Leader optimization in a sequential private-value all-pay contest
= Leader optimization in a sequential private-value all-pay contest
{title2=$b(v)\in\arg\max_b\{vF(b)-b\}$}
With follower distribution $F$, a leader of value $v$ chooses $b\in\arg\max_{0\leq b\leq1}\{vF(b)-b\}$. If $F$ is a <concave function>, the objective is concave and an interior optimum satisfies $vF'(b)=1$. Endpoint derivatives handle zero or maximal effort. Zero effort guarantees nonnegative payoff, so every optimum satisfies $b\leq vF(b)\leq v$.