= Pari-mutuel expected-return allocation
{title2=$x_i=(\sqrt{p_is_i/\tau}-s_i)^+$}
For positive existing stakes $s_i$ and budget $b$, maximizing $\sum_i p_ix_i/(s_i+x_i)$ is equivalent to minimizing $\sum_i p_is_i/(s_i+x_i)$. A positive multiplier $\tau$ chosen so that $\sum_i x_i=b$ gives the displayed allocation. Each coordinate minimizes $p_is_i/(s_i+x)+\tau x$ on $x\ge0$, which proves global optimality by <Lagrangian sufficiency theorem>. Its active set is ordered by $p_i/s_i$. Zero existing stakes require a separate payoff convention and can cause nonattainment; the positive-stake formula should not be extended through undefined ratios.
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