ABSTRACT
Extending the concept of a binary choice system, this chapter considers finite systems of probabilities of qualitative experimental observations. These probabilities are conceived as being assigned to corresponding formal statements (Boolean combinations of basic relational statements). Mixture models are discussed which assume that subjects’ responses are based on underlying finite algebraic structures of a certain type, each of which occurs with a certain probability. It is shown that for each system of probabilities on formal statements the following are equivalent: (i) it is explainable by a mixture model, (ii) it is explainable by a generalized (distribution-free) random utility model, and (iii) it satisfies a finite set of testable linear inequalities. A detailed account is provided of the role of noncoincidence (which may or may not be satisfied by the generalized random utility models considered). A general procedure is described that allows one to derive some simple necessary linear inequalities. In our approach elementary concepts of random utility theory, axiomatic measurement theory and the theory of convex polytopes are connected. Various examples are discussed to illustrate the wide range of possible applications covered by the theorems presented.
