Mean Field Portfolio Games with Consumption

We study mean field portfolio games with consumption. For general market parameters, we establish a one-to-one correspondence between Nash equilibria of the game and solutions to some FBSDE, which is proved to be equivalent to some BSDE. Our approach, which is general enough to cover power, exponential and log utilities, relies on martingale optimality principle in Cheridito and Hu (Stochast Dyn 11(02n03):283–299, 2011) and Hu et al. (Ann Appl Probab 15(3):1691–1712, 2005) and dynamic programming principle in Espinosa and Touzi (Math Financ 25(2):221–257, 2015) and Frei and dos Reis (Math Financ Econ 4:161–182, 2011). When the market parameters do not depend on the Brownian paths, we get the unique Nash equilibrium in closed form. As a byproduct, when all market parameters are time-independent, we answer the question proposed in Lacker and Soret (Math Financ Econ 14(2):263–281, 2020): the strong equilibrium obtained in Lacker and Soret (Math Financ Econ 14(2):263–281, 2020) is unique in the essentially bounded space.