### Solving Stackelberg games with uncertain observability

Recent applications of game theory in security domains use algorithms to solve a Stackelberg model, in which one player (the leader) first commits to a mixed strategy and then the other player (the follower) observes that strategy and best-responds to it. However, in real-world applications, it is hard to determine whether the follower is actually able to observe the leader's mixed strategy before acting. In this paper, we model the uncertainty about whether the follower is able to observe the leader's strategy as part of the game (as proposed in the extended version of Yin et al. [17]). We describe an iterative algorithm for solving these games. This algorithm alternates between calling a Nash equilibrium solver and a Stackelberg solver as subroutines. We prove that the algorithm finds a solution in a finite number of steps and show empirically that it runs fast on games of reasonable size. We also discuss other properties of this methodology based on the experiments.

[1]  C. E. Lemke,et al.  Equilibrium Points of Bimatrix Games , 1964 .

[2]  Eitan Zemel,et al.  Nash and correlated equilibria: Some complexity considerations , 1989 .

[3]  Avrim Blum,et al.  Planning in the Presence of Cost Functions Controlled by an Adversary , 2003, ICML.

[4]  Yoav Shoham,et al.  Run the GAMUT: a comprehensive approach to evaluating game-theoretic algorithms , 2004, Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, 2004. AAMAS 2004..

[5]  Vincent Conitzer,et al.  Mixed-Integer Programming Methods for Finding Nash Equilibria , 2005, AAAI.

[6]  Xiaotie Deng,et al.  Settling the Complexity of Two-Player Nash Equilibrium , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[7]  Vincent Conitzer,et al.  Computing the optimal strategy to commit to , 2006, EC '06.

[8]  Xi Chen,et al.  Computing Nash Equilibria: Approximation and Smoothed Complexity , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[9]  Andrew McLennan,et al.  Gambit: Software Tools for Game Theory , 2006 .

[10]  Vincent Conitzer,et al.  New complexity results about Nash equilibria , 2008, Games Econ. Behav..

[11]  Sarit Kraus,et al.  Using Game Theory for Los Angeles Airport Security , 2009, AI Mag..

[12]  Paul W. Goldberg,et al.  The Complexity of Computing a Nash Equilibrium , 2009, SIAM J. Comput..

[13]  Vincent Conitzer,et al.  Multi-Step Multi-Sensor Hider-Seeker Games , 2009, IJCAI.

[14]  Bernhard von Stengel,et al.  Leadership games with convex strategy sets , 2010, Games Econ. Behav..

[15]  Manish Jain,et al.  Security Games with Arbitrary Schedules: A Branch and Price Approach , 2010, AAAI.

[16]  Vincent Conitzer,et al.  Stackelberg vs. Nash in security games: interchangeability, equivalence, and uniqueness , 2010, AAMAS 2010.

[17]  Milind Tambe,et al.  Security and Game Theory: IRIS – A Tool for Strategic Security Allocation in Transportation Networks , 2011, AAMAS 2011.