[Paper Review] Displacement convexity for first-order mean-field games

Diogo A. Gomes, Tommaso Seneci|arXiv (Cornell University)|Jul 18, 2018

Point processes and geometric inequalities4 citations

TL;DR

This paper establishes displacement convexity for first-order mean-field games (MFGs) by extending optimal transport techniques, proving that $L^q$ norms of the density are convex in time under certain conditions. It derives new a priori bounds, including log-convexity of $L^q$ norms and uniform $L^∞$ bounds for solutions, even in the presence of congestion, via a novel convexity framework applied to the MFG planning problem.

ABSTRACT

Here, we consider the planning problem for first-order mean-field games (MFG). When there is no coupling between players, MFG degenerate into optimal transport problems. Displacement convexity is a fundamental tool in optimal transport that often reveals hidden convexity of functionals and, thus, has numerous applications in the calculus of variations. We explore the similarities between the Benamou-Brenier formulation of optimal transport and MFG to extend displacement convexity methods from to MFG. In particular, we identify a class of functions, that depend on solutions of MFG, that are convex in time and, thus, obtain new a priori bounds for solutions of MFG. A remarkable consequence is the log-convexity of $L^q$ norms. This convexity gives bounds for the density of solutions of the planning problem and extends displacement convexity of $L^q$ norms from optimal transport. Additionally, we prove the convexity of $L^q$ norms for MFG with congestion.

Motivation & Objective

• To extend displacement convexity methods from optimal transport to first-order mean-field games (MFGs), particularly the planning problem.
• To identify a class of functionals depending on MFG solutions that are convex in time, enabling new a priori estimates.
• To establish log-convexity of $L^q$ norms for MFG densities, generalizing a key property from optimal transport.
• To prove uniform $L^\infty$ bounds for MFG solutions under congestion by identifying conditions under which $L^q$ norms are convex.

Proposed method

• Adapts the Benamou-Brenier formulation of optimal transport to the MFG setting, using time-parametrized density paths $\rho^t$.
• Derives second-order time derivatives of $\int U(m(x,t))\,dx$ along MFG trajectories to analyze convexity.
• Applies a weighted convexity framework using parameters $\alpha$ and $\beta$ to control the nonlinearity in the Hamilton-Jacobi and continuity equations.
• Uses matrix trace inequalities and curvature-type estimates to ensure non-negativity of the second derivative, guaranteeing convexity.
• Imposes conditions on $\alpha$ and $\beta$ to ensure the second derivative is non-negative for large $q$, leading to convexity of $L^q$ norms.
• Applies the resulting convexity to derive uniform $L^\infty$ bounds on the density $m(x,t)$, independent of time.

Experimental results

Research questions

• RQ1 Can displacement convexity be extended from optimal transport to first-order mean-field games?
• RQ2 Under what conditions is the $L^q$ norm of the MFG density convex in time?
• RQ3 Does the log-convexity of $L^q$ norms in optimal transport persist in the MFG setting with non-zero interaction terms?
• RQ4 Can uniform $L^\infty$ bounds on the density be derived using convexity-based estimates in the presence of congestion?
• RQ5 What parameter constraints on $\alpha$ and $\beta$ ensure convexity of $L^q$ norms for large $q$ in the congested MFG case?

Key findings

• The $L^q$ norm of the MFG density $m(x,t)$ is convex in time for all $q$ sufficiently large, under appropriate conditions on $\alpha$ and $\beta$.
• Log-convexity of $L^q$ norms is established for first-order MFGs, extending a key property from optimal transport.
• For the congested MFG case, the $L^q$ norm remains convex in time when $\beta \geq 2$ and $\alpha < \frac{2}{\beta-1}$, or $1 < \beta < 2$ and $\alpha < 2$, for large $q$.
• A uniform $L^\infty$ bound on the density is obtained: $\|m(\cdot,t)\|_{L^\infty} \leq \max\{\|m^0\|_{L^\infty}, \|m^T\|_{L^\infty}\}$, valid for all $t \in [0,T]$.
• In dimension $d=1$, the bound holds even when $\alpha = \frac{2}{\beta-1}$ or $\alpha = 2$, due to the absence of the $1/d$ correction term.
• The proof relies on a novel matrix trace inequality to control curvature terms, ensuring non-negativity of the second derivative of the $L^q$ functional.