A Generalization of the Maximum Theorem

The Maximum Theorem and its generalizations' have become one of the most useful tools in economic theory. The theorem first stated and proved, to the best of my knowledge, by Claude Berge [1963, p. 116] gives conditions under which a "maximizing correspondence" (a correspondence which always picks out the set of maximal elements) will be closed, and hence, in many cases, upper hemicontinuous. A further generalization of Berge's theorem is provided here, one which extends the usefulness of the theorem, especially to equilibrium concepts in multi-person decision situations. The generalization is suggested by a natural decomposition of Berge's result into two parts, and by the nature of the domination concept in game theory. Several applications are suggested, probably none of which is a new result, but each of which becomes transparent in the light of the theorem's general form. Thus, the generalized theorem provides a unified treatment of upper hemicontinuity for the kinds of correspondences which occur frequently in the social sciences. The original form of the Maximum Theorem was essentially as follows: