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.

翻译：理想赌博框架为不精确概率理论提供了严格的基础，但其相干公理严重依赖于线性效用。在我们相关的研究中，我们引入了函数相干赌博以容纳非线性效用。然而，当赌博随时间重复进行时——尤其是在奖励以乘法方式复合的跨期选择中——标准的加性组合公理无法捕捉适当的长期评估。在本文中，我们通过放宽加性组合公理并引入一个非线性组合算子来扩展该框架，该算子能有效地在对数域中聚合重复赌博。此算子保留了时间平均（几何）增长率并解决了遍历性问题。我们证明了该算子的关键代数性质，讨论了其对相干性、风险评估和表示的影响，并提供了一系列说明性示例。我们的方法弥合了期望值与时间平均值之间的差距，并将规范理论与经验观察到的非平稳奖励动态统一起来。