International Journal of Game Theory

, Volume 37, Issue 1, pp 93–113 | Cite as

The computational complexity of evolutionarily stable strategies

  • K. Etessami
  • A. Lochbihler
Original Paper


The concept of evolutionarily stable strategies (ESS) has been central to applications of game theory in evolutionary biology, and it has also had an influence on the modern development of game theory. A regular ESS is an important refinement the ESS concept. Although there is a substantial literature on computing evolutionarily stable strategies, the precise computational complexity of determining the existence of an ESS in a symmetric two-player strategic form game has remained open, though it has been speculated that the problem is \(\mathsf{NP}\)-hard. In this paper we show that determining the existence of an ESS is both \({\mathsf{NP}}\)-hard and \({\mathsf{coNP}}\)-hard, and that the problem is contained in \(\Sigma_{2}^{\rm p}\) , the second level of the polynomial time hierarchy. We also show that determining the existence of a regular ESS is indeed \({\mathsf{NP}}\)-complete. Our upper bounds also yield algorithms for computing a (regular) ESS, if one exists, with the same complexities.


Computational complexity Game theory Evolutionarily stable strategies Evolutionary biology Nash equilibria 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abakuks A (1980) Conditions for evolutionarily stable strategies. J Appl Probab 17:559–562CrossRefGoogle Scholar
  2. Blum L, Cucker F, Shub M, Smale S (1998) Complexity and real computation. Springer, HeidelbergGoogle Scholar
  3. Bomze I (1986) Non-cooperative 2-person games in biology: a classification. Int J Game Theory 15:31–59CrossRefGoogle Scholar
  4. Bomze I (1992) Detecting all evolutionarily stable strategies. J Optim Theory Appl 75:313–329CrossRefGoogle Scholar
  5. Bomze I (2002) Regularity versus degeneracy in dynamics, games, and optimization: a unified approach to different aspects. SIAM Rev 44(3):394–414CrossRefGoogle Scholar
  6. Bomze I, Pötscher BM (1989) Game theoretic foundations of evolutionary stability. Springer, HeidelbergGoogle Scholar
  7. Conitzer V, Sandholm T (2003) Complexity results about nash equilibria. In: 18th International joint conference on artificial intelligence (IJCAI), pp 765–771Google Scholar
  8. Etessami K, Lochbihler A (2004) The computational complexity of evolutionarily stable strategies. Electronic Colloquium on Computational Complexity (ECCC) technical report TR04-055Google Scholar
  9. Garey M, Johnson D (1979) Computers and intractability. W. H. Freeman, San FranciscoGoogle Scholar
  10. Gilboa I, Zemel E (1989) Nash and correlated equilibria: some complexity considerations. Games Econ Behav 1:80–93CrossRefGoogle Scholar
  11. Haigh J (1975) Game theory and evolution. Adv Appl Probab 7:8–11CrossRefGoogle Scholar
  12. Hammerstein P, Selten R (1994) Game theory and evolutionary biology. In: Aumann RJ, Hart S (eds) Handbook of game theory, vol. 2, Chap. 2. Elsevier, Amsterdam, pp 929–993Google Scholar
  13. Harsanyi J (1973) Games with randomly distributed payoffs: a new rational for mixed strategy equilibrium points. Int J Game Theory 2:1–23CrossRefGoogle Scholar
  14. Harsanyi J (1973) Oddness of the number of equilibrium points: a new proof. Int J Game Theory 2:235–250CrossRefGoogle Scholar
  15. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeGoogle Scholar
  16. Lancaster P, Tismenetsky M (1985) The theory of matrices, 2nd edn. Academic, New YorkGoogle Scholar
  17. Maynard Smith J (1982) Evolution and the theory of games. Cambridge University Press, CambridgeGoogle Scholar
  18. McNamara JM, Webb JN, Collins EJ, Szekely T, Houston AI (1997) A general technique for computing evolutionarily stable strategies based on dynamics. J Theoret Biol 189:211–225CrossRefGoogle Scholar
  19. Motzkin TS, Straus EG (1965) Maxima for graphs and a new proof of a theorem of Turan. Can J Math 17:533–540Google Scholar
  20. Murty KG, Kabadi SN (1987) Some NP-complete problems in quadratic and non-linear programming. Math Programm 39:117–129CrossRefGoogle Scholar
  21. Nash J (1951) Non-cooperative games. Ann Math 54:286–295CrossRefGoogle Scholar
  22. Nisan N (2006) A note on the computational hardness of evolutionarily stable strategies (DRAFT). Electronic Colloquium on Computational Complexity (ECCC) technical report TR06-076Google Scholar
  23. Osborne MJ, Rubinstein A (1994) A course in game theory. MIT, CambridgeGoogle Scholar
  24. Papadimitriou C (1994) Computational complexity. Addison-Wesley, ReadingGoogle Scholar
  25. Papadimitriou C (2001) Algorithms, games and the internet. In: Proceedings of ACM symposium on theory of computing. pp 249–253Google Scholar
  26. Papadimitriou C, Yannakakis M (1982) The complexity of facets (and some facets of complexity). In: 14th ACM symposium on theory of computing. pp 255–260Google Scholar
  27. Selten R (1983) Evolutionary stability in extensive 2-person games. Math Soc Sci 5:269–363CrossRefGoogle Scholar
  28. Maynard Smith J, Price GR (1973) The logic of animal conflict. Nature (Lond) 246:15–18CrossRefGoogle Scholar
  29. Valiant L (1979) The complexity of computing the permanent. Theor Comput Sci 8:189–201CrossRefGoogle Scholar
  30. van Damme E (1991) Stability and perfection of Nash equilibria, 2nd edn. Springer, HeidelbergGoogle Scholar
  31. Vavasis SA (1990) Quadratic programming is in NP. Inf Process Lett 36(2):73–77CrossRefGoogle Scholar
  32. Weibull J (1997) Evolutionary game theory. MIT, CambridgeGoogle Scholar

Copyright information

© Springer Verlag 2007

Authors and Affiliations

  1. 1.Laboratory for Foundations of Computer Science, School of InformaticsUniversity of EdinburghEdinburghScotland, UK
  2. 2.Fakultät für Mathematik und InformatikUniversität PassauPassauGermany

Personalised recommendations