New Versions of the Minimax Theorem

Abstract

In recent years the minimax theorem of John von Neumann has found numerous new extensions, due to Irle [1985], Kindler [1990], Simons [1990][1991] and others, with the aim to remove from the assumptions the last remnants of linear and convex structures, and to install assumptions of more comprehensive kinds instead. The talk wants to present an extension and unification, due to the author in cooperation with Frank Zartmann, of the recent main contributions. The principal results are a quantitative theorem in the spirit of the concave-convexlike minimax theorem of Ky Fan [1953], and a topological theorem in the spirit of the quasiconcave-convex minimax theorem of Sion [1958]. A further main contribution is to decompose the minimax relation into independent halfs, such that the minimax theorems quoted above — and hence the bulk of the minimax theorems known so far — appear as unions of one-sided theorems, which then can be combined at will to minimax theorems of mixed types in the spirit of Terkelsen [1972].