min 0 . 1 ≤ x ≤ 0 . 4 [ 1 3 u (1) + 2 3 u (0) ] ≥ min 0 . 1 ≤ x ≤ 0 . 4 [( 2 3 − x ) · u (1) + ( 1 3 + x ) · u (0) ] (29b) min 0 . 1 ≤ x ≤ 0 . 4 [ 2 3 u (1) + 1 3 u (0) ] ≥ min 0 . 1 ≤ x ≤ 0 . 4 [( 1 3 + x ) · u (1) + ( 2 3 − x ) · u (0) ] (30) max x ,M b $^{T}$x − M such that (A) ∑ J j $_{=1}$A$_{i,j}$ x$_{j}$ ≤ M for each i ∈ { 1 , 2 , ..., N } (B) 0 ≤ x ≤ Q (31) max x [ b $^{T}$x − max i ∑ J j $_{=1}$A$_{i,j}$ x$_{j}$ ] such that (A’) 0 ≤ x ≤ Q (32) max x min i u [ b $^{T}$x − ∑ J j $_{=1}$A$_{i,j}$ x$_{j}$ ] such that (A’) 0 ≤ x ≤ Q (33) 32 (32) is equivalent to (33). (31) is equivalent to (32). Assume that the feasible set for (31) is not empty. Sending the value of the parameter δ to zero is equivalent to reducing the duality gap along the primal-dual central path in the interior point method. Thus, as δ approaches zero, x ( δ ) should converge to an optimal solution to (31) (Luenberger and Ye, 2008), which we denote x $^{∗}$. If we assume that δ$_{i}$ = δ for ∀ i , (3) is a barrier problem to (31). Equations (29a), (29b) and (30) hold as long as the utility function is non-decreasing. Theorem 1 successfully reconciles the theory with empirical observations. ■ In addition, equation (30) must hold for the DM to prefer lottery BG to RB. 8.2 The Proof of Theorem 2