Linear-quadratic-Gaussian mean field games under high rate quantization

This paper studies discrete time linear-quadratic-Gaussian mean field games (MFGs) with state measurement quantization. In this problem formulation each agent is coupled via both its individual linear stochastic dynamics and its individual quadratic discounted cost function to the average of all agents´ states. In addition, each agent only observes a high rate quantized version of its own state´s local noisy measurement. For this dynamic game problem, the MFG system consisting of a set of coupled deterministic equations is derived which approximates the stochastic system of agents as the population size goes to infinity. In a finite population system each agent only uses (i) the high rate quantization of its own local noisy measurement, and (ii) a function approximating the population effect which is computed offline from the MFG system. It is shown that the resulting set of high rate quantized MFG control strategies possesses an ε-Nash equilibrium property where ε goes to zero as the population size approaches infinity and the individual high rate quantization noise goes to zero.