Abstract
This paper gives wide characterization of n-person non-coalitional games with finite players’ strategy spaces and payoff functions having some concavity or convexity properties. The characterization is done in terms of the existence of two-point-strategy Nash equilibria, that is equilibria consisting only of mixed strategies with supports being one or two-point sets of players’ pure strategy spaces. The structure of such simple equilibria is discussed in different cases. The results obtained in the paper can be seen as a discrete counterpart of Glicksberg’s theorem and other known results about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of players’ pure strategies.
Similar content being viewed by others
References
Bohnenblust HF, Karlin S, Shapley LS (1950) Games with continuous, convex pay-off. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games I, annals of mathematics studies 24. Princeton University Press, Princeton, pp 181–192
Glicksberg I (1952) A further generalization of the Kakutani’s fixed point theorem with applications to Nash equilibrium points. Proc Am Math Soc 38:170–174
Gomez E (1988) Games with convex payoff functions in the first variable. Int J Game Theory 17:201–204
Nash J (1951) Non-cooperative games games. Ann Math 54:286–295
Parthasarathy T, Raghavan TES (1975) Some topics in two-person games. American Elsevier Publishing Company, INC, New York
Połowczuk W (2003) Pure Nash equilibria in finite two-person non-zero-sum games. Int J Game Theory 32:229–240
Połowczuk W (2006) On almost-pure-strategy Nash equilibria in bimatrix games with convexity properties. Appl Math 33:71–84
Radzik T (1991) Saddle point theorems. Int J Game Theory 20:23–32
Radzik T (1993) Nash equilibria of discontinuous non-zero-sum two-person games. Int J Game Theory 21:429–437
Radzik T (2000) Characterization of optimal strategies in matrix games with convexity properties. Int J Game Theory 29:211–227
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Połowczuk, W., Więcek, P. & Radzik, T. On the existence of almost-pure-strategy Nash equilibrium in n-person finite games. Math Meth Oper Res 65, 141–152 (2007). https://doi.org/10.1007/s00186-006-0105-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-006-0105-1