Long time behaviour and turnpike solutions in mildly non-monotone mean field games

TL;DR

This paper analyzes the long-term behavior of mildly non-monotone mean field games, demonstrating well-posedness, turnpike properties, and convergence results without relying on linearization or master equations.

Contribution

It extends the analysis of mean field games to mildly non-monotone couplings, showing well-posedness and long-term properties under less restrictive conditions.

Findings

Well-posedness of systems with mildly non-monotone couplings due to noise effects

Turnpike property and convergence to ergodic solutions in long time horizon

Results applicable to Lipschitz and quadratic Hamiltonians with mild growth

Abstract

We consider mean field game systems in time-horizon $(0,T)$, where the individual cost functional depends locally on the density distribution of the agents, and the Hamiltonian is locally uniformly convex. We show that, even if the coupling cost functions are mildly non-monotone, then the system is still well posed due to the effect of individual noise. The rate of anti-monotonicity (i.e. the aggregation rate of the cost functions) which can be afforded depends on the intensity of the diffusion and on global bounds of solutions. We give applications to either the case of globally Lipschitz Hamiltonians or the case of quadratic Hamiltonians and couplings having mild growth. Under similar conditions, we investigate the long time behavior of solutions and we give a complete description of the ergodic and long term properties of the system. In particular we prove: (i) the turnpike property of solutions in the finite (long) horizon $(0,T)$, (ii) the convergence of the system in $(0,T)$ towards the system in $(0,\infty)$, (iii) the vanishing discount limit of the infinite horizon problem and the long time convergence towards the ergodic stationary solution. This way we extend previous results which were known only for the case of monotone and smoothing couplings; our approach is self-contained and does not need the use of the linearized system or of the master equation.