Ranking Mean-Field Planning Games

TL;DR

This paper analyzes a one-dimensional Mean-Field Planning system with rank-based coupling, establishing solution equivalences, uniqueness, and existence results through potential reformulation and variational methods.

Contribution

It introduces a potential formulation for the ranking MFP system, proving solution equivalence, uniqueness under convexity, and existence via variational inequalities and Minty's method.

Findings

Established equivalence between classical solutions and potential problem.

Proved uniqueness of solutions under strict convexity assumptions.

Demonstrated existence of weak solutions in BV space using variational methods.

Abstract

This paper studies a one-dimensional Mean-Field Planning (MFP) system with a non-local, rank-based coupling. Using a potential formulation, we rewrite the system as an associated scalar partial differential equation. We prove an equivalence between classical solutions to the ranking MFP system with positive density and classical solutions to the associated potential problem, and we derive explicit reconstruction formulas. We then identify a monotonicity structure in the associated operator, which, under strict convexity assumptions, yields uniqueness of classical solutions to the associated problem and, hence, uniqueness of the ranking MFP system up to an additive constant in the value function. Finally, under superlinear growth assumptions, we exploit monotonicity to address existence in a low-regularity setting. By formulating a variational inequality for a q-Laplacian regularized operator, we apply Minty's method to establish the existence of weak solutions in the space of functions of bounded variation for a relaxed potential formulation.