Consider a preference functional of the cumulative prospect theory form (9.4). Let \(\tilde{x}_{1}\) and \(\tilde{x}_{2}\) be two gambles on the same finite probability space:

\[\tilde{x}_{1}=\left\{x_{1}, \ldots, x_{S} ; \pi_{1}, \ldots, \pi_{S}\right\} \quad \text { and } \quad \tilde{x}_{1}=\left\{x_{1}, \ldots, x_{S} ; \pi_{1}^{\prime}, \ldots, \pi_{S}^{\prime}\right\}\]

with \(x_{1}<\ldots