All-pay indifference equation with random entry ($wG(b{)}^{n-1}-b=u$)

For a prize of value $w$, let $G(b)$ be the probability that a rival is absent or enters with effort at most $b$. When rivals act independently and have no atoms at positive efforts, effort $b>0$ wins with probability $G(b{)}^{n-1}$. If a player mixes over an interval of best responses, its payoff on that interval is constant, so $G(b)=((b+u)/w{)}^{1/(n-1)}$. If the absence probability is $q$ and the active effort support starts at zero, then $u=w{q}^{n-1}$.