The concept of random utility is based on the notion that an individual’s preferences are not deterministic but stochastic. For example, if requested to select one from an offered set of alternatives, the subject may not always choose the same alternative. The theory of random utility posits that each alternative is internally represented by a random variable, and that the choice is determined by which of the variables happens to have the maximum value in a given trial. The history of the theory goes back to Thurstone (1927) and, later, to Block and Marschak (1960). An important insight of the latter authors was that the existence of a probability measure over the set of all preferences (or rankings) – the so-called ranking probabilities, i.e., the elementary probabilities assigned to linear orders over the alternatives – is necessary and sufficient to guarantee that observed choices from subsets are due to an underlying random utility distribution.

Random utility theory has been further developed in several disciplines, including mathematical psychology, statistics, and economics. The focus of investigation differs somewhat from one field to the other. In particular, psychologists are mostly interested in the properties of the underlying random variables and their distribution that determine choice behavior. For example, Thurstonian psychophysics studies how stimuli of a certain intensity level are represented internally to produce a response like “tone a is louder than tone b”. On the other hand, economists seek to identify the probability measure on the rankings (i.e., a strict linear order) that lead to selecting a specific alternative, so that the system of choice probabilities is “rationalizable”.

A remarkable result was obtained in Falmagne (1978), deriving necessary and sufficient conditions based on the nonnegativity of the Block–Marschak polynomials – alternating sums of the choice probabilities – for the existence of a random utility representation. Falmagne’s proof consists of recursively constructing a probability measure on the rankings. Later, Fiorini (2004) provided an alternative, non-recursive proof of Falmagne’s theorem by drawing upon a result from graph theory on so-called flow polytopes (see Doignon and Saito, 2023, Turansick, 2022). Recently, Suleymanov (2024) defined a property on the ranking distribution (“branching-independence”) and shows that branching-independence implies exactly the same ranking representation as Falmagne’s construction.

This paper follows yet a different approach to obtaining Falmagne’s result using an explicit construction of ranking probabilities by applying the Shannon maximum entropy theorem from information theory (Jaynes, 1957). The resulting ranking representation is then extended to include the well-known Luce choice model and the generalized extreme value (GEV) model. In this paper, we consider only the case of a finite set of alternatives.

The remainder of the paper is organized as follows: Section 2 recalls Falmagne’s representation problem, pointing out that it can be viewed as a system of linear equations to be solved, whose input data are choice probabilities. It then recalls Falmagne’s representation theorem, together with the Block—Marschak polynomials on which it relies. In Section 3, after showing that the initial linear system based on choice probabilities is equivalent to a linear system expressed in terms of Block–Marschak polynomials, we apply Shannon’s maximum entropy theorem to obtain an explicit particular solution to Falmagne’s representation problem. Section 4 presents two illustrative examples based on a block-structured contingency-table approach for ranking probabilities. The first, with four alternatives, establishes the general solution and two particular solutions corresponding to maximum and minimum entropy. The second, with five alternatives, constructs only the maximum-entropy solution, using two levels of contingency-table blocks built recursively. Section 5, after recalling Fishburn’s classical counterexample to the uniqueness of Falmagne’s representation problem (Fishburn, 1998), discusses the multiplicity of solutions in the most standard random utility models when there are more than three alternatives. In such cases, we emphasize the interior character of the maximum-entropy solution, indicating that a whole neighborhood of solutions exists around it. Finally, Section 6 explores two classical models – the Luce model and the GEV model – and shows how our results are used to construct explicit representations, and discusses how to obtain other solutions by modifying the entropy function used.