Minimax Theorem for U.S.C. (Uppersemicontinuous) - L.S.C. (Lowersemicontinuous) Payoff Functions.

摘要

Our aim is to get a general minimax theorem whose assumptions and conclusions are phrased only in terms of the data of the problem, i.e. the pair of pure strategy sets S and T and the payoff function on S x T. For the assumptions, this means that we want to avoid any assumption of the type there exists a topology (or a measurable structure) on S and (or) T such that... For the conclusions, we are led to require that players have epsilon-optimal strategies with finite support, both because those are the easiest to describe in intrinsic terms, and because in any game where the value would not exist in strategies with finite support, all known general minimax theorems implicitly select as value either the sup inf or the inf sup by in effect restricting either player I or player II arbitrarily to strategies with finite support - so that the resulting value is completely arbitrary and misleading.