Fixed set theory for closed correspondences with applications to self-similarity and games

This paper sketches a highly elementary order-theoretic approach which allows one to derive general existence theorems concerning the largest and minimal fixed and almost-fixed sets of closed (and condensing) families of commuting set-valued self-maps defined on a compact topological (complete and bounded metric, resp.) space. Certain converses of these results are also established, thereby providing new characterizations of compact (complete, resp.) spaces. We also study the asymptotic behavior of fixed sets sequences associated with a uniformly convergent sequence of closed correspondences, and give two applications. The first application provides two theorems on the existence of a compact self-similar set with respect to infinite families of continuous functions. The second application uses the present fixed set theory to derive a result on the existence of rationalizable outcomes in normal-form games with possibly discontinuous payoff functions.