The Yannelis-Prabhakar Theorem on Upper Semi-Continuous Selections in Paracompact Spaces: Extensions and Applications

TL;DR

This paper extends the Yannelis-Prabhakar theorem on upper semi-continuous selections in paracompact spaces, providing necessary and sufficient conditions and applying them to key results in mathematical economics.

Contribution

It offers a new characterization of upper semi-continuous selections and applies this to multiple foundational economic theorems, expanding their scope and understanding.

Findings

Necessary and sufficient conditions for upper semi-continuous selections

Application to Berge's maximum theorem and Gale-Nikaido-Debreu lemma

Extension of selection theorems in economic models

Abstract

In a 1983 paper, Yannelis-Prabhakar rely on Michael's selection theorem to guarantee a continuous selection in the context of the existence of maximal elements and equilibria in abstract economies. In this tribute to Nicholas Yannelis, we root this paper in Chapter II of Yannelis' 1983 Rochester Ph.D. dissertation, and identify its pioneering application of the paracompactness condition to current and ongoing work of Yannelis and his co-authors, and to mathematical economics more generally. We move beyond the literature to provide a necessary and sufficient condition for upper semi-continuous local and global selections of correspondences, and to provide application to five domains of Yannelis' interests: Berge's maximum theorem, the Gale-Nikaido-Debreu lemma, the Gale-McKenzie survival assumption, Shafer's non-transitive setting, and the Anderson-Khan-Rashid approximate existence theorem. The last resonates with Chapter VI of the Yannelis' dissertation.