Before revision:  When combined with an additional restriction, the above conditions are shown to be necessary and sufficient for the canonical Additive Random Utility Model (ARUM) . The sufficiency-proof is constructive, and facilitates nonparametric identification of preference-distributions  .
After revision:  When combined with an additional restriction, the above conditions are equivalent to the canonical Additive Random Utility Model (ARUM) . The sufficiency-proof is constructive, and facilitates nonparametric identification of preference-distributions .