Numerical Representation of Convex Preferences on Anscombe–Aumann Acts

Patrick Cheridito, Freddy Delbaen, Samuel Drapeau and Michael Kupper
In this paper we derive a numerical representation for general complete preferences respecting the fol- lowing two principles: a) more is better than less, b) averages are better than extremes. To be able to distinguish between risk and ambiguity we work in an Anscombe–Aumann framework. Our main result is a quasi-concave numerical representation for a class of preferences wide enough to accommodate Ellsberg- as well as Allais-type behavior. Instead of assuming the usual monotonicity we suppose that our prefer- ences are monotone with respect to first order stochastic dominance. Preference for averages is expressed through convexity. No independence assumptions of any form are made. In general, our preferences in- tertwine attitudes towards risk and ambiguity. But if one assumes a weak form of Savage’s sure thing principle, there is separation between risk and ambiguity attitudes, and the representation decomposes into state dependent preference functionals over the consequences and a quasi-concave functional aggregating the preferences of the decision maker in different states of the world.
Convex preferences, Uncertainty aversion, Allais paradox, Ellsberg paradox
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