A Game-Theoretic Approach to Determining Efficient Patrolling Strategies for Mobile Robots

1. A Game-Theoretic Approach to Determining Efficient Patrolling Strategies for Mobile Robots Francesco Amigoni, Nicola Gatti, Antonio Ippedico

2. Scenario

3. Summary of Contributions • Problem: determination of an efficient patrolling strategy for a mobile robot • Idea: • model the scenario as an extensive-form game played by the patroller and the intruder • solve the game to find the strategy for the patroller

4. The Proposed Model:Assumptions • Time is discrete, players play in turns • Environment with n places • Patroller detects the presence of the intruder (captures the intruder) when it is in the patroller’s current place • Intruder knows the strategy of the patroller • Patroller’s actions: move from one place to another one (incurring in different costs), movements can be between any pairs of places • Intruder’s actions: wait or attempt to enter a place • Entering a place takes d turns • The game ends either when the intruder is captured or has entered a place • Players payoffs are defined according to values attributed to places, to costs for moving between places, and to rewards for capturing the intruder • Intruder can be of different types, each one with different values for places

5. patroller’s action intruder’s action patroller’s action … … … … … … … … The Proposed Model:Extensive-form Game • The intruder knows the patroller’s strategy and the patroller knows it commitment-based strategy for the patroller • Finding an optimal solution is not easy, basically because the environment can dynamically change and because the game is infinite-horizon approximate solution

6. patroller’s action intruder’s action patroller’s action … … … … … … … … Solving the Game:Finding a Patrolling Strategy • Greedy approach: we consider a slice of the extensive-form game as an independent strategic-form game • Solving each slice means finding the next optimal action for the patrolling robot • A slice can be solved by resorting to a multi-LP [Conitzer and Sandholm, EC 2006] or to a MILP [Paruchuri et al., AAMAS 2008] mathematical programming formulation • Solution: mixed strategy for the patrolling robot: {γ1,γ2,…,γn}

7. Experimental Results • The approach scales reasonably well with the number n of places (using the multi-LP formulation) and with the number of intruder’s types (using the MILP formulation) • The approach can be applied to different environments • Linear environment • Ring and star environments • The approach adapts to dynamic changes in the environment 25 nodes, 10 runs, 500 steps each

8. Conclusions • We proposed a game-theoretic approach to determining strategies for patrolling robots • Modeling a patrolling situation as an extensive-form game • Finding an (approximate) solution of the game • Patrolling strategies found with our approach are efficient • Ongoing work • Optimal solutions for the extensive-form game • More realistic scenarios • Implementation on real robots