Numerical Method for Bertrand Mean Field Games Using Multigrid

We study dynamic Bertrand mean field games (MFG) with exhaustible capacities in which companies compete with each other using the price as the strategic variable and the interaction among the competition is through the average price. We consider both continuum mean field games and finite player games. Dynamic continuum mean field games can be modeled as a system of partial differential equations (PDEs) which consists of a backward Hamilton-Jacobi-Bellman (HJB) equation for the value function and a forward Kolmogorov equation for the density function. This system is coupled through the proportion of remaining companies and the average equilibrium price. Finite N-player games can be approximated using the framework of continuum mean field games. Due to the substantial computational cost of solving the HJB equation, we present an efficient multigrid method with Full Approximation Scheme (FAS) to solve this equation. An iterative algorithm is applied to solve the whole MFG problem. Numerical results illustrate the effects of the competition on the proportion of active companies and the average equilibrium price.