An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria. Here we provide such a study that compares a number of approximation methods and exact methods. In particular, we explore the trade-off between the quality of approximate equilibrium and the required running time to find one. We found that the existing library GAMUT, which has been the de facto standard that has been used to test exact methods, is insufficient as a test bed for approximation methods since many of its games have pure equilibria or other easy-to-find good approximate equilibria. We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size $$2000 \times 2000$$. Finally, we provide new close-to-worst-case examples for the best-performing algorithms for finding approximate Nash equilibria.

[1]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[2]  Paul G. Spirakis,et al.  Approximability of Symmetric Bimatrix Games and Related Experiments , 2011, SEA.

[3]  Paul W. Goldberg,et al.  The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions , 2010, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[4]  Aranyak Mehta,et al.  A note on approximate Nash equilibria , 2006, Theor. Comput. Sci..

[5]  Carmine Ventre,et al.  Ranking games that have competitiveness-based strategies , 2010, EC '10.

[6]  Yoav Shoham,et al.  Simple search methods for finding a Nash equilibrium , 2004, Games Econ. Behav..

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

[8]  Rahul Savani,et al.  Polylogarithmic Supports Are Required for Approximate Well-Supported Nash Equilibria below 2/3 , 2013, WINE.

[9]  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..

[10]  Xiaotie Deng,et al.  Settling the complexity of computing two-player Nash equilibria , 2007, JACM.

[11]  Paul G. Spirakis,et al.  An Optimization Approach for Approximate Nash Equilibria , 2007, WINE.

[12]  Paul G. Spirakis,et al.  Well Supported Approximate Equilibria in Bimatrix Games , 2010, Algorithmica.

[13]  Troels Bjerre Lund,et al.  Approximate Well-Supported Nash Equilibria Below Two-Thirds , 2012, SAGT.

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

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

[16]  H. Kuk.  On equilibrium points in bimatrix games , 1996 .

[17]  Paul G. Spirakis,et al.  Performance Evaluation of a Descent Algorithm for Bi-matrix Games , 2008, WINE.

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

[19]  Bruno Codenotti,et al.  An experimental analysis of Lemke-Howson algorithm , 2008, ArXiv.

[20]  Rahul Savani,et al.  Hard‐to‐Solve Bimatrix Games , 2006 .

[21]  Evangelos Markakis,et al.  New algorithms for approximate Nash equilibria in bimatrix games , 2010, Theor. Comput. Sci..

[22]  Troels Bjerre Lund,et al.  Approximate Well-supported Nash Equilibria Below Two-thirds , 2015, Algorithmica.

[23]  Bernhard von Stengel,et al.  Unit Vector Games , 2015, ArXiv.