Formal Power Series Representations in Probability and Expected Utility Theory

TL;DR

This paper develops a broad, flexible theory of coherent preferences that extends classical probability and utility foundations without requiring traditional assumptions like transitivity or continuity, and it generalizes key theorems in the field.

Contribution

It introduces a generalized preference theory that relaxes standard restrictions and proves new representation theorems extending de Finetti, Hölder, and Hahn's results.

Findings

Preferences can be extended to complete systems under coherence.

Preferences can be represented by utility in an ordered field extension.

The theory generalizes and strengthens classical theorems in probability and utility.

Abstract

We advance a general theory of coherent preference that surrenders restrictions embodied in orthodox doctrine. This theory enjoys the property that any preference system admits extension to a complete system of preferences, provided it satisfies a certain coherence requirement analogous to the one de Finetti advanced for his foundations of probability. Unlike de Finetti's theory, the one we set forth requires neither transitivity nor Archimedeanness nor boundedness nor continuity of preference. This theory also enjoys the property that any complete preference system meeting the standard of coherence can be represented by utility in an ordered field extension of the reals. Representability by utility is a corollary of this paper's central result, which at once extends H\"older's Theorem and strengthens Hahn's Embedding Theorem.