On Solutions of Mean Field Games with Ergodic Cost

Abstract

A general class of mean field games are considered where the governing dynamics are controlled diffusions in ${\mathbb{R}}^d$. The optimization criterion is the long time average of a running cost function. Under various sets of hypotheses, we establish the existence of mean field game solutions. We also study the long time behavior of mean field game solutions associated with the finite horizon problem, and under the assumption of geometric ergodicity for the dynamics, we show that these converge to the ergodic mean field game solution as the horizon tends to infinity. Lastly, we study the associated $N$-player games, show existence of Nash equilibria, and establish the convergence of the solutions associated to Nash equilibria of the game to a mean field game solution as $N\to \infty$.

arXiv comment40 pages

References in corpus (9)

• A probabilistic weak formulation of mean field games and applications
• Continuous time finite state mean field games
• Mean field games via controlled martingale problems: Existence of Markovian equilibria
• Subgeometric ergodicity of strong Markov processes
• Mean Field Games and Nonlinear Markov Processes
• Mean Field Games with Ergodic cost for Discrete Time Markov Processes
• Convergence of The Relative Value Iteration for the Ergodic Control Problem of Nondegenerate Diffusions under Near-Monotone Costs
• A Relative Value Iteration Algorithm for Non-degenerate Controlled Diffusions
• A general characterization of the mean field limit for stochastic differential games

Cited by in corpus (2)

• Concentration of ground states in stationary Mean-Field Games systems
• Geometry of Information Structures, Strategic Measures and associated Control Topologies