Random utility models with all choice probabilities expressible as ‘functions’ of the binary choice probabilities

Abstract

The strict utility model (or Luce's choice model) is compatible with an independent Thurstonian random utility model. It also has the property that the choice probabilities (on each finite subset A of a master set T) are representable by a rational function of the relevant binary choice probabilities (between elements of A), with the rational function being of the ‘same form’ for all set sizes. Previously researchers have asked whether the strict utility model is the only random utility model satisfying such a (rational) functional relation. This paper demonstrates that the choice probabilities generated by any independent Thurtonian model can be represented as a ‘function’ of the relevant binary choice probabilities, and that this is also the case for certain non-independent random utility models. However (under certain regularity conditions), the strict utility model is the only representation compatible with an independent Thurstonian model if either i) the ‘function’ for all finite set sizes is in an appropriate sense a serial iterate of the function on sets of size three or ii) the ‘function’ for sets of size three and four is rational, and the ‘function’ for sets of size four is a serial iterate of the function on sets of size three. The paper ends with a discussion of possible extensions of these results: for instance, is the strict utility model implied by the joint assumption of a rational functional relation and an independent random utility model?