Existence of equilibrium actions and of equilibrium : A note on the ‘new’ existence theorems

A lemma by Ky Fan, which asserts the existence of a maximal element relative to some ordering (an equilibrium action) in a compact choice set, is extended to a larger class of orderings. This extension allows one to generalize slightly several existence theorems and seems to be useful for the applications. Kakutani's fixed-point theorem and the existence of an equilibrium in a generalized game are used to exemplify this. In the last case the method of proof may be of some independent interest involving, as it does, the reduction of an n-person game to a corresponding 1-person game.