We characterize preference relations "R" over Anscombe and Aumann acts and give necessary and sufficient conditions that guarantee the existence of a utility function u on consequences and an ambiguity index η on the set of probabilities on the the states of the nature such that, for any acts f and g, f R g iff ∫u(f)dp+η(p)≥ ∫u(g)dp,for any p in Δ. The function u represents the decision maker´s risk attitudes, while the ambiguity index η describes his confidence among the universe of probabilities laws.The axiomatic basis for this class of preference waiver completeness and transitivity, and an interesting property is that cycles are avoided. These preferences include the Knightian decision theory of Bewley as well new class of preferences through specifics ambiguity index. Another contribution in this paper starts from a pair of preferences and provides a novel foundation for variational preferences of Macheroni, Marinacci and Rustichini, based on axioms on these relations that allow a joint representation by a single ambiguity index.
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