Active-set threshold for a proportional contest with outside effort ($\delta <v_j(2-j+{\sum }_{i<j}{v}_j/v_i)$)

For ordered positive valuations, rank $j$ is active exactly when $\delta <v_j(2-j+{\sum }_{i<j}{v}_j/v_i)$. These ranks form a prefix, and the number active is the largest qualifying rank, or zero if none qualifies. This follows by evaluating the decreasing equilibrium equation ${\sum }_i(1-S/v_i{)}_{+}-1+\delta /S=0$ at $S=v_j$. Strict inequality excludes zero-effort boundary players.