Monotone Multispecies Flows

TL;DR

This paper introduces a new concept of $mbda$-monotonicity for multispecies PDE systems, establishing conditions for convergence to steady states, and explores its implications in game theory, mean-field games, and economic models.

Contribution

It extends monotonicity concepts to Wasserstein spaces for multispecies systems, linking it to convergence, Nash equilibria, and displacement convexity, with diverse applications and numerical demonstrations.

Findings

Monotonicity ensures convergence to unique steady states.

Established the relationship between monotonicity and displacement convexity.

Numerical examples demonstrate practical convergence in economic and game models.

Abstract

We present a novel notion of $\lambda$-monotonicity for an $n$-species system of partial differential equations governed by mass-preserving flow dynamics, extending monotonicity in Banach spaces to the Wasserstein-2 metric space. We show that monotonicity implies the existence of and convergence to a unique steady state, convergence of the velocity fields and second moments, and contraction in the Wasserstein-2 metric, at rates dependent on $\lambda$. In the special setting of Wasserstein-2 gradient descent of different energies for each species, we prove convergence to the unique Nash equilibrium of the associated energies and delineate the relationship between monotonicity and displacement convexity. This extends known zero-sum results in infinite-dimensional game theory to the general-sum setting. We provide a number of examples of monotone coupled gradient flow systems, including cross-diffusion, gradient flows with potentials, nonlocal interaction, linear and nonlinear diffusion, and min-max systems, and draw connections to a class of mean-field games. Numerically, we demonstrate convergence of a four-player economic model for service providers and strategic users competing in a market, and a degenerately monotone game.