Uniform-value multi-prize all-pay effort formula
= Uniform-value multi-prize all-pay effort formula
{title2=$R_m=\frac{m(n-m)}{n+1}w(m/n)$}
With $n$ independent uniform types, $m<n$ equal prizes of scale $w(m/n)$ and unit effort costs, a symmetric <Bayesian Nash equilibrium> has total expected effort $R_m=m(n-m)w(m/n)/(n+1)$. The <all-pay effort identity> yields $b(v)=w(m/n)\int_0^v t q_m'(t)dt$. The allocation derivative is a <Beta distribution> density with parameters $n-m,m$, so integration gives the formula.