Decision Making Under Model Uncertainty: Fréchet–Wasserstein Mean Preferences

Abstract

This paper contributes to the literature on decision making under multiple probability models by studying a class of variational preferences. These preferences are defined in terms of Fréchet mean utility functionals, which are based on the Wasserstein metric in the space of probability models. In order to produce a measure that is the “closest” to all probability models in the given set, we find the barycenter of the set. We derive explicit expressions for the Fréchet–Wasserstein mean utility functionals and show that they can be expressed in terms of an expansion that provides a tractable link between risk aversion and ambiguity aversion. The proposed utility functionals are illustrated in terms of two applications. The first application allows us to define the social discount rate under model uncertainty. In the second application, the functionals are used in risk securitization. The barycenter in this case can be interpreted as the model that maximizes the probability that different decision makers will agree on, which could be useful for designing and pricing a catastrophe bond. This paper was accepted by Manel Baucells, decision analysis.

Robust policy selection and harvest risk quantification for natural resources management under model uncertainty

“…where λ are the eigenvalues of (L G +A D ). Using the obtained form of the maximizers in (21) and substituting each HJB calculated term in (18) one obtains…”

“…Then, substituting (34) to ( 22) we obtain the stated result (19) in Proposition 1. Also, combining equations ( 25) and ( 34) and substituting to (21) are obtained the optimal controls stated in (20) in Proposition 1.…”

“…Several approaches have been discussed in the literature where the most recent and quite popular is that of employing the notion of Fréchet mean or Fréchet barycenter [14] in order to determine the mean element even in spaces where the typical notion of euclidean mean is not applicable. A very interesting approach in optimal decision making relying on this concept is proposed in [21] where a very flexible and extendable framework to treat model uncertainty issues is analyzed. Lastly, the issue of model validity is something that someone has to be aware of, especially in cases where the decision making process is performed under a dynamic setting and possibly during the process new evidence become available.…”

“…where d is an appropriate metric in the space of probability measures on R d , P(R d ) e.g., Wasserstein distance [20,21,22,25], and η > 0 is the maximum desired deviation level from model Q 0 in terms of the metric sense that is used. The optimization problem in (4) can be equivalently expressed in a multiplier-preferences formulation instead of the current constraint-preferences representation by…”

“…for an appropriate choice of the aversion parameter γ > 0 (quantifying the manager's aversion preferences from Q 0 ). This concept is further generalized and extended to the very interesting case where a multitude of models is provided for describing the behaviour of random variable Z possibly divergent and with varying probabilities of realization [20,21]. In optimal harvesting problems, different probability models could identify different scenarios about the conditions in the spatio-temporal domain that the natural resources optimal management problem is considered.…”