Minimaxtheoreme und das Integraldarstellungsproblem

Abstract

In the present paper conditions for the strict determinateness of two-person zero-sum games are considered. In order to get such ‘minimax theorems’ we first study games with concave-convex pay-off function. If a game does not have this convexity property one usually passes to a mixed extension where both players are allowed to use probability measures (‘σ-additive randomizations’) or, more generally, probability contents (‘finitely additive randomizations’) as mixed strategies. By means of a very general minimax theorem for such finitely additive randomizations it can be shown that the problem of strict determinateness of σ-additive randomizations is equivalent to an integral representation problem. The latter is investigated in the last paragraph.