Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot

TL;DR

This paper investigates the mathematical foundations of binary relations in economics, focusing on continuity, additivity, and monotonicity, and explores their implications across topology, measure theory, decision theory, and social choice.

Contribution

It provides new necessary and sufficient conditions for the existence of certain binary relations and extends classical results to broader economic and mathematical contexts.

Findings

Conditions for the existence of continuous, anti-symmetric, and transitive relations on topological spaces.

Additivity conditions ensuring transitivity of relations on sigma-algebras.

Connections between mathematical properties and economic decision and social choice theories.

Abstract

This chapter examines how positivity and order play out in two important questions in mathematical economics, and in so doing, subjects the postulates of continuity, additivity and monotonicity to closer scrutiny. Two sets of results are offered: the first departs from Eilenberg's (1941) necessary and sufficient conditions on the topology under which an anti-symmetric, complete, transitive and continuous binary relation exists on a topologically connected space; and the second, from DeGroot's (1970) result concerning an additivity postulate that ensures a complete binary relation on a {\sigma}-algebra to be transitive. These results are framed in the registers of order, topology, algebra and measure-theory; and also beyond mathematics in economics: the exploitation of Villegas' notion of monotonic continuity by Arrow-Chichilnisky in the context of Savage's theorem in decision theory, and the extension of Diamond's impossibility result in social choice theory by Basu-Mitra. As such, this chapter has a synthetic and expository motivation, and can be read as a plea for inter-disciplinary conversations, connections and collaboration.