 Two-condition criterion for evolutionary stability

For mixed-strategy expected payoffs put $A_y=e(x^{\ast },x^{\ast })-e(y,x^{\ast })$ and $B_y=e(x^{\ast },y)-e(y,y)$. The invasion payoff difference is exactly $(1-ϵ)A_y+ϵB_y$. Positivity for all sufficiently small positive $ϵ$ forces $A_y\ge 0$ by taking the limit. When $A_y=0$, it forces $B_y>0$. Conversely $A_y>0$ makes the difference positive for sufficiently small frequency, and $A_y=0,B_y>0$ makes it positive for every positive frequency. Apply this separately to every distinct mutant. This proves the equivalence with an evolutionarily stable strategy; affineness in population composition is essential to the formula.