Abstract
In this paper we study the support sizes of evolutionary stable strategies (ESS) in random evolutionary games. We prove that, when the elements of the payoff matrix behave either as uniform, or normally distributed independent random variables, almost all ESS have support sizes o(n), where n is the number of possible types for a player. Our arguments are based exclusively on the severity of a stability property that the payoff submatrix indicated by the support of an ESS must satisfy. We then combine our normal–random result with a recent result of McLennan and Berg (2005), concerning the expected number of Nash Equilibria in normal–random bimatrix games, to show that the expected number of ESS is significantly smaller than the expected number of symmetric Nash equilibria of the underlying symmetric bimatrix game.
JEL Classification Code: C7 – Game Theory and Bargaining Theory.
This work was partially supported by the 6th Framework Programme under contract 001907 (DELIS).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Durrett, R.: Probability: Theory and Examples, 2nd edn. Duxbury Press (1996)
Etessami, K., Lochbihler, A.: The computational complexity of evolutionary stable strategies. Int. J. of Game Theory (to appear, 2007)
Gillespie, J.H.: A general model to account for enzyme variation in natural populations. III: Multiple alleles. Evolution 31, 85–90 (1977)
Haigh, J.: Game theory and evolution. Adv. in Appl. Prob. 7, 8–11 (1975)
Haigh, J.: How large is the support of an ESS? J. of Appl. Prob. 26, 164–170 (1989)
Karlin, S.: Some natural viability systems for a multiallelic locus: A theoretical study. Genetics 97, 457–473 (1981)
Kingman, J.F.C.: Typical polymorphisms maintained by selection at a single locus. J. of Appl. Prob. 25, 113–125 (1988)
Kontogiannis, S., Spirakis, P.: Counting stable strategies in random evolutionary games. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 839–848. Springer, Heidelberg (2005)
Lewontin, R.C., Ginzburg, L.R., Tuljapurkar, S.D.: Heterosis as an explanation of large amounts of genic polymorphism. Genetics 88, 149–169 (1978)
McLennan, A.: The expected numer of nash equilibria of a normal form game. Econometrica 73(1), 141–174 (2005)
McLennan, A., Berg, J.: The asymptotic expected number of nash equilibria of two player normal form games. Games & Econ. Behavior 51, 264–295 (2005)
Nash, J.: Noncooperative games. Annals of Mathematics 54, 289–295 (1951)
Roberts, D.P.: Nash equilibria of cauchy-random zero-sum and coordination matrix games. Int. J. of Game Theory 34, 167–184 (2006)
Smith, J.M., Price, G.: The logic of animal conflict. Nature 246, 15–18 (1973)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kontogiannis, S.C., Spirakis, P.G. (2007). On the Support Size of Stable Strategies in Random Games. In: Hromkovič, J., Královič, R., Nunkesser, M., Widmayer, P. (eds) Stochastic Algorithms: Foundations and Applications. SAGA 2007. Lecture Notes in Computer Science, vol 4665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74871-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-74871-7_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74870-0
Online ISBN: 978-3-540-74871-7
eBook Packages: Computer ScienceComputer Science (R0)