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What Are Preferences?

An informal presentation of Jeffrey (1983, p. chapter 3) on how decision theory enables us to think of subjective probabilities and preferences as simultaneously derivable from patterns of action.

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Notes

We have relied on notions of belief and desire in considering both philosophical (in Philosophical Theories of Action) and psychological theories (in Goal-Directed and Habitual Processes) of instrumental action and joint action.

But what anchors our understanding, as researchers, of these notions? While some of us might use these words in everyday life, there is probably enough diversity between individuals with different cognitive styles (e.g. Perner & Leekam, 2008), different upbringings (e.g. Morgan et al., 2014) or different cultural backgrounds (e.g. Dixson, Komugabe-Dixson, Dixson, & Low, 2018) that whatever understandings you and I have in everyday life may not entirely overlap. And invoking a philosophical theory does not seem likely to help given the level of agreement that has been reached in this regard over the last 2000 or so years.[1]

An attractive alternative is suggested by Jeffrey:

This book has ‘a philosophical end: elucidation of the notions of subjective probability and subjective desirability or utility.’ (Jeffrey, 1983, p. xi)

In this section we explore how, following Jeffrey, subjective probabilities and preferences can be identified as constructs of decision theory.

Decision theory therefore promises to be an ideal anchor for a shared understanding of these notions.

Inspired by Jeffery (and Davidson, 1990), we might therefore attempt to substitute the informal, poorly understood notions of belief and desire with the theoretical constructs of subjective probabilty and preference.

Required Axioms

‘A binary relation ⪰ on a set A is complete if a⪰b or b⪰a for every a ∈ A and b ∈ A,

reflexive if a⪰a for every a ∈ A, and

transitive if a⪰c whenever a⪰b and b⪰c.

A preference relation is a complete reflexive transitive binary relation’ (Osborne & Rubinstein, 1994, p. 7).

The Continuity Axiom states that if c⪰b⪰a then there is some probability p such that you are indifferent between (i) b happening with certainty and (ii) a happening with probability p and c happening with probability (1-p).

‘Continuity implies that no outcome A is so bad that you would not be willing to take some gamble that might result in you ending up with that outcome, but might otherwise result in you ending up with an outcome (C) that you find to be a marginal improvement on your status quo (B), provided that the chance of A is small enough.’ (Steele & Stefánsson, 2020)

The Independence Axiom states that if b⪰a then for any probability p, {pA,(1−p)C}⪯{pB,(1−p)C}. Put roughly, if you prefer a to b then you should prefer a and c to b and c.

‘Intuitively, this means that preferences between lotteries should be governed only by the features of the lotteries that differ; the commonalities between the lotteries should be effectively ignored.’ (Steele & Stefánsson, 2020)

A preference relation is independent of irrelevant alternatives exactly if ‘no change in the set of candidates (addition to or subtraction from) [can] change the rankings of the unaffected candidates’ (Dixit, Skeath, & Reiley, 2014, p. 600).

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Glossary

anchor : A theory, fact or other thing that is used by a group of researchers to ensure that they have a shared understanding of a phenomenon. An anchor is needed when it is unclear whether different researchers are offering incompatible claims about a single phenomenon or compatible claims about distinct phenomena. For example, we might take decision theory to anchor a shared understanding of belief and desire.
decision theory : I use ‘decision theory’ for the theory elaborated by Jeffrey (1983). Variants are variously called ‘expected utility theory’ (Hargreaves-Heap & Varoufakis, 2004), ‘revealed preference theory’ (Sen, 1973) and ‘the theory of rational choice’ (Sugden, 1991). As the differences between variants are not important for our purposes, the term can be used for any of core formal parts of the standard approaches based on Ramsey (1931) and Savage (1972).
ethically neutral condition : ‘A condition is ethically neutral in relation to a particular agent and a particular consequence if the agent is indifferent between having that consequence when the condition holds and when it fails’ (Jeffrey, 1983, p. 46).
instrumental action : An action is instrumental if it happens in order to bring about an outcome, as when you press a lever in order to obtain food. (In this case, obtaining food is the outcome, lever pressing is the action, and the action is instrumental because it occurs in order to bring it about that you obtain food.)
You may encounter variations on this definition of instrumental in the literature. For instance, Dickinson (2016, p. 177) characterises instrumental actions differently: in place of the teleological ‘in order to bring about an outcome’, he stipulates that an instrumental action is one that is ‘controlled by the contingency between’ the action and an outcome. And de Wit & Dickinson (2009, p. 464) stipulate that ‘instrumental actions are learned’.

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Endnotes

  1. There is a bit more detail on this in some notes for one section of a talk called The Myth of Mindreading. ↩︎