Massimo Marinacci

Differentiating ambiguity and ambiguity attitude

Abstract The objective of this paper is to show how ambiguity, and a decision maker (DM)s response to it, can be modelled formally in the context of a general decision model. We introduce a relation derived from the DMs preferences, called “unambiguous preference”, and show that it can be represented by a set of probabilities. We provide such set with a simple differential characterization, and argue that it is a behavioral representation of the “ambiguity” that the DM may perceive. Given such revealed ambiguity, we provide a representation of ambiguity attitudes. We also characterize axiomatically a special case of our decision model, the “ α -maxmin” expected utility model.

Recursive smooth ambiguity preferences

This paper axiomatizes an intertemporal version of the Smooth Ambiguity decision model developed in Klibanoff, Marinacci, and Mukerji (2005). A key feature of the model is that it achieves a separation between ambiguity, identified as a characteristic of the decision makers subjective beliefs, and ambiguity attitude, a characteristic of the decision makers tastes. In applications one may thus specify/vary these two characteristics independent of each other, thereby facilitating richer comparative statics and modeling flexibility than possible under other models which accomodate ambiguity sensitive preferences. Another key feature is that the preferences are dynamically consistent and have a recursive representation. Therefore techniques of dynamic programming can be applied when using this model.

Ambiguity and the Bayesian Paradigm

This is a survey of some of the recent decision-theoretic literature involving beliefs that cannot be quantified by a Bayesian prior. We discuss historical, philosophical, and axiomatic foundations of the Bayesian model, as well as of several alternative models recently proposed. The definition and comparison of ambiguity aversion and the updating of non-Bayesian beliefs are briefly discussed. Finally, several applications are mentioned to illustrate the way that ambiguity (or “Knightian uncertainty”) can change the way we think about economic problems.

Risk, Ambiguity, and the Separation of Utility and Beliefs

We introduce a general model of static choice under uncertainty, arguably the weakest model achieving a separation of cardinal utility and a unique representation of beliefs. Most of the nonexpected utility models existing in the literature are special cases of it. Such separation is motivated by the view that tastes are constant, whereas beliefs change with new information. The model has a simple and natural axiomatization.Elsewhere (forthcoming), we show that it can be very helpful in the characterization of a notion of ambiguity aversion, as separating utility and beliefs allows us to identify and remove aspects of risk attitude from the decision makers behavior. Here we show that the model allows us to generalize several results on the characterization of risk aversion in betting behavior. These generalizations are of independent interest, as they show that some traditional results for subjective expected utility preferences can be formulated only in terms of binary acts.

Probabilistic sophistication and multiple priors

We show that under fairly mild conditions, a maximin expected utility preference relation is probabilistically sophisticated if and only if it is subjective expected utility.

Mutual absolute continuity of multiple priors

This note provides a behavioral characterization of mutually absolutely continuous multiple priors.

Ultramodular Functions

We study the properties of ultramodular functions, a class of functions that generalizes scalar convexity and that naturally arises in some economic and statistical applications.

A strong law of large numbers for capacities

We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the lower and the upper Choquet integrals of the random variables, provided either the random variables or the capacity are continuous.

Dynamic variational preferences

We introduce and axiomatize dynamic variational preferences, the dynamic version of the variational preferences we axiomatized in [F. Maccheroni, M. Marinacci, A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Mimeo, 2004], which generalize the multiple priors preferences of Gilboa and Schmeidler [Maxmin expected utility with a non-unique prior, J. Math. Econ. 18 (1989) 141–153], and include the Multiplier Preferences inspired by robust control and first used in macroeconomics by Hansen and Sargent (see [L.P. Hansen, T.J. Sargent, Robust control and model uncertainty, Amer. Econ. Rev. 91 (2001) 60–66]), as well as the classic Mean Variance Preferences of Markovitz and Tobin. We provide a condition that makes dynamic variational preferences time consistent, and their representation recursive. This gives them the analytical tractability needed in macroeconomic and financial applications. A corollary of our results is that Multiplier Preferences are time consistent, but Mean Variance Preferences are not.

Revealed Ambiguity and Its Consequences: Updating

We study the updating of beliefs under ambiguity for invariant biseparable preferences. In particular, we show that a natural form of dynamic consistency characterizes the Bayesian updating of these beliefs.