A generalization of expected utility theory. The alternative outcomes are ranked from lowest to highest, and the probabilities of the outcomes are transformed in a way that ensures unlikely extreme outcomes are over-weighted. If (x₁,…, x_n) are the ranked outcomes and (p₁,…, p_n) the probabilities with which they occur, then the rank dependent expected utility function is $U=\sum _{i=1}^n{w}_i(p_1,\dots ,p_n)U(x_i),$ where w_i (p₁, …, p_n), i = 1,…, n, are the weighting functions. The motivation for this construction is to obtain a pay-off function that can explain the choices observed in the Allais paradox and other similar anomalies.