  This paper proposes a new mathematical paradigm to analyze discrete-time
mean-field games. It is shown that finding Nash equilibrium solutions for a
general class of discrete-time mean-field games is equivalent to solving an
optimization problem with bounded variables and simple convex constraints,
called MF-OMO. This equivalence framework enables finding multiple (and
possibly all) Nash equilibrium solutions of mean-field games by standard
algorithms. For instance, projected gradient descent is shown to be capable of
retrieving all possible Nash equilibrium solutions when there are finitely many
of them, with proper initializations. Moreover, analyzing mean-field games with
linear rewards and mean-field independent dynamics is reduced to solving a
finite number of linear programs, hence solvable in finite time. This framework
does not rely on the contractive and the monotone assumptions and the
uniqueness of the Nash equilibrium.
