Stackelberg solution of first-order mean field game with a major player

The paper is concerned with the study of the large system of identical players interacting with the environment. We model the environment as a major (exogenous) player. The main assumption of our model is that the minor players influence on each other and on the major (exogenous) player only via certain averaging characteristics. Such models are called mean field games with a major player. It is assumed that the game is considered in the continuous time and the dynamics of major and minor players is given by ordinary differential equations. We study the Stackelberg solution with the major player playing as a leader, i.e., it is assumed that the major player announces his/her control. The main result of the paper is the existence of the Stackelberg solution in the mean field game with the major player in the class of relaxed open-loop strategies.