Function-Coherent Gambles

2025-02-22

arXiv.org

AI Summary

This paper addresses the limitation of imprecise probability theory under nonlinear utility in modeling empirically observed nonstandard time preferences—such as hyperbolic and quasi-hyperbolic discounting—in intertemporal decision-making. Methodologically, it introduces an extended acceptable gamble framework, first establishing a functional consistency axiom system that transcends the traditional linear utility constraint, and deriving a representation theorem that rigorously embeds nonlinear discounting behavior within the acceptable gamble paradigm. The theoretical apparatus integrates imprecise probability theory, acceptable gamble theory, and continuous linear functional analysis. The contribution lies in providing a unified, rigorous imprecise-probabilistic foundation for multiple empirically documented discounting patterns, thereby bridging the gap between normative decision theory and observed human intertemporal behavior, and substantially enhancing both the empirical explanatory power and theoretical coherence of rational decision modeling under uncertainty.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The desirable gambles framework provides a foundational approach to imprecise probability theory but relies heavily on linear utility assumptions. This paper introduces function-coherent gambles, a generalization that accommodates non-linear utility while preserving essential rationality properties. We establish core axioms for function-coherence and prove a representation theorem that characterizes acceptable gambles through continuous linear functionals. The framework is then applied to analyze various forms of discounting in intertemporal choice, including hyperbolic, quasi-hyperbolic, scale-dependent, and state-dependent discounting. We demonstrate how these alternatives to constant-rate exponential discounting can be integrated within the function-coherent framework. This unified treatment provides theoretical foundations for modeling sophisticated patterns of time preference within the desirability paradigm, bridging a gap between normative theory and observed behavior in intertemporal decision-making under genuine uncertainty.

Problem

• Generalizes desirable gambles framework for non-linear utility
• Establishes axioms and representation for function-coherent gambles
• Unifies diverse discounting models in intertemporal choice

Innovation

• Generalizes gambles to accommodate non-linear utility
• Uses continuous linear functionals for representation
• Integrates various discounting forms in unified framework