Coalition Formation for BearingsOnly Localization in Sensor Networks—A Cooperative Game Approach
ABSTRACT In this paper, formation of optimal coalitions of nodes is investigated for data acquisition in bearingsonly 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 (BFIM). 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 numerical 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.

Conference Paper: Reference node selection for cooperative positioning using coalition formation games
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ABSTRACT: Conference code: 93004, Export Date: 21 December 2012, Source: Scopus, Art. No.: 6268747, doi: 10.1109/WPNC.2012.6268747, Language of Original Document: English, Correspondence Address: Hadzic, S.; Instituto de Telecomunica̧ões, Campus Universitário de Santiago, 3810193, Aveiro, Portugal; email: senka@av.it.pt, References: Lieckfeldt, D., You, J., Timmermann, D., Distributed selection of references for localization in wireless sensor networks Proceedings of the 5th Workshop on Positioning, Navigation and Communication (WPNC), 2008, pp. 3136;, Sponsors: IEEEWPNC'12  Proceedings of the 2012 9th Workshop on Positioning, Navigation and Communication; 01/2012  [Show abstract] [Hide abstract]
ABSTRACT: This paper presents a gametheoretic approach to node activation control in parameter estimation via diffusion least mean squares (LMS). Nodes cooperate by exchanging estimates over links characterized by the connectivity graph of the network. The energyaware activation control is formulated as a noncooperative repeated game where nodes autonomously decide when to activate based on a utility function that captures the tradeoff between individual node's contribution and energy expenditure. The diffusion LMS stochastic approximation is combined with a gametheoretic learning algorithm such that the overall energyaware diffusion LMS has two timescales: the fast timescale corresponds to the gametheoretic activation mechanism, whereby nodes distributively learn their optimal activation strategies, whereas the slow timescale corresponds to the diffusion LMS. The convergence analysis shows that the parameter estimates weakly converge to the true parameter across the network, yet the global activation behavior along the way tracks the set of correlated equilibria of the underlying activation control game.IEEE Journal of Selected Topics in Signal Processing 01/2013; 7(5):821836. · 3.30 Impact Factor  SourceAvailable from: 72.88[Show abstract] [Hide abstract]
ABSTRACT: We present a decentralized adaptive filtering algorithm where each agent acts selfishly to maximize its payoff. Agents are only aware of the actions of other agents within their coalitions and have no knowledge of the actions of agents outside the coalition. We show that the global behavior of the system converges to the set of correlated equilibria. Thus simple behavior by individual agents can result in sophisticated global behavior.Acoustics, Speech and Signal Processing (ICASSP), 2011 IEEE International Conference on; 06/2011
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4322IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
Coalition Formation for BearingsOnly 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 bearingsonly 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 (BFIM). 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—Bearingsonly 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. TaSung Lee.
The authors are with the Department of Electrical and Computer Engi
neering, University of British Columbia, Vancouver, V6T 1Z4, Canada (email:
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 lineofsight 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 (BFIM) 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 NPhard
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]
1053587X/$26.00 © 2010 IEEE
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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGSONLY 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:dutycycling,mobilitybased,anden
ergyefficient data acquisition [2]. Dutycycling 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.Energyefficientdataac
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 gametheoretic scheme for
energyefficient data acquisition in bearingsonly 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 energyefficient data acquisition
problem as a coalition formation game: In Section II,
energyefficient data acquisition in twodimensional bear
ingsonly 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 energyefficient 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 IVB, 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 IVA) 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 rangebased measurement allocation method via
Monte Carlo simulations.
II. FORMULATION OF THE ENERGYEFFICIENT DATA
ACQUISITION PROBLEM
Inthissection,weformulatetheenergyefficientdataacquisi
tion problem for the bearingsonly multitarget localization sce
nario in twodimensional 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 twodimensional 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 energyefficient 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 IIC. 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 bruteforce 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 energyefficient 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 IIIB). We
then propose a distributed dynamic coalition formation algo
rithm in Section IVA 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 IVB, 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” energyefficient data acquisition problem
(see Section IIIB). This approach brings about two main ad
vantages: i) it is performed distributively among the nodes and
eliminates the need for a central decisionmaking 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 IIIA).
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GHAREHSHIRAN AND KRISHNAMURTHY: COALITION FORMATION FOR BEARINGSONLY 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 (PCRLB) theorem
[19] establishes a lower bound on
orem, there exists
given by
. According to this the
(8)
(9)
(10)
such that
(11)
wherethematrixinequalityindicatesthat
itivesemidefinite. Intheaboveequations,
denote the FIM and BFIM, 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 IIIB.
Definition 2.1:Stochastic observability is defined as
, where
In the literature, various matrix means of the BFIM 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 BFIM.
(13)
where
tion. The following proposition provides a closedform 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 BFIM as
(17)
where
tical applications, the PCRLB 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 energyefficient 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. NonSuperadditive Cooperative Games and the Core
The energyefficient 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 realvalued 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
combinatorialenergyefficientdata 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 logdeterminant
to change into a linear function of
then relax (C1) by making this lower bound satisfy the required
localizationaccuracytoderivethecharacteristicfunctionforthe
game. Formally,
. We
(22)