 Uniform-value multi-prize all-pay effort formula

Let $n\ge 2$ players have independent values uniform on $[0,1]$, unit effort costs, and $1\le m<n$ equal prizes of size $w(m/n)>0$. In the symmetric Bayesian Nash equilibrium, total expected effort isThe support $[0,1]$ fixes the scale; a different uniform interval requires adjusting the order-statistic calculation.

One proof uses expected effort in a rank-order contest: only prize layer $m$ is nonzero, so total effort is $mw(m/n)\mathbb{E}{V}_{[m+1]}$. The corresponding descending order statistic of $n$ uniform values has mean $(n-m)/(n+1)$. Equivalently, the all-pay effort identity gives $b(v)=w(m/n){\int }_0^{v}t{q}_m^{\prime }(t)dt$, where the allocation derivative is a Beta distribution density with parameters $n-m,m$.