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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 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

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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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.

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 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

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 NETWORKS 4325

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 recordmea-

’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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4326 IEEE 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 sharefromthat

(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 changeinto a linear function of

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

localizationaccuracytoderivethecharacteristicfunctionforthe

game. Formally,

. We

(22)

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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS 4327

where

logarithm of stochastic observability and refers to the bound

given in (55). Details for the derivation of this lower bound can

be found in Appendix A. The core of the cooperative game with

the characteristic function derived based on (22) corresponds to

the solution of the “relaxed” energy-efficient data acquisition

problem resulted by replacing (C1) with (22).

Let

and denote the target and the coalition localizing

it, respectively. Assuming

to be diagonal, the characteristic

functionsaredefinedas(23)and(24),shownatthebottomofthe

page, where

integer function. In addition, assuming

are given by

denotesthelowerboundof the

and denotes the greatest

, and

(25)

respectively. This function returns the maximum total feasible

sleep time that can be achieved by

grals of . Detailed derivation of the characteristic function is

provided in Appendix B.

We next investigate properties of the function given in

(23) and (24). As can be seen in (23), the first term goes

up as the number of nodes in a specific coalition increases.

This translates to the more sleep time that will be allo-

cated to the nodes in more populated coalitions. However,

if the nodes provide worthless information, the second term

forces (23) to decrease. This worthless information can

be categorized as i) redundant information, and ii) impre-

cise information. Redundant information corresponds to

the case where two or more nodes lie on almost the same

line-of-sight relative to the target. In this case,

for two or more nodes, hence,

However, imprecise information corresponds to the case

where nodes are located far from the target (i.e.,

in which case

in terms of multiple inte-

.

)

, hence,

. On the other hand,

if the prior density shows higher uncertainty in

(i.e.,

), nodes located on

be more informative in reducing uncertainty in that direction.

In this case, since

concluded. Consequently, as it is clear from (24),

the characteristic function also allocates larger sleep times to

the coalitions comprising the nodes with

direction

orwill

and , it can be

or.

From the above discussion, it can be concluded that larger

coalitions do not necessarily guarantee greater characteristic

function values. Therefore, the characteristic function exhibits

the nonsuperadditive property. Finally, the tradeoff between

sleep times allocated to the nodes and localization accuracy

for a specific targetcan also be clearly seen in (23), where

as

goes up, the total sleep time allocated to coalition

reduced.

We now proceed to discuss the constraints that dependence

of the characteristic function on the target index

the game formulation.

is

imposes on

C. Formulation of Constraints

In the context of cooperative games in characteristic form

[24], the basic assumption is that, given a fixed characteristic

function, the coalition value for a particular coalition only de-

pends on coalition members [25]. In our formulation, although

the characteristic function is fixed, its value changes for a spe-

cific coalition as it attempts to localize two different targets.

This problem stems from the fact that the bearings

distances

and required accuracy

of nodes tries to localize different targets. Therefore, values of

coalitions are also dependent on the target index that the coali-

tionattemptstolocalize.Toavoidthisinconsistency,weinclude

targetsasplayersincoalitions withzeropayoffs.Formally,each

coalition is considered as

targetindexandthecoalitionofnodeslocalizingit,respectively.

Singleton coalitions are also denoted by

coalition value is uniquely determined by the members of that

coalition. For our formulation to be well posed, we also need

to disallow targets leaving or jumping between coalitions (the

process of joining and leaving coalitions will be explained in

Section IV-B). This requires imposing the following constraints

on the characteristic function:

1) In order to prevent targets leave coalitions, we set:

for all

to disband and form singleton coalitions when the target

leaves the coalition.

2) Inorder topreventtargets jumpbetween coalitions,we set:

for all

the coalition localizing

, its expected payoff in the new

coalitionwill be

. Thus,

coalition, where it achieves zero payoff.

3) When there exists only one node in a coalition localizing

a target, since no measurement diversity is provided and

triangularizationisimpossible,weset:

, relative

change as a coalition

whereanddenote the

. Hence, each

. This forces the nodes

. Ifjoins

prefers tostay in itscurrent

for

(23)

(24)

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4328 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010

all

to the single node.

4) Finally, to avoid each singleton coalition join other sin-

gleton coalitions, we set:

forall.Indeed,thereisnomotivationforthenodes

which are not localizing any target to cooperate.

Itisworthemphasizingthattheaboveconstraintsareimposed

onlyduetokeepingconsistencyinthegameformulationandare

unrelated to the constraints (C1) and (C2) in (P).

and. Therefore, no sleep time is awarded

IV. DISTRIBUTED DYNAMIC COALITION FORMATION FOR

ENERGY-EFFICIENT DATA ACQUISITION

Having interpreted (P) as a coalition formation game, in this

section we introduce the game-theoretic energy-efficient data

acquisition algorithm for multitarget localization. Adopting

the Bayesian framework, the required localization accuracy

is achieved in consecutive iterations of a Bayesian estimator

where, in each iteration, the optimal coalition of nodes and

number of measurements for each node is obtained through the

distributed dynamic coalition formation algorithm. In general,

any Bayesian estimator can be utilized. Here, we choose the

sequential Markov chain Monte Carlo filter (particle filter) as

an illustration [21].

Below, we first present the main algorithm. The individual

steps are then described in subsequent subsections.

Algorithm 4.1: (Energy-Efficient Data Acquisition for Mul-

tiple Target Localization): Initialization: Set

particles

. Generate

based on for all

the prior density of the target

weight of particle for target

1) Compute

, wheredenotes the

at time .

and

for all.

2) Run the Preprocessing Algorithm (see Section IV-C) for

each target based on

and

accuracy

in period for all

and.Determinethesetof“potential”

nodes

.

3) Run the Distributed Dynamic Coalition Formation Algo-

rithm with initial state

to reach the core

4) Each node , existing in a nonsingleton coalition

, , records

measurements to the corresponding coalition head (CH)

(see Section IV-D) and then enters the sleep mode:

.

5) Run the particle filter

. Compute the achievable

. If , set

and using,,and

.

from target , transmits the

(26)

6) If

: Set and . Go to

Step 1.

In each iteration, the distributed dynamic coalition formation

algorithm (Step 3) determines the optimal coalition structure

and measurement allocations to reach the localization accuracy

, for, by reaching the core of the defined coalition

formation game. Full details are provided in Sections IV-A and

IV-B.Thepreprocessingalgorithm(Step2),aswillbeexplained

in Section IV-C, also determines the maximum achievable ac-

curacy and the set of “potential” nodes such that existence of at

least one absorbing state is guaranteed in the Markov chain un-

derlying the distributed dynamic coalition formation algorithm.

Each node then records a number of measurements based on the

sleep time allocation in Step 3 and the particle filter runs to ob-

tain the posterior distribution statistics.

A. Distributed Dynamic Coalition Formation Algorithm

The following algorithm is being executed independently by

each node in the network. It will be proved in Theorem 4.2 that

if the core of the game is nonempty, this algorithm converges to

the core with probability one where the average sleep time allo-

cated to the nodes is maximized under the relaxed localization

accuracy constraints (22). Further explanation and intuition is

provided in Section IV-B.

Algorithm 4.2: (Distributed Dynamic Coalition Formation):

Let

anddenote the coalition comprising node

the set of blocked nodes at period , respectively.

Initialization: Atselect initial coalition structure and

initialize the sleep time allocation vector. Set

Let

also be fixed for all nodes in the network.

The following steps are done distributively by each node

:

Step 1: Revision Strategy: Take a random draw from the

Bernoulli trial with probability . If the outcome is

“keep strategy”, set

and go to Step 5. Otherwise, go to Step 2.

Step 2: Evaluating the Best Strategy for the Next Period: Let

. Compute

and

.

,

(27)

(28)

Step 3: Experimentation:If

the Bernoulli trial with probability : if the outcome

in is “experiment,” choose

probability

probability

Step 4: Best-reply Process: Set

choose

bility

Step 5: Recursion: Set

In the above algorithm, Step 2 to Step 4 correspond to the

greedy strategy as introduced in (4). Algorithm 4.2 is accompa-

,takearandomdrawfrom

with equal

with equaland

,gotoStep5.Else,gotoStep4.

and

with equal proba-

.

and go to Step 1.

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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS 4329

nied by a procedure to detect the blocked nodes

one has to seek among all

satisfying

for which

to find those and

(29)

This corresponds to coalitions

straint (2) does not hold. This method is referred to as exhaus-

tive search for blocked nodes and requires to check

different combinations of targets and nodes. As the number of

nodes increases, this number goes up exponentially. Further-

more, in order to prevent examining a particular coalition

repeatedly, one needs to keep track of the coalitions for which

(29) has already been checked. Hence, this search scheme im-

poses an immense memory and computational overhead.

Asavariationtotheabovesearchscheme,weproposetocon-

struct a sample set from the set of all possible coalitions and

examine (29) only for this sample set. This sample set is con-

structedbytaking

randomsamplesfrom

with the set of targets

. Throughout, this method is referred to

as randomized search for blocked nodes and the resulting set

of blocked nodes is denoted by

will proved that by replacing

converges to the core of the defined game with probability one.

In what follows, the randomized search method is presented in

pseudo-code format.

for which the fairness con-

andcombining

. Although

with

, it

, Algorithm 4.2 still

Algorithm 4.3: (Randomized Search for Blocked Nodes)

for

to

do

Choose a coalition

probability

from the setwith equal

.

for

to

do

if

then

if

then

end if

end if

end for

end for

Remark 4.1: Using Algorithm 4.3 to detect blocked nodes,

the memory requirement at each node is reduced to

in the exhaustive search method. This is due to the fact

that we do not keep track of the coalitions

already been checked. In addition, the computational costs at

each iteration can be improved to

ever,thisimprovementresultsinslowerconvergencetothecore.

Hence, depending on the specifications of the nodes deployed

in the network, one can compromise between the memory and

computational cost and the convergence rate of Algorithm 4.2

using the size of the sample set .

Finally, Algorithm 4.2 should be accompanied by a mecha-

nism to update the state of the network as it requires

from

for which (29) has

from . How-

at each

period

anism, as well as the search method for blocked nodes, seem

to require a centralized device to accomplish these tasks. How-

ever, as will explained later in Section IV-D, adopting a hierar-

chical network architecture, these tasks can be carried out in a

distributed fashion.

to compute and . This mech-

B. Further Discussion on Algorithm 4.2

In this subsection, detailed explanation and intuition is

provided for the distributed dynamic coalition formation al-

gorithm. Algorithm 4.2 is decentralized in the sense that each

node makes a sequence of decisions independently (without

considering other nodes’ decisions at the current period) which

ultimately results in the whole network converging to the core

of the defined coalition formation game.

1) Myopic Best-Reply Rule: In the context of cooperative

game theory, the basic assumption is that all players are ra-

tional. Rationality means that players always try to maximize

their expected utility taking into account the strategies of their

opponents. Here, it is assumed that the nodes, as players of the

game, are bounded rational. A node which is selected to move,

based on the allocations in the previous period and considering

the feasibility constraints, tries to maximize its payoff only for

the next period.

At each time step, a random subset of the nodes get the

chance to revise their strategies for the next period. Formally,

each node’s opportunity to change its strategy is determined by

a random draw from the Bernoulli trial with probability

outcomes: “revise strategy” and “keep strategy” (Step 1). In the

literature, this trial is referred to as receiving the learn draw

[22]. Each node’s strategic variables are its choice of coalition

and the share of the total sleep time gained by that coalition.

Given that the network is in a specific coalition structure

strategies available to player

for the next period are given by

set

and

,

(30)

Here,

If the outcome of the trial is “revise strategy”, the node de-

cides whether to join any of the existing coalitions

to form singleton coalition and, at the same time, announces its

demand for the next period

the current state of the network

mined greedily by a best-reply rule: a node switches coalition

only if its expected sleep time in the new coalition is strictly

greaterthanitscurrentlyallocatedsleeptimeanditdemandsthe

most it can obtain considering the feasibility constraints. For-

mally, each node determines its maximum expected sleep time

andthecoalitionwhereitcanbeachievedby(27)and(28)(Step

2), respectively.If themaximizer coalition in (28) is not unique,

i.e.,

, the node randomizes between them with

equal probabilities

2) Best-ReplyProcessWithExperimentation: Themaximum

expected sleep time rule described above defines a finite state

denotes the discrete time.

or

. These decisions are based on

and are deter-

.

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4330 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010

Markovchain.ThisMarkovchainisreferredtoasthebest-reply

process. Formally, this Markov chain can be expressed as

(31)

if

if

(32)

where

strategies in order to move from state

denote the coalition to which node

and , respectively. Then,

choosing if it is a member of

expressed as

denotes the set of nodes whichhaveto change their

to . Letand

belongs in state

is the probability of node

in state and can be

if

if.

(33)

where

tively. As stated before,

getthechanceto revisetheir strategies.Therefore,

the probability of node switching to

in state.

Standard results on finite state Markov chains prove that, no

matter where the process starts, the probability that the process

reaches an ergodic state or set of states after

as

tends to infinity [27]. The fact that which of these ergodic

sets (states) will eventually be reached is determined by the

initial state. However, under the best-reply process, absorbing

states do not necessarily guarantee reaching core [18]. The so-

lutiontothisproblemistointroduceperturbations,i.e.,toallow

players choose suboptimal strategies with a small probability.

To let the nodes deviate with suboptimal strategies,

the best-reply process is modified as in (4): in any state

, when there exists a coalition

that

, each node

the best-reply rule with probability

strategy

with probability

(Step 3 and Step 4). The best-reply process modified by this

convention is called best-reply process with experimentation

[18]. Let

denote the set of blocked nodes in state

Then, the Markov chain underlying the best-reply process with

experimentation can be expressed as in (31) and (32) by only

modifying (33) as

anddenotethe“logicaland”and“logicalor”,respec-

denotes the probability that the nodes

is

from

steps tends to one

such

chooses

and chooses each

.

if

if.

(34)

where

node

the feasible sleep time in that coalition

(ifitrandomizesbetweenthemwithequalprobabil-

ities

).However,if

mands

with probability

perimentandwithprobability

the total probability adds up to

It will be shown in Theorem 4.2 that if the core of the game

is nonempty, the above dynamics converges to the core

probability one as time tends to infinity. Algorithm 4.2 follows

the Markov chain Monte Carlo (MCMC) approach in the sense

that a Markov chain is constructed such that the limiting dis-

tribution only assigns probability one to the core state. Having

constructed such a Markov chain, we form a realization of the

chain

and once the convergence is reached, in

the consecutive states the network remains in

only when . Therefore, if,

only joins the maximizer coalitionand demands

with probability 1

,itwilljoin andde-

if it does not ex-

ifitexperiments,hence

.

with

.

C. Preprocessing in Large WSNs

In large WSNs (comprising large number of nodes), to pre-

vent ineffective nodes taking part in the dynamic coalition for-

mation algorithm, a preprocessing algorithm is proposed which

both reduces the memory and computational costs to a great ex-

tent and ensures that the best-reply process with the new set of

nodes reaches an absorbing state. The following theorem states

theconditionunderwhichtheexistenceofatleastoneabsorbing

state is guaranteed.

Theorem 4.1: In a network with the set of nodes given by

trying to localize

targets, there exists at least one absorbing

state in the Markov chain defined by the best-reply process if

,

(35)

Proof: See Appendix A.

Fourparameters,asexplainedinSectionIII-B,affectthetotal

sleeptimeallocatedtoeachcoalition:i)numberofnodes,ii)rel-

ative distances of the nodes, iii) bearings of the nodes relative

to the target, and iv) prior density of the target. Considering

these four parameters, the set of nodes participating in Algo-

rithm 4.2 is contracted as follows: the procedure is initialized

with the two nearest nodes to

sider the set of nodes located inside a circle with radius

centered at . This set is denoted by

, we set

creased until

structural results presented in Section V-A, nodes with the fol-

lowing properties are eliminated form

at the bottom of the page]. The radius

. In each iteration, we con-

. If

. Suppose

. Considering the

is in-

: [see (36), shown

is increased until

even by eliminating the nodes

(36)

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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS4331

characterized in (36) and define

defined to replace

. Then,is

in (23) such that

(37)

is updated in iterationsof theBayesian estimatoruntil

. Here,is defined since the characteristic function with

the original

can result in small coalition values for which

the condition given in (35) cannot be satisfied for any

cases where the prior points to a large uncertainty area, i.e.,

, or the target is required to be localized with

very high accuracy, i.e.,

ducenegativevalues.Hence,theabovepreprocessingalgorithm

is devised to approach the required accuracy

steps of the Bayesian estimator and, at the same time, guarantee

the existence of an absorbing in each step. Note that more ac-

curacy in localizing the targets (larger

steps to reach. Hence, nodes are required to take more mea-

surements which again establishes the aforementioned tradeoff

between

and the average sleep time allocated to the nodes.

The preprocessing algorithm is executed for all targets de-

tected in the network. The resulting smaller set of nodes

will take part in the distributed dynamic coalition

formation algorithm (Algorithm 4.2). The randomized search

for blocked nodes (Algorithm 4.3) can also be refined using the

set of potential nodes

found for each target

lows: for each target , randomly take

set

. If

blocked nodes

.

Remark 4.2: As explained in Remark 4.1, the memory and

computational overhead is closely connected to the number

of nodes being involved in the distributed dynamic coalition

formation algorithm. By reducing the number of nods using

the preprocessing algorithm, the computational overhead for

finding blocked nodes using exhaustive search method is vastly

improved. The mean time before absorption also improves

when the randomized search method is employed.

. In

, (23) and (24) may even pro-

in consecutive

) translates to more

as fol-

from the

to the set of

samples

, add

D. Network Architecture and Implementation Issues

For Algorithm 4.2 to be applicable in real WSNs, it is im-

portant to consider the restrictions imposed by the sensor tech-

nology. In this paper, a hierarchical sensor network is consid-

ered which is composed of i) moderately populated nodes with

limited processing power, memory and battery, and ii) a back-

bone of sparsely spread nodes, assuming the role of coalition

heads (CH), which have more computational power and pro-

vide larger communication ranges. Assume the CHs are able to

communicate with each other. Each nonsingleton coalition is

assigned to a CH which knows the network configuration (i.e.,

locations of other nodes in the network) through an initial setup

process. Each node, existing in a nonsingleton coalition, sets up

a bidirectional communication link with the CH. It is also as-

sumed that the nodes are equipped with passive direction-of-ar-

rival (DOA) detectors and use the Zigbee/IEEE 802.15.4 pro-

tocol to transmit data.

The main computational overhead in Algorithm 4.2 is to de-

tect the blocked nodes. This task is being done by the CHs

collaboratively. The CH, to which the bearing estimations of a

specific target are sent, is responsible for detecting the blocked

nodeslocalizing thattarget. Hence,the CHswill alsobe respon-

sible to inform the blocked nodes of their potential for gaining

larger sleep times in other coalitions. In addition, since the CHs

represent the role of thebase station for each coalition, they will

be in charge for updating the state of the network through com-

municationwithother CHs.Therefore, computing(27) and (28)

canalso beturned overtotheCHs.Otherwise,thenodeshaveto

incur the communication overhead for receiving the following

informationfromCHs:

, and forall.In

the latter case, the nodes also need to experience an initial setup

process to receive the information about the location of all the

other nodes in the network.

Each node, joining a new coalition, sends a message to in-

form the new CH. The former CH will also be informed about

thismovethroughcommunicationwiththenewCHattheendof

each period. However, if a sensor is leaving a coalition to form

a singleton coalition, the former CH should be informed. The

overhead for leaving a coalition and joining a new coalition is

called the switching cost. This cost only includes the commu-

nication overhead for informing either the new or the old CH.

Hence, the switching cost for the node is inexpensive and nodes

can jump between coalitions without expending much energy.

Remark4.3: Theadvantageoftheproposeddistributedarchi-

tecture is that the complexity for integrating an energy-efficient

data routing protocol on top of the data acquisition scheme is

avoided. The premature power depletion of the nodes is also

avoided as CHs are responsible for collecting sensors’ mea-

surements.Inaddition,runningthepreprocessingalgorithm,the

nodes

stay in the sleep mode for the whole period

and do not expend energy in the transceiver. This is in contrast

to the centralized case where nodes need to hear the central-

ized decision-making device to receive the optimal number of

measurements in each iteration of the Bayesian estimator even

if

.

E. Convergence Analysis for Algorithm 4.2

We now proceed to prove that the proposed distributed dy-

namic coalition formation algorithm, accompanied by the ran-

domized search method for blocked nodes, guarantees the max-

imum average sleep time for the nodes conditional on feasi-

bility.Thiswillbeprovedbyshowingthatthebest-replyprocess

with experimentation in Algorithm 4.1.1 converges to the core

of the defined coalition formation game.

Theorem 4.2: Suppose that the randomized search method

(Algorithm 4.3) is employed to detect blocked nodes. Then, if

every node in the network follows Algorithm 4.2 and if the core

ofthegameisnonempty,thebest-replyprocesswithexperimen-

tation converges to the core almost surely, i.e.,

(38)

where

denotes the core of the game.

Proof: See Appendix A.

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4332 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010

Noting that the exhaustive search method is a special case of

Algorithm 4.3, where all the blocked nodes are detected, con-

vergence of the best-reply process with experimentation can be

easily inferred from Theorem 4.2.

In order to study the tradeoff mentioned in Remark 4.1, we

propose to use the mean time before absorption as the metric

to quantify the convergence rate of Algorithm 4.2. Suppose the

core of the game is nonempty. Then, there exists at least one

recurrent state in addition to the transient states in the Markov

chain defined by the best-reply process with experimentation.

By definition, a homogeneous Markov chain with at least one

transient and one recurrent state is called absorbing. The state

space

for an absorbing chain can be decomposed as

, where ’s denote the states containing cores of

the game and

represents the set of all transient states. The

transition probability matrix can also be block-partitioned as

, where denotes the identity matrix with

equaltothenumberofcoresinthegame,

trix with transition probabilities within the transient states, and

contains the probabilities of going from each transient state

to each absorbing state.

Here, we seek to compute mean time before absorption by a

given recurrent class starting from a given transient state. The

expected absorption time from each transient state is given by

the

element of

denotesthesub-ma-

(39)

where

represents a column vector of ones and

time visit to one of the recurrent classes

This measure will be used to compare the convergence rate of

Algorithm 4.2 in Section V-A.

Considering the communication cost required for evaluating

the optimal strategy and establishing membership in a different

coalition (see Section IV-D), an auxiliary optimization problem

can be formulated as the average communication cost until Al-

gorithm 4.2 reaches the core of the game. The proposed algo-

rithm simply constitutes a randomized adaptive search method

whose dynamicis asymptoticallyconsistent and attractedto the

core of the game and, at the same time, is efficient in terms of

the average communication cost for jumping between different

coalition structures. Starting from any state, the proposed adap-

tivesearchplanallowsthenetworktoreachthemaximizercoali-

tion structure with as few jumps between different states of the

Markovchain(31)and(32)aspossiblebyeachnodenotmaking

unnecessary jumps to nonpromising coalitions.

is the fundamental matrix. Here,

denotes the first-

after time 0 [28].

V. NUMERICAL EXAMPLES

In this section, examples are provided to illustrate behavior

and performance of the proposed solution. Throughout this sec-

tion, a standard deviation of 10 degrees is assumed as the mea-

surement error for all nodes, i.e.,

that

and

that

for all . Hence, nodes receive sleep times

in the interval

and it is guaranteed that

.

. It is also assumed

. Furthermore, we assume

A. Structural Results

In this part, behavior of the distributed dynamic coalition for-

mation algorithm is illustrated in a small network comprising 8

nodes. The small size of the network gives insight on how the

prior density of the target and the relative configuration of the

network play a role in the optimal coalition structure

sleep times

allocated to the nodes in the solution to the re-

laxed energy-efficient data acquisition problem.

1) Example 1: Consider the network configuration depicted

in Fig. 1(a). Suppose

is Gaussian with zero mean and

and

covariance matrix

. Equal variances in the

anddirectionareconsideredtoignoretheeffectsoftheprior

density of the target. At this point, we only aim at studying

the role of the relative configuration of the network on the

coalition structure and allocations in the core. Since the pair of

nodes {1, 5}, {2, 6}, {3, 7}, and {4, 8} are located on the same

line-of-sight from the target, they provide the same information

about the bearing of the target. However, information that the

nodes in coalition {1, 2, 3, 8} provide is more accurate due to

being closer to the target. In addition, bearings to nodes 1 and

2 are perpendicular to the ones for nodes 3 and 8, respectively,

which provide the highest diversity in measurements. Hence,

it is expected that the coalition {1, 2, 3, 8} be allocated the

largest total sleep time. If any of the nodes in the set {4, 5, 6,

7} joins this coalition, stochastic observability is no further

improved and the characteristic function allocates less total

sleep time as explained in Section III-B. This is verified by

the numerical results where

and

. As it is

showninFig.1(a),theoptimalcoalitionlocalizingthetargetand

the sleep time allocations in the core are

and

Table I gives the expected time before absorption for two dif-

ferent values of

for both exhaustive and randomized search

methods (Algorithm 4.3). As can be seen, decreasing the proba-

bility thattheset ofblocked nodesgetthechance toexperiment,

the expected time before absorption will increase. The trade off

mentioned between the size of the sample set and the expected

time before absorption can also be observed in Table I.

2) Example 2: In this example, the effect of the prior den-

sity of the target

is investigated on the optimal coalition

structure reached by the Algorithm 4.2. The prior density is as-

sumed to be a zero-mean Gaussian distribution with its covari-

ance given by one of the two following matrices:

, respectively.

(40)

Since

target position, it is expected that the optimal coalition structure

is formed such as to provide more information about that co-

ordinate. Analyzing the network given in Fig. 1(b) reveals that

the node pairs {1, 5} and {3, 7} provide information only on

the

andcoordinates of the target’s position, respectively.

However,

provide information on both coordi-

nates. Since the uncertainty in

to

direction, node 3 may provide redundant information as

nodes {2, 8} reduce the uncertainty in

assumes largeruncertainty onthecoordinateof the

direction is small compared

direction. Fig. 1(b)

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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS 4333

Fig. 1. Effects of: (a) relative configuration of the target and nodes, and (b) prior density of the target on the optimal coalition structure for localization of a single

target. Mean of the target prior distribution and nodes are depicted by the

and

nodes localizing the target and

’s give the sleep times allocated to the nodes in the core in terms of multiples of

the prior and posterior densities of the target, respectively (a)

(b).

signs, respectively. the set of filled squares represent the optimal coalition of

. The solid and dashed ellipses also represent

Fig. 2. Localization of a single target: optimal coalition structure and sleep times allocated to the nodes in the core in terms of multiples of

(b)

. The solid and dashed ellipses depict the prior and posterior densities of the target, respectively.

at: (a)

, and

TABLE I

EXPECTED TIME BEFORE ABSORPTION: EXHAUSTIVE SEARCH METHOD

VERSUS. RANDOMIZED SEARCH METHOD

justifies the above discussion by showing the optimal coalition

localizing the target and sleep time alloca-

tions

since the uncertainty along

axis is increased, we anticipate

that node 3 also takes part in the localization which is verified

by Fig. 1(a).

. Now, considering,

B. Target Localization

In this part, behavior of the energy-efficient data acquisition

for multitarget localization (Algorithm 4.1) is investigated for

the network configuration depicted in Fig. 1(a). Covariance of

the target distribution is assumed to be

the preprocessing algorithm yields

the optimal coalition structure and sleep times allocated to the

nodes when Algorithm 4.2 reaches the core at

quently, each node takes a number of measurements equal to

which results in the posterior distribution depicted by

the dash-dot ellipse. This updated distribution will be used as

the prior for the next measurement interval. Fig. 2(a) and (b)

demonstrates the core state, the prior and the posterior distribu-

tion of the target at

and

seen, although the optimal coalition structure

same from

to , the optimal sleep times

to the nodes in the core change.

at

. Fig. 1(a) shows

. Here,

. Subse-

, respectively. As can be

remains the

allocated

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4334 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010

Fig. 3. Multiple target localization: optimal coalition structure and sleep times

allocated to the nodes in the core at

.

In addition, an example is provided to study the be-

havior of Algorithm 4.1 for multiple target localization in

the network depicted in Fig. 3. Here, it is assumed that

. Running the preprocessing algo-

rithm results in:

and

. These nodes are shown

with the same color as the corresponding target in Fig. 3. Here,

. These nodes will join the coalition,

i.e., either

or , where they can achieve larger sleep times.

Fig. 3 demonstrates the optimal coalition structure

times

allocated to the nodes in the core for

Finally, performance of Algorithm 4.1 is compared with a

scenario where a fixed set of nodes

the target in a Bayesian framework. These nodes are assumed

to be the closest nodes to the target, hence, providing more

accurate observations compared to other nodes. In multitarget

tracking scenarios, if a node

more than one target, i.e.,

and , the node chooses the target to localize randomly.

We refer to this method as range-based measurement alloca-

tion. These nodes are assumed to be awake for the whole pe-

riod and take

measurements, i.e.,

and. Fig. 4(a) shows the average sleep time allocated

to the nodes in each method as a function of the number of

nodes in the network. Here, 100 random network configura-

tions are generated in two-dimensional space with

targets spread uniformly in each network. Nodes are spread

around the targets in a

sities of the targets are assumed to be Gaussian with covariance

and sleep

.

are assigned to localize

is in the closest nodes set for

such that

for all

nodes and

square and the prior den-

matrix

. The core is replaced with the ab-

sorbing state of the best-reply process without experimentation

when the core turns out to be empty. As the number of nodes

in the network increases, uniform distribution of the nodes pro-

vides more diversity in the relative configuration of the target

and nodes, hence, more diverse bearing measurements are col-

lected and the average sleep time allocated to the nodes in-

creases in both methods. However, as can be seen in Fig. 4(a),

the distributed dynamic coalition formation approach (Algo-

rithm4.2)demonstratesasignificantaveragesleeptimeincrease

compared with the range-based method. Particularly, the av-

eragesleeptimeallocatedtothenodesisguaranteedtobe larger

that

.

The tradeoff between the required localization accuracy

and the average sleep time allocated to the nodes is also demon-

strated and compared with the heuristic range-based measure-

ment allocation in Fig. 4(b). Here, 100 random network config-

urations are studied with 10 nodes and one target spread uni-

formly in a

square network. The prior density of

the target is also assumed to be Gaussian with covariance ma-

trixasabove.Fig.4(b)illustratesthatas

sleep time allocated to the nodes decreases in both approaches.

However,theaverageallocatedsleeptimedropsmorerapidlyin

the range-based method. Hence, the distributed dynamic coali-

tion formation approach provides a better tradeoff between

and the average sleep time allocated to the nodes.

goesup,theaverage

VI. CONCLUSION

This paper considered the energy-efficient data acquisition

for multiple target localization with a prespecified accuracy.

The problem was formulated as a coalition formation game

for bearings-only localization in two-dimensional space. The

modified core was proposed as the solution concept for the

cooperative game and it was shown that the coalition structure

and sleep time allocations in the core correspond to the solution

to the energy-efficient data acquisition problem. We proposed a

distributed dynamic coalition formation algorithm and proved

that the core of the game will be reached almost surely given

thateverysensorfollowsthisalgorithmandthecoreofthegame

is nonempty. This algorithm was integrated with a sequential

Bayesianestimatortolocalizetargets,forwhichthesuperiorper-

formanceovertheheuristicrange-basedmeasurementallocation

method was demonstrated through Monte Carlo simulations.

The proposed algorithm can also be employed in range-only

localization, by deriving the appropriate characteristic function,

and tracking slow-moving targets. Due to the random configu-

ration of the nodes relative to the target, determining conditions

that ensure a nonempty core is an open problem.

APPENDIX A

PROOFS

Proof of Proposition 2.1: Since estimation errors

the nodes in a particular coalition

for

are mutually independent,

, where

(41)

Hence,

(42)

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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS 4335

Fig. 4. Average sleep time allocated to the nodes during a localization task versus: (a) number of nodes in the network

.

, and (b) required localization accuracy

Since

sequently, substituting (42) in (10), the FIM

pressed as

’s are zero-mean,. Sub-

can be ex-

(43)

We then substitute

from (43) in (8) and expand

, it is straightforward to. Noting that

derive:

(44)

here

(45)

Subsequently, changing the indexes in the sums using the fol-

lowing equations

(46)

(47)

can finally be written as

(48)

Therefore, Proposition 2.1 is justified.

Proof of Theorem 4.2: Assume that

tions exist in state

each target

(we assume

blocked coalitions is given by

ularly, the probability of detecting all blocked coalitions is

. Therefore, there is a positive probability to

detect blocked coalitions with only checking the sample set.

Note that any two blocked coalitions may comprise overlap-

ping blocked nodes. Hence, since we are interested in detecting

blocked coali-

. Ifsamples are taken from

), probability of detecting

for

. Partic-

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4336IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010

blocked nodes, the probability of detecting all blocked nodes

follows:

(49)

In [18, Theorem 2] it is proved that the vector of sleep times,

allocated in an absorbing state of the best-reply process with

experimentation,coincideswiththesetofcoreallocationsofthe

game. Therefore, it is implied that if

state, then

will be a core allocation for the game.

Finally, we prove that the best-reply process with experimen-

tation will converge to an absorbing state with probability one

as time tends toward infinity when Algorithm 4.1.2 is deployed

for detecting blocked nodes. This is proved by showing that

the process will not get stuck in ergodic sets other than the ab-

sorbing states. Suppose that there exists a nonsingleton ergodic

set

such that . [18, Theorem 2] guarantees that

noneofthestatesin

involveacoreallocation(absorbingstates

are singleton ergodic sets). As a result, for each

ists

such that

some nodes have the incentive to experiment. There is a posi-

tive probability that all the nodes in the blocked coalitions (

nodes)aredetected.Inaddition,thesenodescanexperimentand

formsingletoncoalitionswithsomepositiveprobability.Hence,

which comprises i) singleton coalitions, and ii) nonsin-

gleton coalitions which have no blocked nodes, can be reached

in one step. Since

can be reached from

positive probability, we have

that the core is nonempty and starting with

state

can be reached in one step. All previously

blocked nodes which are in nonsingleton coalitions in

notexperimentand allnodesin singletoncoalitionsexperiment.

Thisoccurswithprobability

, we fix one node denoted by

probability

that all other nodes in

periment in

, joinand demand

is an absorbing

there ex-

. Therefore,

with some

. Now, using the fact

, an absorbing

do

.Now,forevery

. There is a positive

which ex-

. The resulting state

is

tive probability to reach an absorbing state. This contradicts the

assumption that

is an element of an ergodic set and com-

pletes the proof.

Proof of Theorem 4.1: The proof is very similar to the one

presentedin[18].Supposethatfortarget ,forwhich(35)holds,

the grand coalition

is formed and each node achieves a

sleeptime

suchthat

. Now, it is easy to show that there is no in-

centive for any node to leave the grand coalition. Each node

has two choices: i) join other nonsingleton coalitions

and ii) form the singleton coalition. Since

they have no incentive to form nonsingleton coalitions in which

. Furthermore, since

have no incentive to form singleton coalitions. Hence,

constitutes an absorbing state for the best-reply

process.

. Therefore, starting fromthere is a posi-

for all,

, they

APPENDIX B

DERIVATION OF CHARACTERISTIC FUNCTION

In this appendix, the characteristic function presented in (23)

and (24) is derived. In what follows, we benefit from the fol-

lowing inequality:

(50)

This inequality holds due to the concavity property of the loga-

rithm function. As the first step, we remove the expectations in

(14) using the approximation in (17). Then, assuming

diagonal and applying the above inequality in (14)–(16) repeat-

edly, a lower bound can be found as (51), shown at the bottom

of the page.

to be

(51)

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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGS-ONLY LOCALIZATION IN SENSOR NETWORKS4337

(53)

(55)

(56)

(57)

Subsequently, applying the following equality

(52)

one can write (51) as (53), shown on the page. Since the argu-

mentsof

areintegersintheclosedinterval

and due to the concavity property of the logarithm function,

can be lower bounded by

(54)

Consequently, [see (55), shown on the page]. Finally, applying

the relaxed constraint in (22) [see (56), shown on the page],

where

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

Section II-A,

’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.

and

, localizing

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for

their useful comments that helped in improving the quality of

the paper.

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New York:

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

ofTechnology,Tehran,Iran,in2007,andtheM.A.Sc.

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.

V. Krishnamurthy.

Hisresearchinterestsspanstochasticoptimization,

game theory, and learning in games with applications

in wireless communication and sensor networks.

Vikram

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

Krishnamurthy

(S’90–M’91–SM’99–

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.