On a notion of independence proposed by Teddy Seidenfeld

Teddy Seidenfeld has been arguing for quite a long time that binary preference models are not powerful enough to deal with a number of crucial aspects of imprecision and indeterminacy in uncertain inference and decision making. It is at his insistence that we initiated our study of so-called sets of desirable option sets, which we have argued elsewhere provides an elegant and powerful approach to dealing with general, binary as well as non-binary, decision-making under uncertainty. We use this approach here to explore an interesting notion of irrelevance (and independence), first suggested by Seidenfeld in an example intended as a criticism of a number of specific decision methodologies based on (convex) binary preferences. We show that the consequences of making such an irrelevance or independence assessment are very strong, and might be used to argue for the use of so-called mixing choice functions, and E-admissibility as the resulting decision scheme. 1 Context and introduction In much of our earlier work on the foundations of imprecise—or indeterminate [21]—probabilities [1, 33] we availed ourselves of binary preference orders between uncertain rewards to model a subject’s decisions under uncertainty; see [3, 7, 14, 16, 19, 23] for a few representative examples. In the field, the monikers ‘desirability’ and ‘sets of desirable gambles’ are typically used to describe uncertainty models involving such (strict) binary preference orders [2, 14, 19, 22, 34]. In earlier work, Seidenfeld et al. [24] also introduced the term ‘favourability’ for this. Jasper De Bock Foundations Lab for Imprecise Probabilities, Ghent University, Technologiepark-Zwijnaarde 125, 9052 Belgium e-mail: jasper.debock@ugent.be Gert de Cooman Foundations Lab for Imprecise Probabilities, Ghent University, Technologiepark-Zwijnaarde 125, 9052 Belgium e-mail: gert.decooman@ugent.be