  In random expected utility (Gul and Pesendorfer, 2006), the distribution of
preferences is uniquely recoverable from random choice. This paper shows
through two examples that such uniqueness fails in general if risk preferences
are random but do not conform to expected utility theory. In the first,
non-uniqueness obtains even if all preferences are confined to the betweenness
class (Dekel, 1986) and are suitably monotone. The second example illustrates
random choice behavior consistent with random expected utility that is also
consistent with random non-expected utility. On the other hand, we find that if
risk preferences conform to weighted utility theory (Chew, 1983) and are
monotone in first-order stochastic dominance, random choice again uniquely
identifies the distribution of preferences. Finally, we argue that, depending
on the domain of risk preferences, uniqueness may be restored if joint
distributions of choice across a limited number of feasible sets are available.
