Blog
Math of Choice
A blog series exploring the mathematical foundations of discrete choice.
Based on Alfred Galichon’s forthcoming book, Discrete Choice Models: Mathematical Methods, Econometrics, and Data Science, Princeton University Press, April 2026.
About this series
Discrete choice models are the workhorses of modern applied econometrics, used to predict everything from commuter transport choices to consumer product selection. However, the standard textbook treatment often overlooks the rich mathematical and computational properties that govern these models.
In this series, we follow the roadmap of the book and we start with peeling back the layers of the “Random Utility Model” to reveal its deep connections with convex analysis, information theory, optimal transport, extreme value theory, and generalized linear models.
Still following the book, we then use random utility models as a building block for demand inversion methods, models of trade, dynamic pricing and hedonic models, as well as matching models with and without transfers.
Table of Contents
The role of entropy in discrete choice models, part 1: the logit case
January 5, 2026
We begin with the basics. How does random utility translate into social welfare? We explore the classic logit model and derive the famous result that the “randomness” of the individual aggregates into the “entropy” of the group (Gibbs-Shannon entropy). We introduce the Daly-Zachary-Williams theorem and the concept of the representative consumer along the way.
The role of entropy in discrete choice models, part 2: generalized entropy
January 12, 2026
What happens when we leave the logit world? We show that every additive random utility model has an associated “Generalized Entropy of Choice.” We explore how different distributions of noise lead to different convex regularization penalties, and establish the duality between welfare maximization and entropy minimization.
Demand inversion via optimal transport
January 19, 2026
We shift perspective from the theorist to the econometrician. How do we recover unknown preferences from observed market shares? We uncover the geometric nature of this inverse problem and show that demand inversion is equivalent to solving an Optimal Transport problem between the distribution of random shocks and the distribution of choices.
Simulation and estimation via Linear Programming
January 26, 2026
Theory meets practice. When integrals are too hard to calculate, we turn to simulation. We explore how to estimate market shares using Monte Carlo methods and, more importantly, how to invert demand using Linear Programming by applying the optimal transport principles to simulated data.
GEV models and the Pickands representation
Coming Soon
We move beyond independence. We explore Generalized Extreme Value (GEV) models, which allow for correlation between alternatives, and discuss their representation using max-stable distributions and i.i.d. logit factors.