Game-Theoretic Models for Reliable PathLength and Energy-Constrained Routing With
Data Aggregation
-Rajgopal Kannan and S. Sitharama Iyengar
Xinyan Pan
11/22/2004
Main Issues in Sensor Network





Energy-efficiency
Reliability of a data transfer path
Path Length (proportional to energy cost of
transmission
Most prevalent routing algorithms focusing on
minimizing overall energy consumption.
However such routing strategies may result in uneven
energy depletion across sensor nodes and expedite
network partition.
Sensor-centric information routing
strategy





Optimizes energy costs, path reliability and path
length simultaneously.
Energy costs are local
Path reliability and path length are network-wide
metrics
Sensors can be modeled as players in a routing game
with appropriate strategies and utility functions
(payoffs)
Reliable Query Reporting (RQR) Model
Game-Theoretic Framework
Each sensor makes decision taking individual
costs and benefits into account
 Decentralized decision-making
 Self-configuring and adaptive networks
 Identify equilibrium outcomes for reliable
communication

RQR Model Setup



Set of players: S = {sa = s1, …, sn=sq}.
Source node (sa) sends information Va to destination
node (sq).
Information routed through optimally chosen set S’
S of intermediate nodes

Each node can fail with probability 1-pi  (0,1).

Link costs cij >0

Each node forms one link.
RQR Game

Sensor si’s strategy is a binary vector
 li = (li1, …, li,i-1, li,i+1, …, lin)
Where
li1 = 1/0, sensor si sending/not sending a data packet to sensor sj


Each sensor’s strategy is constrained to be nonempty
Strategies resulting in a node linking to its ancestors are not
allowed

A strategy profile
l  (l1 ,..., ln )  in1 Li
defines the outcome of the RQR game.

In a standard non-cooperative game each player cares only about
individual payoffs – therefore behavior is selfish.
Benefit function

For a strategy profile l = (li, l-i) resulting in a tree T
rooted at sq, where l-i denotes the strategy chosen by
all the other players except player i.
 Network
is unreliable and every sensor that receives data
has an incentive in its reaching the query node sq
 The routing protocol includes data aggregation
 Benefit to any sensor si, denoted as Xi, is a function of the
path reliability from si onwards and a function of the
expected value of information that can reach si.
Benefit function


the path reliability from si onwards to sq, denoted as Ri
the expected value of information that can reach si
Vi  g i (v1 ,, vn 1 )  vi   jF (i ) p jV j
where
Vi  expected value of informatio n that can reach si ,
i  1, 2,  , n  1
vi  value of the infmation retrieved at si ,
i  1,2,  , n  1
F (i )  the set of si ' s parents
p j  the probabilit y of s j not failing
Benefit function

Benefit function for si:
X i  g i (v1 , , vn 1 ) Ri  (vi   jF (i ) p jV j ) Ri

Exmaple
X 5  R5 (v5  p1 p3v1  p2 p3v2  p3v3  p4v4 )
*Data aggregation is assumed to be additive
Payoffs

General Payoff Function
 g i (v1 ,, vn 1 ) R i cij
 i (l )  
0
if si  T
otherwise
where
l  strategy profile l  (li , li )
T  tree rooted at sq , resulting from l
RQR Model Properties
Benefits depend on the total reliability of
realized paths. Thus each sensor is induced to
have a cooperative outlook in the game.
 Cost are individually borne and differ across
sensors, thereby capturing the tradeoffs
between global reliability and individual
sensor costs

RQR Model Properties

Definition:
A strategy li is said to be a best response of player i to l-i if
0   i (li' , li )   i (li , li ) for all li'  Li

Let BRi(l-i) denote the set of player i’s best response
to l-i. A strategy profile l = (l1, …, ln) is said to be an
optimal RQR tree T if li  BRi (li ), i.e., sensors are
playing a Nash equilibrium.
Optimal RQR Computation in
Geographically Routed Sensor Networks


Let Di = {si ,si ,…,si } be the set of downstream next-hop
neighbors of si.
For each sij in Di, let expected values of incoming
information be divided into Nij disjoint consecutive intervals
1
2
l
I1 j , I 2j ,, I Nji , where  t I t j  (0, vr )
i


i
ij
t
i
ij
t
i
j
B( I ) and E( I ) -- the left and right endpoints
i
R( I t ) -- optimal path reliability from si j onwards for
information of expected value in the given interval I ti
i
 i (i j , vi , I t ) -- payoff to sensor si on sending information of
i
value vi  I t j to downstream neighbor si
j
j

j
j
Optimal RQR Computation in
Geographically Routed Sensor Networks

Lemma:
- if  i (i j , vi , I t j )   i (ik , vi , I tik ) for vi  inf[ I t j  I uik ],
i
i
then  i (i j , vi , I t j )   i (ik , vi , I tik ) for all vi  [ I t j  I uik ].
i
i
- If the two payoffs are equal at the fixed point, then
 i (i j , vi , I t )   i (ik , vi , I ti ) throughou t the intersecti on iff
ij
R( I t j )  R( I uik ).
i
k
Optimal RQR Computation in
Geographically Routed Sensor Networks




To compare two different intervals, we only need to
evaluate their payoff at the smallest point.
Algorithm of optimal-next-neighbor at each node
enables computation of the optimal RQR path.
Assume that upstream and downstream neighbors of
each node are known a priori
The output of the algorithm is the set of disjoint and
contiguous information value intervals at si along
with the reliability and next hop neighbor on the
optimal path from si to sq for each interval
Algorithm of Optimal-next-neighbor
References


R. Kannan, S.S. Iyengar, Game-theoretic models for
reliable path-length and energy-constrained routing
with data aggregation in wireless sensor networks,
IEEE J. Selected Areas Comm., Vol. 22, No. 6,
August 2004, 1141-1150
R. Kannan, S. Sarangi, S.S. Iyengar, Sensor-centric
energy-constrained reliable query routing for wireless
sensor networks, J. Parallel Distrib. Comput. 64
(2004), 839-852