What is risk aversion?

For more information about Professor Bradley, please see: http://personal.lse.ac.uk/bradleyr/

ABSTRACT: Risk aversion is characterised by an inclination to choose any action over one with the same expected benefit but with greater variance in its consequences `(a mean-preserving spread’ of the action). In the orthodox treatment of risk, an agent’s degree of risk aversion is associated with the curvature of the von Neumann and Morgenstern (vNM) utility function representing her preferences over the goods serving as outcomes of lotteries. A common criticism is that this treatment collapses two distinct attitudes: to marginal increases in the quantity of the good in question and to risk per se. The criticism is simply senseless however unless utility can be cardinalised independently of the rationality conditions on risk preferences built into the vNM framework. A number of recent theories take up this challenge by deriving a risk function on probabilities and a separate utility function on outcomes from preferences satisfying weaker conditions than the vNM ones. In this paper I explore a different (and more conservative) way of doing so: using Bayesian decision theory to provide the required cardinalisation of utility. Our crucial postulate is that chances of outcomes, being objective features of the world, can figure both as contents of our belief and desire attitudes and as possible consequences of actions. The application of Bayesian decision theory to preferences over acts with consequences that include both `ordinary’ outcomes and chances of such outcomes then yields a cardinalisation of the utilities of both without imposing any constraints on how the utilities of chances are related to the utilities of the outcomes of which they are chances of. Within such a framework it is possible to separate attitudes to marginal increases in quantities of a good from attitudes to marginal differences in the chances of a (fixed quantity of) a good. Agent’s preferences over lotteries will then reflect both factors. For instance, risk neutrality, qua indifference between an act and mean-preserving spreads of it, can result from concave utilities for both the good and for chances of the good, or from convex utilities for both, as well as, of course, linear utilities for both. I show that the typical patterns of preferences observed in both the Allais and Ellsberg paradoxes can be explained within such a broadly Bayesian framework, in terms of the dependence of the utility of marginal differences in chances of goods on the chances of other outcomes.