Continuity Postulates and Solvability Axioms in Economic Theory and in Mathematical Psychology: A Consolidation of the Theory of Individual Choice

TL;DR

This paper establishes four theorems linking continuity conditions in economics to solvability axioms in mathematical psychology, providing a unified framework for understanding individual choice models.

Contribution

It introduces new theorems that connect continuity and solvability, expanding the theoretical foundation for representation theorems in economic and psychological choice theory.

Findings

Theorem 1 links various continuity notions with solvability under completeness and transitivity.

Theorem 2 characterizes full continuity of relations via separately-continuous functions.

Theorem 3 establishes equivalences between restricted solvability and continuity notions.

Abstract

This paper presents four theorems that connect continuity postulates in mathematical economics to solvability axioms in mathematical psychology, and ranks them under alternative supplementary assumptions. Theorem 1 connects notions of continuity (full, separate, Wold, weak Wold, Archimedean, mixture) with those of solvability (restricted, unrestricted) under the completeness and transitivity of a binary relation. Theorem 2 uses the primitive notion of a separately-continuous function to answer the question when an analogous property on a relation is fully continuous. Theorem 3 provides a portmanteau theorem on the equivalence between restricted solvability and various notions of continuity under weak monotonicity. Finally, Theorem 4 presents a variant of Theorem 3 that follows Theorem 1 in dispensing with the dimensionality requirement and in providing partial equivalences between solvability and continuity notions. These theorems are motivated for their potential use in representation theorems.