1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal...
-
date post
21-Dec-2015 -
Category
Documents
-
view
226 -
download
5
Transcript of 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal...
![Page 1: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/1.jpg)
1
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
![Page 2: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/2.jpg)
2
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
![Page 3: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/3.jpg)
3
Cooperative Systems: Natural and Virtual
![Page 4: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/4.jpg)
4
Sensor Coverage Problem( Cassandras and Li 2005 )
1x
2x
Function Reward : )(xR
SpaceMission :
![Page 5: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/5.jpg)
5
Sensor Coverage Problem( Cassandras and Li 2005 )
Sensor Model
Example :
isx
),( : ProbDetection ii sxp
i-th sensor location
point of interest
iii sxp exp0
![Page 6: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/6.jpg)
6
Sensor Coverage Problem( Cassandras and Li 2005 )
is
)( ii s
)(
when
0),(
ii
i
sx
sxp
Sensor Model
Limited Coverage :
![Page 7: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/7.jpg)
7
Sensor Coverage Problem( Cassandras and Li 2005 )
Given sensors at locations
Joint Detection Probability at : Nsss ,...,: 1
N
iii sxpsxP
1
),(11),(
x
N
![Page 8: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/8.jpg)
8
Sensor Coverage Problem( Cassandras and Li 2005 )
Optimize the expected total reward
by choosing the sensor locations
x
sxPxRsU ),()()(
Nsss ,...,1
![Page 9: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/9.jpg)
9
Sensor Coverage Problem( Cassandras and Li 2005 )
• Pictorially, place the circles to maximize the total weighted coverage
![Page 10: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/10.jpg)
10
Dynamic Sensor Coverage Problem
)()1( tsRts iii
Sensor Mobility Model
![Page 11: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/11.jpg)
11
Dynamic Sensor Coverage Problem
iiiiii sRssRs ~~
Sensor Mobility Model
• Reversibility :
• Feasibility : For any
ii ss ~
ii ss ~,
![Page 12: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/12.jpg)
12
Dynamic Sensor Coverage Problem
)())(,(
ii si tssU
))(( tsRs iii
Local Information Model
At time t, sensors i can compute
for any
![Page 13: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/13.jpg)
13
Dynamic Sensor Coverage Problem
)(tsi
i
s
))(( tsR ii
)( i
si
![Page 14: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/14.jpg)
14
Dynamic Sensor Coverage Problem
?max )(lim st
sUtsU
Question
How should the sensors update
so that
)(,...,1 ts(t)s N
![Page 15: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/15.jpg)
15
Game Theory Formulation
• Sensors = Selfish Decision Makers
• Sensor i maximizes its own reward
which is private and localized to sensor i.
),( max iiis
ssUi
![Page 16: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/16.jpg)
16
Agreeable Sensor Locations:Nash Equilibrium
• Sensor locations
form an equilibrium if, for each sensor i,
*** ,...,1 Nsss
*** , , iiiii ssUssUi
![Page 17: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/17.jpg)
17
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
![Page 18: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/18.jpg)
18
Aligned Sensor Rewards
)()( aUsU i
• For every sensor i ,
• Not localized ( global information required)
• Low SNR (Wolpert et al. 2000)
![Page 19: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/19.jpg)
19
Wonderful Life Utility (Wolpert et al. 2000)
• Marginal contribution of sensor i :
• Localized
• SNR maximized
)" 0 " :()()( ii ssUsUsU
OFF
![Page 20: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/20.jpg)
20
Wonderful Life Utility (Wolpert et al. 2000)
• Aligned :
• Potential Game with potential
),( ),~(
),( ),~(
iiii
iiiiii
ssUssU
ssUssU
)(sU
![Page 21: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/21.jpg)
21
A Misaligned Reward Structure
• Equally Shared Rewards :
xn
sxPxRsU
ii sxi
)( )(
),()()(
# of sensors covering x
![Page 22: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/22.jpg)
22
A Misaligned Reward Structure
• Looks aligned :
• But, optimum may not be agreeable
• An equilibrium may not exists at all !
i
i sUsU )()(
&
![Page 23: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/23.jpg)
23
Negotiation Algorithms
?max )(lim st
sUtsU
How should the sensors update
so that
)(,...,1 ts(t)s N
![Page 24: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/24.jpg)
24
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
![Page 25: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/25.jpg)
25
Selective Spatial Adaptive Play ( SSAP )
• Updating sensor i updates its location
with high probability, if
))1(())1(,~( tsUtssU iiii
ii sts ~)(
![Page 26: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/26.jpg)
26
•
• For potential games, SSAP induces
1-
) )1(,~ () )1( (1
exp1
}~)({
tssUtsU
stsP
iiii
ii
s
sU
sU
k e
esksP
/)(
/)(
)( lim
SSAP
![Page 27: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/27.jpg)
27
SSAP
• As , we have
• Therefore,
0
1/)(
/)(
s
sU
sU
e
e
)(maxargfor sUs
1)(maxarg)( lim0
sUsP
![Page 28: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/28.jpg)
28
Simulations
![Page 29: 1 A Game Theoretic Formulation of the Dynamic Sensor Coverage Problem Jason Marden ( UCLA ) Gürdal Arslan ( University of Hawaii ) Jeff Shamma ( UCLA )](https://reader036.fdocuments.in/reader036/viewer/2022062320/56649d615503460f94a42375/html5/thumbnails/29.jpg)
29
THANK YOU !