Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A variational mean field game of controls with free final time and pairwise interactions

Guilherme Mazanti, Laurent Pfeiffer, Saeed Sadeghi Arjmand

TL;DR

This work addresses a mean field game of controls with pairwise interactions and a free final time, motivated by crowd-motion scenarios. It develops a variational framework on abstract Polish spaces, showing that equilibria correspond to critical points of a potential $\mathcal{J}$ and proving existence, with strong/weak equilibrium equivalence under suitable conditions. The abstract theory is then applied to a Cucker–Smale–type crowd model, incorporating a free exit time and a numerical demonstration to illustrate the equilibria in a two-population setting. The results provide a rigorous variational path to existence and computation for complex MFGs with free horizons and pairwise interactions, and suggest viable numerical strategies via the potential structure.

Abstract

This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.

A variational mean field game of controls with free final time and pairwise interactions

TL;DR

This work addresses a mean field game of controls with pairwise interactions and a free final time, motivated by crowd-motion scenarios. It develops a variational framework on abstract Polish spaces, showing that equilibria correspond to critical points of a potential and proving existence, with strong/weak equilibrium equivalence under suitable conditions. The abstract theory is then applied to a Cucker–Smale–type crowd model, incorporating a free exit time and a numerical demonstration to illustrate the equilibria in a two-population setting. The results provide a rigorous variational path to existence and computation for complex MFGs with free horizons and pairwise interactions, and suggest viable numerical strategies via the potential structure.

Abstract

This article considers a mean field game model inspired by crowd motion models in which agents aim at reaching a given target set and wish to minimize a cost consisting of an individual running cost, an individual cost depending on the arrival time at the target set, and an interaction running cost, which takes the form of pairwise interactions with other agents through both positions and velocities. We subsume this game under a more general class of games on abstract Polish spaces with pairwise interactions, and prove that the latter games have a variational structure (in the sense that their equilibria can be characterized as critical points of some potential functional) and admit equilibria. We also discuss two a priori distinct notions of equilibria, providing a sufficient condition under which both notions coincide. The results for the games in abstract Polish spaces are applied to our mean field game model, and a numerical illustration concludes the paper.
Paper Structure (9 sections, 24 theorems, 101 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 24 theorems, 101 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mean field game model with pairwise interactions
  3. A general non-atomic game with pairwise interactions
  4. First properties
  5. Potential game
  6. Existence of equilibria
  7. Strong equilibria
  8. Application to the mean field game model with pairwise interactions
  9. Numerical illustration

Key Result

Lemma 3.6

Assume that Hypo-XY-Polish, Hypo-LH, and Hypo-H-leq-L are satisfied. Let $C > 0$ be the constant from Hypo-H-leq-L and $\mathcal{L}$ and $\mathcal{H}$ be defined as in eq:def-mathcal-L and eq:def-mathcal-H. Then, for every $(Q, \widetilde{Q}) \in \mathcal{P}(X) \times \mathcal{P}(X)$, we have

Figures (1)

  • Figure 5.1: Trajectories of the agents of the game described in Section \ref{['sec:illustration']} at times (a) $t = 0$, (b) $t = 0.24$, (c) $t = 0.36$, and (d) $t = 0.76$. Each population is represented by a different color, trajectories are represented by solid lines, and circles represent the current positions of the agents.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 3.6
  • Corollary 3.7
  • Remark 3.8
  • Lemma 3.9
  • ...and 48 more