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First take $s_i>0$, the hypothesis required for the printed ratios and their optimization formula to be well-defined. Since $\sum p_i=1$,The maximizing and minimizing problems therefore have exactly the same solutions.

Let $\tau >0$ and minimize the separable optimization LagrangianEach summand is convex, and its derivative is $-p_i{s}_i/(s_i+x_i{)}^2+\tau$. Its minimizer is $x_i(\tau )=(\sqrt{p_i{s}_i/\tau }-s_i{)}^{+}$. The sum of these coordinates is continuous in $\tau$, decreases from infinity to zero, and is strictly decreasing wherever positive. Thus there is a unique $\tau$ for which the sum is $b$. For every feasible $y$, coordinatewise minimization gives $L(y,\tau )\ge L(x(\tau ),\tau )$; both constraint terms vanish. The Lagrangian sufficiency theorem proves global optimality, not just a necessary stationary condition.

Put $r_i=p_i/s_i$ in decreasing order. A positive bet occurs exactly when $r_i>\tau$. Hence the positive bets form an initial block. To interpret the maximal index in the question, one may include any subsequent indices tied at $r_i=\tau$: their bets are zero but their marginal-return equality still holds. Take $k=\max \{i:r_i\ge \tau \}$. ThenFor this block, $\sqrt{\tau }=({\sum }_{i\le k}\sqrt{p_i{s}_i})/(b+{\sum }_{i\le k}{s}_i)$, with the threshold inequalities determining $k$. Zero-probability horses receive zero bets.

Let $H$ be the set attaining $\rho =\max r_i$, and write $S_H={\sum }_{i\in H}{s}_i$, $P_H={\sum }_{i\in H}{p}_i=\rho S_H$. For sufficiently small $b$, the solution uses only $H$, with $x_i=b{s}_i/S_H$ there and $\tau =\rho [S_H/(S_H+b){]}^2$. If the next ratio is $r_{\ast }<\rho$, this remains valid when $b\le S_H(\sqrt{\rho /r_{\ast }}-1)$; if all outside ratios are zero there is no finite threshold. Put $S={\sum }_i{s}_i$. The fixed pool is $S+b$, so the expected gross payout and net gain areTied best horses receive bets proportional to their existing stakes; betting all on one tied horse is not generally optimal for finite $b$. This is pari-mutuel expected-return allocation.

The literal allowance $s_i=0$ needs qualification. At $s_i=x_i=0$ the printed objective contains $0/0$ and the ordered ratios are undefined. Even defining an unbacked horse's payout to be zero does not always yield an attained optimum. For example, take $b=1$, $p_1=p_2=1/2$, $s_1=0$, $s_2=1$. For $x_1>0$, the normalized expected payout is $1/2+(1/2)(1-x_1)/(2-x_1)$, tending to $3/4$ as $x_1\downarrow 0$ but never attaining it; at $x_1=0$ that zero-payout convention gives $1/4$. Thus the proved allocation formula requires positive existing stakes, or an explicitly modified market model. The zero-stake case cannot be justified by silently dividing by zero.