For simple prospects of the kind routinely used for certainty equivalent elicitation, random expected utility preferences imply a conditional expectation function that can mimic deterministic rank dependent preferences. That is, an agent with random expected utility preferences can have mean certainty equivalents that look exactly like rank dependent probability weighting functions of the inverse-s shape discussed by Quiggin (1982) and later advocated by Tversky and Kahneman (1992) and other scholars. It seems that certainty equivalents cannot nonparametrically identify preferences, at least not in every relevant sense, since their conditional expectation depends on assumptions concerning the source and nature of their variability.
Working paper version
Wilcox, N.T. (2016). Random expected utility and certainty equivalents: Mimicry of probability weighting functions. ESI Working Paper 16-14. Retrieved from http://digitalcommons.chapman.edu/esi_working_papers/192
Wilcox, N.T. J Econ Sci Assoc (2017) 3: 161. https://doi.org/10.1007/s40881-017-0042-1