Schedule - Parallel Session 7 - Models of Uncertainty

In this paper, we model ambiguity as the anticipated receipt of different objective lotteries obtained in different states. Using Savage’s acts that map the set of states to the set of simple objective probability measures as the ambiguous lotteries faced by the decision maker, we develop preference conditions to separate “ambiguity” from “risk” to obtain an additive model of decision making under uncertainty consisting of two parts, i.e., the risk part and ambiguity part. This model explicitly captures the tradeoff between the magnitude of risk and the magnitude of ambiguity when making choice over ambiguous lotteries. The risk of an ambiguous lottery is evaluated by von Neumann and Morgenstern’s expected utility model; the ambiguity is evaluated by a nested utility model similar to the smooth ambiguity model developed by Klibanoff, Marinacci, and Mukerji (2005). A measure that ranks lotteries in terms of the magnitude of ambiguity can also be obtained using this separation. Using Taylor expansion, we show that this ambiguity measure is the variance of the compound lottery based on the second order subjective probability over possible objective lotteries. By applying our model to asset pricing under uncertainty, we show that the equity premium can be decomposed into two parts: the risk premium and the ambiguity premium, where the risk premium takes the standard form in the literature and the ambiguity premium is proportional to the ambiguity measure. Further, combining this model with the standard risk-value model developed by Jia and Dyer (1996), we obtain a risk-ambiguity-value preference model. This model is consistent with the mean-variance-ambiguity approximation of the smooth ambiguity model (Maccheroni, Marinacci, and Ruffino. 2013), but we do not need to rely on an approximation argument to obtain the additive form.

One way of avoiding the Ellsberg paradox is to relax the Independence Axiom of Anscombe et al. (1963) (AA). This approach yielded two of the most well-known models of ambiguity: Schmeidler’s (1989) Choquet Expected Utility (CEU), and Gilboa and Schmeidler’s (1989) Maximin Expected Utility (MEU). Under both approaches the decision maker acts as if he maximizes utility with respect to a set of priors. In this paper we identify necessary assumptions on the preference relation to obtain a representation under which the decision maker maximizes utility with respect to a set of priors, called Local Prior Expected Utility (LEU). Moreover, we show that the prior used to evaluate a certain act is equal to the gradient of some appropriately defined utility mapping. We argue that the equality is not a mere technicality but coherent with the qualitative interpretation of a probabilistic belief. Based on this result we provide a unified approach to MEU, CEU, and AA and characterize the respective sets of priors. A preference relation on acts that satisfies the standard axioms Weak Order, Monotonicity, Continuity, Risk Independence, and Non-Degeneracy can be represented by a functional D that maps vectors of statewise utilities to “overall” utilities. We show that Monotonicity of D guarantees the existence of its Gâteaux derivative almost everywhere relying on a deep mathematical theorem by Chabrillac et al. (1987). Using only the Gâteaux derivative of D as an analytical tool, we prove characterization results corresponding to LEU, CEU, MEU, and AA. Within our approach we clearly identify the structure of the set of priors needed in these environments. As a basis for our approach, we identify an axiom called Independence of Certainty Equivalents that is weaker than Gilboa and Schmeidler’s (1989) C-Independence and weaker than Schmeidler’s (1989) Comonotonic Independence, but together with Weak Order, Monotonicity, Continuity, Risk Independence, Non-Degeneracy, and Uncertainty Aversion induces the same restrictions on the preference relation on acts as the Gilboa-Schmeidler axioms. We then show that without assuming Uncertainty Aversion we can represent a decision maker’s preferences by taking expectations of an affine utility function u with respect to a (possibly) different prior for every act, which we call LEU. The prior used is equal to the gradient of D at the vector of utilities induced by the act if the Gâteaux derivative exists. Moreover, we show that relaxing the Independence Axiom further by requiring invariance with respect to translations but not invariance to rescaling still results in a prior representation, where the gradient is take at a possibly different location. In case the Gâteaux derivative does not exist at a particular vector, we can approximate it by the Gâteaux derivatives of nearby acts.