6 research outputs found
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision making. We provide these choice functions with a clear interpretation in terms of desirability, use this interpretation to derive a set of basic coherence axioms, and show that this notion of coherence leads to a representation in terms of sets of strict preference orders. By imposing additional properties such as totality, the mixing property and Archimedeanity, we obtain representation in terms of sets of strict total orders, lexicographic probability systems, coherent lower previsions or linear previsions
Evenly convex sets, and evenly quasiconvex functions, revisited
Since its appearance, even convexity has become a remarkable notion in convex analysis. In the fifties, W. Fenchel introduced the evenly convex sets as those sets solving linear systems containing strict inequalities. Later on, in the eighties, evenly quasiconvex functions were introduced as those whose sublevel sets are evenly convex. The significance of even convexity relies on the different areas where it enjoys applications, ranging from convex optimization to microeconomics. In this paper, we review some of the main properties of evenly convex sets and evenly quasiconvex functions, provide further characterizations of evenly convex sets, and present some new results for evenly quasiconvex functions.This research has been partially supported by MINECO of Spain and ERDF of EU, Grants PGC2018-097960-B-C22 and ECO2016-77200-P
Coherent and Archimedean choice in general Banach spaces
I introduce and study a new notion of Archimedeanity for binary and
non-binary choice between options that live in an abstract Banach space,
through a very general class of choice models, called sets of desirable option
sets. In order to be able to bring an important diversity of contexts into the
fold, amongst which choice between horse lottery options, I pay special
attention to the case where these linear spaces don't include all `constant'
options.I consider the frameworks of conservative inference associated with
Archimedean (and coherent) choice models, and also pay quite a lot of attention
to representation of general (non-binary) choice models in terms of the
simpler, binary ones.The representation theorems proved here provide an
axiomatic characterisation for, amongst many other choice methods, Levi's
E-admissibility and Walley-Sen maximality.Comment: 34 pages, 7 figure
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that appear in imprecise-probabilistic
decision making. We provide these choice functions with a clear interpretation
in terms of desirability, use this interpretation to derive a set of basic
coherence axioms, and show that this notion of coherence leads to a
representation in terms of sets of strict preference orders. By imposing
additional properties such as totality, the mixing property and Archimedeanity,
we obtain representation in terms of sets of strict total orders, lexicographic
probability systems, coherent lower previsions or linear previsions.Comment: arXiv admin note: text overlap with arXiv:1806.0104
Exposing some points of interest about non-exposed points of desirability
We study the representation of sets of desirable gambles by sets of probability mass functions. Sets of desirable gambles are a very general uncertainty model, that may be non-Archimedean, and therefore not representable by a set of probability mass functions. Recently, Cozman (2018) has shown that imposing the additional requirement of even convexity on sets of desirable gambles guarantees that they are representable by a set of probability mass functions. Already more that 20 years earlier, Seidenfeld et al. (1995) gave an axiomatisation of binary preferencesâon horse lotteries, rather than on gamblesâthat also leads to a unique representation in terms of sets of probability mass functions. To reach this goal, they use two devices, which we will call âSSKâArchimedeanityâ and âSSKâextensionâ. In this paper, we will make the arguments of Seidenfeld et al. (1995) explicit in the language of gambles, and show how their ideas imply even convexity and allow for conservative reasoning with evenly convex sets of desirable gambles, by deriving an equivalence between the SSKâArchimedean natural extension, the SSKâextension, and the evenly convex natural extension