Function-Coherent Gambles with Non-Additive Sequential Dynamics

• URL: http://arxiv.org/abs/2503.02889v1
• Date: Sat, 22 Feb 2025 14:58:20 GMT
• Title: Function-Coherent Gambles with Non-Additive Sequential Dynamics
• Authors: Gregory Wheeler,
• Abstract summary: We relax the additive combination axiom and introduce a nonlinear combination operator that effectively aggregates repeated gambles in the log-domain.<n>Our approach bridges the gap between expectation values and time averages and unifies normative theory with empirically observed non-stationary reward dynamics.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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