Revisiting Risk and Rationality: a reply to Pettigrew and Briggs

Abstract

I have claimed that risk-weighted expected utility (REU) maximizers are rational, and that their preferences cannot be captured by expected utility (EU) theory. Richard Pettigrew and Rachael Briggs have recently challenged these claims. Both authors argue that only EU-maximizers are rational. In addition, Pettigrew argues that the preferences of REU-maximizers can indeed be captured by EU theory, and Briggs argues that REU-maximizers lose a valuable tool for simplifying their decision problems. I hold that their arguments do not succeed and that my original claims still stand. However, their arguments do highlight some costs of REU theory.

Keywords:

• Risk-weighted expected utility
• expected utility
• decision theory
• sure-thing principle
• risk
• accuracy
• Allais paradox

Notes

1. Throughout, except where it is an explicit topic of discussion, I remain neutral on whether the relevant attitudes exist independently of preferences or are constructed from them.

2. More precisely, the REU of an act f = {E₁, x₁; …; E_n, x_n}, where u(x₁) ≤ … ≤ u(x_n), is . In other words, This is equivalent to . We can see REU as a generalization of Quiggin’s (1982) anticipated utility to subjective rather than objective probabilities; or as a restriction of Gilboa’s (1987) (and, recast, Schmeidler’s 1989) Choquet EU to weightings that are a function of probabilistic credences; or as a similar generalization of objective cumulative prospect theory or restriction of subjective cumulative prospect theory (see Kahneman and Tversky 1979; Tversky and Kahneman 1992), with the additional stipulation that gains and losses are not treated differently. See (Buchak 2013, Ch. 1 and 2) for a discussion of the relationship of REU theory to other non-EU theories.

3. REU = u(x₁) + r(p(E₂) + p(E₄))(u(x₂) − u(x₁)) + r(p(E₄))(u(x₄) − u(x₂)) = 3 + (0.3 + 0.4)²(5 − 3) + 0.4²(6 − 5) = 4.14.

4. More precisely, since the only real facts about utility functions are difference ratios, the utility difference between two pairs of outcomes is the same if these two pairs constitute equal tradeoffs in acts with the same structure.

5. See Strotz (1955), Hammond (1988), and McClennen (1990).

6. See Strotz (1955), McClennen (1990, 1997), and Gauthier (1997).

7. This coheres with a view recently put forth by Hedden (2015), a proponent of the time-slice picture, of what an option is: ‘your options are all and only the decisions you are presently able to make’ (95).

8. The usual game-theoretic terminology is ‘strategy’ for an individual’s choice and ‘strategy pair’ for the pair of strategies chosen by the two agents. However, since I follow Briggs in using ‘strategy’ to refer to the pair of acts chosen by the agent at the earlier time and at the later time, the interpersonal analogue of which is the pair of acts chosen by the two agents, I will instead use ‘act’ and ‘act pair.’

9. This isn’t quite the right way to interpret Briggs herself, because the argument of mine that Briggs cites against premise (1) would actually concern premise (2’). Nonetheless, this argument is at least strongly suggested by her discussion.

10. A recent proponent of this view is Hedden (2015), who argues for ‘time-slice rationality,’ according to which ‘how you rationally ought to be at a time directly depends only on your mental states at that time, not on how you (or time-slices psychologically continuous with you) were in the past or will be in the future’ (7). Moss (2015) argues for a similar view in epistemology, which she dubs ‘time-slice epistemology.’

11. See, for example, McClennen (1997), who explicitly makes the connection between intrapersonal decisions and the interpersonal decisions, and argues that resolute choice can resolve problems within both. Gauthier (1997) denies that the two types of decisions are analogous: other people’s concerns always ought to be taken seriously but my past concerns can cease to matter when I no longer embrace them. Furthermore, he holds that sophisticated choice rather than resolute choice ought to be employed in typical intrapersonal situations; however, when sophisticated choice would be sub-optimal, resolute choice ought to be employed. Thus, insofar as he holds that landing on a sub-optimal strategy pair is irrational, he holds that resolute choice is appropriate.

12. Notice that if there are two states with the same utility, then a contiguous event with this utility as the minimum or maximum value can include one state, the other state, or both. This won’t pose the same kind of problem as it does for the comonotonic individuation (discussed in Section 2), since we are not trying to come up with a utility contribution for an event independent of the ordering under consideration (i.e. independent of the particular coarse-graining we consider).