On Quasi-stationary Mean Field Games Models
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We explore a mechanism of decision-making in mean field games with myopic players. At each instant, agents set a strategy which optimizes their expected future cost by assuming their environment as immutable. As the system evolves, the players observe the evolution of the system and adapt to their new environment without anticipating. With a specific cost structures, these models give rise to coupled systems of partial differential equations of quasi-stationary nature. We provide sufficient conditions for the existence and uniqueness of classical solutions for these systems, and give a rigorous derivation of these systems from N-players stochastic differential games models. Finally, we show that the population can self-organize and converge exponentially fast to the ergodic mean field games equilibrium, if the initial distribution is sufficiently close to it and the Hamiltonian is quadratic.
KeywordsMean field games Quasi-stationary models Nonlinear coupled PDE systems Long time behavior Self-organization N-person games Nash equilibria Myopic equilibrium
Mathematics Subject Classification35Q91 49N70 35B40
This work was supported by LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program ”Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and partially supported by project (ANR-16-CE40-0015-01) on mean field games. The author would like to thank Martino Bardi and Pierre Cardaliaguet for fruitful discussions.
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