Risk, rationality and expected utility theory

Abstract

There are decision problems where the preferences that seem rational to many people cannot be accommodated within orthodox decision theory in the natural way. In response, a number of alternatives to the orthodoxy have been proposed. In this paper, I offer an argument against those alternatives and in favour of the orthodoxy. I focus on preferences that seem to encode sensitivity to risk. And I focus on the alternative to the orthodoxy proposed by Lara Buchak’s risk-weighted expected utility theory. I will show that the orthodoxy can be made to accommodate all of the preferences that Buchak’s theory can accommodate.

Keywords:

• Risk
• decision theory
• expected utility theory
• rational choice theory
• accuracy
• scoring rules

Notes

This paper began life as my contribution to the Author Meets Critics session on Lara Buchak’s Risk and Rationality at the Pacific APA in San Diego in April 2014. I received extremely helpful feedback from Lara at that point and then later, when I came to write up the paper for publication. I would also like to thank Jason Konek, Rachael Briggs, Ralph Wedgwood, Greg Wheeler, and Ben Levinstein for further comments on this paper. The work on this paper was supported by the European Research Council Seventh Framework Program (FP7/2007-Ð2013) Starting Researcher Grant (308961-EUT), Epistemic Utility Theory: Foundations and Applications.

1 A finite set X of subsets of a set S is an algebra if (i) S is in X; (ii) if Z is in X, then its complement is in X; (iii) if , are in X, then their union is in X.

2 The names should be considered labels only. I do not take them to imply that one sort of attitude can be observed directly, while the other sort is knowable only by inference.

3 As we will see below, one of Buchak’s central contentions is that there is a third type of internal attitude with which decision theory deals, namely, attitudes to risk. In my alternative to Buchak’s theory, I will incorporate such attitudes into the utilities on the outcomes. So, while these internal attitudes to risk will be present in my account, they will be a component of the utilities, not separate attitudes.

4 A technical note on the definition of Bregman divergences; what follows is not essential to the rest of the argument. Suppose C is a closed, convex subset of the real numbers. And suppose is a continuously differentiable and strictly convex function. Then the Bregman divergence generated by is defined as follows: . That is, is the difference between the value of at x and the value at x of the tangent to taken at y. is the Bregman divergence generated by .

5 See also (D’Agostino and Dardanoni 2009), where the original mathematical result is stated and proved.

6 Recall: like a set, a multiset is unordered, so that . Unlike a set, it allows repetitions, so that .

7 Note that Buchak (2013, Section 4.4) considers a redescription strategy that is very close to the one I describe in this section. However, she notes that it is ill-defined. The strategy that I describe here does not suffer from this problem.

8 In general, for , there are many sequences with such that , if there are any.

9 If C is a finite set of vectors in a vector space V over the real numbers, the convex hull of C is written and defined as follows: is the smallest convex set that includes C, where a set is convex if it contains every mixture of two vectors whenever it contains those vectors; alternatively,

Additional information

Funding

This work was supported by the European Research Council (ERC), [grant number 308961-EUT].