  We provide a unified approach to find equilibrium solutions for
time-inconsistent problems with distribution dependent rewards, which are
important to the study of behavioral finance and economics. Our approach is
based on {\it equilibrium master equation}, a non-local partial differential
equation on Wasserstein space. We refine the classical notion of derivatives
with respect to distribution and establish It$\hato$'s formula in the sense of
such refined derivatives. Our approach is inspired by theories of Mckean-Vlasov
stochastic control and mean field games, but is significantly different from
both in that: we prohibit marginal distribution of state to be an input of
closed loop control; we solve the best reaction to individual selves in an
intra-person game instead of the best reaction to large populations as in mean
field games. As applications, we reexamine the dynamic portfolio choice problem
with rank dependent utility based on the proposed novel approach. We also
recover the celebrated extended HJB equation when the reward of the problem has
a nonlinear function of expectation while reformulating and weakening the
assumptions needed. Most importantly, we provide a procedure to find an
equilibrium solution of a dynamic mean-ES portfolio choice problem, which is
completely new to the literature.
