Abstract
In Richard Bradley’s book, Decision Theory with a Human Face (2017), we have selected two themes for discussion. The first is the Bolker-Jeffrey (BJ) theory of decision, which the book uses throughout as a tool to reorganize the whole field of decision theory, and in particular to evaluate the extent to which expected utility (EU) theories may be normatively too demanding. The second theme is the redefinition strategy that can be used to defend EU theories against the Allais and Ellsberg paradoxes, a strategy that the book by and large endorses, and even develops in an original way concerning the Ellsberg paradox. We argue that the BJ theory is too specific to fulfil Bradley’s foundational project and that the redefinition strategy fails in both the Allais and Ellsberg cases. Although we share Bradley’s conclusion that EU theories do not state universal rationality requirements, we reach it not by a comparison with BJ theory, but by a comparison with the non-EU theories that the paradoxes have heuristically suggested.
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Notes
The Boolean operations can of course be replaced by the standard set-theoretic operations on some set. Jeffrey’s followers use this handier notation.
As Bolker (1967, p. 335) himself writes, “we blur the often useful distinction among acts, consequences and events”. This critical point is a major reason why mathematical decision theorists are generally unattracted to the BJ theory (see, e.g., Fishburn 1981, p. 194). Bradley (2017, p. 159) notes this disinclination, in contrast with the high consideration of the theory among philosophers.
Note that this claim literally concerns the preference axioms, not the domain assumptions, to be discussed in the next paragraph.
Up to an isomorphism, it is identical to the set of measurable subsets of the unit interval, when two subsets that differ by a set of measure zero are identified with each other; see Halmos (1974, p. 173). Singletons, which are measurable subsets of the unit interval, disappear from consideration as they are identified with the empty set.
There are other variants with the same purpose, each being based on a different set of assumptions; a recent example has appeared in Mongin and Pivato (2015).
With relevant technical differences, this move is already performed in Bradley (2007). Joyce (1999) also considers the possibility of giving internal structure to Jeffrey’s propositions, but his final move consists in partitioning the algebra of these propositions in different ways, with each partition corresponding to a particular interpretation of a decision-theoretic concept (i.e., a state, a consequence or an act). Bradley also uses partitions of the set of propositions with this semantic purpose; see the example in the paragraph following the next.
There are more details on these theorems in Bradley and Stefansson’s (2017) article. This article takes an even stronger stand against the view that a rational agent should behave as EU theories prescribe.
In particular, this definition should handle the role of singletons appropriately; see fn. 4.
Notice incidentally that this propositional rendering of an act only holds for simple acts, i.e., those which have a finite number of distinct values. Savage’s formal concept of an act is not so restricted.
This invariance problem has a clear conceptual underpinning. When the standard uniqueness property of SEU representations holds, the agent’s beliefs and desires are well identified and moreover separated from each other. But Bolker's uniqueness conditions are too weak to fulfil this purpose, as, e.g., Joyce (1999, p. 136) notes.
Jeffrey’s (1965–1983, p. 142) suggestion to match the unboundedness of desirability with a preference condition is obscure and usually omitted from the ensuing literature.
Bradley (2017, pp. 84–85) states Joyce’s version of Bolker’s representation theorem and adds that he will draw on it. Whether he effectively does in the sequel would need to be clarified.
The paradox has given rise to a large literature, which is covered in part by Mongin (2019).
“Outcome” is a more common term than “consequence” in the VNM context, but we use the latter for uniformity.
While the explanation suggested by Allais may thus be used as a motivation, Allais himself would not have condoned the move described next. He advocated instead the development of a proper alternative to VNM theory. That being clarified, the first elaborate occurrence of the move may be in Raiffa (1968, pp. 85–86), who actually contemplates it without endorsing it. Like Savage (1954–1972, pp. 101–103), Raiffa interprets the paradoxical pair of choices as an irrationality that needs to be corrected.
See, among others, the theories developed for regret avoidance in Bell (1982) and Loomes and Sugden (1982), and for disappointment avoidance in Bell (1985) and Loomes and Sugden (1986). All these theories significantly depart from the VNM one. On regret theory, see also the retrospective by Bleichrodt and Wakker (2015).
This representation is popular among philosophers of decision theory when they discuss the Allais paradox; see Buchak (2013, ch. 4) for an example and a list of previous references. It is actually inherited from Savage (1954–1972, pp. 101–103), who used it in his rebuttal of the Allais paradox. However, Savage’s rebuttal did not amount to using the redefinition strategy, which is these writers’ focus of attention.
Heuristically, there is more to VNM independence than to the STP, and this is confirmed by a step in Savage’s proof of his representation theorem, in which he derives the VNM lottery framework and axiom system with a view of using the VNM representation theorem as a lemma (see Savage 1954–1972, pp. 73–76, and Fishburn 1970, pp. 203–206). This derivation requires Savage’s full set of postulates P1–P6.
The principle is more commonly said of “reduction of compound lotteries”, a slightly misleading phrase because it operates in both directions, from the compound form to the reduced one, and vice versa. It is actually contained in the mathematical representation of a lottery as a probability measure on the set of consequences X, because this representation automatically identifies a convex combination of probability measures on X (hence a compound lottery) with a probability measure on X (hence a reduced form lottery).
The difference between a SEU formula and the more recent formulas of probabilistic sophistication hinges on the fact that the latter may be non-linear in the probabilities; e.g., they may involve distorting the latter. For a review of the Ellsberg paradox and the surrounding literature, see Machina and Siniscalchi (2014).
This corresponds to Table 5 in Bradley's (2016) article, which gives more details on his restatement of the Ellsberg paradox. The analysis reported next would also follow, had Bradley replaced each monetary consequence x by (x; β), with β the chance, given the redefined state, of receiving x (β generally differing from α, the associated chance of winning the bet).
Buchak (2013, pp. 121–122) distinguishes between “local” and “global” redefinitions of consequences. Bradley’s suggestion is of the latter type, while ours is of the former. Buchak is generally critical of the redescription literature, but perhaps more so when it takes the "global" form.
See Klibanoff et al. (2005), Nau (2006) and Ergin and Gul (2009). The general principle is to have first-order uncertainty represented by a EU functional whose values serve as arguments for a non-linear transformation of the EU functional that represents second-order uncertainty. There are of course many alternative explanations for the Ellsberg paradox in the ambiguity literature, and we do not touch on them here.
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Acknowledgements
The authors thank Orri Stefánsson for inviting this paper and giving them the opportunity to discuss Richard Bradley’s thought-provoking work. They also thank two anonymous reviewers for their comments on a first draft.
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Appendix
Appendix
The set of objects in Bolker’s (1966) representation theorem as well as in the BJ theory generally speaking (see Jeffrey 1965–1983) is a Boolean algebra Ω, whose elements will be termed propositions, using the analogy with propositional logic that is central to Jeffrey’s interpretation. (Bradley says “prospects”.) The usual ingredients of a Boolean algebra will be designated here in standard logical notation: ˅ and ˄ for the sup and inf operations, ⊨ for the induced relation, ⊤ and ⊥ for the maximum and minimum elements. By a preference relation, we mean a weak ordering ≿ on Ω, with ≻ and ∼ denoting its asymmetric (or strict preference) and symmetric (or indifference) parts respectively.
Bolker’s representation theorem. Let Ω be a complete atomless Boolean algebra, less the proposition ⊥, and let ≿ be a preference relation on Ω that is continuous and satisfies the following two conditions:
-
(i)
Averaging condition. If α and β are disjoint propositions of Ω, then
$$\alpha > \beta \implies >\alpha \vee \beta >\beta, \text{and}$$$$\alpha\sim\beta \implies\alpha\sim\alpha\vee\beta\sim\beta;$$ -
(ii)
Impartiality. If α, β, γ are pairwise disjoint propositions of Ω, α ∼ β, not α ∼ γ, and α ˅ γ ∼ β ˅ γ, then for all γ’ in Ω that is disjoint from α and β,
$$\alpha \vee \upgamma{\text{'}} \sim \beta \vee \upgamma{\text{'}}.$$
Then, there exists a probability measure P and function V (a “desirability function”) such that for all propositions α, β of Ω,
and if α and β are disjoint,
Moreover, given the normalization V(⊤) = 0, the probability measure P’ and the function V’ can replace P and V in the above equations if and only if there exist a > 0 and c such that for all propositions α’ of Ω, c V(α’) + 1 > 0, with the following properties: for all propositions α of Ω,
For finite Boolean algebras, Domotor (1978), and recently, Gravel, Marchant and Sen (2018) derive the existence part of the theorem from rather similar axioms, plus richness or continuity conditions put on the preference relation. These variants have no uniqueness result, which is unsurprising given the techniques of proof appropriate for finite sets. Ahn’s (2008) version departs from Bolker’s and the above writers’ reliance on Boolean algebras, as he defines his preference objects as sets of lotteries and, crucially, he exploits the topological structure of these sets. The axiom set includes a form of Averaging (there called Disjoint Set Betweenness) and a form of Impartiality (called Balancedness). The representation theorem has existence and uniqueness conclusions that are closely related to those of Bolker, whose mathematical contribution is put to work in the proof, but the utility representation is more precise as it takes the form of a conditional SEU formula.
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Mongin, P., Baccelli, J. Expected utility theory, Jeffrey’s decision theory, and the paradoxes. Synthese 199, 695–713 (2021). https://doi.org/10.1007/s11229-020-02691-3
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DOI: https://doi.org/10.1007/s11229-020-02691-3