Function-coherent gambles with non-additive sequential dynamics

Gregory Wheeler
Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications, PMLR 290:296-306, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v290-wheeler25b, title = {Function-coherent gambles with non-additive sequential dynamics}, author = {Wheeler, Gregory}, booktitle = {Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications}, pages = {296--306}, year = {2025}, editor = {Destercke, Sébastien and Erreygers, Alexander and Nendel, Max and Riedel, Frank and Troffaes, Matthias C. M.}, volume = {290}, series = {Proceedings of Machine Learning Research}, month = {15--18 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v290/main/assets/wheeler25b/wheeler25b.pdf}, url = {https://proceedings.mlr.press/v290/wheeler25b.html}, 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.} }
Endnote
%0 Conference Paper %T Function-coherent gambles with non-additive sequential dynamics %A Gregory Wheeler %B Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications %C Proceedings of Machine Learning Research %D 2025 %E Sébastien Destercke %E Alexander Erreygers %E Max Nendel %E Frank Riedel %E Matthias C. M. Troffaes %F pmlr-v290-wheeler25b %I PMLR %P 296--306 %U https://proceedings.mlr.press/v290/wheeler25b.html %V 290 %X 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.
APA
Wheeler, G.. (2025). Function-coherent gambles with non-additive sequential dynamics. Proceedings of the Fourteenth International Symposium on Imprecise Probabilities: Theories and Applications, in Proceedings of Machine Learning Research 290:296-306 Available from https://proceedings.mlr.press/v290/wheeler25b.html.

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