Computing Stackelberg Equilibrium with Memory in Sequential Games

Stackelberg equilibrium is a solution concept that describes optimal strategies to commit: Player 1 (the leader) first commits to a strategy that is publicly announced, then Player 2 (the follower) plays a best response to the leader’s commitment. We study the problem of computing Stackelberg equilibria in sequential games with finite and indefinite horizons, when players can play history-dependent strategies. Using the alternate formulation called strategies with memory, we establish that strategy profiles with polynomial memory size can be described efficiently. We prove that there exist a polynomial time algorithm which computes the Strong Stackelberg Equilibrium in sequential games defined on directed acyclic graphs, where the strategies depend only on the memory states from a set which is linear in the size of the graph. We extend this result to games on general directed graphs which may contain cycles. We also analyze the setting for approximate version of Strong Stackelberg Equilibrium in the games with chance nodes. JEL Classification: C61; C73; D83

[1]  Branislav Bosanský,et al.  Sequence-Form Algorithm for Computing Stackelberg Equilibria in Extensive-Form Games , 2015, AAAI.

[2]  Branislav Bosanský,et al.  Finite State Machines Play Extensive-Form Games , 2020, EC.

[3]  Bo An,et al.  Deploying PAWS: Field Optimization of the Protection Assistant for Wildlife Security , 2016, AAAI.

[4]  Branislav Bosanský,et al.  Tackling Sequential Attacks in Security Games , 2019, GameSec.

[5]  Stefano Coniglio,et al.  Methods for Finding Leader-Follower Equilibria with Multiple Followers: (Extended Abstract) , 2016, AAMAS.

[6]  H. Stackelberg,et al.  Marktform und Gleichgewicht , 1935 .

[7]  Sven Schewe,et al.  Making the Best of Limited Memory in Multi-Player Discounted Sum Games , 2014, GandALF.

[8]  Haifeng Xu,et al.  Mitigating the Curse of Correlation in Security Games by Entropy Maximization , 2018, AAMAS.

[9]  Tuomas Sandholm,et al.  Limited Lookahead in Imperfect-Information Games , 2015, IJCAI.

[10]  Vincent Conitzer,et al.  Commitment to Correlated Strategies , 2011, AAAI.

[11]  Viliam Lisý,et al.  Hardening networks against strategic attackers using attack graph games , 2019, Comput. Secur..

[12]  Nicola Basilico,et al.  Leader-follower strategies for robotic patrolling in environments with arbitrary topologies , 2009, AAMAS.

[13]  Anshul Gupta,et al.  Equilibria in finite games , 2016 .

[14]  Vincent Conitzer,et al.  Computing optimal strategies to commit to in extensive-form games , 2010, EC '10.

[15]  Bo An,et al.  PAWS - A Deployed Game-Theoretic Application to Combat Poaching , 2017, AI Mag..

[16]  Bo An,et al.  Stackelberg Security Games: Looking Beyond a Decade of Success , 2018, IJCAI.

[17]  Vojtech Rehák,et al.  Automatic Synthesis of Efficient Regular Strategies in Adversarial Patrolling Games , 2018, AAMAS.

[18]  Sarit Kraus,et al.  Playing games for security: an efficient exact algorithm for solving Bayesian Stackelberg games , 2008, AAMAS.

[19]  Vincent Conitzer,et al.  Computing Optimal Strategies to Commit to in Stochastic Games , 2012, AAAI.

[20]  Hanif D. Sherali,et al.  A Multiple Leader Stackelberg Model and Analysis , 1984, Oper. Res..

[21]  Haifeng Xu,et al.  Exploring Information Asymmetry in Two-Stage Security Games , 2015, AAAI.

[22]  Rabah Amir,et al.  Stackelberg versus Cournot Equilibrium , 1999 .

[23]  Branislav Bosanský,et al.  Computation of Stackelberg Equilibria of Finite Sequential Games , 2015, WINE.

[24]  Branislav Bosanský,et al.  Using Correlated Strategies for Computing Stackelberg Equilibria in Extensive-Form Games , 2016, AAAI.

[25]  Vincent Conitzer,et al.  Complexity of Computing Optimal Stackelberg Strategies in Security Resource Allocation Games , 2010, AAAI.

[26]  Tuomas Sandholm,et al.  Robust Stackelberg Equilibria in Extensive-Form Games and Extension to Limited Lookahead , 2017, AAAI.

[27]  Toshihiro Matsumura.  Stackelberg Mixed Duopoly with a Foreign Competitor , 2003 .

[28]  B. Stengel,et al.  Leadership with commitment to mixed strategies , 2004 .

[29]  Alexander S. Poznyak,et al.  Stackelberg security games: Computing the shortest-path equilibrium , 2015, Expert Syst. Appl..

[30]  Joshua Letchford,et al.  Computational Aspects of Stackelberg Games , 2013 .

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