Mean Field Games of Control and Cryptocurrency Mining

This paper studies Mean Field Games (MFGs) in which agent dynamics are given by jump processes of controlled intensity, with mean-field interaction via the controls and affecting the jump intensities. We establish the existence of MFG equilibria in a general discrete-time setting, and prove a limit theorem as the time discretization goes to zero, establishing equilibria in the continuous-time setting for a class of MFGs of intensity control. This motivates numerical schemes that involve directly solving discrete-time games as opposed to coupled Hamilton-Jacobi-Bellman and Kolmogorov equations. As an example of the general theory, we consider cryptocurrency mining competition, modeled as an MFG both in continuous and discrete time, and illustrate the effectiveness of the discrete-time algorithm to solve it.