Equilibrium existence results for simple economies and dynamic games

The purpose of this paper is to bring some simplicity and generality to the investigation of equilibrium existence in certain "simple" dynamic games. The way we deal with these objects is, essentially, by reducing them to what we term "simple economies", these latter being, from the viewpoint of equilibrium existence results, at once more general and less cumbersome. In §1, we exhibit some topological facts about a certain rather general class of simple economies, including the fact (1.7, Main Theorem) that they each possess a non-empty and compact set of equilibria. Then, in §2, we define the simple dynamic games of interest, immediately reducing them (2.5, Reduction Lemma) to the simple economies studied. In §3, we show (3.0) that the simple dynamic games in question have non-empty and compact sets of equilibria, using the fact (1.7) that the simple economies to which they reduce have such sets of equilibria. We then illustrate by example (3.3) that a general class of discrete-time, deterministic games with convex performance criteria is covered by the equilibrium existence results just described. This class includes dynamic games for which certain non-linearities in the next-State map are allowed, and for which controls are restricted to compact regions, these regions themselves varying as a function of state. Of course, we do not intend that results particularized to this quite special example be understood as the main thrust of the paper.