Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

Approximations of Bimatrix Nash Equilibria

  • Paul Pavlos Spirakis
  • Paul Pavlos Spirakis
  • Paul Pavlos Spirakis
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_30

Years and Authors of Summarized Original Work

  • 2003; Lipton, Markakis, Mehta

  • 2006; Daskalaskis, Mehta, Papadimitriou

  • 2006; Kontogiannis, Panagopoulou, Spirakis

Problem Definition

Nash [15] introduced the concept of Nash equilibria in noncooperative games and proved that any game possesses at least one such equilibrium. A well-known algorithm for computing a Nash equilibrium of a 2-player game is the Lemke-Howson algorithm [13]; however, it has exponential worst-case running time in the number of available pure strategies [18].

Daskalakis et al. [5] showed that the problem of computing a Nash equilibrium in a game with 4 or more players is PPAD-complete; this result was later extended to games with 3 players [8]. Eventually, Chen and Deng [3] proved that the problem is PPAD-complete for 2-player games as well.

This fact emerged the computation of approximateNash equilibria. There are several versions of approximate Nash equilibria that have been defined in the literature; however, the...

Keywords

ε-Nash equilibria ε-Well-supported Nash equilibria 
This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 1.
    Althöfer I (1994) On sparse approximations to randomized strategies and convex combinations. Linear Algebr Appl 199:339–355MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bosse H, Byrka J, Markakis E (2007) New algorithms for approximate Nash equilibria in bimatrix games. In: Proceedings of the 3rd international workshop on internet and network economics (WINE 2007), San Diego, 12–14 Dec 2007. Lecture notes in computer scienceGoogle Scholar
  3. 3.
    Chen X, Deng X (2005) Settling the complexity of 2-player Nash-equilibrium. In: Proceedings of the 47th annual IEEE symposium on foundations of computer science (FOCS’06), Berkeley, 21–24 Oct 2005Google Scholar
  4. 4.
    Chen X, Deng X, Teng S-H (2006) Computing Nash equilibria: approximation and smoothed complexity. In: Proceedings of the 47th annual IEEE symposium on foundations of computer science (FOCS’06), Berkeley, 21–24 Oct 2006Google Scholar
  5. 5.
    Daskalakis C, Goldberg P, Papadimitriou C (2006) The complexity of computing a Nash equilibrium. In: Proceedings of the 38th annual ACM symposium on theory of computing (STOC’06), Seattle, 21–23 May 2006, pp 71–78Google Scholar
  6. 6.
    Daskalakis C, Mehta A, Papadimitriou C (2006) A note on approximate Nash equilibria. In: Proceedings of the 2nd workshop on internet and network economics (WINE’06), Patras, 15–17 Dec 2006, pp 297–306Google Scholar
  7. 7.
    Daskalakis C, Mehta A, Papadimitriou C (2007) Progress in approximate Nash equilibrium. In: Proceedings of the 8th ACM conference on electronic commerce (EC07), San Diego, 11–15 June 2007Google Scholar
  8. 8.
    Daskalakis C, Papadimitriou C (2005) Three-player games are hard. In: Electronic colloquium on computational complexity (ECCC TR 05-139)Google Scholar
  9. 9.
    Fearnley J, Goldberg PW, Savani R, Bjerre Sørensen T (2012) Approximate well-supported Nash equilibria below two-thirds. In: SAGT 2012, Barcelona, pp 108–119Google Scholar
  10. 10.
    Kannan R, Theobald T (2007) Games of fixed rank: a hierarchy of bimatrix games. In: Proceedings of the ACM-SIAM symposium on discrete algorithms, New Orleans, 7–9 Jan 2007Google Scholar
  11. 11.
    Kontogiannis S, Panagopoulou PN, Spirakis PG (2006) Polynomial algorithms for approximating Nash equilibria of bimatrix games. In: Proceedings of the 2nd workshop on internet and network economics (WINE’06), Patras, 15–17 Dec 2006, pp 286–296Google Scholar
  12. 12.
    Kontogiannis S, Spirakis PG (2010) Well supported approximate equilibria in bimatrix games. Algorithmica 57(4):653–667MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Lemke CE, Howson JT (1964) Equilibrium points of bimatrix games. J Soc Indust Appl Math 12:413–423MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lipton RJ, Markakis E, Mehta A (2003) Playing large games using simple startegies. In: Proceedings of the 4th ACM conference on electronic commerce (EC’03), San Diego, 9–13 June 2003, pp 36–41Google Scholar
  15. 15.
    Nash J (1951) Noncooperative games. Ann Math 54:289–295MathSciNetCrossRefGoogle Scholar
  16. 16.
    von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, PrincetonzbMATHGoogle Scholar
  17. 17.
    Papadimitriou CH (1991) On inefficient proofs of existence and complexity classes. In: Proceedings of the 4th Czechoslovakian symposium on combinatorics 1990, PrachaticeGoogle Scholar
  18. 18.
    Savani R, von Stengel B (2004) Exponentially many steps for finding a Nash equilibrium in a bimatrix game. In: Proceedings of the 45th annual IEEE symposium on foundations of computer science (FOCS’04), Rome, 17–19 Oct 2004, pp 258–267Google Scholar
  19. 19.
    Tsaknakis H, Spirakis P (2007) An optimization approach for approximate Nash equilibria. In: Proceedings of the 3rd international workshop on internet and network economics (WINE 2007). Lecture notes in computer science. Also in J Internet Math 5(4):365–382 (2008)Google Scholar
  20. 20.
    Tsaknakis H, Spirakis PG (2010) Practical and efficient approximations of Nash equilibria for win-lose games based on graph spectra. In: WINE 2010, Stanford, pp 378–390Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Paul Pavlos Spirakis
    • 1
  • Paul Pavlos Spirakis
    • 2
  • Paul Pavlos Spirakis
    • 3
  1. 1.Computer Engineering and InformaticsResearch and Academic Computer Technology Institute, Patras UniversityPatrasGreece
  2. 2.Computer ScienceUniversity of LiverpoolLiverpoolUK
  3. 3.Computer Technology Institute (CTI)PatrasGreece