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

TL;DR

This paper formalizes the von Neumann-Morgenstern utility theorem in Lean 4, providing machine-verified proofs of core axioms, utility existence, and representation, bridging theoretical decision models and computational applications.

Contribution

It offers the first comprehensive mechanized proof of the vNM theorem, including detailed formalization of axioms and utility representation in a proof assistant.

Findings

Verified the existence and uniqueness of utility functions under vNM axioms

Formalized the independence axiom and mixture lotteries

Validated the formalization through computational experiments

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.