Multiplayer matrix games provide a convenient framework to model the logic of decision making in social dilemmas involving free-riding, coordination, or congestion. Their analysis, however, can be intrinsically more complex than that of their two-person counterparts~\citep{Broom1997,Gokhale2010}. In order to harness this complexity, much previous work has resorted to lengthy algebra or numerical methods~\citep{Boyd1988,Dugatkin1990,Pacheco2009,Souza2009}. We unify, simplify and expand such previous work by deriving general results applicable to a large class of multi-person, symmetric, two-strategy games~\citep{Pena2014}. We do so by making use of several shape-preserving properties of polynomials in Bernstein form~\citep{Lorentz1986,Farouki2012},
well appreciated in the fields of approximation theory~\citep{DeVore1993} and computer aided geometric design~\citep{Farin2002,Prautzsch2002}, but relatively less well known in evolutionary game theory. Such shape-preserving properties imply a tight link between the sign pattern of the gains from switching on the one hand and the number of evolutionarily stable states on the other hand. We show how previous results available 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 markedly simplifies the analysis of otherwise involved evolutionary multi-person games.

evolutionary game theory, multiplayer games, polynomials in Bernstein form