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

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 ) DM1 DM3 DM2 DM4 DM5

3 Cooperative Systems: Natural and Virtual

4 Sensor Coverage Problem ( Cassandras and Li 2005 )

5 Sensor Model Example : i-th sensor location point of interest

6 Sensor Coverage Problem ( Cassandras and Li 2005 ) Sensor Model Limited Coverage :

7 Sensor Coverage Problem ( Cassandras and Li 2005 ) Given sensors at locations Joint Detection Probability at :

8 Sensor Coverage Problem ( Cassandras and Li 2005 ) Optimize the expected total reward by choosing the sensor locations

9 Sensor Coverage Problem ( Cassandras and Li 2005 ) Pictorially, place the circles to maximize the total weighted coverage

10 Dynamic Sensor Coverage Problem Sensor Mobility Model

11 Dynamic Sensor Coverage Problem Sensor Mobility Model Reversibility : Feasibility : For any

12 Dynamic Sensor Coverage Problem Local Information Model At time t, sensors i can compute for any

13 Dynamic Sensor Coverage Problem

14 Dynamic Sensor Coverage Problem Question How should the sensors update so that

15 Game Theory Formulation Sensors = Selfish Decision Makers Sensor i maximizes its own reward which is private and localized to sensor i.

16 Agreeable Sensor Locations: Nash Equilibrium Sensor locations form an equilibrium if, for each sensor i,

17 Design of Sensor Rewards ( Ideal ) Alignment : – Only optimal sensor locations should be agreeable Relaxed alignment ( Wolpert et al ) : – Optimal sensor locations are always agreeable

18 Aligned Sensor Rewards For every sensor i, Not localized ( global information required) Low SNR (Wolpert et al. 2000)

19 Wonderful Life Utility (Wolpert et al. 2000) Marginal contribution of sensor i : Localized SNR maximized OFF

20 Wonderful Life Utility (Wolpert et al. 2000) Aligned : Potential Game with potential

21 A Misaligned Reward Structure Equally Shared Rewards : # of sensors covering

22 A Misaligned Reward Structure Looks aligned : But, optimum may not be agreeable An equilibrium may not exists at all !

23 Negotiation Algorithms How should the sensors update so that

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.

25 Selective Spatial Adaptive Play ( SSAP ) Updating sensor i updates its location with high probability, if

26 For potential games, SSAP induces SSAP

27 SSAP As, we have Therefore,

28 Simulations

29 THANK YOU !