Abstract
This paper describes an algorithm for calculating approximately mixed Nash equilibria (NE) in bimatrix games. The algorithm fuzzifies the strategies with normalized possibility distributions. The fuzzification takes advantage of the piecewise linearity of possibility distributions and transforms the NE problem of bimatrix games into a reduced form. The algorithm is guaranteed to find approximate NE in bimatrix games and ensures that the approximate NE is the saddle point of expected payoff functions in the reduced form. The algorithm provides a method of determining how close an approximate NE is to a solution during computation. Numerical results show that the new algorithm is approximately seven-time faster than the Lemke–Howson (LH) algorithm when the game size is 96, and the value of approximation deviation can be as small as 0.1.
Similar content being viewed by others
References
Bector CR, Chandra S (2005) Fuzzy mathematical programming and fuzzy matrix games. Springer, Berlin
Bosse H, Byrka J, Markakis E (2010) New algorithms for approximate Nash equilibria in bimatrix games. Theor Comput Sci 411(1):164–173
Campos L, Gonzalez A, Vila MA (1992) On the use of the ranking function approach to solve fuzzy matrix games in direct way. Fuzzy Set Syst 49:193–203
Chen X, Deng X, Teng SH (2009) Setting the complexity of coomputing two-player Nash equilibria. J ACM 56(3):1–57
Daskalakis C, Mehta A, Papadimitriou C (2009) A note on approximate Nash equilibria. Theor Comput Sci 410:1581–1588
Dijkman JG, ven Haeringen H, DE Lange SJ (1983) Fuzzy numbers. J Math Anal Appl 92:301–341
Downey R, Fellows MR (2013) Fundamentals of parameterized complexity. Springer, Berlin
Dubois D, Prade H (2015) Practical methods for constructing possibility distributions. Int J Intell Syst 31(3):215–239
Fearnley J, Goldberg PW, Savani R, Sorensen TB (2012) Approximate well-supported Nash equilibria below two-thirds. In: International symposium on algorithmic game theory, SAGT: algorithmic game theory, pp 108–119
Gao L (2017) The application of possibility distribution for solving standard quadratic optimization problems. Comput Inf Sci. https://doi.org/10.5539/cis.v10n3p60
Gao L (2012) The discussion of applications of the fuzzy average to matrix game theory. In: The proceeding of CCECE2012
Gao L (1999) The fuzzy arithmetic mean. Fuzzy Sets Syst 107:335–348
Kacher F, Larbanib M (2006) Solution concept for a non-cooperative game with fuzzy parameters. Int Game Theory Rev 8:489–498
Kaufmann A, Gupta MM (1998) Fuzzy mathematical models in engineering and management science. Elsevier, Amsterdam
Kontogiannis SC, Spirakis PG (2007) Efficient algorithms for constant well supported approximate equilibria in bimatrix games. In: Proceeding of ICALP, pp 595-606
Larbani M (2009) Non cooperative fuzzy games in normal form: a survey. Fuzzy Sets Syst 160:3184–3210
Li C, Zhang Q (2011) Nash equilibrium strategy for fuzzy non-cooperative games. Fuzzy Sets Syst 176:46–55
Li DF (2016) Linear programming models and methods of matrix games with payoffs of triangular fuzzy numbers, studies in fuzziniss and soft computing 328. Springer, Berlin
Lipton RJ, Markakis E, Mehta A (2003) Playing large games using simple strategies. In: Proceeding of the fourth ACM conference on electronic commerce. New York, USA, pp 36–41
McKelvey RD, McLennan A (1996) Chapter 2, computation of equilibria in finite games. Handb Comput Econ 1:87–142
McKelvey RD, McLennan A, Turocy TL (2014) Gambit: software tools for game theory. Version 16:1. http://www.gambit-project.org/
Nishizaki I, Sakawa M (2001) Fuzzy and multiobjective games for conflict resolution. Springer, Berlin
Ortiz LE, Irfan MT (2017) Tractable algorithms for approximate Nash equilibria in generalized graphical Games with tree structure. In: The Proceeding of the thirty-first conference on artificial intelligence
Rubinstein A (2016) Settling the complexity of computing approximate two-player Nash equilibria. In: Proceeding of FOCS, pp 258–265
Vijay V, Chandra S, Bector CR (2005) Matrix games with fuzzy goals and fuzzy payoffs. Omega Int J Manag Sci 33:425–429
Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1:3–28
Zimmermann H-J (2011) Fuzzy set theory and its application, 4th edn. Kluwer Academic Publishers, Berlin
Acknowledgements
The author thanks anonymous reviewers and Gao J and appreciates their constructive suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no financial disclosures of conflicts to report.
Additional information
Communicated by V. Loia.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gao, L. An algorithm for finding approximate Nash equilibria in bimatrix games. Soft Comput 25, 1181–1191 (2021). https://doi.org/10.1007/s00500-020-05213-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-020-05213-y