Mean Field Portfolio Games with Epstein-Zin Preferences

We study mean field portfolio games under Epstein-Zin preferences, which naturally encompass the classical time-additive power utility as a special case. In a general non-Markovian framework, we establish a uniqueness result by proving a one-to-one correspondence between Nash equilibria and the solutions to a class of BSDEs. A key ingredient in our approach is a necessary local stochastic maximum principle, applied to log-wealth, tailored to Epstein-Zin utility, and a nonlinear transformation. In the deterministic setting, we further derive an explicit closed-form solution for equilibrium investment and consumption policies. The strength of our approach is further illustrated by two special cases: (i) in the power utility setting without consumption, we obtain the same one-to-one correspondence as in Fu and Zhou [22] under exactly the same assumption, but without invoking the dynamic programming principle in Espinosa and Touzi [16]; and (ii) in the power utility setting with both investment and consumption, we strengthen the correspondence result of Fu [18], by proving a genuine one-to-one relation in the BMO space, where both the equilibrium strategy and the associated BSDE components belong to BMO.