Complexity of Rational and Irrational Nash Equilibria

  • Vittorio Bilò
  • Marios Mavronicolas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6982)


We introduce two new decision problems, denoted as ∃ RATIONAL NASH and ∃ IRRATIONAL NASH, pertinent to the rationality and irrationality, respectively, of Nash equilibria for (finite) strategic games. These problems ask, given a strategic game, whether or not it admits (i) a rational Nash equilibrium where all probabilities are rational numbers, and (ii) an irrational Nash equilibrium where at least one probability is irrational, respectively. We are interested here in the complexities of ∃ RATIONAL NASH and ∃ IRRATIONAL NASH.

Towards this end, we study two other decision problems, denoted as NASH-EQUIVALENCE and NASH-REDUCTION, pertinent to some mutual properties of the sets of Nash equilibria of two given strategic games with the same number of players. NASH-EQUIVALENCE asks whether the two sets of Nash equilibria coincide; we identify a restriction of its complementary problem that witnesses ∃ RATIONAL NASH. NASH-REDUCTION asks whether or not there is a so called Nash reduction (a suitable map between corresponding strategy sets of players) that yields a Nash equilibrium of the former game from a Nash equilibrium of the latter game; we identify a restriction of it that witnesses ∃ IRRATIONAL NASH.

As our main result, we provide two distinct reductions to simultaneously show that (i) NASH-EQUIVALENCE is co-\(\cal NP\)-hard and ∃ RATIONAL NASH is \(\cal NP\)-hard, and (ii) NASH-REDUCTION and ∃ IRRATIONAL NASH are \(\cal NP\)-hard, respectively. The reductions significantly extend techniques previously employed by Conitzer and Sandholm [6, 7].


Nash Equilibrium Decision Problem Mixed Strategy Surjective Mapping Positive Instance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abbott, T., Kane, D., Valiant, P.: On the Complexity of Two-Player Win-Lose Games. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Sciences, pp. 113–122 (October 2005)Google Scholar
  2. 2.
    Austrin, P., Braverman, M., Chlamtáč, E.: Inapproximability of NP-complete Variants of Nash Equilibrium, arXiv:1104.3760v1, April 19 (2001)Google Scholar
  3. 3.
    Borgs, C., Chayes, J., Immorlica, N., Tauman Kalai, A., Mirrokni, V., Papadimitriou, C.H.: The Myth of the Folk Theorem. Games and Economic Behavior 70(1), 34–43 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, X., Deng, X., Teng, S.H.: Settling the Complexity of Computing Two-Player Nash Equilibria. Journal of the ACM 56(3) (2009)Google Scholar
  5. 5.
    Codenotti, B., Stefanovic, D.: On the Computational Complexity of Nash Equilibria for (0, 1) Bimatrix Games. Information Processing Letters 94(3), 145–150 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Conitzer, V., Sandholm, T.: Complexity Results about Nash Equilibria. In: Proceedings of the 18th Joint Conference on Artificial Intelligence, pp. 765–771 (August 2003)Google Scholar
  7. 7.
    Conitzer, V., Sandholm, T.: New Complexity Results about Nash Equilibria. Games and Economic Behavior 63(2), 621–641 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The Complexity of Computing a Nash Equilibrium. SIAM Journal on Computing 39(1), 195–259 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Etessami, K., Yannakakis, M.: On the Complexity of Nash Equilibria and Other Fixed Points. SIAM Journal on Computing 39(6), 2531–2597 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fiat, A., Papadimitriou, C.H.: When Players are not Expectation Maximizers. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds.) Algorithmic Game Theory. LNCS, vol. 6386, pp. 1–14. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: Proceedings of the 8th Annual ACM Symposium on Theory of Computing, pp. 10–22 (1976)Google Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability — A Guide to the Theory of \({\cal NP}\)-Completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  13. 13.
    Gilboa, I., Zemel, E.: Nash and Correlated Equilibria: Some Complexity Considerations. Games and Economic Behavior 1(1), 80–93 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Koutsoupias, E.: Personal communication during. In: 2nd International Symposium on Algorithmic Game Theory, Paphos, Cyprus (October 2009)Google Scholar
  15. 15.
    Mavronicolas, M., Monien, B., Wagner, K.K.: Weighted Boolean Formula Games. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 469–481. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    McLennan, A., Tourky, R.: Simple Complexity from Imitation Games. Games abd Economic Behavior 68(2), 683–688 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nash, J.F.: Equilibrium Points in n-Person Games. Proceedings of the National Academy of Sciences of the United States of America 36, 48–49 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nash, J.F.: Non-Cooperative Games. Annals of Mathematics 54(2), 286–295 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Papadimitriou, C.H.: On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence. Journal of Computer and System Sciences 48(3), 498–532 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Vittorio Bilò
    • 1
  • Marios Mavronicolas
    • 2
  1. 1.Dipartimento di Matematica ”Ennio De Giorgi”Università del SalentoLecceItaly
  2. 2.Department of Computer ScienceUniversity of CyprusNicosiaCyprus

Personalised recommendations