Mean Field Games with Major and Minor Players

Abstract

This thesis extends the standard mean field game theory in two directions. The first direction is mean field games with major and minor players (MMMFG), where we assume that there exists a unique major player and a large number of minor players. In this model, the effect of the major player will not be averaged out when the number of minor players goes to infinity. We are able to identify the limiting problem as a two-player stochastic differential game, in which the control problem faced by the major player is of conditional McKean-Vlasov type, while the optimization problem faced by the representative minor player is a standard control problem. A matching procedure then follows the solution of the two-player game, which gives a FBSDE of McKean-Vlasov type as a characterization of the solution of the limiting problem. The other direction is mean field stopping games (MFSG) and mean field stopping games with major and minor players (MMMFSG), where players are solving optimal stopping problems instead of stochastic control problems as in the standard mean field games. With the help of randomized stopping times, we are able to prove the existence of solutions for the limiting games. We also develop an algorithm to search for the solutions. Finally, as an application of the MMMFSG theory we developed, we study a model for callable convertible bonds, where the major player is the bond issuer deciding when to call the bonds, and the minor players are the bond holders deciding when to convert their bonds into equities.