Representing Choice Probabilities by Ranking Probabilities Via Entropy Maximization

Falmagne's representation problem is revisited by maximizing Shannon entropy applied to ranking probabilities, under the linear constraints imposed by choice probabilities. Unlike Falmagne's recursive construction, our method leads directly to an explicit solution, obtained after transforming the initial system into an equivalent one via alternating sums, in the spirit of Block-Marschak polynomials. We compute this solution for the Luce model and the generalized extreme value model, and show that, as soon as there are at least four alternatives, the construction based on Shannon entropy is only one among infinitely many possible representations. Other solutions could be obtained by maximizing alternative entropy functions, further highlighting the potential role of information theory in enriching the analysis of stochastic choice.