Abstract
The basic axioms or formal conditions of decision theory, especially the ordering condition put on preferences and the axioms underlying the expected utility (EU) formula, are subject to a number of counter-examples, some of which can be endowed with normative value and thus fall within the ambit of a philosophical reflection on practical rationality. Against such counter-examples, a defensive strategy has been developed which consists in redescribing the outcomes of the available options in such a way that the threatened axioms or conditions continue to hold. We examine how this strategy performs in three major cases: Sen's counterexamples to the binariness property of preferences, the Allais paradox of EU theory under risk, and the Ellsberg paradox of EU theory under uncertainty. We find that the strategy typically proves to be lacking in several major respects, suffering from logical triviality, incompleteness, and theoretical insularity (i.e., being cut off from the methods and results of decision theory). To give the strategy more structure, philosophers have developed “principles of individuation”; but we observe that these do not address the aforementioned defects. Instead, we propose the method of checking whether the strategy can overcome its typical defects once it is given a proper theoretical expansion (i.e., it is duly developed using the available tools of decision theory). We find that the strategy passes the test imperfectly in Sen's case and not at all in Allais's. In Ellsberg's case, however, it comes close to meeting our requirement. But even the analysis of this more promising application suggests that the strategy ought to address the decision problem as a whole, rather than just the outcomes, and that it should extend its revision process to the very statements it is meant to protect. Thus, by and large, the same cautionary tale against redescription practices runs through the analysis of all three cases. A more general lesson, simply put, is that there is no easy way out from the paradoxes of decision theory.
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Notes
Savage (1954–1972) uses the word “consequence” to refer to what results from an uncertain prospect in a given state; Luce and Raiffa (1957) use “outcome” to refer to what results from a lottery when this lottery is drawn. This is a reasonably well-established distinction; but we will use the word “outcome” throughout, both for simplicity and to follow the antecedent literature on redescription.
The redescription-of-outcomes strategy has sometimes attracted philosophers' attention for reasons other than the defence of axiomatic conditions against counter-examples. Thus, Weirich (1986 and 2019) argues that a proper theory of risk and risk-attitude should include the risk of a lottery in the description of its outcomes. Following a different line, Verbeek (2001) discusses the proper definition of outcomes in connection with the decision-theoretic principle of consequentialism, a principle defended by Hammond (1988) and rejected by Machina (1989). Consequentialism is an important topic both for decision theory and its philosophical appraisal; but we will not discuss it here.
Formally, let us denote the choice function by C(.) and two subsets of options by A, B. Property α then says that if B ⊂ A and x ∈ C(A), then x ∈ C(B).
As Sen (1982, ch. 1) explains, when the choice function is singleton-valued, property α is necessary and sufficient for the choice function to originate in an ordering; generally, though, it is only necessary. The definition of binariness adopted in this work by Sen, and taken up here, puts no restriction on the binary relation in the maximization of which the choice function originates. Elsewhere, Sen (1993, p. 500) considers a definition that requires the choice function to originate in the maximization of a particular relation (the “revealed preference” relation). A fortiori, violations of property α contradict this stronger sense of binariness.
A famous early variant of Sen's second example appears in Luce and Raiffa (1957, p. 288).
Suggested by Neumann (2007).
Mongin (2019) also covers some of the vast secondary literature.
To see that this resolution violates VNM independence more specifically, define the auxiliary lottery l giving outcome 500 with probability 10/11 and outcome 0 with probability 1/11. Under the ordering condition, by VNM independence, choosing p1 over q1 implies choosing 100 over l, while choosing q2 over p2 implies choosing l over 100–a contradiction.
This may not be Allais's own evaluation of the two arguments; but it certainly is that of Machina (1987) and other technical followers of Allais.
This pattern of response to the Allais and related paradoxes goes back as far as Raiffa (1968, pp. 85–86), who, however, did not mean to endorse it. It has occasionally reappeared in later decision theory, albeit allusively. It has been actively discussed by philosophers; see the references in fn 11.
Proponents of the strategy do not seem to have ever considered redescriptions based on coarsenings rather than refinements of the original outcomes.
In this statement and the ones that follow, “preference” has its ordinary language sense, i.e., it is synonymous with “strict preference” in decision theory.
To use philosophical terminology that is currently widespread, this makes decision theory a brand of “Humeanism”. The terminology adopted here of “thin” versus “thick” rationality comes from the methodology of the social sciences. It resembles the traditional philosophical distinction between “formal” and “substantial” rationality.
See also Buchak (2013, pp. 128–129). Bradley (2017, p. 23) dismisses PIJ, though for the different reason that the preferences involved in a principle of individuation can be normatively determined without being subject to a rationality constraint. In still another critical move, Bermudez (2009, p. 103–108) stresses that it is difficult to provide an independent grounding for the rationality concept involved in PIJ. Bermudez understands Broome as being after a thick rationality concept.
Buchak (2013, pp. 131–132) also discusses this problem of finding empirical evidence, though she chooses to do that in connection with PIP rather than PIA.
That PIJ can play this auxiliary role seems to be the grain of truth in Broome's own position. One may interpret him as reacting to the difficulty of finding empirical evidence that would make PIA operative. He does, however, overreact by promoting PIJ as if it were a self-sufficient principle.
This requirement should not be interpreted more strongly than intended. Each paradox may be decomposed in antecedent and consequent preferences. What we envisage here is that given the antecedent preferences, the relevant principle of rationality imply the consequent preferences. That is naturally not to say that rationality should mandate, rather than merely allow for, the antecedent preferences themselves.
Bhattacharyya, Pattanaik, and Xu (2011) survey the main contributions.
Baigent and Gaertner (1996) formalize the politeness explanation of the first example differently from Bossert and Suzumura. They interpret the agent's choices as being driven by the rule “never choose the uniquely largest”, and axiomatize compliance with this rule in terms of properties of choice functions. The relevant properties are interestingly close to the standard consistency conditions. Heuristically, this work accords with the redescription that takes the two choice sets to be {next-to-largest piece, largest piece} and {smallest piece, next-to-largest piece, largest}. This treatment is unfortunately only of limited interest to our enquiry because it is restricted to a special class of choice problems and does not illuminate the effect of following a norm generally.
Typically, these criteria replace minimax by expected value at the final stage of the deliberation, and include an evaluation of the outcome itself besides that of the regret attached to the outcome; see, e.g., Bell (1982). They do not help recover the Allais paradox because they satisfy the sure-thing principle. If Loomes and Sugden's (1982) regret theory does not satisfy this principle, and can indeed recover the Allais paradox, this is because they make the special assumption of independent variation, according to which the prospects being compared give rise to stochastically independent drawings.
This seems to be well enough to dispose of the redescription-of-outcomes strategy in the Allais case. But we could have followed another direction of theoretical expansion by investigating whether other counter-examples proposed by Allais and followers could similarly suggest an explanation in terms of negative feelings. The short answer is that this is not always the case; see the established list of counter-examples in Larsson and MacCrimmon (1979), Machina (1987), and (with a philosophy of science perspective) Mongin (2009).
Probabilistic sophistication has become a focus of attention in decision theory after Machina and Schmeidler (1992) axiomatized in a more general form than the special form it takes in Savage's SEU representation theorem.
We state this connection informally, but it is actually implied, as an intermediary result, by Savage's postulates.
See Machina and Siniscalchi (2014) for a review of the ambiguity literature.
In more mathematical terms, a Cartesian product representation.
In the theories of second-order expected utility, the STP holds non-trivially at each level of uncertainty. This is necessary for these theories to obtain a full-fledged SEU representation at each level. This is one of the senses in which these theories push the analysis of “orthogonal” representations further than the analysis made in the previous paragraph.
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Acknowledgements
For helpful comments and suggestions, the authors are grateful to two reviewers, Associate Editor Blake Roeber, and the 2020 Spring Fellows at the Center for Philosophy of Science, University of Pittsburgh.
Funding
Both authors gratefully acknowledge funding from the ANR-DFG project ColAForm. The first author additionally acknowledges funding from the University of Pittsburgh and the Deutsche Forschungsgemeinschaft (DFG project 426170771).
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Baccelli, J., Mongin, P. Can redescriptions of outcomes salvage the axioms of decision theory?. Philos Stud 179, 1621–1648 (2022). https://doi.org/10.1007/s11098-021-01723-z
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DOI: https://doi.org/10.1007/s11098-021-01723-z