  Evolutionary game theory is a mathematical approach to studying how social
behaviors evolve. In many recent works, evolutionary competition between
strategies is modeled as a stochastic process in a finite population. In this
context, two limits are both mathematically convenient and biologically
relevant: weak selection and large population size. These limits can be
combined in different ways, leading to potentially different results. We
consider two orderings: the wN limit, in which weak selection is applied before
the large population limit, and the Nw limit, in which the order is reversed.
Formal mathematical definitions of the Nw and wN limits are provided. Applying
these definitions to the Moran process of evolutionary game theory, we obtain
asymptotic expressions for fixation probability and conditions for success in
these limits. We find that the asymptotic expressions for fixation probability,
and the conditions for a strategy to be favored over a neutral mutation, are
different in the Nw and wN limits. However, the ordering of limits does not
affect the conditions for one strategy to be favored over another.
