Optimal and Equilibrium Transport: Theory and Applications to Economics and Data Science

New York University, New York NY

Investigators

Abstract

This project is about developing some of the novel mathematical tools that are required for the new peer-to-peer economy. Recent years have witnessed the astonishing emergence of online matching platforms (such as Uber, Odesk, Airbnb, OkCupid), where agents seek to team up with each other to exchange various goods and services or form various partnerships. These platforms are provided with massive data on the populations of users, and need to take real-time decisions regarding market clearing, such as imposing surge prices, or suggesting assignments. The goal of this project is to expand the boundaries of a central tool: optimal transport theory, which was initiated in the 18th century but experienced a surge of research activity over the last decade, and which is very relevant for problems in matching, pricing, and data analysis. The project is divided into three separate projects; the first two are related, and the third one is further apart. The first project will define the framework of equilibrium transportation, which extends optimal transport to equilibrium situations. It will focus on the semi-discrete case, where one of the distributions is continuous and the other one is discrete. In this setting, it will derive analytical properties of the solution such as existence, uniqueness, monotonicity, and lattice structure. This framework is applied to the solution of a problem on the labor market. In another application, it is extended to the case of equilibrium flows on networks. The second project will demonstrate the usefulness of the equilibrium transportation framework for computing equilibrium in markets such as the taxi market, which cannot be modeled as an optimal transport problem since prices are fixed; in this type of markets, demand and supply are balanced by a congestion cost induced by limited availability. The project will derive the properties of the equilibrium, and use the framework to study the surge pricing problem: when demand and supply are uncertain, which price should be set to minimize the expected market inefficiency? The third project will focus on computational challenges brought by the introduction (due to the PI and collaborators) of the notion of vector quantile regression in the statistical literature. The project will focus on iterated projection methods (which have been very successfully applied to classical optimal transport problems) for this new class of problems.