Decision Theory for Agents with Incomplete Preferences

Abstract

Orthodox decision theory gives no advice to agents who hold two goods to be incommensurate in value because such agents will have incomplete preferences. According to standard treatments, rationality requires complete preferences, so such agents are irrational. Experience shows, however, that incomplete preferences are ubiquitous in ordinary life. In this paper, we aim to do two things: (1) show that there is a good case for revising decision theory so as to allow it to apply non-vacuously to agents with incomplete preferences, and (2) to identify one substantive criterion that any such non-standard decision theory must obey. Our criterion, Competitiveness, is a weaker version of a dominance principle. Despite its modesty, Competitiveness is incompatible with prospectism, a recently developed decision theory for agents with incomplete preferences. We spend the final part of the paper showing why Competitiveness should be retained, and prospectism rejected.

Keywords:

• Decision theory
• incommensurate value
• practical reason
• incomplete preferences
• dominance

Notes

¹ E.g. Ariely [2008], Brafman and Brafman [2009], Schwartz [2003], Thaler and Sunstein [2008].

² Ariely is particularly explicit in this regard [2008: xxix–xxx].

³ E.g. Broome [1991: 92–3]; Savage [1954: 21].

⁴ Here we follow, and revise slightly Broome [2004: 21].

⁵ Some prefer to use the term ‘incomparability’ to describe this value relation. See, e.g. Hsieh [2008]. Ruth Chang [2002] claims that, in addition to the possibility of incomparability—understood as the absence of any comparative relation—there is also the possibility of the relation of ‘parity’, which is a fourth comparative relation that might be described as ‘rough equality’. In this paper, we ignore the alleged distinction between parity and incomparability, and assume that they can be treated identically.

⁶ A terminological note: one might want to refer to an agent who lacks a preference between choices in trivial cases, such as the example above, as ‘indifferent’. Standardly, however, decision theorists use the term ‘indifference’ to refer to the state of holding two things to be equally preferable. As we hope the above cases illustrate, these two varieties of ‘indifference’ are crucially different, and we will—with one exception—avoid the term to minimize confusion.

⁷ For those seeking further persuasion, Gigerenzer, Todd, and the ABC Research Group [1999: 76] argue that Bayesian conditionalization is computationally intractable, and therefore not possible for humans. Simon [1955: 101] suggests a similar connection as we do between human cognitive limitations and the normativity of rational choice theory.

⁸ See Mandler [2005: 261] for formal characterization of the necessary assumptions.

⁹ In particular, it has been suggested that agents who experience preference reversals will need to be capable of making and committing to plans in order to avoid money pumps. Perhaps such resoluteness will also allow agents with incomplete preferences to avoid money pumps. See McClennen [1990] or Bratman [1998] for a discussion of such views of rational agency. Alternatively, perhaps backward-looking agents that consider their past decisions when making their current ones will be able to avoid money pumps. See Mandler [2005] for a discussion.

A referee has pointed out to us that a parallel money-pump concern has been raised for defenders of imprecise credences [Elga 2010; White 2010]. To that end, the defender of incomplete preferences may be able to take inspiration from James Joyce's [2010] responses to those objections.

¹⁰ Prospectism is not entirely new. An extremely similar idea has been defended by Paul Weirich [2004: 70], and Weirich cites a predecessor of the idea due to I. J. Good [1952]. We take our criticisms of Hare's proposal to generalize to Weirich's account also.

¹¹ This sort of reasoning has clear echoes of the Reflection principle—though it seems less vulnerable to the sorts of problem cases that have been raised against Reflection, given those cases tend to exploit future states in which the agent is not rational [Christensen 1991], or is in receipt of very special information that is inaccessible to others [Elga 2000]. Neither of those sorts of concerns seems relevant here.

A further, indirect, consideration in favour of deferring to the preferences of my future self is that this idea features heavily in Krister Bykvist's [2006] elegant account of how to make prudent decisions in anticipation that one's future preferences will depend upon one's present choices.

¹² Although deferentialism respects Competitiveness, one reservation we have is that it violates Strong Competitiveness. Discussing why, and how this feature might best be redressed, would take us too far afield, however.

¹³ Hare allows that the relevant credences will be different, depending upon whether one favours causal decision theory or evidential decision theory. Thus his defence of prospectism remains neutral on that dispute. For all the examples discussed in this paper, the dispute between evidentialists and causalists is irrelevant.

¹⁴ The idea of Competitiveness is essentially the thought captured by Hare's principle of ‘Recognition’ [ibid.: 241], though Hare does not identify how closely it follows the idea of dominance reasoning, extended into the domain of incomplete preferences.

In a different context, Amartya Sen [1997, 2000] has argued that consequentialists do better to adopt a ‘maximizing’ form of consequentialism, rather than an ‘optimizing’ form. The maximizing consequentialist merely seeks to bring about outcomes that are no worse than any alternative. The optimizing consequentialist seeks to bring about an outcome that is at least as good as all alternatives. If there are incommensurate goods, or there is some other failure of the assumptions of completeness and transitivity in the betterness relation, then the optimizing goal may be impossible—there may be no outcome that is at least as good as all others—but the maximizing goal remains viable. The adoption of a rule permitting ‘competitive’ actions as opposed to a rule permitting only dominant actions complements, in decision theory, Sen's proposal for ethics more generally.

¹⁵ For those who lack our anti-prospectist intuitions, but would like to take our advice in distancing themselves from Hare's highly sensitive form of prospectism, the following, non-probabilistic principles might seem to be a better way to formalize the intuitive idea behind the second argument:

P1. If none of the possible outcomes associated with a given action A₁ are worse than any of the possible outcomes associated with any alternative action A_i, then A₁ is rationally permissible.

P2. If one of the possible outcomes associated with a given action A₁ is worse than one of the possible outcomes associated with an alternative action A_i, and no possible outcomes associated with A₁ are better than the possible outcomes associated with A_i, then A₁ is rationally prohibited.

These principles would entail that taking the right box is rationally required in variants of Two Opaque Boxes where the credences are not equal. However, these principles would also implausibly entail that no decision is rationally permissible in some cases (as there will be scenarios where P2 will imply that all decisions are rationally prohibited). As such, this alternative theory is also unsatisfactory.

¹⁶ Using terminology due to Larry Temkin [2012: 238–44], we can say that such theories satisfy the first principle of equivalence. The impossibility proofs that Temkin considers following his introduction of this principle could be used, with slight changes, to illustrate the tension between Competitiveness and the first principle of equivalence.

¹⁷ Thanks to Wylie Breckenridge, Caspar Hare, Jakob Hohwy, Morgan Luck, Miriam Schoenfield, and Alastair Wilson for helpful discussions regarding this paper. We also benefited from sharing earlier versions of this paper with audiences at MIT, The University of Melbourne, and the Frontiers of Rationality Conference, held at the University of Groningen in 2012.