Simple equilibria in finite games with convexity properties

Tadeusz Radzik¹ ; Piotr Więcek¹

¹ Institute of Mathematics Wrocław University of Technology Wybrzeże Wyspiańskiego 27 50-370 Wrocław, Poland

Applicationes Mathematicae, Tome 42 (2015) no. 1, pp. 83-109.

Résumé

This review paper gives a characterization of non-coalitional zero-sum and non-zero-sum games with finite strategy spaces and payoff functions having some concavity or convexity properties. The characterization is given in terms of the existence of two-point Nash equilibria, that is, equilibria consisting of mixed strategies with spectra consisting of at most two pure strategies. The structure of such simple equilibria is discussed in various cases. In particular, many of the results discussed can be seen as discrete counterparts of classical theorems about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of pure strategies. The paper provides many examples illustrating the results presented and ends with four related open problems.

Détail

DOI : 10.4064/am42-1-6

Keywords: review paper gives characterization non coalitional zero sum non zero sum games finite strategy spaces payoff functions having concavity convexity properties characterization given terms existence two point nash equilibria equilibria consisting mixed strategies spectra consisting pure strategies structure simple equilibria discussed various cases particular many results discussed seen discrete counterparts classical theorems about existence pure almost pure nash equilibria continuous concave convex games compact convex spaces pure strategies paper provides many examples illustrating results presented ends related problems