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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 !