Team-Maxmin Equilibrium: Efficiency Bounds and Algorithms

The Team-maxmin equilibrium prescribes the optimal strategies for a team of rational players sharing the same goal and without the capability of correlating their strategies in strategic games against an adversary. This solution concept can capture situations in which an agent controls multiple resources-corresponding to the team members-that cannot communicate. It is known that such equilibrium always exists and it is unique (unless degeneracy) and these properties make it a credible solution concept to be used in real-world applications, especially in security scenarios. Nevertheless, to the best of our knowledge, the Team-maxmin equilibrium is almost completely unexplored in the literature. In this paper, we investigate bounds of (in)efficiency of the Team-maxmin equilibrium w.r.t. the Nash equilibria and w.r.t. the Maxmin equilibrium when the team members can play correlated strategies. Furthermore, we study a number of algorithms to find and/or approximate an equilibrium, discussing their theoretical guarantees and evaluating their performance by using a standard testbed of game instances.

[1]  Ido Erev,et al.  The Effect of Repeated Play in the IPG and IPD Team Games , 1994 .

[2]  Milind Tambe,et al.  Defender (Mis)coordination in Security Games , 2013, IJCAI.

[3]  Richard J. Lipton,et al.  Simple strategies for large zero-sum games with applications to complexity theory , 1994, STOC '94.

[4]  Vincent Conitzer,et al.  Fast Equilibrium Computation for Infinitely Repeated Games , 2013, AAAI.

[5]  Eyal Winter,et al.  Experimental study of repeated team-games , 1996 .

[6]  Steve Alpern,et al.  The Symmetric Rendezvous-Evasion Game , 1998 .

[7]  Shmuel Zamir,et al.  Cooperation in Intergroup, N-Person, and Two-Person Games of Chicken , 1997 .

[8]  Paul G. Spirakis,et al.  Equilibrium Points in Fear of Correlated Threats , 2008, WINE.

[9]  Stein W. Wallace.  Algorithms and model formulations in mathematical programming , 1989 .

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

[11]  Vincent Conitzer,et al.  Role Assignment for Game-Theoretic Cooperation , 2016, IJCAI.

[12]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[13]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[14]  Tamás Vinkó,et al.  A comparison of complete global optimization solvers , 2005, Math. Program..

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

[16]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

[17]  Tim Roughgarden,et al.  Algorithmic Game Theory , 2007 .

[18]  Nikolaos V. Sahinidis,et al.  A polyhedral branch-and-cut approach to global optimization , 2005, Math. Program..

[19]  Kristoffer Arnsfelt Hansen,et al.  Approximability and Parameterized Complexity of Minmax Values , 2008, WINE.

[20]  B. Stengel,et al.  Team-Maxmin Equilibria☆ , 1997 .

[21]  Eilon Solan.  Absorbing Team Games , 2000, Games Econ. Behav..

[22]  Xiaotie Deng,et al.  On the complexity of equilibria , 2002, STOC '02.

[23]  S. Wallace.  Algorithms and Model Formulations in Mathematical Programming , 1989, NATO ASI Series.

[24]  Thomas R. Palfrey,et al.  A strategic calculus of voting , 1983 .

[25]  Wei Shi Lim,et al.  A rendezvous-evasion game on discrete locations with joint randomization , 1997, Advances in Applied Probability.

[26]  Adam Tauman Kalai,et al.  The myth of the folk theorem , 2008, Games Econ. Behav..

[27]  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..

[28]  Itai Ashlagi,et al.  On the Value of Correlation , 2005, UAI.

[29]  Xiaofeng Wang,et al.  Reinforcement Learning to Play an Optimal Nash Equilibrium in Team Markov Games , 2002, NIPS.