Mean Field Games of Control and Cryptocurrency Mining
Nicolas Garcia, Ronnie Sircar, H. Mete Soner
TL;DR
This work develops a general framework for Mean Field Games where jump dynamics are controlled via the jump intensity and interact through controls, establishing finite-horizon discrete-time existence and a limit theorem to continuous time. A Kakutani-fixed-point construction yields MFG equilibria in the discrete setting, and the continuous-time equilibria are obtained as limits of discrete-time equilibria using relaxed controls and Skorokhod representations. The authors instantiate the theory in a cryptocurrency mining MFG, solving the discrete-time version with damped fixed-point iterations and demonstrating wealth-dynamics similar to PDE-based analyses, while also proving continuous-time existence and, under mild conditions, sharp equilibria. Overall, the paper provides a practical, scheme-based approach to MFGs with intensity control and strengthens the connection between discrete-time computations and continuous-time equilibria in jump-driven systems, with concrete relevance to crypto mining competition models.
Abstract
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.