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In addition, equation (30) must hold for the DM to prefer lottery BG to RB. 

<img src='content_image/310697.jpg'>

Equations (29a), (29b) and (30) hold as long as the utility function is non-decreasing. Theorem 1 successfully reconciles the theory with empirical observations. ■ 

## 8.2 The Proof of Theorem 2 

If we assume that δ$_{i}$ = δ for ∀ i , (3) is a barrier problem to (31). 

<img src='content_image/310698.jpg'>

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 $^{∗}$. 

(31) is equivalent to (32). 

(32) is equivalent to (33). 

<img src='content_image/310699.jpg'>

<img src='content_image/310700.jpg'>

