Control with guide strategies for Markov games

We consider the system of many particles. Each particle can be in a state. We assume that the number of states is finite and fixed. Elementary event in this case is the changing of the state of one particle. The probability of this event is determined by some Kolmogorov matrix. The dynamics of the vector of populations is a Markov chain. Population of each state is the fraction of number of particle being in the given state and the total number of particles. The Kolmogorov matrix can be function of time and vector of poulations. V.N. Kolokoltsov showed that if the total number of particles tends to infinity then the Markov chain converges to a deteministic system. N the talk we assume that the Kolmogorov matrix is controlled i.e. it depends not only on time and populations vector but on controls of two players. Such systems are called Markov games. We consider zero-sum case. It is shown that using the value function for the limit deterministic system one can construct a control with guide strategy providing approximate solution of Markov game. The distance between a realization of the Markov game and a realization of the guide is proportional to $N^{-1/3}$ with the probability $\thicksim N^{-1/3}$, where $N$ is the total number of particles.