Formal Power Series Representations in Probability and Expected Utility Theory

2025-07-31

AI Summary

Traditional preference theory relies heavily on strong assumptions—transitivity, the Archimedean property, boundedness, and continuity—limiting its applicability to realistic, often intransitive or unbounded preferences. Method: We develop a generalized coherent preference theory under weakened assumptions, introducing a novel coherence condition independent of classical axioms. We extend Hölder’s theorem to non-Archimedean ordered structures and strengthen the Hahn embedding theorem. Using ordered field extensions, formal power series representations, and abstract algebraic tools, we integrate de Finetti-style probabilistic coherence. Contribution/Results: We prove that any preference system satisfying our weak coherence condition admits a completion into a total preorder and admits a utility representation within an ordered field extension of the reals. This yields the first unified utility characterization accommodating non-transitive, non-Archimedean, and unbounded preferences—establishing foundational representability without requiring standard continuity or Archimedeanity.

Technology Category

Knowledge Representation and Reasoning: PreferencesReasoning under Uncertainty: Decision/Utility TheoryGame Theory and Economic Paradigms: Cooperative Game Theory

Application Category

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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 Hölder's Theorem and strengthens Hahn's Embedding Theorem.

Problem

• Develops a general theory of coherent preference without orthodox restrictions
• Extends preference systems to completeness under coherence requirements
• Represents coherent preferences via utility in ordered field extensions

Innovation

• General coherent preference theory without orthodox restrictions
• Complete preference system representation in ordered field
• Extends Hölder's Theorem and strengthens Hahn's Theorem