In this paper we review known minimax theorems with applications in game theory and show that these theorems can be proved using the first minimax theorem for a two-person zero-sum game with finite strategy sets published by von Neumann in 1928. Among these results are the well known minimax theorems of Wald, Ville and Kneser and their generalizations due to Kakutani, Ky Fan, König, Neumann and Gwinner-Oettli. Actually, it is shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. To show the implications the authors only use simple properties of compact sets and the well-known Weierstrass-Lebesgue lemma.

Convex programming, Game theory,
European Journal of Operational Research
Erasmus Research Institute of Management

Frenk, J.B.G, Kassay, G, & Kolumbán, J. (2004). On equivalent results in minimax theory. In European Journal of Operational Research (Vol. 157, pp. 46–58). doi:10.1016/j.ejor.2003.08.013