= Active-set threshold for a proportional contest with outside effort
{title2=$\delta<v_j(2-j+\sum_{i<j}v_j/v_i)$}
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.
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