Decision theory and the"almost implies near"phenomenon

2025-02-10

AI Summary

This paper addresses the tension between the strict axiomatic foundations of classical decision theory and empirically observed deviations from rational behavior. Specifically, it asks: when choice behavior only approximately satisfies standard axioms—such as completeness, transitivity, and independence—how closely can its utility representation approximate the classical expected utility form? Method: We introduce a novel framework that quantifies the degree of axiom violation by formalizing axiom satisfaction as a measurable metric; within a suitably defined metric space, we establish a precise quantitative relationship between satisfaction level and approximation error. Under the expected utility framework, we prove that small violations of the axioms imply Lipschitz-continuous approximate utility representations. Contribution: We provide the first rigorous approximation guarantee: “almost satisfying the axioms” implies “existence of a close classical utility representation.” This bridges behavioral anomalies and canonical representations, yielding a robust, empirically testable axiomatic foundation for behavioral decision models.

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
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Security and Privacy: Large-scale security measurements

Abstract

We propose to relax traditional axioms in decision theory by incorporating a measurement, or degree, of satisfaction. For example, if the independence axiom of expected utility theory is violated, we can measure the size of the violation. This measure allows us to derive an approximation guarantee for a utility representation that aligns with the unmodified version of the axiom. Almost satisfying the axiom implies, then, a utility that is near a utility representation. We develop specific examples drawn from expected utility theory under risk and uncertainty.

Problem

• Develops framework for measuring axiom violations in decision theory
• Links approximate axiom satisfaction to near-canonical utility representations
• Applies theory to risk, uncertainty, and intertemporal choice domains

Innovation

• Approximate axiom satisfaction measurement framework
• Quantifiable error bounds for utility approximations
• Applied in risk, uncertainty, intertemporal choice domains