= Expected effort in a rank-order contest
{title2=$\mathbb E\sum_i b(V_i)=\sum_{k=1}^{n-1} k(w_k-w_{k+1})\mathbb E V_{[k+1]}$}
Let $V_{[1]}\geq\cdots\geq V_{[n]}$ be descending <order statistics>, with $w_n=0$ and nonnegative decreasing prizes. Decompose the prize vector into awards of $w_k-w_{k+1}$ to each of the best $k$ players. The corresponding truthful multi-unit auction charges each winner the next value $V_{[k+1]}$. Its total payment is $k(w_k-w_{k+1})V_{[k+1]}$. <Revenue equivalence> transfers the expected payment to the <all-pay contest>, because the interim allocations and lowest-type utilities coincide. Summing the layers proves the formula.
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