A discrete fixed point theorem and its applications

Abstract

This paper proves a fixed point theorem on a discrete set which exploits the “contiguous convexity” of the set and the “direction preserving-ness” of correspondence. Then, applying this result, we study a general equilibrium model with indivisible commodities. A set of sufficient conditions for the existence of a Walrasian equilibrium price is given, in terms of properties of the excess demand function. We also apply it to a non-cooperative game on discrete strategy sets, and show some conditions for the existence of a Nash equilibrium point. A new notion of “contiguity” is proposed as a discrete analog of continuity.