@conference{Benavoli2017c,
title = {A polarity theory for sets of desirable gambles},
author = {Benavoli, Alessio and Facchini, Alessandro and Vicente-Perez, Jose and Zaffalon, Marco},
url = {https://arxiv.org/abs/1705.09574},
year = {2017},
date = {2017-05-07},
booktitle = {Proc. ISIPTA'17 Int. Symposium on Imprecise Probability: Theories and Applications, },
pages = {1-12},
publisher = {PJMLR},
abstract = {Coherent sets of almost desirable gambles and credal sets are known to be equivalent models.
That is, there exists a bijection between the two collections of sets preserving the usual operations,
e.g. conditioning. Such a correspondence is based on the polarity theory for closed convex cones.
Learning from this simple observation, in this paper we introduce a new (lexicographic) polarity
theory for general convex cones and then we apply it in order to establish an analogous correspondence
between coherent sets of desirable gambles and convex sets of lexicographic probabilities.},
keywords = {desirability, lexicographic},
pubstate = {published},
tppubtype = {conference}
}

Coherent sets of almost desirable gambles and credal sets are known to be equivalent models.
That is, there exists a bijection between the two collections of sets preserving the usual operations,
e.g. conditioning. Such a correspondence is based on the polarity theory for closed convex cones.
Learning from this simple observation, in this paper we introduce a new (lexicographic) polarity
theory for general convex cones and then we apply it in order to establish an analogous correspondence
between coherent sets of desirable gambles and convex sets of lexicographic probabilities.

@conference{Benavoli2017,
title = {Accepting and Rejecting Gambles: A Logical Point of View},
author = { Benavoli, Alessio and Colacito, Almudean and Facchini, Alessandro and Zaffalon, Marco
},
year = {2017},
date = {2017-03-30},
booktitle = {Progic 2017: The 8th Workshop on Combining Probability and Logic, Munich 2017},
abstract = {A powerful theory of uncertainty is that of coherent sets of desirable gambles (or desirability).
It encompasses, in a uniform way, the Bayesian theory of probability as well as Bayesian robustness and many other theories of uncertainty. In recent years, several attempts have been carried out to explicitly formulate desirability as a logical system; for our purposes, the one by Gillett, Scherl and Shafer is particularly relevant. The goal of this paper is first to provide an appropriate semantics, with the aim to obtain a full completeness result for the logical system of Gillett, Scherl and Shafer. The second goal is to study, from a logical point of view, a generalisation of desirability that allows rejecting
gambles to be possible too. Thus we enrich the system by adding a rejection operator, and we formulate some additional deductive rules concerning this operation. The obtained systems are investigated, both from a syntactical and a semantical point of view.},
keywords = {desirability},
pubstate = {published},
tppubtype = {conference}
}

A powerful theory of uncertainty is that of coherent sets of desirable gambles (or desirability).
It encompasses, in a uniform way, the Bayesian theory of probability as well as Bayesian robustness and many other theories of uncertainty. In recent years, several attempts have been carried out to explicitly formulate desirability as a logical system; for our purposes, the one by Gillett, Scherl and Shafer is particularly relevant. The goal of this paper is first to provide an appropriate semantics, with the aim to obtain a full completeness result for the logical system of Gillett, Scherl and Shafer. The second goal is to study, from a logical point of view, a generalisation of desirability that allows rejecting
gambles to be possible too. Thus we enrich the system by adding a rejection operator, and we formulate some additional deductive rules concerning this operation. The obtained systems are investigated, both from a syntactical and a semantical point of view.