Tropical linear regression and mean payoff games: or, how to measure the distance to equilibria

TL;DR

This paper introduces a tropical linear regression framework to approximate points with tropical hyperplanes, establishes duality with tropical polyhedra, and links the problem to mean payoff games, with applications in auction theory.

Contribution

It develops a polynomial-time equivalent formulation of tropical regression as mean payoff games and applies it to measure market distances to equilibrium in auction settings.

Findings

Strong duality theorem for tropical regression.

Polynomial-time equivalence to mean payoff games.

Application to quantifying market distance to equilibrium.

Abstract

We study a tropical linear regression problem consisting in finding the best approximation of a set of points by a tropical hyperplane. We establish a strong duality theorem, showing that the value of this problem coincides with the maximal radius of a Hilbert's ball included in a tropical polyhedron. We also show that this regression problem is polynomial-time equivalent to mean payoff games. We illustrate our results by solving an inverse problem from auction theory. In this setting, a tropical hyperplane represents the set of equilibrium prices. Tropical linear regression allows us to quantify the distance of a market to the set of equilibria, and infer secret preferences of a decision maker.