Consider the Rank-dependent Expected Utility model (Yaari, Quiggin), under which the individual, besides "perturbing" money x by a utility function u(x), perturbs probability p of winning by a "probability perception function" g(p). Can enough "pessimism" (g below the identity function) displayed by g offset "greediness" (non-concavity) displayed by u and still render the individual averse to risk? The answer depends on the notion of risk under study. It is negative for Martingale dilation (u must be concave, g must be convex - characterized by Chew, Karni and Safra), but positive for the mere notion of preference for fair certainty over randomness. The answer is positive as well for the dispersive order of Bickel & Lehmann (characterized by CCM). The talk will describe current attempts by CCM to characterize the preference for fair certainty.