Rank-order expected prize allocation
= Rank-order expected prize allocation
{title2=$a(v)=\sum_{k=1}^n w_k\binom{n-1}{k-1}(1-F(v))^{k-1}F(v)^{n-k}$}
For $n$ players with independent valuations having a continuous <distribution function> $F$, a type $v$ occupies rank $k$ when exactly $k-1$ rivals have higher values. Multiplying this <binomial distribution> by the rank prize $w_k$ and summing gives $a(v)$. Ties occur only on <zero-probability events> when the valuation law is atomless.