Function-Coherent Gambles with Non-Additive Sequential Dynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917946246299648 |
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| author | Wheeler, Gregory |
| author_facet | Wheeler, Gregory |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02889 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Function-Coherent Gambles with Non-Additive Sequential Dynamics Wheeler, Gregory Theoretical Economics Artificial Intelligence Computational Engineering, Finance, and Science Probability 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. |
| title | Function-Coherent Gambles with Non-Additive Sequential Dynamics |
| topic | Theoretical Economics Artificial Intelligence Computational Engineering, Finance, and Science Probability |
| url | https://arxiv.org/abs/2503.02889 |