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Nonlinear desirability theory
We show how Allais paradox finds a solution in the new theory, and discuss the role of sets of probabilities in the theory.
Lexicographic choice functions
We investigate a generalisation of the coherent choice functions considered by Seidenfeld et al. (2010), by sticking to the convexity axiom but imposing no Archimedeanity condition.
Desirability and the birth of incomplete preferences
On this basis, we obtain new results and insights: in particular, we show that the theory of incomplete preferences can be developed assuming only the existence of a worst act---no best act is needed---, and that a weakened Archimedean axiom suffices too; this axiom allows us also to address some controversy about the regularity assumption (that probabilities should be positive---they need not), which enables us also to deal with uncountable possibility spaces; we show that it is always possible to extend in a minimal way a preference relation to one with a worst act, and yet the resulting relation is never Archimedean, except in a trivial case; we show that the traditional notion of state independence coincides with the notion called strong independence in imprecise probability---this leads us to give much a weaker definition of state independence than the traditional one; we rework and uniform the notions of complete preferences, beliefs, values; we argue that Archimedeanity does not capture all the problems that can be modelled with sets of expected utilities and we provide a new notion that does precisely that.
Irrelevant and independent natural extension for sets of desirable gambles
The independent natural extension of maximal coherent sets of desirable gambles allows us to define the strong product of sets of desirable gambles.
Conservative Inference Rule for Uncertain Reasoning under Incompleteness
In this paper we formulate the problem of inference under incomplete information in very general terms.
