 There is however substantial evidence that the linearity of indifference curves isn’t any more empirically adequate that their parallelism (see Camerer &amp; Ho 1994 for a survey) and a number of models of probabilistically sophisticated preferences give up on Betweenness too. The best known of these is undoubtedly Rank Dependent Utility (RDU), a version of which was first proposed by Quiggin  (1982).[8]  To present the proposal in functional form, it will be assumed that the subscripts associated with each outcome in \(\mathcal{X}\) indicate increasing order of preference, so that \(x_1\preceq x_2\preceq \ldots \preceq x_n\) and hence \(\bigcup\limits_{j=i}^{n} E^{f}_{j}\) is the event given which \(f\) yields an outcome at least as preferable as \(x_i\). RDU proposes: 