  Logit dynamics are evolution equations that describe transitions to
equilibria of actions among many players. We formulate a pair-wise logit
dynamic in a continuous action space with a generalized exponential function,
which we call a generalized pair-wise logit dynamic, depicted by a new
evolution equation nonlocal in space. We prove the well-posedness and
approximability of the generalized pair-wise logit dynamic to show that it is
computationally implementable. We also show that this dynamic has an explicit
connection to a mean field game of a controlled pure-jump process, with which
the two different mathematical models can be understood in a unified way.
Particularly, we show that the generalized pair-wise logit dynamic is derived
as a myopic version of the corresponding mean field game, and that the
conditions to guarantee the existence of unique solutions are different from
each other. The key in this procedure is to find the objective function to be
optimized in the mean field game based on the logit function. The monotonicity
of the utility is unnecessary for the generalized pair-wise logit dynamic but
crucial for the mean field game. Finally, we present applications of the two
approaches to fisheries management problems with collected data.
