  In this paper we unify, simplify, and extend previous work on the
evolutionary dynamics of symmetric $N$-player matrix games with two pure
strategies. In such games, gains from switching strategies depend, in general,
on how many other individuals in the group play a given strategy. As a
consequence, the gain function determining the gradient of selection can be a
polynomial of degree $N-1$. In order to deal with the intricacy of the
resulting evolutionary dynamics, we make use of the theory of polynomials in
Bernstein form. This theory implies a tight link between the sign pattern of
the gains from switching on the one hand and the number and stability
properties of the rest points of the replicator dynamics on the other hand.
While this relationship is a general one, it is most informative if gains from
switching have at most two sign changes, as it is the case for most
multi-player matrix games considered in the literature. We demonstrate that
previous results for public goods games are easily recovered and extended using
this observation. Further examples illustrate how focusing on the sign pattern
of the gains from switching obviates the need for a more involved analysis.
