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Ranking Mean-Field Planning Games

Ali Almadeh, Tigran Bakaryan, Diogo Gomes, Melih Ucer

TL;DR

A monotonicity structure in the associated operator 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.

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.

Ranking Mean-Field Planning Games

TL;DR

A monotonicity structure in the associated operator 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.

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.
Paper Structure (13 sections, 20 theorems, 132 equations)

This paper contains 13 sections, 20 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Classical Solutions
  3. Assumptions and Preliminary Identities
  4. The Linking Lemma
  5. Uniqueness of Classical Solutions
  6. Weak Solutions
  7. Preliminaries and Assumptions
  8. Monotone Operator Theory
  9. Functions of Bounded Variation
  10. Growth Assumptions
  11. Operator Formulation
  12. Existence of Approximate Solutions
  13. A Priori Estimates and Convergence

Key Result

Theorem 1.2

Suppose Assumptions assump:H and assump:positivity hold. Then Problem associated problem admits at most one classical solution $\varphi$ with $\varphi_{x} > 0$.

Theorems & Definitions (51)

  • Remark 1.1: On the Choice of Coupling
  • Theorem 1.2: Uniqueness
  • Corollary 1.3
  • Definition 1.4: Weak Solution
  • Remark 1.5: Boundary conditions and Minty formulation
  • Theorem 1.6: Existence of Weak Solutions
  • Remark 2.1: Properties of $L$ and $F$
  • Lemma 2.2
  • proof
  • Lemma 2.3: Linking Lemma
  • ...and 41 more