= Median-density threshold for the leader in an all-pay contest
{title2=$v>1/F'(F^{-1}(1/2))$}
For increasing concave $F$, write $q=F^{-1}(1/2)$. Under the smallest-maximizer convention, the leader wins with probability greater than $1/2$ exactly when $v>1/F'(q)$. The sign of $vF'(q)-1$ places the maximizer before or after $q$. At equality, concavity alone permits a flat interval of optima; strict concavity or an explicit selection is needed for the strict comparison at that type.
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