From Axioms to Algorithms: Mechanized Proofs of the vNM Utility Theorem
Li Jingyuan·June 08, 2025
Summary
The paper formalizes the vNM expected utility theorem in Lean 4, addressing encoding and lottery representation challenges. It provides a complete formalization, rigorous proofs, and applications in management science, AI, and computational economics, advancing decision theory's practical implementation. A formal framework ensures machine-checked proofs for learned models' rationality and safety in intelligent systems, mechanizing reward learning verification and offering end-to-end formal guarantees. The text discusses safe exploration strategies, proving the existence of a unique α* for safe exploration, and formalizes AI alignment using utility theory, defining rational, human-aligned preferences and proving independence.
Introduction
Background
Overview of the von Neumann-Morgenstern (vNM) Expected Utility Theorem
Importance of formalizing decision theory in theorem provers like Lean 4
Objective
Objective of formalizing the vNM theorem in Lean 4
Goals: encoding challenges, lottery representation, rigorous proofs, and applications
Method
Encoding and Lottery Representation
Challenges in encoding the vNM theorem in Lean 4
Techniques for representing lotteries and utilities in the formal system
Formalization Process
Detailed steps in formalizing the vNM theorem
Use of Lean 4's features for mathematical proofs
Applications
Integration of formalized decision theory in management science
Applications in AI and computational economics
Advancements in practical implementation of decision theory
Applications and Implications
Decision Theory in Practice
Case studies demonstrating the use of formalized decision theory
Benefits in real-world decision-making processes
Intelligent Systems and Safety
Formal framework for ensuring machine-checked proofs in intelligent systems
Verification of learned models' rationality and safety
Mechanizing Reward Learning
Formalization of reward learning verification
End-to-end formal guarantees for intelligent systems
Formal Framework for Intelligent Systems
Safe Exploration Strategies
Formal proof of the existence of a unique α* for safe exploration
Strategies ensuring safe and efficient exploration in AI
AI Alignment and Utility Theory
Formalization of AI alignment using utility theory
Definition of rational, human-aligned preferences
Proof of independence in preference relations
Conclusion
Summary of Contributions
Recap of the formalization process and its implications
Future Directions
Potential areas for further research and development
Opportunities for expanding the formal framework in decision theory
Basic info
papers
theoretical economics
computational finance
artificial intelligence
Advanced features