ball, the person chooses a blue ball), and let X denote the set of consequences (e.g., the person wins a dollar). Subsets of S are called events. Let Σ denote the algebra of subsets of the state space X . We are interested in the decision maker’s preference over different simple acts : A simple act is a Σ-measurable function f : S −→ X that is finite-valued (Ghirardato e al, 2004). Let F denote the set of all simple acts. Suppose that the binary relations ≽ and ≻ characterize the decision maker’s preference over different acts: f ≽ ( ≻ ) g if and only if the decision maker (strictly) prefers the simple act f to the simple act g . Finally, let u : X → R denote the decision maker’s utility function. Theorem 1 (Decision Making under Ambiguity) (Gilboa and Schmeidler, 1989; Ghi- rardato et al., 2004) The decision maker’s preference relation ≽ satisfies the set of six behavioral axioms 2 if and only if there exists a unique set Ψ of probabilities on ( S, Σ) such that (1) holds for ∀ f,g ∈ F . The set Ψ is weakly compact, convex and nonempty. 3 Proof. See Gilboa and Schmeidler (1989) or Ghirardato et al. (2004) . The set of probabilities Ψ in Theorem 1 encapsulates the decision maker’s (hereafter called the DM) perception of ambiguity (Ghirardato et al., 2004). Recall that a DM facing ambiguity cannot attach a single probability distribution to the unknown prospect. Instead, the DM has a set of candidates Ψ that he/she believes are fairly accurate predictions of the future (Ghirardato et al., 2004). In other words, the DM has a set of multiple priors (Gilboa and Schmeidler, 1989). The size of Ψ represents the extent to which the DM feels ambiguous toward the unrealized future outcome (Ghirardato et al., 2004). A large size of Ψ implies that the DM cannot easily narrow down the set of reasonable probability distributions because he/she is too ambiguous about the future outcome (Ghirardato et al., 2004). Among the set of multiple priors, the DM is exclusively concerned with the worst possible scenario. First, for each P ∈ Ψ, the DM calculates the expected utility ∫ u ( f ) dP from the unknown prospect assuming that P is the true description of the future. Second, the DM finds the distribution that results in the lowest level of utility. Third, when comparing one act with another, the DM $^{2}$Please see Ghirardato et al. (2004) for the set of six behavioral axioms. $^{3}$Ghirardato et al. (2004) presents three different versions of the theorem depending on the DM’s attitude toward ambiguity. However, we only work with the version that assumes aversion to ambiguity. Please refer to Ghirardato et al. (2004) for a more rigorous formal definition of aversion to ambiguity. 5 f ≽ g ⇔ min P ∈ Ψ ∫ u ( f ) dP ≥ min P ∈ Ψ ∫ u ( g ) dP (1)