  We study zero-sum stochastic differential games where the state dynamics of
the two players is governed by a generalized McKean-Vlasov (or mean-field)
stochastic differential equation in which the distribution of both state and
controls of each player appears in the drift and diffusion coefficients, as
well as in the running and terminal payoff functions. We prove the dynamic
programming principle (DPP) in this general setting, which also includes the
control case with only one player, where it is the first time that DPP is
proved for open-loop controls. We also show that the upper and lower value
functions are viscosity solutions to a corresponding upper and lower Master
Bellman-Isaacs equation. Our results extend the seminal work of Fleming and
Souganidis [15] to the McKean-Vlasov setting.
