SUMMARY:

This course will deal with the state-of-the-art in the theory of Pareto-optimal (re)insurance design under model uncertainty and/or non-Expected-Utility preferences, as well as provide an introduction to Pareto optimality in problems of peer-to-peer collaborative insurance. Specifically:

• We will start with some background material on the theory of decision-making under uncertainty, from the classical work on Expected-Utility Theory (EUT) of von Neumann and Morgenstern, Savage, and De Finetti, to the more recent work on ambiguity and probability distortions (Quiggin, Yaari, Schmeidler, Gilboa, Amarante, Maccheroni-Marinacci).
• We will cover some required mathematical tools, such as non-additive measure theory, probability distortions, Choquet integration, the theory of equimeasurable rearrangements, as well as risk measures, their properties, and their representations. We will pay special attention to distortion risk measures and spectral risk measures.
• We will then formally introduce a general model of the insurance market, following the work of Carlier and Dana [8], and study the existence and characterization of Pareto optima in this general setup. As a special case, we will consider the classical formulation of the optimal (re)insurance problem due to Arrow, in the framework of EUT, as well as more recent work extending Arrow’s setting to situations of belief heterogeneity between the two agents.
• We will then proceed to formulating several problems that extend the insurance model above to more general setting with non-EUT preferences, distortion risk measures, or situations of model uncertainty.
• Finally, we will go over a brief introduction to conditional mean risk sharing and peer-to-peer collaborative insurance.