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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