Mean Field Game Problem with Kantorovich Potential Coupling: An Attrition model
Abstract
In this work we address a class of Mean-Field Game (MFG) systems on the flat torus ๐d in which individual agent incentives arise from macroscopic, distribution-dependent objectives governed by the Wasserstein-2 (W2) distance. Taking the first variation (Lions derivative) of the Wasserstein-based macroscopic functional yields a coupled forward-backward system of Hamilton--Jacobi--Bellman (HJB) and Fokker--Planck (FP) equations featuring a nonlocal coupling driven by the Kantorovich optimal transport potential. As a primary case study, we formulate a large-scale attacker-defender attrition model aimed at protecting High-Value Units (HVUs). By defining population-wise attrition rates via the W2 metric and casting the defense strategy as a Pareto-optimal McKean--Vlasov control problem, we derive a concrete HJB--FP system with a Kantorovich potential coupling. We prove the local-in-time existence of classical solutions for this system using Schauder's fixed-point theorem. Finally, we compute the MFG dynamics by combining the Sinkhorn algorithm with ResNet-based Physics-Informed Neural Networks (PINNs), and simulate two-dimensional attacker-defender engagements across diverse attrition parameters, control costs, and multi-HVU configurations.
Related
- Asymptotic analysis of an optimal control problem for a phase field tumor growth model with hyperbolic relaxation of the chemical potential
- Stability of Kantorovich potentials via heat kernel regularization
- A coupling normal form for reversible center-center Hamiltonians with application to the restricted three-body problem
- Operator preconditioning of the Costabel FEM-BEM coupling for the time-harmonic Maxwell transmission problem
- Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance
BibTeX
@misc{arakelyan2026mean,
title = {{Mean Field Game Problem with Kantorovich Potential Coupling: An Attrition model}},
author = {Avetik Arakelyan and Davit Alaverdyan and Naira Hovakimyan and Isaac Kaminer and Tigran Bakaryan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01616},
eprint = {2611.01616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01616},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Mean Field Game Problem with Kantorovich Potential Coupling: An Attrition model
AU - Avetik Arakelyan
AU - Davit Alaverdyan
AU - Naira Hovakimyan
AU - Isaac Kaminer
AU - Tigran Bakaryan
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01616
UR - https://arxiv.org/abs/2611.01616
AB - In this work we address a class of Mean-Field Game (MFG) systems on the flat torus ๐^d in which individual agent incentives arise from macroscopic, distribution-dependent objectives governed by the Wasserstein-2 (W_2) distance. Taking the first variation (Lions derivative) of the Wasserstein-based macroscopic functional yields a coupled forward-backward system of Hamilton--Jacobi--Bellman (HJB) and Fokker--Planck (FP) equations featuring a nonlocal coupling driven by the Kantorovich optimal transport potential. As a primary case study, we formulate a large-scale attacker-defender attrition model aimed at protecting High-Value Units (HVUs). By defining population-wise attrition rates via the W_2 metric and casting the defense strategy as a Pareto-optimal McKean--Vlasov control problem, we derive a concrete HJB--FP system with a Kantorovich potential coupling. We prove the local-in-time existence of classical solutions for this system using Schauder's fixed-point theorem. Finally, we compute the MFG dynamics by combining the Sinkhorn algorithm with ResNet-based Physics-Informed Neural Networks (PINNs), and simulate two-dimensional attacker-defender engagements across diverse attrition parameters, control costs, and multi-HVU configurations.
ER -