= Two-condition criterion for evolutionary stability
{title2=$A_y\geq0,\quad A_y=0\Longrightarrow B_y>0$}
For mixed-strategy expected payoffs put $A_y=e(x^*,x^*)-e(y,x^*)$ and $B_y=e(x^*,y)-e(y,y)$. The invasion payoff difference is exactly $(1-\epsilon)A_y+\epsilon B_y$. Positivity for all sufficiently small positive $\epsilon$ forces $A_y\geq0$ 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.
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