Pari-mutuel expected-return allocation ($x_i=(\sqrt{p_i{s}_i/\tau }-s_i{)}^{+}$)

For positive existing stakes $s_i$ and budget $b$, maximizing ${\sum }_i{p}_i{x}_i/(s_i+x_i)$ is equivalent to minimizing ${\sum }_i{p}_i{s}_i/(s_i+x_i)$. A positive multiplier $\tau$ chosen so that ${\sum }_i{x}_i=b$ gives the displayed allocation. Each coordinate minimizes $p_i{s}_i/(s_i+x)+\tau x$ on $x\ge 0$, 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.