Numerical Methods for Finite-State Mean-Field Games Satisfying a Monotonicity Condition

Abstract

Here, we develop numerical methods for finite-state mean-field games (MFGs) that satisfy a monotonicity condition. MFGs are determined by a system of differential equations with initial and terminal boundary conditions. These non-standard conditions make the numerical approximation of MFGs difficult. Using the monotonicity condition, we build a flow that is a contraction and whose fixed points solve both for stationary and time-dependent MFGs. We illustrate our methods with a MFG that models the paradigm-shift problem.

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Correspondence to Diogo A. Gomes.

D. Gomes was partially supported by KAUST baseline and start-up funds and KAUST SRI, Center for Uncertainty Quantification in Computational Science and Engineering. J. Saúde was partially supported by FCT/Portugal through the CMU-Portugal Program.

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Gomes, D.A., Saúde, J. Numerical Methods for Finite-State Mean-Field Games Satisfying a Monotonicity Condition. Appl Math Optim 83, 51–82 (2021). https://doi.org/10.1007/s00245-018-9510-0

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Keywords

• Mean-field games
• Finite state problems
• Monotonicity methods

• 91A13
• 91A10
• 49M30