Representing choice probabilities by ranking probabilities via entropy maximization

Falmagne's representation problem is revisited, where a unique and explicit probability measure on rankings is obtained by maximizing Shannon entropy under linear constraints ensuring the reproduction of choice probabilities. Unlike Falmagne's recursive construction, we propose an explicit construction of ranking probabilities by applying the Shannon maximum entropy theorem. By transforming the initial system of constraints into an equivalent one via alternating sums, as used in Block and Marschak polynomials, we obtain an explicit analytical expression for ranking probabilities. We derive this representation for the Luce model and generalized extreme value models and show that, for these models, when n ≥ 4, this construction is only one of infinitely many possible representations. Other representations could be obtained by maximizing alternative entropy measures, such as Rényi entropy, opening the possibility of constructing new representations and further highlighting the relevance of this approach. Thus, our paper establishes a promising connection between stochastic choice and information theory.