SIPTA Seminar by Gregory Wheeler: Function-coherent gambles

ABSTRACT: The desirable gambles framework provides a foundational approach to imprecise probability but relies on linear utility assumptions. I present recent work extending this framework to accommodate non-linear utility while preserving coherence. Function-coherent gambles maintain rationality axioms while allowing arbitrary strictly increasing utility functions. A representation theorem shows that acceptable gambles are characterized via continuous linear functionals in the transformed utility space. I apply this framework to two domains: First, various forms of discounting in intertemporal choice (hyperbolic, quasi-hyperbolic, scale-dependent, state-dependent) are integrated within the function-coherent framework. Second, I address sequential decisions with multiplicative dynamics by introducing a non-linear combination operator that preserves coherence while capturing compound growth. This naturally handles the ergodicity problem and unifies time-average growth optimization with normative decision theory. Unlike approaches that take preferences as primitive, this framework maintains coherence axioms that distinguish between rational preference changes (e.g., horizon-dependent discounting under multiplicative dynamics) and genuinely irrational reversals, providing normative foundations for dynamic choice within the desirability paradigm. This talk is part of a series of seminars on imprecise probabilities that are organized by SIPTA, the "Society for Imprecise Probabilities: Theories and Applications". We also organize conferences and schools, provide documentation and maintain a mailing list and blog. More information is available at http://sipta.org. Info on the SIPTA seminars in particular is available at http://sipta.org/events/sipta-seminars Contents 00:00 - Start 03:45 - Ergodicity breaking 10:33 - Varieties of discounting 19:35 - Function coherence 32:41 - Representation and risk measures 48:03 - Conclusion