Cooperative Content Retrieval in Nomadic SensorNetworks
Ioannis Koukoutsidis, Eva Jaho, Ioannis Stavrakakis
Dept. Informatics and TelecommunicationsNational & Kapodistrian University of AthensIlissia, 157 84 Athens, GreeceEmail:
{
i.koukoutsidis,ejaho,ioannis
}
@di.uoa.gr
Abstract
—A nomadic sensor network consists of: a) sensornodes, that are ﬁxed at some points and collect informationabout states or variables of the environment, and b) mobilenodes that collect and disseminate this information. Mobile nodesusually belong to different classes, and are thus interested indifferent subsets of sensor node information. In such networks,dissemination of information content at smaller costs can beachieved if mobile nodes are cooperative and collect and carryinformation not only in their own interest, but also in the interestof other mobile nodes. A speciﬁc modeling scenario is consideredin this paper where the network has the form of a graph; sensornodes are located on the vertices of the graph and Unodes movealong the edges according to a random waypoint model. Wepresent a gametheoretic analysis to ﬁnd conditions under whicha cooperative equilibrium can be sustained.
I. I
NTRODUCTION
A
nomadic sensor network
is a recently introduced networking paradigm [1]. It consists of: a) simple, tiny sensordevices (Tnodes) ﬁxed at some points, whose purpose is tocollect information about states or variables of the environmentand b) more complex mobile devices that are carried byusers (Unodes), that collect and disseminate this information.Compared to traditional sensor networks where communication to endusers is realized in a multihop fashion, thisparadigm exploits user mobility to conserve limited sensorenergy, prolonging the lifetime of the network and makingit more costefﬁcient.A Unode can collect sensor data either from the sourceTnode or from an encountered Unode who has previouslyacquired the data. Collecting data from other Unodes mayincur a smaller access cost (in time or energy), especiallyif the source Tnode is faraway. Exploiting mobility in thisway to reduce content retrieval costs requires each Unode toshow some kind of cooperative behavior. However, Unodesare highly autonomous and intelligent devices; different Unodes may belong to different classes, and thus be interestedin different sensor node information. Acquiring unwantedinformation incurs a cost in time or energy, thus a Unodewould be cooperative and collect information of potentialinterest to other Unodes only in anticipation of the same
This work has been supported by the European Commission under EUproject BIONETS (ISTFETSACFP6027748, www.bionets.eu) and the NoECONTENT (ISTFP60384239, www.istcontent.eu).
U2T1T2T3T4U1
Fig. 1. Network graph
behavior by other Unodes. The purpose of this paper is toidentify requirements and conditions under which cooperativebehavior may emerge in such a network. We present a gametheoretic analysis to ﬁnd conditions under which a cooperativeequilibrium exists.The general modeling scenario that we study is as follows.We consider a graph model of the network (see Fig. 1).Tnodes are located at the vertices of the graph and Unodes move randomly along the edges of the graph collectinginformation when they reach a Tnode or upon meeting anotherUnode somewhere on the graph. Nodes move according toa Markovian waypoint model on the graph, with a constantspeed
v
. Waypoints are set to be the vertices of the graph (Tnodes). No pause times at waypoints are considered. Transitions from one waypoint to the next are governed by a Markovchain, i.e. a Unode moves from waypoint
T
i
to waypoint
T
j
with probability
p
(
T
i
,T
j
)
. This is a special case of the socalled “space graph” model in [2].In order to simplify the analysis, instead of studying thenetwork as a whole, we decompose the problem by studyingpossibilities of cooperation on each leg (edge) of the graph.We say that two Unodes coming from opposite directionsand meeting somewhere on a leg
(
T
i
,T
j
)
are cooperative on
(
T
i
,T
j
)
, if and only if each one copies the content of itssrcin Tnode (e.g., in Fig. 1,
U
1
would copy
T
1
and
U
2
wouldcopy
T
2
), even if they are not interested in it. We shall ﬁnd
conditions under which the following strategy of each Unoderesults in an equilibrium. This strategy is made up of twoactions:
Initially, a Unode is cooperative and copies unwanted content. However, if it meets a selﬁsh Unode somewhere ona leg, it will only transmit its acquired content with a certain probability.
This strategy may easily be applied, provided thatupon meeting each other, Unodes exchange messages thatcontain the list of information objects stored in their memory.The equilibrium conditions depend mainly on the probability of meeting other Unodes on the leg. Under the setting wediscuss in this paper, satisfying equilibrium conditions for eachleg and between each pair of Unodes means that a cooperativeequilibrium is achieved for the whole network. Note that inmaking this decomposition, we are implicitly assuming thatUnodes base their decision only on their interest for contentthat is in distance of one leg. The analysis becomes morecomplicated when this “decision horizon” is more than one,and is not attempted here. (For example in Fig. 1, if node
U
1
that starts from
T
1
is interested in both
T
2
and
T
3
and isabout to follow the path
{
T
1
,T
2
,T
3
}
, it should consider theprobability of meeting Unodes that previously acquired datafrom
T
2
or
T
3
, on all segments of this path.)On what concerns the knowledge each Unode has, thefollowing assumptions are made. Each Unode knows thetopology of the network, the number of other Unodes and thedistances between each pair of Tnodes. They are also awareof the information content at each Tnode. Furthermore, Unodes know that other Unodes move also according to therandom waypoint model on the graph. On the other hand,the (instantaneous) rate at which the information content oneach Tnode is updated is a function of time unknown to theUnodes. (Thus a Unode may return to an already visited Tnode to get updated information.) In addition, each Unodedoes not know the interests of other Unodes for sensor dataand has no memory of previous encounters with them.Several hypotheses that facilitate the analysis are also made.First it is assumed that Unodes are homogeneous devices,have the same processing and communication costs and havethe same movement parameters. Secondly, Unodes have inﬁnite (in practice this means sufﬁciently large) storage space.Thus they do not have to consider possible replacement policies (for example, throwing away their less valuable content)and respective costs. Thirdly, that each Tnode generates asingle type of information content, and information contentis not depreciated in time or space. If it was depreciated, aUnode should have to include in its decision the (possiblysubjective) value of each information object at the moment of its acquisition. Examples of nondepreciated content includestatistical samples (e.g., measurements of physical quantities).On the other hand, information with limited temporal orspatial scope, or software modules that are subject to updatesare considered to be depreciating in time or space. We alsoconsider that Unodes make decisions for their best interest,but are not malicious, i.e., they don’t perform actions fromwhich they have no material gain, only to hurt others. Finally,we assume that data exchanged between Unodes or betweena U and Tnode consist only of a few bits; thus they aretransmitted within an inﬁnitesimal time interval, which is notconsidered in the analysis.II. A
NALYTICAL
M
ODEL
In the analysis that follows, we consider a number
N
of Unodes, and calculate the expected cost for a Unode to followa certain strategy on an arbitrary leg
(
T
i
,T
j
)
. Because of thesymmetry of our model, the cost is the same for all Unodes.A strategy consists of a sequence of actions, concerning thedecisions to collect, carry and transmit information that is notof interest to a Unode. Actions are either of a cooperative orselﬁsh nature. A strategy is itself called cooperative if all theactions it is composed of are cooperative.We consider that the game is played between a Unodestarting from the srcin sensor
T
i
and
N
−
1
other Unodesit may meet before reaching the destination sensor
T
j
. Ourworking hypothesis is that a Unode is not interested inthe content of the origin sensor, but only in that of thedestination. A priori, it assumes the same for the other Unodes. This will produce conditions for cooperation in a worstcase scenario, since otherwise if a Unode is also interestedin the content of the srcin sensor, it has greater incentive tocooperate. The analogous assumption for the other Unodescan be partially justiﬁed by the lack of any information aboutthe identities of the encountered nodes and their interests forsensor information.Costs can be expressed in time or energy units. Here weinterpret this cost as the delay to retrieve information content.The length of a leg
(
T
i
,T
j
)
is denoted by
d
(
T
i
,T
j
)
. Thecommunication and processing cost of a Unode to acquirecontent from a Tnode and then transmit it to a Unode isa constant
c
, translated in time units. Consistently with ourassumption of inﬁnite storage, we do not consider any costfor carrying the information. Finally, not to complicate theanalysis, the transmission ranges of both Unodes and Tnodesare set to be zero.Consider the process
X
(
t
)
of the position of the Unodeon the graph at each time instant
t
. We ﬁnd its stationarydistribution as follows. First consider the embedded Markovchain of waypoints visited sequentially by a Unode. Under thestationary distribution of this chain, the fraction of transitionsto waypoint
T
i
is denoted by
π
i
. Then if we have constantvelocity, the probability that the Unode is on any segment of length
x
on leg (
T
i
,
T
j
) in direction from
T
i
to
T
j
is
π
i
p
(
T
i
,T
j
)
x
T
i
T
j
=
T
i
π
i
p
(
T
i
,T
j
)
d
(
T
i
,T
j
)
.
(1)That is, it is the fraction of time the Unode spends on thesegment of length
x
while moving from
T
i
to
T
j
.Suppose that the Unode, hereafter called
U
i
, is at waypoint
T
i
at
t
= 0
and decides to go towards waypoint
T
j
. Conﬁne thestrategy of each player to take values in the set
S
=
{
C,S
}
(
C
:cooperative,
S
: selﬁsh). By choosing strategy
C
,
U
i
copies,carries and transmits the content of
T
i
to an encountered Unode that is interested in it, whereas by following strategy
S
it ignores it. In order to calculate the expected cost by following a certain strategy,
U
i
should estimate the probability of meeting another cooperative Unode coming from
T
j
towards
T
i
, at a certain distance
x
from
T
i
. Suppose there are
k
other
(
k < N
)
cooperative Unodes. To meet another Unode withina distance
x
, the latter must be at a distance of at most
2
x
at
t
= 0
and be headed towards
T
i
. Since the move processesof the Unodes are mutually independent as well as jointlystationary, the instant
t
= 0
is an arbitrary instant at which
U
i
at
T
i
observes the position of the other Unodes. Thereforeit observes the other Unodes in their stationary distribution.Consequently, if
d
(
T
i
,T
j
)
≥
2
x
the probability of a meetingwith at least one cooperative Unode within a distance
x
is
F
k
(
x
) = 1
−
(1
−
π
j
p
(
T
j
,T
i
)2
x
T
i
T
j
=
T
i
π
i
p
(
T
i
,T
j
)
d
(
T
i
,T
j
))
k
.
(2)If
d
(
T
i
,T
j
)
<
2
x
, we must also include the event that anotherUnode is on a leg or direction different from
(
T
j
,T
i
)
at
t
=0
but can meet with
U
i
in time less than
x/v
. Consider allsuch legs
m
and the starting points
T
m
of the Unodes onthese legs. Let
{
T
m
,...,T
j
}
denote all the paths that maybe followed to reach
T
j
before the meeting, and for which
p
(
T
m
,
·
)
···
p
(
·
,T
j
)
>
0
, where
·
denote intermediate states.Then this meeting probability equals
1
−
(1
−
m
π
m
p
(
T
m
,
·
)
···
p
(
·
,T
j
)
p
(
T
j
,T
i
)2
x
T
i
T
j
=
T
i
π
i
p
(
T
i
,T
j
)
d
(
T
i
,T
j
) )
k
.
The summation in the numerator is over all possible pathsthat can be followed. This probability again equals (2), since
k
π
k
p
(
T
k
,
·
)
···
p
(
·
,T
j
) =
π
j
from the balance equations inthe embedded Markov chain.Hence the distribution function
F
k
(
x
)
gives the probabilitythat a meeting with another cooperative Unode takes place atdistance
≤
x
, for any
x
such that
0
< x < d
(
T
i
,T
j
)
.Suppose that there are only two nodes in the network,
U
i
and
U
j
, and that
U
i
meets with
U
j
at distance
x
from
T
i
(
x <d
(
T
i
,T
j
)
). If they follow strategies
s
i
,
s
j
respectively (
s
i
,s
j
∈S
), then the cost of
U
i
, denoted as a function
C
(
T
i
,T
j
)
i
(
s
i
,s
j
,x
)
of
U
i
, where
C
(
T
i
,T
j
)
i
:
S ×S ×
[0
,d
(
T
i
,T
j
)]
→
R
, is
C
(
T
i
,T
j
)
i
(
C,C,x
) =
c
+
x/v
C
(
T
i
,T
j
)
i
(
S,C,x
) =
x/v
C
(
T
i
,T
j
)
i
(
C,S,x
) =
c
+
d
(
T
i
,T
j
)
/v
C
(
T
i
,T
j
)
i
(
S,S,x
) =
d
(
T
i
,T
j
)
/v .
(3)We denote by
α
(
T
j
,T
i
)
the probability that
U
i
meets witha
speciﬁc
other Unode before meeting
T
j
, i.e.:
α
(
T
j
,T
i
)
2
π
j
p
(
T
j
,T
i
)
d
(
T
i
,T
j
)
T
i
T
j
=
T
i
π
i
p
(
T
i
,T
j
)
d
(
T
i
,T
j
)
.
(4)As it can be seen, this meeting probability increases withthe length of a leg. Additionally,
U
i
has a higher meetingprobability for greater
π
j
p
(
T
j
,T
i
)
, that is if the destinationnode
T
j
is more frequently visited, or if Unodes at
T
j
havean increased probability of heading towards
T
i
.We will derive the expected cost of
U
i
to take action
s
i
,when
k
other Unodes are cooperative (
0
≤
k
≤
N
−
1
). Wedenote this as
C
(
T
i
,T
j
)
i
(
s
i

k
)
. For
1
≤
k
≤
N
−
1
, we havethat
C
(
T
i
,T
j
)
i
(
s
i

k
) =
d
(
T
i
,T
j
)0
C
(
T
i
,T
j
)
i
(
s
i
,C,x
)
dF
k
(
x
)+ (1
−
α
(
T
j
,T
i
))
k
C
(
T
i
,T
j
)
i
(
s
i
,C,d
(
T
i
,T
j
))
.
(5)We arrive at the following expressions for different combinations of followed strategies:
C
(
T
i
,T
j
)
i
(
C

k
) =
c
+
d
(
T
i
,T
j
)
v
1
−
(1
−
α
(
T
j
,T
i
))
k
(
k
+ 1)
α
(
T
j
,T
i
)
C
(
T
i
,T
j
)
i
(
S

k
) =
d
(
T
i
,T
j
)
v
1
−
(1
−
α
(
T
j
,T
i
))
k
(
k
+ 1)
α
(
T
j
,T
i
)
C
(
T
i
,T
j
)
i
(
C

0) =
c
+
d
(
T
i
,T
j
)
/v
C
(
T
i
,T
j
)
i
(
S

0) =
d
(
T
i
,T
j
)
/v .
(6)III. G
AME
T
HEORETIC
A
NALYSIS
We observe from (6) that
C
(
T
i
,T
j
)
i
(
C

k
)
>
C
(
T
i
,T
j
)
i
(
S

k
)
∀
i,k
. Furthermore it can be shown that
C
(
T
i
,T
j
)
i
(
C

k
)
,
C
(
T
i
,T
j
)
i
(
S

k
)
are decreasing functions of
k
.The game is a Bayesian analog of the Nperson prisoner’sdilemma (see [3]). (Since
U
i
does not know the number oridentity of other Unodes it may meet on
(
T
i
,T
j
)
.) Whenseen as a noncooperative game, it is evident from the aboveexpressions that there exists only one equilibrium, in whichevery node is selﬁsh. (Since
S
strongly dominates
C
for everyplayer.) However, it may not be the best solution; player
U
i
can have a beneﬁt by cooperating on
(
T
i
,T
j
)
, when
k
otherUnodes are cooperative, if
C
(
T
i
,T
j
)
i
(
C

k
)
<
C
(
T
i
,T
j
)
i
(
S

0)
.
(7)That is, if the cost for
U
i
when
k
other Unodes are cooperativeis smaller than its cost when all Unodes are selﬁsh. Thiscondition also complies with individual rationality of
U
i
, since
C
(
T
i
,T
j
)
i
(
S

0)
is the minimax value
U
i
can guarantee for itself.From (6), this inequality is satisﬁed if
1
−
1
−
(1
−
α
(
T
j
,T
i
))
k
(
k
+ 1)
α
(
T
j
,T
i
)
> cvd
(
T
i
,T
j
)
.
(8)It must always hold that
c <
d
(
T
i
,T
j
)
v
; that is, our initialassumption must be that the cost (expressed in time units)for a Unode to acquire and transmit unwanted content mustbe smaller than the time to reach
T
j
starting from
T
i
.We ﬁnd an approximate condition for the above inequalityto be satisﬁed. The Taylor polynomial of
(1
−
α
)
k
at
α
= 0
is,up to a second order approximation,
1
−
kα
+
k
(
k
−
1)
α
2
2
(
k >
1
).Substituting this approximate expression in inequality (8), we
0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1
T h r e s h o l d v a l u e o f k
α
d/v=10d/v=50d/v=100
(a)
c
= 3
0 20 40 60 80 100 120 0 0.2 0.4 0.6 0.8 1
T h r e s h o l d v a l u e o f k
α
c=3c=10c=30
(b)
d/v
= 50
Fig. 2. Approximate threshold values of the number of other cooperativenodes
k
above which cooperation on a directed leg is better for
U
i
than fullselﬁshness, for different values of
α
and
d/v
,
c
.
get the condition
k >
2
cvd
(
T
i
,T
j
)
α
(
T
j
,T
i
)
−
2(1 +
α
(
T
j
,T
i
))
α
(
T
j
,T
i
)(
k
+ 1) + 2
,
which is satisﬁed when
k >
2
cvd
(
T
i
,T
j
)
α
(
T
j
,T
i
) + 2
.
(9)Based on (9), in Fig. 2 we show approximate threshold valuesof the number of other cooperative nodes
k
above whichcooperation for
U
i
on
(
T
i
,T
j
)
is more beneﬁcial than the casewhen all Unodes are selﬁsh.These conﬁrm that from the point of view of
U
i
, a smallernumber of cooperative nodes is required on longdistanceroutes or when the destination node
T
j
is visited more often.On the other hand, a higher number is required when Unodesare moving at a high speed or if the cost
c
is higher.The situation in which each Unode is cooperative onboth directions of a leg
(
T
i
,T
j
)
will be identiﬁed as “fullcooperation” on
(
T
i
,T
j
)
. The inverse situation in which eachUnode is selﬁsh, will be called “full selﬁshness”. When fullcooperation is achieved as a strategic equilibrium in a leg of the graph, then we will say that a cooperative equilibriumexists on this leg. If this can be achieved on all legs, then acooperative equilibrium exists in the whole network. We nextproceed to ﬁnd a strategy of each Unode and conditions underwhich a cooperative equilibrium can be achieved.First pay attention to the fact that even is full cooperationis beneﬁcial for Unodes in the network, this is not enough tosustain an equilibrium. For an equilibrium to exist, there musteither be some punishment to selﬁsh nodes or some form of contract, in which all Unodes would agree to be cooperative,threatening to be selﬁsh if such contract is not signed byeveryone [4]. Given that the arrival to such an agreement isdifﬁcult in an unstructured network with autonomous nodes,we consider the following scheme that is easily applicable.A requirement of the scheme is that Unodes, upon meetingeach other, ﬁrst exchange the lists of information objects intheir memory. These lists contain metadata regarding the typeof information and the source Tnode. Then each Unodeexecutes this strategy: initially it is generous and collects andcarries unwanted content; however if on a certain leg of thenetwork it meets a selﬁsh Unode, it will only transmit thiscontent with a probability
p
, called the cooperation probability(
p <
1
). (If a Unode does not communicate its list of objects,it can be considered selﬁsh and the same strategy applies.)We proceed to write a condition under which this strategyis preferable for
U
i
on
(
T
i
,T
j
)
, and thus can lead to anequilibrium. Given that there are
N
−
1
k
different combinationsof Unodes where exactly
k
other Unodes are cooperative, thiscondition is
C
(
T
i
,T
j
)
i
(
C

N
−
1)
≤
N
−
1
k
=0
p
k
(1
−
p
)
N
−
k
−
1
N
−
1
k
C
(
T
i
,T
j
)
i
(
S

k
)
.
(10)That is, the expected cost for
U
i
when all Unodes arecooperative must be smaller or equal to the expected costwhen
U
i
is selﬁsh and
k
other Unodes are cooperative withprobability
p
,
k
= 0
,...,N
−
1
. It is evident that (7) isa special case of this condition, where
p
= 0
and
k
= 0
(admitting
0
0
= 1
).Substituting from (6), we have that (in the following weomit the parameters in
d
(
·
)
,
α
(
·
)
for notational convenience)
c
+
dv
1
−
(1
−
α
)
N
−
1
Nα
≤
dv
(1
−
p
)
N
−
1
+
N
−
1
k
=1
p
k
(1
−
p
)
N
−
1
−
k
N
−
1
k
1
−
(1
−
α
)
k
(
k
+ 1)
α
(11)The righthand side (rhs) of this inequality becomes
dv
(1
−
p
)
N
−
1
1 + 1
a
N
−
1
k
=1
p
1
−
p
k
N
−
1
k
1
k
+ 1
−
N
−
1
k
=1
p
(1
−
α
)1
−
p
k
N
−
1
k
.
It can be derived that
N
−
1
k
=1
p
1
−
p
k
N
−
1
k
1
k
+1
=
N
−
1
k
=1
p
1
−
p
k
1
N
N k
+1
=
1
Np
(1
−
p
)
N
−
1
−
1
−
pNp
−
1
and
N
−
1
k
=1
p
(1
−
α
)1
−
p
k
N
−
1
k
=
1
−
αp
1
−
p
N
−
1
−
1
. Therefore (11)
0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20
Ν
p
=0.2f
lhs
(N)=Nf
rhs
(N,a=0.3,c=5,d/v=100)f
rhs
(N,a=0.3,c=5,d/v=10)f
rhs
(N,a=0.4,c=5,d/v=100)f
rhs
(N,a=0.3,c=20,d/v=100)
(a)
2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20
Ν
p
=0.3f
lhs
(N)=Nf
rhs
(N,a=0.3,c=5,d/v=100)f
rhs
(N,a=0.3,c=5,d/v=10)f
rhs
(N,a=0.4,c=5,d/v=100)f
rhs
(N,a=0.3,c=20,d/v=100)
(b)
2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20
Ν
p
=0.5f
lhs
(N)=Nf
rhs
(N,a=0.3,c=5,d/v=100)f
rhs
(N,a=0.3,c=5,d/v=10)f
rhs
(N,a=0.4,c=5,d/v=100)f
rhs
(N,a=0.3,c=20,d/v=100)
(c)
2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20
Ν
p
=0.9f
lhs
(N)=Nf
rhs
(N,a=0.3,c=5,d/v=100)f
rhs
(N,a=0.3,c=5,d/v=10)f
rhs
(N,a=0.4,c=5,d/v=100)f
rhs
(N,a=0.3,c=20,d/v=100)
(d)Fig. 3. Graphical illustration of cooperative equilibrium conditions, for different values of the cooperation probability
p
and model parameters: the crossoverpoints of
f
rhs
with the diagonal
N
show approximate threshold values of
N
above which a Unode is cooperative in a directed leg in our model.
becomes
c
+
dv
1
−
(1
−
α
)
N
−
1
Nα
≤
dv
(1
−
p
)
N
−
1
1+ 1
α
1
Np
(1
−
p
)
N
−
1
−
1
−
pNp
−
1
−
αp
1
−
p
N
−
1
.
Applying the second order Taylor polynomial approximationto
(1
−
α
)
N
at
α
= 0
, we ﬁnally obtain the condition
N
≥
1 + 2
α
+ 2(1
−
α
)
α
1
αN
−
(1
−
p
)
N
−
1
−
1
aNp
1
−
(1
−
p
)
N
+ (1
−
αp
)
N
−
1
+
cvd
.
(12)In Fig. 3 we show graphically the required number of nodes in the network that would satisfy this condition, fordifferent values of the meeting probability
α
and
d/v
. (Wedraw the rhs of (12), called
f
rhs
and the diagonal
N
, called
f
lhs
.) It can be deduced from the graphs that a cooperativeequilibrium on a leg is easier to achieve (i.e., we need asmaller number of Unodes) for smaller values of
c
and largervalues of
d/v
. The behavior with respect to
α
is less intuitive:when the cooperation probability of other Unodes is small, ahigher meeting probability leads a Unode actually having lessincentive to cooperate and appearing more selﬁsh, whereas if the cooperation probability of other Unodes is high, it leads aUnode to actually be cooperative. Finally, as the probabilityof cooperation decreases, a smaller number
N
of nodes in thenetwork is required for an equilibrium to be achieved. We canexpect this result, since a cooperative Unode “punishes” morea selﬁsh Unode by giving its acquired content with a smallerprobability. Therefore Unodes refrain from being selﬁsh.Note that in the model we develop in this paper we donot have to consider a “stricter” condition for a cooperativeequilibrium to exist (i.e., a condition that would call for ahigher
N
). This is because we have assumed in the beginningthat each Unode thinks, a priori, that all the other Unodes areinterested in the content it collects; thus we have excluded thecase where a Unode would not collect unwanted informationbecause it might think that other Unodes would also notbe interested in it (and hence also might not collect data attheir respective srcin points). Therefore if such cooperativeequilibrium conditions are satisﬁed in all directed legs simultaneously, a cooperative equilibrium exists in the whole network.IV. R
ELATED
W
ORKS
Previous applications of gametheoretic methods in examining cooperation between mobile nodes have focused mainlyon traditional adhoc networks, where cooperation consists of