Mathematical Economics and Descriptive Set Theory1

The purpose of this paper is first to show that for any integer n ≥ 2, there exist no Borel measurable necessary and sufficient conditions on n lower semi-continuous payoff functions defining a non-cooperative n-person game in strategic form which can assert the existence of non-cooperative Nash equilibria (in either pure or mixed strategies). Second we show that there exist no Borel measurable necessary and sufficient conditions on an exchange economy with preference relations that are represented by lower semi-continuous utility functions which can assert the existence of Walrasian equilibria. And third we show that there exist no Borel measurable necessary and sufficient conditions on a triple of a lower semi-continuous one-period return function, a compact-valued continuous constraint correspondence, and a discount factor defining a deterministic discrete infinite horizon macroeconomic model which can assert the existence of optimal plans starting at some point