A 1/n Nash equilibrium for non-linear Markov games of mean-field-type on   finite state space
  We investigate mean field games for players, who are weakly coupled via their empirical measure. To this end we investigate time-dependent pure jump type propagators over a finite space in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics which leads to the well-known Mckean-Vlasov dynamics in the limit as the number N of players goes to infinity. The case where one player has an individual preference different to the ones of the remaining players is also covered. The limiting system represents a 1/N-Nash Equilibrium for the approximating system of N players. 
Rani Basna, Astrid Hilbert, and Vassili N. Kolokoltsov
math.OC math.DS math.PR