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We study random utility models in which heterogeneity of preferences is modeled using an ordered collection of utilities, or types. The paper shows that these models are particularly amenable when combined with domains in which the alternatives of each decision problem are ordered by the structure of the types. We enhance their applicability by: (i) working with arbitrary domains composed of such decision problems, i.e., we do not need to assume any particularly rich data domain, and (ii) making no parametric assumption, i.e., we do not need to formulate any par- ticular assumption on the distribution over the collection of types. We characterize the model by way of two simple properties and show the applicability of our result in settings involving decisions under risk. We also propose a goodness-of-fit measure for the model and prove the strong consistency of extremum estimators defined upon it. We conclude by applying the model to a dataset on lottery choices.
