An approximate Nash equilibrium for pure jump Markov games of mean-field-type on continuous state space

Abstract

We investigate mean-field games from the point of view of a large number of indistinguishable players, which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markovian semi groups. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean–Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-Nash Equilibrium for the N-player games.

Keywords:

• Mean-field games
• pure jump Markov process
• dynamic programing
• optimal control
• non-linear Markov processes
• Koopman dynamics
• -Nash equilibrium

Acknowledgements

The authors are indebted to the unknown referee for structural and technical suggestions, which improved the paper substancially. We deeply enjoyed scientific discussions on control theory and mean-field games with Sigurd Assing, Fabio Bagagiolo, Rainer Buckdahn, Christine Grün, Juan Li, and Marianna Troeva. Rani Basna and Astrid Hilbert gratefully acknowledge financial support by the Royal Swedish Academy of Sciences.

Notes

No potential conflict of interest was reported by the authors.