On equivalent results in minimax theory

Abstract

In this paper we review known minimax results with applications in game theory and show that these results are easy consequences of the first minimax result 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 these implications the authors only use simple properties of compact sets and the well-known Weierstrass Lebesgue lemma.

On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory

“…Refs. [2,17]), showed that relation (10) holds. Applying the result that any continuous function on a compact set in a metric space is uniformly continuous (cf.…”

“…with λ ∈ P F (A) and μ ∈ P F (B) listed in relations (2) and (3). To extend the function f to the larger domain P F (A) × P(B) we always assume that the function f (a, .)…”

“…[1,2]) introducing different proof techniques in combination with sufficient conditions on the function f and the sets A and B to verify relation (1). Among the used techniques are fixed point theorems, techniques from topology (cf.…”

“…with λ represented by relation (2). Finally, if we extend the function f to the largest domain P(A) × P(B), we assume that the function f belongs for every μ ∈ P(B) and λ ∈ P(A) to the set L 1 λ⊗μ (A × B) of Borel measurable functions on A × B (measurable with respect to the product σ -algebra A ⊗ B), which are Lebesgue absolutely integrable with respect to the Borel product measure λ ⊗ μ.…”

Introduction to Convex and Quasiconvex Analysis

“…For a survey on some equivalent minimax theorems related to game theory and proved by finite dimensional separation of convex sets the reader should consult [24].…”