aversion. We present a polynomial-time algorithm to solve the optimization problem. Our paper may contribute to facilitating the adoption of a pari-mutuel mechanism in the trading community. As Robert Shiller once noted, a pari-mutuel mechanism can be particularly useful in launching a wide variety of innovative derivatives markets, thereby enabling investors to hedge a new class of fundamental risks (Baron and Lange, 2007). Suppose that there is an urn that contains red, blue and green balls, of which there are 90 in total. While there are 30 red balls in the urn, the exact number of either blue balls or green balls is unknown to the DM. Suppose that five lotteries are available. Lottery R pays $1 if and only if the DM draws a red ball from the urn. Lotteries B and G pay $1 if and only if he/she draws a blue ball and a green ball, respectively. Similarly, lottery RB pays $1 if and only if either a red ball or a blue ball is drawn. Lottery BG pays $1 if and only if either a blue ball or a green ball is drawn. Empirical studies show that most people prefer lottery R to either lottery B or G. Moreover, most people prefer lottery BG to RB. It is well known that this empirical result contradicts Savage’s theory of utility maximization with subjective probability (Savage, 1954). Let us reformulate the DM’s problem in the language of Theorem 1. The set of all possible states S is { red, blue, green } . The set of consequences X is { 0 , 1 } , expressed in dollars. The DM is interested in five different acts: f$_{R}$ , f$_{B}$ , f$_{G}$ , f$_{RB}$ , and f$_{BG}$ . The act f$_{R}$ : S −→ X is a mapping such that f$_{R}$ ( red ) = 1, f$_{R}$ ( blue ) = 0 and f$_{R}$ ( green ) = 0. We define the other four acts similarly. Without knowing the exact number of either blue or green balls, the DM cannot attach a single probability distribution to S . Suppose that the DM’s set of candidates is Ψ = { (1 / 3 , x, 2 / 3 − x ) ∈ R $^{3}$| 0 . 1 ≤ x ≤ 0 . 4 } , where 1 / 3 , x, and 2 / 3 − x are the chances of drawing red, blue and green balls, respectively. Let u : X −→ R denote the DM’s utility function. Equations (29a) and (29b) should hold for the DM to prefer lottery R to the other two lotteries. min 0 . 1 ≤ x ≤ 0 . 4 [ 1 3 u (1) + 2 3 u (0) ] ≥ min 0 . 1 ≤ x ≤ 0 . $_{4}$[ x · u (1) + (1 − x ) · u (0)] (29a) 31 8 Appendix 8.1 Illustration of the Theory of Decision Making Under Uncertainty