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Odd Choices: On the Rationality of Some Alleged Anomalies of Decision and Inference

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Abstract

This paper presents a number of apparent anomalies in rational choice scenarios, and their translation into the logic of everyday reasoning. Three classes of examples that have been discussed in the context of probabilistic choice since the 1960s (by Debreu, Tversky and others) are analyzed in a non-probabilistic setting. It is shown how they can at the same time be regarded as logical problems that concern the drawing of defeasible inferences from a given information base. I argue that initial appearances notwithstanding, these cases should not be classed as instances of irrationality in choice or reasoning. One way of explaining away their apparent oddity is to view certain aspects of these examples as making particular options salient. The decision problems in point can then be solved by ‘picking’ these options, although they could not have been ‘chosen’ in a principled way, due to ties or incomparabilities with alternative options.

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Notes

  1. I will take the asymmetric relation < as primitive in this paper (i.e., < is not taken to be derived from a primitive reflexive relation ≤).

  2. I will neglect in the present paper the fact that in doxastic applications of rational choice theory, the "character" of a person encoded by a choice function or by rationalizing preferences is not so persistent and stable after all. It will typically change in response to the experiences the person makes, or more precisely, to the pieces of information ϕ 1, ϕ 2, ϕ 3, … that happen to come in. On the plurality of methods to change qualitative doxastic preferences, see Rott (2009).

  3. The likening to chess computers is kind of flattering, as many machines have become stronger than any human chess player now. Kant (1996, p. 218) famously talked about the “freedom of a turnspit [Freiheit eines Bratenwenders], which, when once it is wound up, also accomplishes its movements of itself”.

  4. Locke (1975, section II.xxi.48, p. 264).

  5. Cf. Sen (1971) and Moulin (1985).

  6. Here it is of course still assumed that all two-element sets {x, y} are in the domain of σ.

  7. A relation < is modular if and only if x < y implies that for every z, either x < z or z < y.

  8. (IV) is stronger than (III) if the choice set σ(S') is always required to be non-empty.

  9. Rott (2004) gives a numerical reconstruction that makes the example plausible for a comparatively wide variety of weights that may be given to logic (in relation to metaphysics). Critical discussions of this example have been provided by Hill (2008), Stalnaker (2009) and Arló-Costa and Pedersen (2010).

  10. There are other important differences. The options involved are identical, since we are talking about the same persons across the different scenarios. So there must be some variation concerning the relevant choice functions or preferences.

  11. See Lindström (1991), Rott (1993, 1994, 2001), Schlechta (1997), Freund (1998), Lehmann (2001), and more recently Bonanno (2009) and Arló-Costa and Pedersen (2010).

  12. Inf can also take finite sets of sentences, if they are conjoined by the conjunction \( \wedge \). An extension to infinite sets of sentences is non-trivial.

  13. As long as we stick to finite sets of premises and identify them with their conjunctions, this implies that a subset X of a premise set Y may have consequences that Y itself does not have. Concerning the use of variables in this paper, a, b, c etc. stand for particular sentences occurring in our examples, while ϕ and ψ are schematic for arbitrary sentences.

  14. On induction see Levi (2005) and Spohn (2005). On nonmonotonic logics see Ginsberg (ed., 1987) and Makinson (2005).

  15. Compare the first chapter of Rott (2001).

  16. Note that (a ∨ b ∨ c) ∧ (a ∨ b) is Cn-equivalent with a ∨ b, and that it should therefore allow the same inferences as a ∨ b.

  17. Here and in the following, this formulation is supposed to mean that the premise ϕ contains all the information available to the person. This point is crucial to the understanding of nonmonotonic reasoning.

  18. Unfortunately, this latter condition is rather hard to understand. It can be replaced by the condition that ϕ → ψ is not among the most implausible elements of {ϕ → ψ, ϕ → ¬ψ}. For the motivation of both conditions see Rott (2001), pp. 172–181. There are bridge principles showing that the concept of a “best possible world” has the same theoretical power as the concept of a “weakest belief”. The following bridge principles are suitable to bear this out: A best ϕ-world is one at which ϕ and all except the least plausible Cn-consequences of ϕ are true. Conversely, a sentence χ in a set of sentences M is least plausible in M if and only if among the best M-falsifying worlds (i.e., best possible worlds at which at least one element of M is false) there is a world at which χ is false (cf. Rott 2001, pp. 208–213). Here I am not presuming that a choice function needs to be rationalizable by a preference relation.

  19. The condition of Weak Disjunctive Rationality is unfortunately not very intuitive. It says that \( Inf(\phi \vee \psi ) \subseteq Cn(Inf(\phi ) \cup \{ \psi \} ) \cup Cn(Inf(\psi ) \cup \{ \phi \} ) \)

  20. We can leave it open here whether this research is meant to imply that every individual has a specifiable propensity to choose a certain option in a given single situation.

  21. Debreu (1960). This three-page review of Luce (1959) was seminal for the theme of the present paper. The bone of contention, Luce’s Choice Axiom, is worth reproducing once again, in a variant that is sometimes called the Constant Ratio Rule: If A and B are elements of a menu S and P S (A) denotes the probability that the element A is chosen from the menu S, then P S (A)/P S (B) = P{A,B}(A)/P{A,B}(B). Luce’s Axiom formulates a sort of “independence of irrelevant alternatives” (cf. Luce 2008). The qualitative analogue is this: If S ⊆ S′ and σ(S′) ∩ S ≠ Ø, then σ(S) = σ(S′) ∩ S. This condition is known as Arrow's Axiom, and it is equivalent with the conjunction of (I) and (IV) (cf. Rott 2001, p. 154). Debreu uses an example of a menu with two similar recordings of the eighth symphony of Beethoven and one recording of a string quartet of Debussy. Important papers for the history of the problem of similar options were Chipman (1960), Tversky (1972) and McFadden (1974). In the literature the problem is widely discussed by means of red and blue busses as vehicles that are not different in any significant respect (the red bus/blue bus problem).

  22. Simonson (1989).

  23. Huber et al. (1982), Simonson and Tversky (1992), Shafir et al. (1993), and Shafir and Tversky (1995). The effect of asymmetrically dominated options is also called attraction effect and decoy effect in the literature.

  24. Being relatives does not exclude being competitors.

  25. We pretend for the purposes of this example that good metaphysics is possible without logic, and good logic without metaphysics. It is crucial for the examples adduced in this paper that there are different factors or criteria that are, and are thought of as, independent of each other.

  26. The original appointment example of Rott (1994) may also be described as a compromise case. Here, however, the effect of the presence of the third candidate Cortez in the race even reverses the choice between A and B—at least in the opinion of the person informed by the dean.

  27. The situation is actually worse. Still supposing that only a single person can be employed, the union of Inf(a ∨ b) and Inf(a ∨ c) even has a among its Cn-consequences.

  28. Notice that (a ∨ b ∨ b) ∧ (a ∨ b) is Cn-equivalent with a ∨ b and should therefore license the same inferences as the latter.

  29. A notable exception is Ruth Chang (1997) who would probably subsume our type of incomparability under the heading "diversity of values" or "bidirectionality"—and deny that such things give rise to genuine incomparability.

  30. The talk about dimensions is supposed to suggest that the various aspects of evaluation are independent of each other. The recognition of multiple criteria or dimensions opens up completely new questions. The very wide field of social choice theory gets immediately relevant if the preferences or evaluations according to various criteria involved in a decision are treated like the preferences or evaluations of different members in a social community. The impossibility theorems of Arrow and others have to be kept in mind. Cf. Arrow et al. (2002) and Figueira et al. (2005).

  31. Giacomo Bonanno (personal communication) has suggested that context effects may exist even in cases when A and B are tied rather than incomparable. Suppose for instance that A and B are applicants who are exactly alike in all relevant respects, but that A is female and B is male. If a third candidate C enters the competition, who is inferior to both A and B, his or her sex (a property that everyone agrees should be irrelevant for the appointment) may still tip the scales in favour of A and B. This is a very interesting hypothesis which, if true, I would not know how to explain. I do not know whether it is empirically valid.

  32. Cf. Simonson and Tversky (1992) and Tversky and Simonson (1993).

  33. See Ullmann-Margalit and Morgenbesser (1977).

  34. See the excellent survey of Rieskamp et al. (2006). It was only after the present paper was essentially finished that I became aware that a few psychological papers of the last 10 years juxtapose examples illustrating the similarity, compromise and dominance effects in a way that is similar to the presentation here.

  35. See, for instance, the first chapter, "The Truth about Relativity", of Ariely (2008). I thank Jim Delgrande for pointing me to this book.

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    Hans Rott

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Rott, H. Odd Choices: On the Rationality of Some Alleged Anomalies of Decision and Inference. Topoi 30, 59–69 (2011). https://doi.org/10.1007/s11245-010-9084-1

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