We study identifying assumptions in the random utility model, the standard empirical paradigm in modern economics. Our main result characterizes those ex-ante restrictions which lead to identification. Our characterization utilizes a simple class of mass preserving swaps. These swaps take in two preferences which share a common upper and lower contour set but disagree on their ordering within these two sets. The output of this procedure is two preferences which are created by swapping the matching between the ordering of the upper and lower contour sets of the two input preferences. Any two distributions over preferences are observationally equivalent if and only if one can be recovered from the other by a finite sequences of such swaps. It follows that a random utility model is identified if it does not contain any two such distributions over preferences.

We readdress the problem of nonparametric statistical testing of random utility models proposed in Kitamura and Stoye [2018]. Although their test is elegant, it is subject to computational constraints which leaves execution of the test infeasible in many applications. Testing the random utility hypothesis is equivalent to testing whether the observed data lies within a polyhedral cone. Polyhedral cones can be defined through their vertices or through their facets. In higher dimensions and when the vertices of the cone fail to be linearly independent, the number of vertices and number of faces of a polyhedral cone can differ. We note that much of the computational burden in Kitamura and Stoye's test is due to their test defining a polyhedral cone through its vertices rather than its facets.