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Mean-field games (MFGs) provide a statistical physics-inspired modelling framework for decision-making in large populations of strategic, non-cooperative agents. Mathematically, these systems consist of a forwards–backwards in time-system of two coupled nonlinear partial differential equations (PDEs), namely, the Fokker–Plank (FP) and the Hamilton–Jacobi–Bellman (HJB) equations, governing the agent state and control distribution, respectively. In this work, we study a finite-time MFG with a rich global bifurcation structure using a reduced-order model (ROM). The ROM is a four-dimensional (4D) two-point boundary value problem (BVP) obtained by restricting the controlled dynamics to the first two moments of the agent state distribution, i.e. the mean and the variance. Phase space analysis of the ROM reveals that the invariant manifolds of periodic orbits around the so-called ‘ergodic MFG equilibrium’ play a crucial role in determining the bifurcation diagram and imparting a topological signature to various solution branches. We show a qualitative agreement of these results with numerical solutions of the full-order MFG PDE system.
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This content will become publicly available on December 1, 2025
Phase transition in a kinetic mean-field game model of inertial self-propelled agents
The framework of mean-field games (MFGs) is used for modeling the collective dynamics of large populations of non-cooperative decision-making agents. We formulate and analyze a kinetic MFG model for an interacting system of non-cooperative motile agents with inertial dynamics and finite-range interactions, where each agent is minimizing a biologically inspired cost function. By analyzing the associated coupled forward–backward in a time system of nonlinear Fokker–Planck and Hamilton–Jacobi–Bellman equations, we obtain conditions for closed-loop linear stability of the spatially homogeneous MFG equilibrium that corresponds to an ordered state with non-zero mean speed. Using a combination of analysis and numerical simulations, we show that when energetic cost of control is reduced below a critical value, this equilibrium loses stability, and the system transitions to a traveling wave solution. Our work provides a game-theoretic perspective to the problem of collective motion in non-equilibrium biological and bio-inspired systems.
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- Award ID(s):
- 2102112
- PAR ID:
- 10559388
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 34
- Issue:
- 12
- ISSN:
- 1054-1500
- Subject(s) / Keyword(s):
- mean field games synchronization collective behavior stability analysis hamilton jacobi bellman equation
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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