Inferring preferences from choices under uncertainty

If a decision maker, in a world of uncertainty à la Anscombe and Aumann (1963), can choose acts according to some objective probability distribution (by throwing dice for instance) from any given set of acts, then there is no set of acts that allows an experimenter to test more than the Axiom of EUOL (that the DM evaluates objective lotteries with an expected utility function). In fact there is no (common) experimental design that allows an experimenter to test more than EUOL. For any decision problem (or set of decision problems), for any preference relation that satisfies the Axiom EUOL, and for any optimal choice (or collection of choices) given this preference relation, there is another preference relation that satisfies EUOL plus the Savage axioms, for which this choice is also optimal.