From Axioms to Algorithms: Mechanized Proofs of the vNM Utility Theorem

2025-06-08

AI Summary

How to rigorously prove that any preference relation satisfying completeness, transitivity, continuity, and independence must be representable by an expected utility function, per the von Neumann–Morgenstern (vNM) Expected Utility Theorem? Method: We introduce a fine-grained formalization of the independence axiom and employ a constructive proof strategy, integrating real analysis and decision theory within Lean 4 via its standard library and a formal algebraic model of probabilistic lotteries. Contribution/Results: (1) The first fully machine-checked, end-to-end formalization of the vNM theorem—covering both existence and uniqueness of the utility representation; (2) a more precise characterization of equivalence at decision boundaries, refining the classical statement; and (3) a reusable, foundational formalization of decision theory, designed for applications in economic modeling, AI alignment, and trustworthy decision systems.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

This paper presents a comprehensive formalization of the von Neumann-Morgenstern (vNM) expected utility theorem using the Lean 4 interactive theorem prover. We implement the classical axioms of preference-completeness, transitivity, continuity, and independence-enabling machine-verified proofs of both the existence and uniqueness of utility representations. Our formalization captures the mathematical structure of preference relations over lotteries, verifying that preferences satisfying the vNM axioms can be represented by expected utility maximization. Our contributions include a granular implementation of the independence axiom, formally verified proofs of fundamental claims about mixture lotteries, constructive demonstrations of utility existence, and computational experiments validating the results. We prove equivalence to classical presentations while offering greater precision at decision boundaries. This formalization provides a rigorous foundation for applications in economic modeling, AI alignment, and management decision systems, bridging the gap between theoretical decision theory and computational implementation.

Problem

• Formalizing vNM utility theorem with Lean 4 for machine-verified proofs
• Implementing vNM axioms to prove utility representation existence and uniqueness
• Bridging decision theory and computation for economic and AI applications

Innovation

• Formalized vNM utility theorem using Lean 4
• Verified utility existence and uniqueness proofs
• Granular implementation of independence axiom