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Function-Coherent Gambles with Non-Additive Sequential Dynamics

Gregory Wheeler

TL;DR

The paper addresses the ergodicity problem in sequential decision making with multiplicative dynamics by relaxing linear utility and introducing function-coherence. It defines a log-domain nonlinear operator $f \oplus g = (1+f)(1+g) - 1$ with $L(f)=\log(1+f)$, proving a representation theorem that preserves coherence in the transformed space and aligns sequential aggregation with geometric growth. By generalizing to $f \oplus_u g = u^{-1}(u(f)+u(g))$, it provides conditions for well-behaved operators, a taxonomy of utility classes (power, exponential, logarithmic), and induced risk measures that capture multiplicative risk and time-average growth. The framework unifies ergodicity considerations, risk assessment, and non-stationary reward dynamics, with practical implications for portfolio management and long-horizon decisions where volatility drag and asymmetric gains/losses matter. Overall, it offers a principled, mathematically rigorous bridge between expectation values and time averages in dynamic, non-linear contexts.

Abstract

The desirable gambles framework provides a rigorous foundation for imprecise probability theory but relies heavily on linear utility via its coherence axioms. In our related work, we introduced function-coherent gambles to accommodate non-linear utility. However, when repeated gambles are played over time -- especially in intertemporal choice where rewards compound multiplicatively -- the standard additive combination axiom fails to capture the appropriate long-run evaluation. In this paper we extend the framework by relaxing the additive combination axiom and introducing a nonlinear combination operator that effectively aggregates repeated gambles in the log-domain. This operator preserves the time-average (geometric) growth rate and addresses the ergodicity problem. We prove the key algebraic properties of the operator, discuss its impact on coherence, risk assessment, and representation, and provide a series of illustrative examples. Our approach bridges the gap between expectation values and time averages and unifies normative theory with empirically observed non-stationary reward dynamics.

Function-Coherent Gambles with Non-Additive Sequential Dynamics

TL;DR

The paper addresses the ergodicity problem in sequential decision making with multiplicative dynamics by relaxing linear utility and introducing function-coherence. It defines a log-domain nonlinear operator with , proving a representation theorem that preserves coherence in the transformed space and aligns sequential aggregation with geometric growth. By generalizing to , it provides conditions for well-behaved operators, a taxonomy of utility classes (power, exponential, logarithmic), and induced risk measures that capture multiplicative risk and time-average growth. The framework unifies ergodicity considerations, risk assessment, and non-stationary reward dynamics, with practical implications for portfolio management and long-horizon decisions where volatility drag and asymmetric gains/losses matter. Overall, it offers a principled, mathematically rigorous bridge between expectation values and time averages in dynamic, non-linear contexts.

Abstract

The desirable gambles framework provides a rigorous foundation for imprecise probability theory but relies heavily on linear utility via its coherence axioms. In our related work, we introduced function-coherent gambles to accommodate non-linear utility. However, when repeated gambles are played over time -- especially in intertemporal choice where rewards compound multiplicatively -- the standard additive combination axiom fails to capture the appropriate long-run evaluation. In this paper we extend the framework by relaxing the additive combination axiom and introducing a nonlinear combination operator that effectively aggregates repeated gambles in the log-domain. This operator preserves the time-average (geometric) growth rate and addresses the ergodicity problem. We prove the key algebraic properties of the operator, discuss its impact on coherence, risk assessment, and representation, and provide a series of illustrative examples. Our approach bridges the gap between expectation values and time averages and unifies normative theory with empirically observed non-stationary reward dynamics.

Paper Structure

This paper contains 23 sections, 6 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Motivation
  3. The Sequential Decision Problem
  4. Challenge of Sequential Decisions
  5. Function-Coherent Gambles
  6. Basic Framework
  7. Representation Theorem
  8. Discussion and Domain Restriction
  9. Non-Additive Sequential Dynamics
  10. The Nonlinear Combination Operator
  11. Preservation of Function-Coherence
  12. Risk Assessment Under Sequential Dynamics
  13. Illustrative Examples
  14. Basic Nonlinear Combination
  15. Mixed Outcomes and Growth Rates
  16. ...and 8 more sections

Key Result

Theorem 3.1

Under the conditions above, there exists a nonzero continuous linear functional $\ell: V\to\mathbb{R}$, unique up to multiplication by a positive scalar, such that for every gamble $f\in X$, Equivalently, defining the evaluation functional we have

Figures (1)

  • Figure 1: Comparison of ensemble averaging versus time averaging of Option A in Example \ref{['example:seq-gamble']} over a fixed period of 30 iterations. The dashed line represents the expectation value $\mathbb{E}_p(f)$ under additive accumulation, while the solid line depicts the time-average growth rate, $\mathbb{E}_t(f)$, under multiplicative growth. The divergence between these measures highlights the ergodicity problem in multiplicative processes.

Theorems & Definitions (13)

  • Example 2.1
  • Definition 2.1: Discounted Utility
  • Theorem 3.1: Representation Theorem
  • Theorem 3.2: Upward Closure Under Domain Restriction
  • Theorem 4.1: Log-Domain Additivity
  • proof
  • Theorem 4.2: Function-Coherence Preservation
  • proof
  • Theorem 6.1: Properties of Generalized Combination
  • proof
  • ...and 3 more