 Leader optimization in a sequential private-value all-pay contest

With follower distribution $F$, a leader of value $v$ chooses $b\in \arg {\max }_{0\le b\le 1}\{vF(b)-b\}$. If $F$ is a concave function, the objective is concave and an interior optimum satisfies $v{F}^{\prime }(b)=1$. Endpoint derivatives handle zero or maximal effort. Zero effort guarantees nonnegative payoff, so every optimum satisfies $b\le vF(b)\le v$.