= All-pay effort identity
{title2=$b(v)=\int_{\underline v}^v t a'(t)dt$}
Suppose values are nonnegative, the <rank-order expected prize allocation> $a$ is increasing and differentiable, and $a(\underline v)=0$. With unit effort cost and zero effort at the lowest type, the symmetric equilibrium effort is $b(v)=v a(v)-\int_{\underline v}^v a(t)dt$. A true type $v$ imitating type $z$ receives utility $v a(z)-b(z)$, whose derivative in $z$ is $(v-z)a'(z)$. It increases up to $v$ and decreases afterwards, proving the <best response> property. This is the <interim payment identity> specialized to an <all-pay contest>.
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