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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).

I. INTRODUCTION

A

power. Energy expenditure in WSNs can be categorized under

i) data transmission, ii) data processing, and iii) data acquisition

(sensing).Experimentalmeasurementshaveshownthatdataac-

quisition and transmission consume significantly more energy

than data processing [1]. 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 [2].

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

ManuscriptreceivedJune24,2009;acceptedApril08,2010.Dateofpublica-

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:

omidn@ece.ubc.ca; vikramk@ece.ubc.ca).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

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

sensormeasurements.Asanexample,whentwonodeslieonthe

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-

ablelevel,andii)capturingthespatialcorrelationofsensorobser-

vations. The abstract formulation we consider is a nonsuperad-

ditive cooperative game. The term nonsuperadditive means that

thegrandcoalition(thecoalitioncomprisingallnodes)isnotop-

timal.Thisismainlyduetothetradeoffbetweenbatterylifeand

thevarianceofestimatesmentionedabove.Nodes ineachcoali-

tion share measurements to localize a particular target and, as a

result,arerewardedwithsleeptimes.Twoquestionsthatariseare

asfollows: i) What are theoptimal coalitionstructures for local-

izing multiple targets with a prespecified accuracy? ii) How can

nodesdynamicallyformoptimalcoalitionstoensurethattheav-

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., [3] and [4]), 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

problem.Thisisbecauseoneneedstosearchamongallpossible

coalition structures which is given by the

in a network constituted of

sensors.

1) Why Cooperative Games? Cooperative game theory pro-

videsanexpressiveandflexibleframeworkformodelingcollab-

oration in multiagent systems. This is appropriate for bearings-

only localization where localization is essentially achieved by

Bell number [5]

1053-587X/$26.00 © 2010 IEEE

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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS4323

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

coalitionsofsensors,eachlocalizingaparticulartarget.Consid-

eringthespatialcorrelationofnodesobservations,acooperative

game analysis allows us to optimize these coalitions in terms of

energy consumption.

2) Related Work: Energy conservation methods in sensor net-

workscanbeclassifiedas:duty-cycling,mobility-based,anden-

ergy-efficient data acquisition [2]. Duty-cycling and mobility-

based techniques [6]–[10] focus on the networking subsystem

andattempttoreducetheenergyconsumptionintheradiotrans-

ceiver.In[11] and[12],noncooperativegame theoreticmethod-

ologies have been developed for decentralized activation of the

radiotransceiverinnetworkedsensors.Energy-efficientdataac-

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 [13] and [14] 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 [15]–[17] that in-

vestigatethecoalitionformationasadynamicprocess.In[16],a

dynamic social learning model is studied where the focus is on

allocations and completely abstracts from coalition formation

process. In [17], a generic approach is proposed for coalition

formation through simple merge and split operations. This ap-

proach, unlike [16], can be utilized in both supperadditive and

nonsuperadditive games. However, [17] 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 [18] and focusses on both the allocations

and coalition formation for both supperadditive and nonsuper-

additivegames. Our work generalizes[18] in thesense that con-

vergencetothecoreofthegameisestablishedassumingthatfull

information about the blocked players is not available at each

iteration.

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.

• Numericalexamples:InSectionV,numericalexamplesare

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

ACQUISITION PROBLEM

Inthissection,weformulatetheenergy-efficientdataacquisi-

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.

Notation: Let

nodes. Any subset

is called a coalition and can be iden-

tified with a vector

denote the set of sensor

, where

if

if

(1)

Those subsets which only contain one node are called singleton

coalitions, i.e.,

. The set of sensors localizing a particular

target

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-

work. Thesetcomprisingallcoalitionsinthenetwork(bothsin-

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 [5].

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4324IEEE 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.

Each target

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

and

. Hence,

measurements that each node

measurements. Each node attempts to reduce its energy expen-

diturebymaximizing .Inthispaper,weseektheoptimalcoali-

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

.

(P)

(C1)

(C2)

where

and

coalition

the objective function is the average sleep time allocated to the

nodes which has to be maximized over the set of all possible

coalition structures

. The constraints (C1) also guarantee the

required accuracy is achieved for all targets in the network.

This formulation establishes a tradeoff between the required

localization accuracy

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

haveformedacoalitionstructure

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

fashion. Formally,

denotesthesleeptimeallocationvector,

denotes the stochastic observability for

which will be elaborated in Section II-C. In (P),

andareallocatedsleeptimes

is called fair if no group

(2)

where

in coalition

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

totalsleeptimethatcanbeobtainedby

localization accuracy. Hence,

thatcanbedividedamongthesensorsand,astheresult,increase

returns the maximum total sleep time achievable

such that the prespecified localization accuracy

is always greater that the

subjecttotherequired

provides no surplus sleep time

their currently allocated sleep times. Formally,

expressed as

can be

(3)

We set

Therefore, by solving (P) subject to (2), although the sum of

thefeasibletotalsleeptimesfor allcoalitions

is maximized, the total sleep time achievable by each coalition

is divided among the coalition members in a fair fashion. In

trackingapplications,asthetargetmoves,theoptimumcoalition

structureandsleeptimeallocationsevolveovertime.Hence,the

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

manner.

1) Outline of the Main Result: The energy-efficient data ac-

quisition problem is interpreted as a coalition formation game

with

constituting the set of players. The characteristic func-

tion1

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

In addition,

only when there exists a coalition

comprising nodesuch 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

ii)ineachiteration,nodessolvethenoncombinatorial optimiza-

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.

such that

de-

.

. 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-

tion III-A).

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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS4325

ticular target by each node recording noisy bearing measure-

ments of the target relative to a coordinate frame. Let

denote the target position vector that the coalition aims to esti-

mate. Each node then records a noisy measurement

(5)

where

estimation error for node , respectively. In addition,

notes the prior density of the target. Suppose that the target is

stationary. Given that the true position of the target and position

of the

node are denoted by

spectively,

anddenote the estimated bearing and

de-

and, re-

(6)

If sensor is allocated sleep time

surements from the target. Here, it is assumed that the measure-

ment intervals are sufficiently long so that

independent between the intervals.

Assuming thatthetargetposition

prior density

, we adopt a sequential Bayesian framework.

Ideally, the localization accuracy metric must be defined based

on the covariance matrix of the posterior distribution. However,

bounds on attainable performance are considered of significant

interest since they provide a baseline whereby to compare the

performance of candidate techniques [19]. The most popular

bound is the Cramér–Rao bound, which derives much of its at-

tractiveness from its analytic tractability and is widely used in

the literature [4], [20], [21]. Let

measurements by the nodes in

posterior target distribution

, it will record mea-

’s are statistically

is astochasticvectorwith

denote the set of bearing

. Then, the covariance of the

is

(7)

The posterior Cramér–Rao lower bound (P-CRLB) theorem

[19] establishes a lower bound on

orem, there exists

given by

. According to this the-

(8)

(9)

(10)

such that

(11)

wherethematrixinequalityindicatesthat

itivesemi-definite. Intheaboveequations,

denote the FIM and B-FIM, respectively. Throughout, it is as-

sumed that the prior density of the target is approximated by a

Gaussian distribution with covariance

ispos-

and

. Hence,

(12)

This assumption helps to reduce computations in evaluating the

characteristic function in Section III-B.

Definition 2.1:Stochastic observability is defined as

, where

In the literature, various matrix means of the B-FIM have

been used as the estimation accuracy metric among which we

can refer to trace and determinant [3], [22]. The choice of de-

terminant is justified as it can be attributed to how accurate an

estimate is by noting that it determines the volume of the

confidence ellipsoid around the estimate [19]. This boundary is

defined as points

that satisfy

denotes the B-FIM.

(13)

where

tion. The following proposition provides a closed-form expres-

sion for the stochastic observability.

Proposition 2.1: Consider the measurement model adopted

in (5). For a specific coalition

can be expressed as

denotes the mean of the posterior target distribu-

, stochastic observability

(14)

Here

(15)

(16)

and

tance of the

Proof: See Appendix A.

Proposition 2.1 will be used in Section III to derive the char-

acteristic function for the coalition formation game. Expecta-

tions in (14) cannot be evaluated analytically. Although one can

denotes the relative dis-

node to the target.

with probability

with probability

(4)

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4326IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010

utilize Monte Carlo methods, a simpler approach to avoid com-

puting expectations is to approximate the B-FIM as

(17)

where

tical applications, the P-CRLB is approximated by (17). As an

instance, the covariance of the estimate is approximated in the

same way in the extended Kalman filter [21]. In this paper, as

willbeseeninSectionIII,(17)willbeusedtoevaluatecoalition

values in the game.

denotes mean of the prior density. In many prac-

III. THE COALITION FORMATION GAME

In this section, the energy-efficient data acquisition problem

(P) is interpreted as a coalition formation game. The advantage

of such an interpretation is that one can use dynamic coalition

formation algorithms to compute the solution. As will be seen

later in this section, the characteristic function is defined such

that larger coalitions of nodes do not necessarily ensure larger

sleep times. This is mainly due to the fact that the stochastic

observability,dependingonbothrelativeanglesanddistancesof

the nodes, does not necessarily improve as the number of nodes

in a coalition increases. We propose the modified core [23] as

the solution concept for this game.

A. Non-Superadditive Cooperative Games and the Core

The energy-efficient data acquisition problem (P) can be in-

terpreted as a cooperative game with transferable utility (TU)

[23] defined by the set of nodes

teristic function

with any nonempty coalition the maximum total sleep time that

can be gained by that coalition such that the required localiza-

tion accuracy is achieved. Simply put,

as the reward for nodes collaboration in localizing a particular

target. The payoff for each node

it claims from the coalition to which it belongs and tries to

maximize it.Coalition formation games encompass cooperative

games where the coalition structure plays a major role and are

not generally superadditive. A game is called superadditive if

and a real-valued charac-

. This function associates

can be interpreted

is a share fromthat

(18)

In superadditive games, collaboration is always beneficial,

hence, the grand coalition

structure. We refer to [23] for extensive textbook treatment of

cooperative game theory.

Inoursetup,sincetheoptimalcoalitionstructureshouldcom-

priseseveraldisjointcoalitionsofsensors(eachlocalizingaspe-

cific target), it is natural to adopt the coalition formation game

formulation. Each node

is encouraged to join a nonsingleton

coalition if it can achieve a sleep time larger than its reserva-

tion payoff

. Considering the constraints in (C2), we set

. Hence, nodes’ sleep times in nonsingleton coali-

tions are restricted to the integers in the interval

Throughout, this set is denoted by

nodes with sleep times equal to

chastic observability, hence, are not considered as a member in

nonsingleton coalitions.

forms the optimal coalition

.

. Here,

do not contribute to the sto-

is removed since

In this paper, the modified core is formulated as the solution

concept for the defined game which relies on the modified defi-

nition for feasibility in nonsuperadditive games.

Definition 3.1: In nonsuperadditive TU games, an allocation

is called feasible if

(19)

and is called efficient if the equality holds [18].

In other words, sum of the sleep times of all nodes cannot ex-

ceed the maximum total sleep time achievable under the most

desirable coalition structure. This definition generalizes feasi-

bility in supperadditive games where the basic assumption is

that the grand coalition always forms. In supperadditive games,

an allocation

is called feasible if:

be considered as a special case of the above definition noting

that, in superadditive games,

Suppose that an allocation

has been proposed by the nodes.

Ifagroupofnodescanformacoalitionwhichprovidesitsmem-

bers higher sleep times, this coalition will block the proposal.

Formally, a coalition

will block an allocation if

. This is because the current sleep time allocations to the

nodes

can be improved by forming the new coalition

and dividing the surplus

cation is in the core if it is both feasible and nonblocking. The

followingdefinitionextendsthecoreinsupperadditivegamesto

nonsuperadditive games.

Definition 3.2: An allocation

tion if it satisfies the following conditions:

. This can

.

. Finally, an allo-

is called a core alloca-

(20)

(21)

The modified core can be considered as an equilibrium point

in the game in the sense that reaching a core allocation and the

coalition structure corresponding to it, no sensor can achieve

largersleeptimesbydeviatingfromit.Hence,definingthechar-

acteristicfunction

suchthat(C1)issatisfied,themodifiedcore

for the coalition formation game

combinatorialenergy-efficientdata acquisitionproblem (P).We

now proceed to derive the characteristic function for the game.

is the solution to the

B. Characteristic Function

As the first step to derive the characteristic function for the

game, a lower bound is found for the stochastic observability

using the results in Proposition 2.1. As it can be seen in (16),

stochastic observability is a bilinear function of the number of

measurements

. This motivates to use the log-determinant

to change into a linear function of

then relax (C1) by making this lower bound satisfy the required

localizationaccuracytoderivethecharacteristicfunctionforthe

game. Formally,

. We

(22)