# American Institute of Mathematical Sciences

September  2019, 9(3): 571-605. doi: 10.3934/mcrf.2019026

## Optimal control and zero-sum games for Markov chains of mean-field type

 1 Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden 2 Learning & Game Theory Laboratory, New York University, 19 Washington Square North New York, NY 10011, USA

* Corresponding author: H. Tembine

Received  October 2017 Revised  November 2018 Published  April 2019

We establish existence of Markov chains of mean-field type with unbounded jump intensities by means of a fixed point argument using the total variation distance. We further show existence of nearly-optimal controls and, using a Markov chain backward SDE approach, we suggest conditions for existence of an optimal control and a saddle-point for respectively a control problem and a zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by such Markov chains of mean-field type.

Citation: Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control & Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026
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