4322IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
Coalition Formation for Bearings-Only Localization
in Sensor Networks—A Cooperative Game Approach
Omid Namvar Gharehshiran, Student Member, IEEE, and Vikram Krishnamurthy, Fellow, IEEE
Abstract—In this paper, formation of optimal coalitions of
nodes is investigated for data acquisition in bearings-only target
localization such that the average sleep time allocated to the
nodes is maximized. Targets are required to be localized with a
prespecified accuracy where the localization accuracy metric is
defined to be the determinant of the Bayesian Fisher information
matrix (B-FIM). We utilize cooperative game theory as a tool to
devise a distributed dynamic coalition formation algorithm in
which nodes autonomously decide which coalition to join while
maximizing their feasible sleep times. Nodes in the sleep mode do
not record any measurements, hence, save energy in both sensing
and transmitting the sensed data. It is proved that if each node
operates according to this algorithm, the average sleep time for
the entire network converges to its maximum feasible value. In nu-
merical examples, we illustrate the tradeoff between localization
accuracy and the average sleep time allocated to the nodes and
demonstrate the superior performance of the proposed scheme via
Monte Carlo simulations.
Index Terms—Bearings-only localization, distributed dynamic
coalition formation, lifetime maximization, nonsuperadditive co-
operative games, wireless sensor network (WSN).
power. Energy expenditure in WSNs can be categorized under
i) data transmission, ii) data processing, and iii) data acquisition
quisition and transmission consume significantly more energy
than data processing . In tracking applications, due to the
dense deployment of nodes, sensor observations are highly cor-
related in the space domain. This spatial correlation results in
unneeded sensed data which is unnecessary to be transmitted to
the sink. Hence, benefits from developing efficient data sensing
protocols which capture this spatial correlation is twofold: i) by
taking less measurements, it reduces energy consumption when
the sensors are power hungry, and ii) it reduces the unneeded
communications even if the cost of sensing is negligible .
In this paper, we consider a WSN that is deployed to localize
multiple targets based on noisy bearing (angle) measurements
at individual nodes. Since estimating the position of a target in
CRUCIAL issue in the design of wireless sensor net-
works (WSN) is the efficient utilization of the battery
tion April 29, 2010; date of current version July 14, 2010. The associate editor
coordinating the review of this manuscript and approving it for publication was
Dr. Ta-Sung Lee.
The authors are with the Department of Electrical and Computer Engi-
neering, University of British Columbia, Vancouver, V6T 1Z4, Canada (e-mail:
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/TSP.2010.2049201
two dimensions needs at least two angle measurements (to per-
form triangularization), it is natural for the nodes to form co-
operative coalitions. There exists an inherent tradeoff between
battery power and sensing accuracy such that if too few sensors
form a coalition, the variance of their collaborative estimate is
high. On the other hand, if too many sensors form a coalition,
excessive energy is consumed due to the spatial correlation of
same line-of-sight from the target, they record the same bearing
information. This redundant data can be avoided by putting one
of the nodes in the sleep mode. Sensor nodes in the sleep mode
do not record observations, hence, conserve energy in both data
acquisition and transmitting the sensed data.
Given that localization requires nodes cooperation, the main
idea of this paper is to develop a novel coalition formation and
sleep time allocation scheme to reduce the number of measure-
ments by i) keeping the localization accuracy within an accept-
vations. The abstract formulation we consider is a nonsuperad-
ditive cooperative game. The term nonsuperadditive means that
tion share measurements to localize a particular target and, as a
asfollows: i) What are theoptimal coalitionstructures for local-
izing multiple targets with a prespecified accuracy? ii) How can
erage sleep time allocated to the nodes is maximized?
The above questions can be addressed nicely within the
framework of coalition formation in a cooperative game. As is
commonly used in the tracking literature (e.g.,  and ), we
utilize the determinant of the Bayesian Fisher information ma-
trix (B-FIM) as the metric of estimation accuracy. Throughout
the paper, this measure is referred to as stochastic observability.
Since stochastic observability depends on both the angle of
measurements and distances of nodes to the target, it is clear
that the optimal coalition does not necessarily comprise the
nearest nodes to the target. The optimal coalition structure
would typically have some sort of diversity amongst angle
measurements of the nodes. In general, determining the optimal
coalition structure for tracking multiple targets is an NP-hard
coalition structures which is given by the
in a network constituted of
1) Why Cooperative Games? Cooperative game theory pro-
oration in multiagent systems. This is appropriate for bearings-
only localization where localization is essentially achieved by
Bell number 
1053-587X/$26.00 © 2010 IEEE
GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS 4323
triangularization. Coalition formation games, as a main branch
of cooperative games, study the complex interactions among
agents when the equilibrium state comprises several disjoint
coalitions. Hence, it conforms to the framework in multitarget
tracking where the optimal network structure comprises several
game analysis allows us to optimize these coalitions in terms of
2) Related Work: Energy conservation methods in sensor net-
ergy-efficient data acquisition . Duty-cycling and mobility-
based techniques – focus on the networking subsystem
ceiver.In and,noncooperativegame theoreticmethod-
ologies have been developed for decentralized activation of the
quisition schemes nevertheless achieve energy conservation by
minimizing the energy expenditure in both data transmission
and sensing. Thealgorithmsinthisclass aremostlyapplication-
tailored. As examples, we refer to  and  which consider
the adaptive sampling problem in a flood warning system and
environmental monitoring scenario, respectively.This paper fo-
cuses on a distributed cooperative game-theoretic scheme for
energy-efficient data acquisition in bearings-only localization
which, to the best of our knowledge, has not been investigated.
In literature, there exist only a few works – that in-
dynamic social learning model is studied where the focus is on
allocations and completely abstracts from coalition formation
process. In , a generic approach is proposed for coalition
formation through simple merge and split operations. This ap-
proach, unlike , can be utilized in both supperadditive and
nonsuperadditive games. However,  departs from the work
presented here in the sense that it focuses on the coalition struc-
ture generation process and does not investigate the bargaining
process. The algorithm devised in this paper is based on the ap-
proach presented in  and focusses on both the allocations
and coalition formation for both supperadditive and nonsuper-
additivegames. Our work generalizes in thesense that con-
information about the blocked players is not available at each
1) Main Results and Outline: Our main results are summa-
rized as follows.
• Formulation of the energy-efficient data acquisition
problem as a coalition formation game: In Section II,
energy-efficient data acquisition in two-dimensional bear-
ings-only localization is formulated as a maximization
problem for the average sleep time allocated to the nodes
subject to a fairness criteria. In Section III, this problem
is formulated as a coalition formation game where nodes
share measurements within coalitions and, as the payoff,
achieve sleep time. The modified core is proposed as the
solution concept for this game, which corresponds to the
solution to the energy-efficient data acquisition problem.
• Distributed dynamic coalition formation algorithm: In
Section IV, a distributed dynamic coalition formation
algorithm (Algorithm 4.2) is proposed where each node
greedily maximizes its expected sleep time for the next
period by choosing the optimal coalition whenever it
gets the opportunity to revise its strategy. This algorithm
simply forms a randomized adaptive search method on the
set of all possible coalition structures. In Section IV-B, it
is proved that if all the nodes follow the proposed algo-
rithm, the entire network eventually reaches the maximum
feasible average sleep time. Finally, the implementation
issues are addressed and it is demonstrated how this
algorithm can be employed in a sequential Bayesian
framework to localize multiple targets (Algorithm 4.1).
• Randomized search for blocked nodes: Considering
the large computational and memory overhead (see
Section IV-A) to search for all blocked nodes, i.e., poten-
tial nodes for gaining larger sleep times in other coalitions,
a randomized search method (Algorithm 4.3) is proposed
which reduces the aforementioned overhead. Convergence
to the core is established taking into account the fact that,
employing the randomized search method, the full set of
blocked nodes may not be available at each iteration of
the distributed dynamic coalition formation algorithm. It
is shown that a tradeoff can be achieved between compu-
tational cost at each iteration and the convergence rate of
the algorithm using the proposed search scheme.
provided to illustrate the behavior of the proposed algo-
rithm. We demonstrate its superior performance over the
heuristic range-based measurement allocation method via
Monte Carlo simulations.
II. FORMULATION OF THE ENERGY-EFFICIENT DATA
tion problem for the bearings-only multitarget localization sce-
nario in two-dimensional space and elaborate on the measure-
ment model and introduce stochastic observability as the metric
of localization accuracy.
nodes. Any subset
is called a coalition and can be iden-
tified with a vector
denote the set of sensor
Those subsets which only contain one node are called singleton
. The set of sensors localizing a particular
form a coalitionand sensors which are not assigned
the localization task form singleton coalitions. In addition,
denotes the set of target indices detected in the net-
gleton and nonsingleton) is also denoted by
coalition structure.By definition,each coalitionstructure forms
a partition on
. Finally, the set of all possible coalition struc-
tures, i.e., the set of all possible partitions on
with the cardinality given by the
and is called the
, is denoted by
Bell number .
4324 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
A. Network Average Lifetime Maximization Problem
Consider a scenario in which
tions to localize
targets in a field in two-dimensional space.
is required to be localized with a prespecified ac-
curacy denoted by
. All nodes in a particular coalition
share bearing measurements to localize target
ward, receive some sleep time denoted by
the time required by each node to record a single measurement
measurements that each node
measurements. Each node attempts to reduce its energy expen-
tion structure of nodes and sleep time allocations such that the
average sleep time that the nodes obtain is maximized and, at
the same time, all the targets are localized with the required ac-
curacy. In addition, to prevent premature power depletion of the
nodes, each node is guaranteed a minimum sleep time of
The coalition formation problem for energy-efficient data ac-
quisition can then be formulated as
sensors have to form coali-
and, as the re-
. Here, denotes
determines the number of
records from a maximum of
the objective function is the average sleep time allocated to the
nodes which has to be maximized over the set of all possible
. The constraints (C1) also guarantee the
required accuracy is achieved for all targets in the network.
This formulation establishes a tradeoff between the required
for each target and the average sleep
time allocated to the sensors in the localization task.
In addition to (C1) and (C2), we introduce a fairness criteria
on the sleep times allocated to the nodes. Suppose the nodes
given by . The allocation vector
of sensors can improve their allocated sleep times by forming
a new coalition
. This implies that all the nodes are
satisfied with their current allocations and the total sleep time
achieved by the coalitions is divided among the nodes in a fair
denotes the stochastic observability for
which will be elaborated in Section II-C. In (P),
is called fair if no group
is achieved for target . Here, we denote the new coalition by
to differentiate with the coalition formed to localize target
in , i.e.,. This means that the sum of the current
allocations in the new coalition
localization accuracy. Hence,
returns the maximum total sleep time achievable
such that the prespecified localization accuracy
is always greater that the
provides no surplus sleep time
their currently allocated sleep times. Formally,
Therefore, by solving (P) subject to (2), although the sum of
is maximized, the total sleep time achievable by each coalition
is divided among the coalition members in a fair fashion. In
above nonlinear combinatorial optimization problem should be
solved repeatedly. Nevertheless, there exists no obvious way of
relaxing the problem such that one can apply existing method-
ologies for solving standard combinatorial optimization prob-
lems. One natural solution to solve (P) is the brute-force search
on the set of all possible coalition structures and sleep time al-
locations that incurs an immense computational overhead and,
considering the limited power and computational resources of
the sensors in WSNs, has to be accomplished in a centralized
1) Outline of the Main Result: The energy-efficient data ac-
quisition problem is interpreted as a coalition formation game
constituting the set of players. The characteristic func-
for this game is defined as the maximum total sleep
time that can be achieved by a particular coalition
a relaxed version of (C1) is satisfied (see Section III-B). We
then propose a distributed dynamic coalition formation algo-
rithm in Section IV-A where, in each iteration, each node as
a myopic optimizer chooses among the existing coalitions to
greedily maximize its expected sleep time for the next period
as [see (4), shown at the bottom of the next page]. Here,
denotes the state of the network and Uniform
notes discrete uniform distribution on the elements of set
only when there exists a coalition
comprising node such that
be explained in Section IV-B, the randomization among the ex-
isting coalitions, which happens with probability , prevents
the nodes being stuck in nonoptimal coalition structures. It will
be proved in Theorem 4.2 that if each node follows (4), itera-
tions of the above algorithm eventually converges to the solu-
tion to the “relaxed” energy-efficient data acquisition problem
(see Section III-B). This approach brings about two main ad-
vantages: i) it is performed distributively among the nodes and
eliminates the need for a central decision-making device, and
tion problem in (4) for which the computational cost is linear in
the number of nonsingleton coalitions
if the feasible set in (3) is empty.
. As will
B. Stochastic Observability
With the above formulation and outline of the main result,
we now fill in the details of the measurement model and sto-
chastic observability. Consider a coalition
localizing a par-
1The term characteristic function is as used in cooperative games (see Sec-
GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS4337
Subsequently, applying the following equality
one can write (51) as (53), shown on the page. Since the argu-
and due to the concavity property of the logarithm function,
can be lower bounded by
Consequently, [see (55), shown on the page]. Finally, applying
the relaxed constraint in (22) [see (56), shown on the page],
andare as given in (24). The right-hand side in
(56) gives the maximum total sleep time that can be achieved
by a coalition
subject to the required localization accuracy
. Here, the aim is to minimize the energy consumption by
maximizing the average sleep time allocated to the sensors.
Hence, we equate the sum of the sleep times to the upper
bound provided by the right-hand side. However, as defined in
’s are positive integer numbers. Hence, the sum
on the left-hand side should also be confined to
. Thus, [see
(57), shown on the page], where
denotes the greatest integer function. This function gives
the maximum feasible sleep time for a coalition
target , and hence is considered as the characteristic function
for the game defined in Section III-A.
The authors would like to thank the anonymous reviewers for
their useful comments that helped in improving the quality of
 V. Raghunathan, C. Schurgers, S. Park, and M. Srivastava, “Energy-
aware wireless microsensor networks,” IEEE Signal Process. Mag.,
vol. 19, no. 2, pp. 40–50, Mar. 2002.
 G. Anastasi, M. Conti, M. Francesco, and A. Passarella, “Energy con-
servation in wireless sensor networks: A survey,” Ad Hoc Netw., vol.
7, no. 3, pp. 537–568, May 2009.
 Y. Bar-Shalom, X. Li, and T. Kirubarajan, Estimation With Applica-
tions to Tracking and Navigation.
 C. Hue, J. L. Cadre, and P. Perez, “Posterior Cramér-Rao bounds for
multi-target tracking,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, pp.
37–49, Jan. 2006.
 L. Lovasz, Combinatorial Problems and Exercises.
AMS Chelsea Pub, 2007.
 P. Santi, “Topology control in wireless ad hoc and sensor networks,”
ACM Comput. Surv., vol. 37, no. 2, pp. 164–194, Jun. 2005.
 V. Rajendran, K. Obraczka, and J. Garcia-Luna-Aceves, “Energy-ef-
ficient, collision-free medium access control for wireless sensor net-
works,” Wireless Netw., vol. 12, no. 1, pp. 63–78, Feb. 2006.
New York: Wiley, 2001.
4338IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
 W. Ye, J. Heidemann, and D. Estrin, “Medium access control with co-
ordinated adaptive sleeping for wireless sensor networks,” IEEE/ACM
Trans. Netw., vol. 12, no. 3, pp. 493–506, Jun. 2004.
bility for energy efficient data collection in wireless sensor networks,”
Mobile Netw. Appl., vol. 11, no. 3, pp. 327–339, Jun. 2006.
 E. Ekici, Y. Gu, and D. Bozdag, “Mobility-based communication in
wireless sensor networks,” IEEE Commun. Mag., vol. 44, no. 7, pp.
56–62, Jul. 2006.
 V. Krishnamurthy, “Self-configuration in dense sensor networks via
global games,” IEEE Trans. Signal Process., vol. 56, no. 10, pp.
4936–4950, Oct. 2008.
 V. Krishnamurthy, M. Maskery, and G. Yin, “Decentralized activation
in a zigBee-enabled unattended ground sensor network: A correlated
56, no. 12, pp. 6086–6101, Dec. 2008.
 J. Zhou and D. De Roure, “Floodnet: Coupling adaptive sampling
with energy aware routing in a flood warning system,” J. Comput. Sci.
Technol., vol. 22, no. 1, pp. 121–130, Jan. 2007.
 M. Rahimi, M. Hansen, W. Kaiser, G. Sukhatme, and D. Estrin,
“Adaptive sampling for environmental field estimation using robotic
sensors,” presented at the IEEE/RSJ Int. Conf. Intelligent Robots
Systems (IROS), Alberta, Canada, Aug. 2005.
 H. Konishi and D. Ray, “Coalition formation as a dynamic process,” J.
Econom. Theory, vol. 110, no. 1, pp. 1–41, May 2003.
 M. Agastya, “Perturbed adaptive dynamics in coalition form games,”
J. Econom. Theory, vol. 89, no. 2, pp. 207–233, Dec. 1999.
sented at the Int. Workshop Computational Social Choice (COMSOC),
Amsterdam, The Netherlands, Dec. 2006.
 T. Arnold and U. Schwalbe, “Dynamic coalition formation and the
core,” J. Econom. Behav. Org., vol. 49, no. 3, pp. 363–380, Nov. 2002.
 H. L.Van Trees, Detection, EstimationandModulation Theory.
York: Wiley, 1968.
 L. Kaplan, “Global node selection for localization in a distributed
sensor network,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, pp.
113–135, Jan. 2006.
 B. Ristic and S. Arulampalam, Beyond the Kalman Filter: Particle Fil-
ters for Tracking Applications.
 J. Helferty and D. Mudgett, “Optimal observer trajectories for bear-
ings-only tracking by minimizing the trace of the Cramér-Rao lower
bound,” presented at the 32nd IEEE Conf. Decision Control, San An-
tonio, TX, Dec. 1993.
 G. Owen, Game Theory. San Diego, CA: Academic, 1995.
 J. Von Neumann and O. Morgenstern, Theory of Games and Economic
Behavior.Princeton, NJ: Princeton Univ. Press, 1947.
plications.Amsterdam, The Netherlands: North Holland, 1994, vol.
Norwood, MA: Artech House, 2004.
 G. Nöldeke and L. Samuelson, “An evolutionary analysis of backward
and forward induction,” Games Econom. Behav., vol. 5, no. 3, pp.
425–454, Jul. 1993.
 J. G. Kemeny and J. L. Snell, Finite Markov Chains.
 P. Bremaud, Markov Chains: Gibbs fields, Monte Carlo Simulation,
and Queues. New York: Springer, 1999.
Omid Namvar Gharehshiran (S’09) was born in
Tehran, Iran, in 1985. He received the Bachelor’s de-
gree in electrical engineering from Sharif University
degree in electrical and computer engineering from
the University of British Columbia, Vancouver, BC,
Canada, in 2010, where he is currently working to-
wards the Ph.D. degree under the supervision of Dr.
game theory, and learning in games with applications
in wireless communication and sensor networks.
F’05) was born in 1966. He received the Bachelor’s
degree from the University of Auckland, New
Zealand, in 1988 and the Ph.D. degree from the
Australian National University, Canberra, in 1992.
He is currently a Professor and holds the Canada
Research Chair at the Department of Electrical Engi-
neering, University of British Columbia, Vancouver,
Canada. Prior to 2002, he was a chaired professor
at the Department of Electrical and Electronic Engi-
neering, University of Melbourne, Australia, where
he also served as deputy head of department. His current research interests in-
clude computational game theory, stochastic dynamical systems for modeling
of biological ion channels, and stochastic optimization and scheduling.
Dr. Krishnamurthy has served as Associate Editor for several jour-
nals, including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the
IEEE TRANSACTIONS ON SIGNAL PROCESSING, the IEEE TRANSACTIONS
ON AEROSPACE AND ELECTRONIC SYSTEMS, the IEEE TRANSACTIONS ON
NANOBIOSCIENCE, and Systems and Control Letters. In 2009 and 2010, he has
been serving as Distinguished Lecturer for the IEEE Signal Processing Society.
Beginning in 2010, he has served as Editor-in-Chief of the IEEE JOURNAL
SELECTED TOPICS IN SIGNAL PROCESSING.