  The pairwise comparison dynamic is a forward ordinary differential equation
in a Banach space whose solution is a time-dependent probability measure to
maximize utility based on a nonlinear and nonlocal protocol. It contains a wide
class of evolutionary game models, such as replicator dynamics and its
generalization. We present an inverse control approach to obtain a
replicator-type pairwise comparison dynamic from the large discount limit of a
mean field game (MFG) as a coupled forward-backward system. This methodology
provides a new interpretation of replicator-type dynamics as a myopic
perception limit of the dynamic programming. The cost function in the MFG is
explicitly obtained to derive the generalized replicator dynamics. We present a
finite difference method to compute these models such that the conservation and
nonnegativity of the probability density and bounds of the value function can
be numerically satisfied. We conduct a computational convergence study of a
large discount limit, focusing on potential games and an energy management
problem under several conditions.
