Discounting Desirable Gambles

Gregory Wheeler
Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications, PMLR 147:331-341, 2021.

Abstract

The desirable gambles framework offers the most comprehensive foundations for the theory of lower previsions, which in turn affords the most general account of imprecise probabilities. Nevertheless, for all its generality, the theory of lower previsions rests on the notion of linear utility. This commitment to linearity is clearest in the coherence axioms for sets of desirable gambles. This paper considers two routes to relaxing this commitment. The first preserves the additive structure of the desirable gambles framework and the machinery for coherent inference but detaches the interpretation of desirability from the multiplicative scale invariance axiom. The second strays from the additive combination axiom to accommodate repeated gambles that return rewards by a non-stationary processes that is not necessarily additive. Unlike the first approach, which is a conservative amendment to the desirable gambles framework, the second is a radical departure. Yet, common to both is a method for describing rewards called discounted utility.

Cite this Paper

BibTeX
@InProceedings{pmlr-v147-wheeler21a, title = {Discounting Desirable Gambles}, author = {Wheeler, Gregory}, booktitle = {Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications}, pages = {331--341}, year = {2021}, editor = {Cano, Andrés and De Bock, Jasper and Miranda, Enrique and Moral, Serafı́n}, volume = {147}, series = {Proceedings of Machine Learning Research}, month = {06--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v147/wheeler21a/wheeler21a.pdf}, url = {https://proceedings.mlr.press/v147/wheeler21a.html}, abstract = {The desirable gambles framework offers the most comprehensive foundations for the theory of lower previsions, which in turn affords the most general account of imprecise probabilities. Nevertheless, for all its generality, the theory of lower previsions rests on the notion of linear utility. This commitment to linearity is clearest in the coherence axioms for sets of desirable gambles. This paper considers two routes to relaxing this commitment. The first preserves the additive structure of the desirable gambles framework and the machinery for coherent inference but detaches the interpretation of desirability from the multiplicative scale invariance axiom. The second strays from the additive combination axiom to accommodate repeated gambles that return rewards by a non-stationary processes that is not necessarily additive. Unlike the first approach, which is a conservative amendment to the desirable gambles framework, the second is a radical departure. Yet, common to both is a method for describing rewards called discounted utility.} }
Endnote
%0 Conference Paper %T Discounting Desirable Gambles %A Gregory Wheeler %B Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications %C Proceedings of Machine Learning Research %D 2021 %E Andrés Cano %E Jasper De Bock %E Enrique Miranda %E Serafı́n Moral %F pmlr-v147-wheeler21a %I PMLR %P 331--341 %U https://proceedings.mlr.press/v147/wheeler21a.html %V 147 %X The desirable gambles framework offers the most comprehensive foundations for the theory of lower previsions, which in turn affords the most general account of imprecise probabilities. Nevertheless, for all its generality, the theory of lower previsions rests on the notion of linear utility. This commitment to linearity is clearest in the coherence axioms for sets of desirable gambles. This paper considers two routes to relaxing this commitment. The first preserves the additive structure of the desirable gambles framework and the machinery for coherent inference but detaches the interpretation of desirability from the multiplicative scale invariance axiom. The second strays from the additive combination axiom to accommodate repeated gambles that return rewards by a non-stationary processes that is not necessarily additive. Unlike the first approach, which is a conservative amendment to the desirable gambles framework, the second is a radical departure. Yet, common to both is a method for describing rewards called discounted utility.
APA
Wheeler, G.. (2021). Discounting Desirable Gambles. Proceedings of the Twelveth International Symposium on Imprecise Probability: Theories and Applications, in Proceedings of Machine Learning Research 147:331-341 Available from https://proceedings.mlr.press/v147/wheeler21a.html.

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