Abstract
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.
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Notes
- 1.
How to also deal with ‘NOT’ in this and related languages, was studied in quite some detail by one of us in an earlier collaboration (Quaeghebeur et al. 2015).
- 2.
We will use the language of events, rather than propositions, to express the things we are uncertain about, but the difference is immaterial for what we have in mind.
- 3.
Our results can be developed using horse lotteries, but we opt here for a simplified version.
- 4.
In both his hand-outs and the above-mentioned paper (Seidenfeld et al. 2010), Seidenfeld argues, similarly, for what he calls the inadmissibility of the third option.
- 5.
- 6.
- 7.
It is also customary in much of the literature to furthermore remove from a choice set C(A) those options that are dominated by other options in A for the point-wise ordering ≥ of options. We will leave this implicit here, as an operation that can always be performed afterwards.
- 8.
Strictly speaking, Levi only introduced, and argued for, this criterion in a context where he required the set \(\mathcal {P}\) to be convex. We will still use the term ‘E-admissibility’ even when \(\mathcal {P}\) is not convex. As mentioned before, we also leave the removal of ≥-dominated options implicit, as something that can be done afterwards.
- 9.
Again, we leave the removal of dominated options for the point-wise ordering ≥ implicit, as something that can be done afterwards.
- 10.
There may arise, due to Walley’s (1991, 2000) perhaps unfortunate introduction of this terminology, some confusion in the reader’s mind about the use of ‘strict’. In most accounts of preference relations, the term ‘strict preference’ refers to ‘(weak) preference without indifference’, and it is also in this sense that we have used the term ‘strict preference’ in the Introduction. Walley uses the moniker ‘strict’ for a stronger requirement, which is essentially based on some lower (or linear) prevision being strictly positive. We maintain this rather unhappy use of terminology here merely for historical reasons, but insist on warning the reader about the possible confusion this may entail.
- 11.
In this sense, it would perhaps be preferable to call it an ‘unmixing property’, as the term ‘mixing’ is better suited for an approach that favours choice over rejection. Nevertheless, we have decided to stick to ‘mixing’, for reasons of consistency with the terminology introduced in Seidenfeld et al. (2010).
- 12.
Although it is related to what we call a variable in spirit, we will refrain from using the term ‘random variable’, as that is typically associated with precise and countable additive probability models, and typically comes with a measurability requirement.
- 13.
An engaged reader will be able to verify further on that, despite our insistence on a finitary approach, this doesn’t really matter from a mathematical point of view. In particular, none of our proofs—even the ones that establish sufficient conditions for S-irrelevance or S-independence for variables—will actually require the restriction that the gambles s E—or s G—should be simple.
- 14.
Since ‘factorisation’ of this kind for multiple variables leads to a version of the law of large numbers (De Cooman and Miranda 2008; De Cooman et al. 2011), it doesn’t seem too farfetched to envision extensions of S-irrelevance and S-independence from two to multiple variables that allow us to prove similar laws of large numbers.
- 15.
This notation uses the implicit convention that gambles with domain \(\mathcal {Y}\) are considered as special instances of gambles with domain \(\mathcal {X}\times \mathcal {Y}\).
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Acknowledgements
We would like to thank Teddy Seidenfeld for the many discussions, throughout the years, on so many issues related to imprecise probabilities and the foundations of decision-making. This paper, and our related earlier work on choice functions, would not have existed without his constructive and destructive criticism of our earlier work on binary choice.
We would also like to thank the editors of this Festschrift for giving us the opportunity to contribute to it, and two anonymous reviewers for their valuable and constructive feedback.
Jasper De Bock’s work was partially supported by his BOF Starting Grant “Rational decision making under uncertainty: a new paradigm based on choice functions”, number 01N04819.
As with most of our joint work, there is no telling, after a while, which of us two had what idea, or did what, exactly. An irrelevant coin flip may have determined the actual order we are listed in.
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De Bock, J., de Cooman, G. (2022). On a Notion of Independence Proposed by Teddy Seidenfeld. In: Augustin, T., Cozman, F.G., Wheeler, G. (eds) Reflections on the Foundations of Probability and Statistics. Theory and Decision Library A:, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-031-15436-2_11
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