derived a probability that
corresponds to a symmetrical mixed strategy Nash Equi-
librium for the one-stage clustering game. So, for a set of
nodes that can hear each other, there is a natural incentive
to cooperate and form a cluster when every node becomes
clusterhead with probability p. This result is very important,
since it proves that there is no need to use a cooperation
enforcement mechanism, since cooperation arises naturally
from the rules of the game.
3.4 Expected Payoffs and Price of Anarchy
In this section we compare the expected payoffs correspond-
ing to the Nash Equilibrium probability to the maximum
possible expected payoff in the framework of the game and
also against the global optimum. The latter will permit us
calculate the Price of Anarchy.
The average payoff of an arbitrary node i is given by:
P = (v − c) · Pr{s
i
= D} +
v · Pr{s
i
= ND
\
∃ j s.t. s
j
= D, j 6= i} (15)
= (v − c) · Pr{s
i
= D} +
v · Pr{s
i
= ND} · Pr{∃ j s.t. s
j
= D, j 6= i} (16)
= (v − c) · p +
v · (1 − p) · (1 − Pr{s
j
= ND, ∀ j ∈ N, j 6= i}) (17)
= (v − c)p + v(1 − p)(1 − (1 − p)
N−1
) =⇒
P = v − cp − v(1 − p)
N
. (18)
Substituting in Eq. (18) the equilibrium probability p
∗
,
the average payoff of the equilibrium strategy P
NE
is
P
NE
= v − cp
∗
− v(1 − p
∗
)
N
=
= v − c
µ
1 −
³
c
v
´
1/(N−1)
¶
−
v
µ
1 −
µ
1 −
³
c
v
´
1/(N−1)
¶¶
N
= (19)
= v − c + c
³
c
v
´
1/(N−1)
− v
³
c
v
´
N/(N−1)
=⇒
P
NE
= v − c. (20)
However, this is not the maximum average payoff the
users may gain. We can easily compute the probability p
0
that maximizes P by setting the derivative of P equal to zero.
The corresponding probability p
0
is equal to:
p
0
= 1 −
³
c
Nv
´
1/(N−1)
(21)
and the maximum average payoff P
max
is then:
P
max
= v − cp
0
− v(1 − p
0
)
N
=
= v − c
µ
1 −
³
c
Nv
´
1/(N−1)
¶
−
v
µ
1 −
µ
1 −
³
c
Nv
´
1/(N−1)
¶¶
N
= (22)
= v − c + c
³
c
Nv
´
1/(N−1)
− v
³
c
Nv
´
N/(N−1)
=⇒
P
max
= v − c + c
µ
1 −
1
N
¶
³
c
Nv
´
1/N−1
. (23)
Nevertheless, the maximum average payoff we have com-
puted is probably not the optimum solution. The latter cor-
responds to the case where each time just one player plays
D and all the others play ND. Since we do not differenti-
ate the players, the optimum solution (within a period of N
rounds) corresponds to the case where each player plays D
in just only one round, and no other player does so. In this
optimum case the average payoff is
P
opt
=
(N − 1)v + (v − c)
N
=
Nv − c
N
=⇒
P
opt
= v −
c
N
. (24)
It is interesting to observe that P
opt
is not equal to the P
max
,
which means that probability p
0
that optimizes the expected
payoff of a user playing the game is suboptimal with respect
to the global optimum setting.
Once we have computed the P
NE
and P
opt
, it is easy to
compute the price of anarchy (PoA), which corresponds to
the payoff loss due to the suboptimum choice of the equilib-
rium probability p
∗
:
PoA = P
opt
− P
NE
=
= v −
c
N
−
·
v − c + c
³
c
v
´
1/(N−1)
− v
³
c
v
´
N/(N−1)
¸
=⇒
PoA = c −
c
N
=
N − 1
N
c. (25)
The conclusion of this analysis is that the PoA depends
only on the cost level of playing D and not on the gain v
of managing to join a cluster. The minimum value of PoA
corresponds to a two-player game (N = 2). On the contrary,
when the number of nodes tends to infinity, an upper bound
is reached:
PoA
min
= c/2 (26)
PoA
upper bound
= c. (27)
7
As a consequence, the PoA is upper and lower bounded and
its values are always within the interval [c/2, c) regardless
the number of nodes.
Let us now investigate the case where the cost c is not
constant but it depends on the total number of players N and
the number of players that play D at the same round of the
game N
CH
. For simplicity we assume that if the number of
CH in a round is greater than one, the rest of the nodes are
distributed evenly between the formed clusters. In specific,
we adopt the following form of the cost function:
c = c(N, N
CH
) =
N
N
CH
c
1
+ c
0
. (28)
The above expression incorporates the basic facts the cost
function is supposed to have. When the total number of play-
ers N increases, the cost increases too. On the other hand,
when the number of players that play D increases, the cost
decreases. In particular, the cost decreases with the increas-
ing number of N
CH
and the decrease is faster for small val-
ues of N
CH
. The rationale behind this is that the cost of be-
ing a clusterhead is distributed between the declared clus-
terheads. Thus, when the clusterheads are small in number,
their increase has greater impact in the decrease of the cost
that every clusterhead should bear than in the case where the
number of clusterheads is high, so an additional clusterhead
would only slightly reduce the cost of every single cluster-
head.
Before evaluating the new equilibrium probability, we
need to find the expected cost, as it is not constant like be-
fore. In order to do that, we need the expected number of
CHs. The probability the number of self-elected clusterheads
in a round is equal to n is given by the binomial distribution:
P{N
CH
= n} =
µ
N
n
¶
p
n
(1 − p)
N−n
, (29)
and its expected value is N p. Hence, the expected cost will
be a function of the probability p:
c = c(p) =
N
N p
c
1
+ c
0
=
c
1
p
+ c
0
. (30)
Now, based on Eq. (3) we may recompute the new equi-
librium probability as:
v − c(p) = v ·
¡
1 − (1 − p)
N−1
¢
=⇒
c(p) = v(1 − p)
N−1
=⇒
c
1
p
+ c
0
= v(1 − p)
N−1
=⇒
c
1
v
+
c
0
v
p = p(1 − p)
N−1
. (31)
The above equation has no trivial solution and it is not
guaranteed that there is a probability p ∈ [0,1] that satisfies
it. Obviously, for c
1
= 0 the above equation can be easily
solved and the solution is the probability of Eq. (4). If we
assume that c
0
= 0, then Eq. (31) takes the following form:
c
1
v
= p(1 − p)
N−1
. (32)
whose solution is not guaranteed to exist. The maximum
value of the right part of the above equation is maximized
for N = 2 and the maximum is 0.25. Hence, if c
1
/v < 1/4
then the gain of delivering the data is so high that the node
prefers to play D, no matter what all the other players play.
This threshold decreases, as the number of players increases,
because of the extra cost induced by the increased number
of players.
Let us now recompute the expected payoff based on Eq.
(18), taking into account the dependence of the cost on the
equilibrium probability.
P = v − c(p)p − v(1 − p)
N
=
= v − (
c
1
p
+ c
0
)p − v(1 − p)
N
=⇒
P = v − c
1
− c
0
p − v(1 − p)
N
. (33)
Thus, the expected payoff for the equilibrium probability
p
∗
is now:
P
NE
= v − c
1
− c
0
p
∗
− v(1 − p
∗
)
N
=⇒
P
NE
= v − (
c
1
p
∗
+ c
0
) = v − c(p
∗
). (34)
The maximum expected payoff is achieved for probabil-
ity p
0
, which is computed as:
p
0
= 1 −
³
c
0
Nv
´
1/(N−1)
(35)
and the maximum average payoff P
max
is then:
P
max
= v − c
1
− c
0
p
0
− v(1 − p
0
)
N
=⇒ (36)
P
max
= v − c
1
− c
0
− c
0
µ
1 −
1
N
¶
³
c
0
Nv
´
1/N−1
. (37)
4 A new clustering mechanism
In this section we will try to exploit the results of the pre-
vious analysis to design a simple clustering algorithm that
will be based on the natural incentive for cooperation as an-
alyzed before. Let us summarize first the results obtained
previously. We have resulted into an expression of the equi-
librium probability p a sensor is self-declared as cluster-
head. This means that no sensor has any incentive to deviate
from this probability. Since the parameters v, c and
δ
are
known to every node, so does the parameter
ω
. So, the re-
maining problem is only the calculation of the total number
of players, that is the total number of nodes participating
in the clustering game. Following the assumption made for
the LEACH protocol [9], we will assume that every sensor
may hear the transmission from every other sensor. This is of
course not very realistic, however it will permit us evaluate
this simple method with respect to LEACH, which is a very
popular clustering mechanism for sensor networks. It could
also be considered as the first level of a clustering proce-
dure with many levels of hierarchy. Let us mention here that
8
if we need to be very strict, we have to examine if a node
has an incentive not to declare its existence to its neighbors.
Since a node is interested in using another node to send its
packets on behalf of it and thus save energy, if it does not
declare its existence then it will be unable to pass its data
to a clusterhead and so it will have to send the data to sink
by himself, which is undesirable. Therefore, all nodes will
make their existence known to all others and since we as-
sumed that there is no sensor out of range of any other, all
nodes will compute the same number of players participat-
ing in the clustering game.
Using Eq. (14), a node computes the probability of be-
coming a CH at the first round of the clustering game. Due to
this random procedure, some of the nodes will declare them-
selves clusterheads and send beacons, so that every other
node will be able to select the closest CH. Then, the CH will
collect the data from the members of their clusters and then
the data aggregation is performed at each CH. Finally, ev-
ery CH sends the aggregated data to the sink. The question
that we have to answer know is what should be the prob-
ability p for the nodes on the second round. In order for
the energy expenditure of the clusterhead role to be evenly
distributed among the nodes, it is a good idea to set p = 0
for those nodes that have been clusterheads in the previous
round. If N
CH
(1) is the number of clusterheads at the first
round, then the game at the beginning of the second round
should be played among N
play
(2) = N − N
CH
(1) nodes. In
general, at round j + 1, the number of players playing the
clustering game should be N
play
( j + 1) = N −
∑
j
k=1
N
CH
(k).
To achieve this, we set an following rule:
Zero Probability Rule (ZPR): Every node that has served as
clusterhead sets the probability p to zero (p = 0) until all
its neighbors have also served as clusterheads. Then, it
switches back to the normal way of computing probabil-
ity p, according to Eq. (14).
ZPR has a very interesting property: The nodes that have
served as clusterheads have no reason to deviate since any
probability p > 0 would result in lower payoff, as there is
a positive probability that it is self-declared as CH again
and thus consume more energy. What is more, the nodes
that have not served as CHs yet, have no reason to deviate
from ZPR either, since they know that the number of players
have been reduced to N
play
( j + 1) and thus the equilibrium
probability is given by Eq. (14). When all nodes will have
served as CHs, a ”reset” takes place and all nodes initialize
the number of players to N. If a node’s energy have been
depleted, though, the total number of players is reduced to
adapt to this change.
By playing the game in rounds we actually define a re-
peated clustering game. In order to analyze this repeated
game, we should use an appropriate definition of the util-
ities of the nodes and a discount factor to model the pa-
tience of the nodes in receiving their payoffs. Due to the
ZPR, which bounds the number of the stages of the repeated
game, and assuming that the sensors are too impatient, the
previous analysis is still valid.
To sum up, we have defined a clustering mechanism,
that we call Clustered Routing Of Selfish Sensors (CROSS),
whose critical characteristic is the random rotation of the
role of clusterhead for energy balancing reasons, based on
the rationality and selfishness of the sensor nodes participat-
ing in the network. There is no assumption that the nodes
will cooperate with each other to follow the rules of a tra-
ditional clustering protocol. What is more, there is no guar-
antee that after a specific number of rounds all nodes will
have served as CHs for exactly one time. However, we ex-
pect that the randomness of the choices and the selfishness
of the nodes will finally result in the desired performance.
5 Performance Evaluation
In order to evaluate CROSS, we run a number of simulations
for several values of parameter
ω
. In an area 50mx50m we
randomly placed N = 100 nodes that remained stationary
throughout the simulation time. The sink was placed at po-
sition (25,125), thus it was located at least 75m from the
closest node of the network. The parameters for energy con-
sumption were those of Table 2 and the initial energy of
all nodes was set to E
init
= 0.5J, while the packet size had
a fixed length of k=2000bits. The path loss exponent for
short range transmissions (between sensors) was 2, while
for long distance transmissions it was considered equal to
4. The range of every node was set to be large enough, so
that any node could reach the sink in one hop. We compared
CROSS to LEACH, based on the methodology described in
[9], as LEACH is a very well-known algorithm, it is dis-
tributed and most of the works in this area use it as refer-
ence. What is more, in [16] it is mentioned that when the
nodes are assumed to be able to reach any other node, orig-
inal LEACH achieves increased network lifetime compared
to both HEED and a generalized and energy-aware version
of LEACH. For LEACH, the average number of clusters per
round is set to the proposed by the authors value of 5% (thus
5 nodes per round for 100 nodes it total) and after 20 rounds
all nodes should have served as clusterheads only once. At
the beginning of each round each sensor selects individually
the probability of declaring itself as clusterhead according to
the algorithm followed. Then, the rest of the nodes select the
closest clusterhead and join its cluster. After collecting the
data packets from all its members, every clusterhead fuses
them into a single packet that is sent to the sink directly. The
procedure continues, even if a node has exhausted all its en-
ergy. In this case, the rest of the nodes follow the process ig-
noring that node. We used 6 different values for
ω
, namely
0.05, 0.1, 0.3, 0.5, 0.7 and 0.9. All metrics were averaged
over 100 independent simulation runs, which was found to
9
be a number large enough to guarantee statistically confident
values.
The most important metric that reveals the performance
of any clustering technique is the network lifetime. Here, we
use the most common definition (although alternative defi-
nitions exist), i.e. the network lifetime is the lifespan of the
node that first among all the others depletes its energy. We
assume that a node’s energy is exhausted if 99.9% of its ini-
tial energy has been consumed. The results for several val-
ues of
ω
are presented in Fig. 3. The curve for LEACH is
straight because it is independent of the parameter
ω
and we
used the same value repeatedly for illustration purposes. It
is clear from this graph that CROSS outperforms LEACH in
almost all cases. Only when
ω
= 0.05, the lifetime achieved
by LEACH is slightly higher than that achieved by CROSS.
Another interesting observation is that the network lifetime
under CROSS seems to achieve a maximum value at
ω
=
0.5. For smaller or larger values, the lifetime is decreased.
However, the reduction is not greater than 200 rounds, which
is around 10% of the maximum network lifetime. A general
conclusion is that the selfishness of the sensors seems to lead
to a performance at least as good as LEACH, which is a pro-
tocol that requires cooperation between nodes and expects
from each one of them to follow it without deviations.
In order to show the performance of the protocols in
more detail, we provide in Fig. 4 a graph showing the num-
ber of nodes that are still alive with respect to the number of
rounds. Since there are initially 100 nodes, this graph shows
actually the percentage of alive nodes as time passes. We
plotted the results for both LEACH and CROSS. For the lat-
ter, we selected only two representative values of the param-
eter
ω
, 0.1 and 0.9, in order to compare two extreme cases
and the graph is still easily readable. The results confirm the
previous ones. Although for the small value of
ω
the first
node ”dies” sooner than for the large value of
ω
, the opposite
is true for the last node that spends all of its energy. Another
interesting observations is the time instance when the curves
of LEACH and CROSS intersect. It is the instance when
under both protocols, the same number of ”alive” nodes.
For
ω
=0.1, this happens when approximately 70% of the
nodes have exhausted their energy, while for
ω
=0.9 this hap-
pens when only 60% of them are still alive. In any case, the
curves corresponding to CROSS and LEACH intersect after
the time instance when half of the nodes have lost all their
energy. This means that, if we consider as a critical point in
the network’s operation the point in time when half of the
nodes consume all their energy, CROSS would still be a bet-
ter choice than LEACH. This is why we argue that CROSS
has a superior performance when compared to LEACH. As
a final comment, the curve corresponding to
ω
=0.9 is much
steeper than the one corresponding to
ω
=0.1. This could be
explained by the smoothness of the curves referring to the
probabilities p and P
A
, as they were presented in Figs. 1 and
0.05 0.1 0.3 0.5 0.7 0.9
1500
1600
1700
1800
1900
2000
Network Lifetime (rounds)
Parameter ø
LEACH
CROSS
Fig. 3 Network Lifetime under CROSS for different values of param-
eter
ω
and comparison with LEACH.
2. The more smooth the curve for the two probabilities (and
especially that of p) the more smooth the decrease of the
number of alive nodes as the time passes, in Fig. 4.
Although the network lifetime metric is important, since
it provides the knowledge of the instance where sensors be-
gin to die due to loss of energy, the lifespan of the last node
whose energy is depleted is of some importance too. We call
this metric the maximum node lifetime and the results for
both CROSS and LEACH are depicted in Fig. 5. Now, the
performance of the two protocols has been inverted. LEACH
achieves higher maximum node lifetime compared to CROSS,
regardless of the value of
ω
used by the latter. This can be
explained by the fact that, when the number of active nodes
in the network decreases, the probability of a node declaring
itself as clusterhead increases, as Eq. (14) shows. That being
the case, as more and more nodes run out of energy, the re-
maining ones increase their probabilities and declare them-
selves as clusterheads more and more frequently. Hence,
they begin to consume energy with an increasing rate, re-
sulting in higher energy consumption rates. Fig. 1 indicates
that the higher the
ω
the more smooth the curve is, thus we
would expect that for higher values of
ω
, the maximum life-
time would be decreasing. This is in accordance with the
results showed on Fig. 5, where the curve tends to decrease
when
ω
increases.
Finally, we measured the average number of clusterheads
per round in the case of both CROSS and LEACH. As ex-
pected, LEACH maintains this average at a value close to
the targeted one, namely 5 clusterheads per round. The rea-
son why the value is a bit lower than the targeted one is that
we average the number of clusters over the maximum node
lifetime (as described above) and not over the network life-
time. On the other hand, the number of clusterheads under
CROSS strongly depends on the parameter
ω
. The higher
the
ω
the less clusterheads per round. This behavior can be
explained if we recall the dependence of the probability p
10
1700 1800 1900 2000 2100 2200 2300 2400
0
10
20
30
40
50
60
70
80
90
100
Number of Í odes Álive (nodes)
Number of rounds (rounds)
LEACH
CROSS (ù =0.1)
CROSS (ù =0.9)
Fig. 4 Number of nodes with positive remaining energy under LEACH
and CROSS for
ω
= 0.1 and
ω
= 0.9, versus the number of rounds.
0.05 0.1 0.3 0.5 0.7 0.9
2000
2100
2200
2300
2400
2500
Maximum Node Lifetime (rounds)
Parameter ø
LEACH
CROSS
Fig. 5 Maximum Node Lifetime under CROSS for different values of
parameter
ω
and comparison with LEACH.
0.05 0.1 0.3 0.5 0.7 0.9
0
1
2
3
4
5
6
Average Number of Clusters
Parameter ø
LEACH
CROSS
Fig. 6 Average Number of CHs per round under CROSS for different
values of parameter
ω
and comparison with LEACH.
on the number of nodes and the value of
ω
. The clustering
game we described admits an equilibrium when only a sin-
gle node in the entire network declares itself as clusterhead
and the calculated probabilities attempt to achieve exactly
this objective. We can see that this objective is almost ful-
filled for the majority of the values of the parameter
ω
. Only
for small values of
ω
the average number of clusterheads per
round in increased up to approximately 3 clusterheads per
round. We remind here that the smaller the
ω
, the higher
the probability of self-declaring, assuming the number of
users constant. Hence, a higher number of clusterheads per
round is expected. In game-theoretic terms, a low value for
ω
means that v >> c, in other words, the benefit of sending
the data to sink is much greater in this case than the cost
of being a clusterhead, and thus the node is prevented from
risking not to send its data to the sink at all by not declar-
ing. For this reason, the more the parameter
ω
tends to 1,
the more a node risks by not declaring, resulting in reduced
number of clusters per round. As a final comment, it seems
that under this simulation settings, a total number of cluster-
heads for LEACH smaller that 5 would be more beneficial
for the network lifetime. Indeed, simulations with a target
value smaller that 5 showed an increase network lifetime.
However, the performance achieved by LEACH in this case
is still worse than that of CROSS for
ω
= 0.5 and the nearby
values. We need to mention here that the average number of
clusterheads for CROSS changes from round to round, and
strongly depends on the number of nodes that did not acted
as CHs yet.
The purpose of the previous performance evaluation was
to verify that a protocol as simple as CROSS, which uses
the results of a theoretical analysis of selfish behavior can
achieve a performance similar to algorithms specifically de-
signed for a purpose (clustering a network). At the same
time, the designed algorithm has one strong point with re-
spect to the competition: it can be applied to networks where
node behave selfishly, since it can ensure a robust network
performance similar to that of networks where the entities
are ”a priori” assumed to cooperate.
We were also interested in evaluating the performance of
a CROSS algorithm following the equilibrium probability
(referred as CROSS1) with respect to a CROSS algorithm
that computes the probability that maximizes the expected
payoff based on Eq. (21) (referred as CROSS2). Therefore,
we run a number of simulations with the same parameters as
above, except from the simulation area (100m x 100m), the
sink position ((50,175)) and the initial energy (1J). The per-
centage of nodes declaring themselves as CHs under LEACH
in every round was set to 5%, a value that results in a close
to optimum behavior (unlike the previous case of the small
network). Finally, for both CROSS1 and CROSS2
ω
was
fixed to 0.5, since CROSS was shown to achieve near opti-
mal performance for this value.
First, we varied the number of nodes from 60 to 150, in
order to observe the relation between LEACH, CROSS1 and
CROSS2. The results for the network lifetime, which is the
11
60 80 100 120 150
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
Network Lifetime (rounds)
Number of Nodes (nodes)
LEACH
CROSS1
CROSS2
Fig. 7 Network Lifetime for LEACH, CROSS1 and CROSS2 for var-
ious number of nodes.
40 50 60 80 100 max
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
LEACH
CROSS1
CROSS2
Network Lifetime (rounds)
Communication Range (m)
Fig. 8 Network Lifetime for LEACH, CROSS1 and CROSS2 for var-
ious radio ranges.
most critical metric in these networks, are shown in Fig. 7.
As it was expected, CROSS2 achieves higher lifetime val-
ues than CROSS1, due to the more appropriate calculation
of the probability of self-declaring. CROSS1 is also outper-
formed by LEACH. It is also interesting to observe that, for
large number of nodes, CROSS2 seems to perform better
than LEACH.
In order to investigate the performance of CROSS1 and
CROSS2 for communication ranges less than the maximum,
we simulated networks with lower ranges up to 40m. The
results are quite interesting and are presented in Fig. 8. Nei-
ther CROSS1 nor CROSS2 are capable of maintaining the
performance they achieve for maximum range. In particular,
as the radius is reduced, their performance degrades, with
CROSS2’s curve being more smooth than CROSS1. The
reason of this degradation is that the equilibrium probabil-
ity is increased when the number of nodes decreases, which
happens when the radius becomes smaller and the number
of neighbors, resulting in higher energy expenditure. In gen-
eral, when the number of nodes gets smaller, the probability
of self-declaring becomes drastically higher. Thus, for small
ranges or small number of nodes cross does not perform
well, because the nodes’ selfishness leads to high probability
of self-declaration. When the number of nodes is high and
the radius is large then this probability is reduced, leading
to networks with small number of clusterheads and higher
lifetimes.
6 Conclusions
In this work we provided a model appropriate to catch the
basic rationality behind selfish nodes that form clusters in
order to save energy. We have shown that the Nash Equilib-
rium of pure strategies corresponds to the case where only
one node declares to be a clusterhead and all others restrain
from declaring themselves. Although there is no symmetri-
cal equilibrium in the pure strategies case, such an equilib-
rium exists for the mixed strategies case. We provide a for-
mula for calculating the mixed strategy equilibrium proba-
bilities taking into consideration some more particular char-
acteristics of the clustering game, namely the energy con-
sumption. The initial model was then extended so that it
incorporates the dependency of the cost on the number of
clusterheads and the efficiency of the Nash Equilibrium was
evaluated by calculating the price of anarchy. We also showed
that we may use the equilibrium probability in a real sensor
network so as to evenly distribute the energy consumption
among the sensors by randomly rotating the role of cluster-
head among the sensor nodes. In comparison with LEACH,
the proposed method achieves similar network lifetime val-
ues for most of the cases. In the future, we intend to investi-
gate the iterative game, where the node maintains informa-
tion about the history of the game and act according to the
knowledge obtained, in case the nodes are not too impatient.
Acknowledgements This work was performed within the framework
of the project PENED 2003 (Grant No. 636), co-financed by Euro-
pean Union-European Social Fund (75%) and the Greek Ministry of
Development-General Secretariat for Research and Technology (25%).
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