Median-density threshold for the leader in an all-pay contest ($v>1/F^{\prime }(F^{-1}(1/2))$)

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 $v>1/F^{\prime }(q)$. The sign of $v{F}^{\prime }(q)-1$ 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.