Separately Convex and Separately Continuous Preferences: On Results of Schmeidler, Shafer, and Bergstrom-Parks-Rader

TL;DR

This paper establishes necessary and sufficient conditions for the openness of correspondences in finite-dimensional spaces, revisiting foundational work and introducing separate convexity to connect convexity and continuity in choice theory.

Contribution

It introduces the concept of separate convexity for correspondences and relates it to classical continuity, addressing open questions from foundational economic theory.

Findings

Provided conditions for the openness of correspondences in Euclidean spaces.

Linked separate convexity with classical notions of continuity.

Clarified implications for choice theory and decision-making models.

Abstract

We provide necessary and sufficient conditions for a correspondence taking values in a finite-dimensional Euclidean space to be open so as to revisit the pioneering work of Schmeidler (1969), Shafer (1974), Shafer-Sonnenschein (1975) and Bergstrom-Rader-Parks (1976) to answer several questions they and their followers left open. We introduce the notion of separate convexity for a correspondence and use it to relate to classical notions of continuity while giving salience to the notion of separateness as in the interplay of separate continuity and separate convexity of binary relations. As such, we provide a consolidation of the convexity-continuity postulates from a broad inter-disciplinary perspective and comment on how the qualified notions proposed here have implications of substantive interest for choice theory.