Abstract
To explain inconsistency in choice experiments, where a subject on repeated presentations of one particular subset of alternatives does not always select the same alternative, random utility theory models the subject's evaluation of a stimulus by a random variable sampled at each presentation of the stimulus. The problem addressed in this entry is the characterization of random utility theory in its most general form (i.e., with an arbitrary joint distribution of the random variables) in terms of the testable restrictions it imposes on the choice data. While for the experimental paradigm where choices are obtained for every subset of alternatives this characterization problem has been solved, it is still open for the case of binary choice probabilities, where just all 2-element subsets are offered. For this case, the problem turns out to be equivalent to finding a linear description of the linear ordering polytope, which constitutes the convex hull of all linear orders of the alternatives, identifying these orders with their indicator functions (0-1 vectors). It is illustrated that many necessary conditions for a random utility representation can be found by applying graph-theoretic techniques, but also that with increasing the number of alternatives there is a combinatorial explosion of such necessary conditions with no apparent structural regularities. The problem of a complete characterization for an arbitrary number of alternatives seems intractable at the moment. Finally it is shown how this characterization problem for binary choice probabilities generalizes to other instances of probabilistic measurement.
