Computing Optimal Ex Ante Correlated Equilibria in Two-Player Sequential Games

We investigate the computation of equilibria in extensive-form games when ex ante correlation is possible, focusing on correlated equilibria requiring the least amount of communication between the players and the mediator. Motivated by hardness results on normal-form correlated equilibria, we investigate whether it is possible to compute normal-form coarse correlated equilibria efficiently. We show that an optimal (e.g., social welfare maximizing) normal-form coarse correlated equilibrium can be computed in polynomial time in two-player games without chance moves, and that in general multi-player games (including two-player games with chance) the problem is NP-hard. For the two-player case, we provide both a polynomial-time algorithm based on the ellipsoid method and a column generation algorithm based on the simplex method which can be efficiently applied in practice. We also show that the pricing oracle employed in the column generation procedure can be extended to games with two players and chance.

[1]  Yoav Shoham,et al.  Multiagent Systems - Algorithmic, Game-Theoretic, and Logical Foundations , 2009 .

[2]  Tuomas Sandholm,et al.  Safe and Nested Subgame Solving for Imperfect-Information Games , 2017, NIPS.

[3]  F. Forges.  Correlated Equilibrium in Games with Incomplete Information Revisited , 2006 .

[4]  Tuomas Sandholm,et al.  Ex ante coordination and collusion in zero-sum multi-player extensive-form games , 2018, NeurIPS.

[5]  Michael Buro,et al.  Alpha-Beta Pruning for Games with Simultaneous Moves , 2012, AAAI.

[6]  S. Ross.  GOOFSPIEL -- THE GAME OF PURE STRATEGY , 1971 .

[7]  Haifeng Xu,et al.  Algorithmic Bayesian persuasion , 2015, STOC.

[8]  Bernhard von Stengel,et al.  Extensive-Form Correlated Equilibrium: Definition and Computational Complexity , 2008, Math. Oper. Res..

[9]  Tuomas Sandholm,et al.  Combining local search techniques and path following for bimatrix games , 2012, UAI.

[10]  Kristoffer Arnsfelt Hansen,et al.  Finding Equilibria in Games of No Chance , 2007, COCOON.

[11]  Noam Brown,et al.  Superhuman AI for heads-up no-limit poker: Libratus beats top professionals , 2018, Science.

[12]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[13]  Edoardo Amaldi,et al.  Improving Cutting Plane Generation with 0-1 Inequalities by Bi-criteria Separation , 2010, SEA.

[14]  J. Vial,et al.  Strategically zero-sum games: The class of games whose completely mixed equilibria cannot be improved upon , 1978 .

[15]  Bernhard von Stengel,et al.  Computing an Extensive-Form Correlated Equilibrium in Polynomial Time , 2008, WINE.

[16]  Edoardo Amaldi,et al.  Coordinated cutting plane generation via multi-objective separation , 2014, Math. Program..

[17]  Christos H. Papadimitriou,et al.  Computing correlated equilibria in multi-player games , 2005, STOC '05.

[18]  F. Forges,et al.  Five legitimate definitions of correlated equilibrium in games with incomplete information , 1993 .

[19]  Nicola Basilico,et al.  Coordinating Multiple Defensive Resources in Patrolling Games with Alarm Systems , 2017, AAMAS.

[20]  Martin Grötschel,et al.  The ellipsoid method and its consequences in combinatorial optimization , 1981, Comb..

[21]  R. Aumann.  Subjectivity and Correlation in Randomized Strategies , 1974 .

[22]  Stefano Coniglio,et al.  On the Generation of Cutting Planes which Maximize the Bound Improvement , 2015, SEA.

[23]  Katrina Ligett,et al.  Finding any nontrivial coarse correlated equilibrium is hard , 2015, SECO.

[24]  Nicola Basilico,et al.  Team-Maxmin Equilibrium: Efficiency Bounds and Algorithms , 2016, AAAI.

[25]  B. Stengel,et al.  Efficient Computation of Behavior Strategies , 1996 .

[26]  Nicola Gatti,et al.  Computational Results for Extensive-Form Adversarial Team Games , 2017, AAAI.

[27]  Éva Tardos,et al.  No-Regret Learning in Bayesian Games , 2015, NIPS.

[28]  Tim Roughgarden,et al.  Intrinsic Robustness of the Price of Anarchy , 2015, J. ACM.

[29]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[30]  S. Hart,et al.  A simple adaptive procedure leading to correlated equilibrium , 2000 .

[31]  Nicola Gatti,et al.  Local search techniques for computing equilibria in two-player general-sum strategic-form games , 2010, AAMAS.

[32]  John N. Tsitsiklis,et al.  Introduction to linear optimization , 1997, Athena scientific optimization and computation series.

[33]  F. Forges.  Published by: The , 2022 .

[34]  Kevin Leyton-Brown,et al.  Polynomial-time computation of exact correlated equilibrium in compact games , 2010, EC '11.

[35]  R. Myerson.  MULTISTAGE GAMES WITH COMMUNICATION , 1984 .

[36]  Jean R. S. Blair,et al.  PERFECT RECALL AND PRUNING IN GAMES WITH IMPERFECT INFORMATION 1 , 1996, Comput. Intell..