A New Look at Local Expected Utility

We revisit the classical local Expected Utility analysis of Machina [30] and show which is its global behavioral foundation, as an illustration, we compute the local utilities for the Prospect Theory model. JEL classi…cation: D81 Keywords: Local utilities, Risk aversion, Multi–utility representation, Prospect theory 1 Introduction The seminal paper of Machina [30] showed that the global risk aversion analysis carried on for the Expected Utility (brie‡y, EU) model naturally extends to a local risk aversion analysis in the case of non-EU binary relations % that can be represented via a Frechet di¤erentiable utility function V . In the non-EU realm the role of the "single global" utility of the standard setting is taken by "multiple local" utilities, which, for example, are all concave if and only if % is averse to Mean Preserving Spreads. Despite the recognized importance of the local approach, the global role of these local utilities, their preferential counterpart, remained unexplained. In this paper, we show how the set of local utilities has a natural interpretation in terms of behavior: it represents the largest part of % that is consistent with the EU axioms. More formally, we study binary relations % on D: the space of probability distributions on a closed interval I. We interpret % as capturing the preferences of a Decision Maker (brie‡y, DM) in a situation of choice under risk and we assume that % can be represented by a continuous utility function V . Given %, we consider the auxiliary binary relation % de…ned by F % G () F + (1 )H % G+ (1 )H 8 2 (0; 1] ;8H 2 D: We interpret F % G as saying that the DM is sure that F is weakly better than G. In fact, no matter how F is mixed/hedged with a third prospect H, the mixture with F dominates the mixture with G. In Lemma 1, we show that % satis…es all the assumptions of EU, with the potential exception of completeness, and it is the largest subrelation of % with these properties. In other words, % collects the largest portion of % which is consistent with the EU paradigm. If V is further assumed to be di¤erentiable, then our Theorem 1 shows that F % G () Z