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A Game Theoretic Formulation of the
Dynamic Sensor Coverage Problem
Jason Marden ( UCLA )
Gürdal Arslan ( University of Hawaii )
Jeff Shamma ( UCLA )
AFOSR / MURI & Lockheed Martin
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Cooperative Systems Design
• Optimize a global objective via selfish DMs
• Design Problem:
–Utility Design ( tell DMs what to optimize )
–Negotiation Algorithm Design ( tell DMs how to optimize )
DM1DM1
DM3DM3
DM2DM2
DM4DM4
DM5DM5
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Cooperative Systems: Natural and Virtual
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Sensor Coverage Problem( Cassandras and Li 2005 )
1x
2x
Function Reward : )(xR
SpaceMission :
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Sensor Coverage Problem( Cassandras and Li 2005 )
Sensor Model
Example :
isx
),( : ProbDetection ii sxp
i-th sensor location
point of interest
iii sxp exp0
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Sensor Coverage Problem( Cassandras and Li 2005 )
is
)( ii s
)(
when
0),(
ii
i
sx
sxp
Sensor Model
Limited Coverage :
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Sensor Coverage Problem( Cassandras and Li 2005 )
Given sensors at locations
Joint Detection Probability at : Nsss ,...,: 1
N
iii sxpsxP
1
),(11),(
x
N
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Sensor Coverage Problem( Cassandras and Li 2005 )
Optimize the expected total reward
by choosing the sensor locations
x
sxPxRsU ),()()(
Nsss ,...,1
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Sensor Coverage Problem( Cassandras and Li 2005 )
• Pictorially, place the circles to maximize the total weighted coverage
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Dynamic Sensor Coverage Problem
)()1( tsRts iii
Sensor Mobility Model
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Dynamic Sensor Coverage Problem
iiiiii sRssRs ~~
Sensor Mobility Model
• Reversibility :
• Feasibility : For any
ii ss ~
ii ss ~,
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Dynamic Sensor Coverage Problem
)())(,(
ii si tssU
))(( tsRs iii
Local Information Model
At time t, sensors i can compute
for any
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Dynamic Sensor Coverage Problem
)(tsi
i
s
))(( tsR ii
)( i
si
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Dynamic Sensor Coverage Problem
?max )(lim st
sUtsU
Question
How should the sensors update
so that
)(,...,1 ts(t)s N
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Game Theory Formulation
• Sensors = Selfish Decision Makers
• Sensor i maximizes its own reward
which is private and localized to sensor i.
),( max iiis
ssUi
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Agreeable Sensor Locations:Nash Equilibrium
• Sensor locations
form an equilibrium if, for each sensor i,
*** ,...,1 Nsss
*** , , iiiii ssUssUi
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Design of Sensor Rewards
• ( Ideal ) Alignment :– Only optimal sensor locations should be agreeable
• Relaxed alignment ( Wolpert et al. 2000 ) :
– Optimal sensor locations are always agreeable
),( ),~(
),( ),~(
iiii
iiiiii
ssUssU
ssUssU
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Aligned Sensor Rewards
)()( aUsU i
• For every sensor i ,
• Not localized ( global information required)
• Low SNR (Wolpert et al. 2000)
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Wonderful Life Utility (Wolpert et al. 2000)
• Marginal contribution of sensor i :
• Localized
• SNR maximized
)" 0 " :()()( ii ssUsUsU
OFF
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Wonderful Life Utility (Wolpert et al. 2000)
• Aligned :
• Potential Game with potential
),( ),~(
),( ),~(
iiii
iiiiii
ssUssU
ssUssU
)(sU
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A Misaligned Reward Structure
• Equally Shared Rewards :
xn
sxPxRsU
ii sxi
)( )(
),()()(
# of sensors covering x
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A Misaligned Reward Structure
• Looks aligned :
• But, optimum may not be agreeable
• An equilibrium may not exists at all !
i
i sUsU )()(
&
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Negotiation Algorithms
?max )(lim st
sUtsU
How should the sensors update
so that
)(,...,1 ts(t)s N
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Selective Spatial Adaptive Play ( SSAP )
• At each step, only 1 sensor, say, sensor i
is given the chance to update its location.
• Updating sensor i randomly picks
with uniform probability.
))1((~ tsRs iii
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Selective Spatial Adaptive Play ( SSAP )
• Updating sensor i updates its location
with high probability, if
))1(())1(,~( tsUtssU iiii
ii sts ~)(
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•
• For potential games, SSAP induces
1-
) )1(,~ () )1( (1
exp1
}~)({
tssUtsU
stsP
iiii
ii
s
sU
sU
k e
esksP
/)(
/)(
)( lim
SSAP
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SSAP
• As , we have
• Therefore,
0
1/)(
/)(
s
sU
sU
e
e
)(maxargfor sUs
1)(maxarg)( lim0
sUsP
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Simulations
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THANK YOU !
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