Function-coherent gambles

TL;DR

The paper addresses the limitation of linear utility in the desirable gambles framework by introducing function-coherent gambles, which relax linearity while preserving normative coherence through axioms and a representation via a continuous linear functional $\ell$ applied to a non-linear utility $u$. The core contribution is a representation theorem linking acceptable gambles to a risk functional $\rho(f)=\ell(u(f))$, together with a unified treatment of intertemporal discounting forms—hyperbolic, quasi-hyperbolic, generalized hyperbolic, scale-dependent, state-dependent, and hybrid—within the function-coherent framework. This approach explains time-preference phenomena like present bias and magnitude effects as rational under non-linear utility, while maintaining coherence and separating preferences from beliefs. The framework thus provides a rigorous, versatile foundation for modeling sophisticated discounting under uncertainty, with implications for normative theory, empirical analysis, and future computational methods.

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.

Figures (3)

• Figure 1: Quasi-hyperbolic ($\beta$-$\delta$) discounting with discrete time periods. The left panel shows the effect of varying the present bias parameter $\beta$ (0.6-0.9) while holding the long-run discount factor $\delta$ constant at 0.95. The right panel demonstrates the impact of varying $\delta$ (0.90-0.98) while maintaining $\beta=0.7$. The discontinuity at $t=0$ captures immediate gratification, while subsequent periods follow exponential discounting.
• Figure 2: Visualization of standard and generalized hyperbolic discounting functions. The top panel shows standard hyperbolic discounting with varying k values (left) and generalized hyperbolic discounting with varying p values (right). The middle row displays a direct comparison of different parameter combinations. The bottom panel demonstrates parameter interaction effects by varying k values at two different p values (left) and varying p values at two different k values (right). Higher k values produce steeper initial declines, while higher p values affect the tail behavior of the discount function.
• Figure 3: Three types of generalized discounting. Top panel shows scale-dependent discounting, where larger rewards are discounted less steeply (left) and their present values on a log scale (right). Center panel illustrates state-dependent discounting across different economic conditions, with steeper discounting during recessions. Bottom panel demonstrates hybrid discounting as a mixture of exponential and hyperbolic functions, showing how different weights create intermediate discounting patterns between the two pure forms.

Theorems & Definitions (15)

• Definition 2.1: Discounted Utility
• Proposition 3.1: Non-Triviality
• Proposition 3.2: Upward Closure
• Proposition 3.3: Transform Convexity
• Proposition 3.4: Transform Invariance
• Theorem 4.1: Representation
• Theorem 4.2: Closure Under Limits
• Example 5.1
• Example 6.1: Investment Choice with Time-Driven Preference Reversal
• Example 6.2: Commitment Savings