The authors consider two interacting populations P and Q. The strategy choices of P and Q are indexed by finite sets I and J, respectively. When an i strategist from P meets a j strategist from Q the payoffs are constants \(A_{ij}\), \(B_{ij}\) to the P and Q players, respectively. If the current states of P and Q are given by distributions p and q (over I resp. J), then the average payoff to an i strategist is \(A_{iq}=\Sigma_ JA_{ij}q_ j\) and the average payoff for the population as a whole is \(A_{pq}=\Sigma_{I,J}p_ iA_{ij}q_ j\), with similar definitions using \(B_{ij}\) for population Q. If the current generation is in state p then it is assumed that the weight of strategy i in the next generation will be more or less than \(p_ i\) according to whether - in the current environment - the payoff \(A_{iq}\) is more or less than the mean payoff \(A_{pq}\). (Similarly for population Q). A coevolutionary process is defined as a discrete time dynamical system satisfying \(sgn(\Delta p_ i)=sgn(A_{iq}-A_{pq}) (1>p_ i>0)\), \(sgn(\Delta q_ j)=sgn(B_{pj}-B_{pq}) (1>q_ j>0)\), and certain boundary conditions. Under some nondegeneracy and smoothness assumptions the authors show that a locally stable equilibrium of a coevolutionary process can occur only at a vertex (i.e. pure strategies for P and Q).