Completeness and Transitivity of Preferences on Mixture Sets

TL;DR

This paper investigates how certain continuity and Archimedean properties of preferences in mixture sets imply that preferences must be either complete or transitive, revealing fundamental constraints on preference structures.

Contribution

It establishes that under specific non-falsifiable properties, preferences cannot be both incomplete and intransitive, extending classical results to generalized mixture sets.

Findings

Preferences with Archimedean and mixture-continuity are either complete or transitive.

Specialized results for Herstein-Milnor mixture sets.

Links to foundational work by Eilenberg, Sonnenschein, and Schmeidler.

Abstract

In this paper, we show that the presence of the Archimedean and the mixture-continuity properties of a binary relation, both empirically non-falsifiable in principle, foreclose the possibility of consistency (transitivity) without decisiveness (completeness), or decisiveness without consistency, or in the presence of a weak consistency condition, neither. The basic result can be sharpened when specialized from the context of a generalized mixture set to that of a mixture set in the sense of Herstein-Milnor (1953). We relate the results to the antecedent literature, and view them as part of an investigation into the interplay of the structure of the choice space and the behavioral assumptions on the binary relation defined on it; the ES research program due to Eilenberg (1941) and Sonnenschein (1965), and one to which Schmeidler (1971) is an especially influential contribution.