  Evolutionary game theory is an abstract and simple, but very powerful way to
model evolutionary dynamics. Even complex biological phenomena can sometimes be
abstracted to simple two-player games. But often, the interaction between
several parties determines evolutionary success. Rather than pair-wise
interactions, in this case we must take into account the interactions between
many players, which are inherently more complicated than the usual two-player
games, but can still yield simple results. In this manuscript we derive the
composition of a many-player multiple strategy system in the mutation-selection
equilibrium. This results in a simple expression which can be obtained by
recursions using coalescence theory. This approach can be modified to suit a
variety of contexts, e.g. to find the equilibrium frequencies of a finite
number of alleles in a polymorphism or that of different strategies in a social
dilemma in a cultural context.
