Function-Coherent Gambles with Non-Additive Sequential Dynamics
This work addresses a foundational issue in decision theory for researchers and practitioners dealing with non-stationary reward dynamics, though it is incremental as it builds on existing function-coherent gambles.
The paper tackles the problem of evaluating repeated gambles over time in imprecise probability theory, where standard additive combination fails for non-linear utility and multiplicative rewards, by introducing a nonlinear combination operator that preserves geometric growth rates and addresses ergodicity, bridging expectation values with time averages.
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.