Local Turnpike Stability for Two-Population Second-Order Mean-Field Game Systems
Abstract
This paper establishes a local turnpike property for a two-population, second-order mean-field game system with a quadratic Hamiltonian, in which the running cost for each population may depend on both population densities. In the case of two populations, imposing a scalar stability condition separately on each species is not sufficient because it neglects cross-interaction effects. The appropriate requirement is instead a matrix-valued coercivity condition on vector perturbations (μ1,μ2), which involves the symmetric part of the linearized interaction matrix together with weighted Poincaré inequalities associated with the two stationary measures. Consequently, in the multi-population framework, diffusion still acts independently on each population, but the interactions are controlled by the full coupling matrix. A naive scalar energy functional fails to capture mixed second-variation terms arising from costs that depend on both densities. We prove that matrix coercivity is precisely equivalent to the strict positivity of this second variation, and we provide a spectral characterization in terms of the principal spectral value of the coupled stationary linearized operator. For linear models with cross-interactions, this yields explicit algebraic conditions that separate stable parameter ranges from unstable ones. In the non-potential framework, the skew-symmetric part of the interaction matrix leaves the coercivity threshold unchanged, but it can induce transient oscillatory dynamics. Moreover, the scalar turnpike theory extends in a natural, though nontrivial, manner to multi-population systems, where stability is governed by the positivity of a coupled matrix-valued second variation that encodes both self-interactions and cross-interactions.
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BibTeX
@misc{ghattassi2026local,
title = {{Local Turnpike Stability for Two-Population Second-Order Mean-Field Game Systems}},
author = {Mohamed Ghattassi and Fanze Kong and Nader Masmoudi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17103},
eprint = {2610.17103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17103},
}
RIS
TY - UNPB
TI - Local Turnpike Stability for Two-Population Second-Order Mean-Field Game Systems
AU - Mohamed Ghattassi
AU - Fanze Kong
AU - Nader Masmoudi
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17103
UR - https://arxiv.org/abs/2610.17103
AB - This paper establishes a local turnpike property for a two-population, second-order mean-field game system with a quadratic Hamiltonian, in which the running cost for each population may depend on both population densities. In the case of two populations, imposing a scalar stability condition separately on each species is not sufficient because it neglects cross-interaction effects. The appropriate requirement is instead a matrix-valued coercivity condition on vector perturbations (μ_1,μ_2), which involves the symmetric part of the linearized interaction matrix together with weighted Poincaré inequalities associated with the two stationary measures. Consequently, in the multi-population framework, diffusion still acts independently on each population, but the interactions are controlled by the full coupling matrix. A naive scalar energy functional fails to capture mixed second-variation terms arising from costs that depend on both densities. We prove that matrix coercivity is precisely equivalent to the strict positivity of this second variation, and we provide a spectral characterization in terms of the principal spectral value of the coupled stationary linearized operator. For linear models with cross-interactions, this yields explicit algebraic conditions that separate stable parameter ranges from unstable ones. In the non-potential framework, the skew-symmetric part of the interaction matrix leaves the coercivity threshold unchanged, but it can induce transient oscillatory dynamics. Moreover, the scalar turnpike theory extends in a natural, though nontrivial, manner to multi-population systems, where stability is governed by the positivity of a coupled matrix-valued second variation that encodes both self-interactions and cross-interactions.
ER -