A Game-Theoretic Approach to Determining
Efficient Patrolling Strategies for Mobile
Robots
Francesco Amigoni, Nicola Gatti, Antonio Ippedico
Scenario
2
F. Amigoni, N. Gatti, A, Ippedico
Summary of Contributions
3
• Problem:
determination of an efficient patrolling strategy for a mobile robot
• Idea:
1. model the scenario as an extensive-form game played by the
patroller and the intruder
2. solve the game to find the strategy for the patroller
F. Amigoni, N. Gatti, A, Ippedico
The Proposed Model:
Assumptions
4
• 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
F. Amigoni, N. Gatti, A, Ippedico
The Proposed Model:
Extensive-form Game
5
patroller’s action
intruder’s action
patroller’s action
…
… …
… …
… …
…
• 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
F. Amigoni, N. Gatti, A, Ippedico
Solving the Game:
Finding a Patrolling Strategy
6
• Greedy approach: we consider a slice of the extensive-form game as
an independent strategic-form game
patroller’s action
intruder’s action
patroller’s action
…
… …
… …
… …
…
• 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}
F. Amigoni, N. Gatti, A, Ippedico
Experimental Results
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• 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
25 nodes,
method
utility
10 runs, 500 steps each
our approach (multi-LP)
79.0
uniform
52.1
proportional
69.5
random
48.0
• Ring and star environments
• The approach adapts to dynamic changes in the environment
F. Amigoni, N. Gatti, A, Ippedico
Conclusions
8
• 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
F. Amigoni, N. Gatti, A, Ippedico