A gambler at a horse race has an amount $b>0$ to bet. The gambler assesses $p_{i}$, the probability that horse $i$ will win, and knows that $s_{i} \geq 0$ has been bet on horse $i$ by others, for $i=1,2, \ldots, n$. The total amount bet on the race is shared out in proportion to the bets on the winning horse, and so the gambler's optimal strategy is to choose $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ so that it maximizes

$$\sum_{i=1}^{n} \frac{p_{i} x_{i}}{s_{i}+x_{i}} \quad \text { subject to } \sum_{i=1}^{n} x_{i}=b, \quad x_{1}, \ldots, x_{n} \geq 0$$

where $x_{i}$ is the amount the gambler bets on horse $i$. Show that the optimal solution to (1) also solves the following problem:

$$\text { minimize } \sum_{i=1}^{n} \frac{p_{i} s_{i}}{s_{i}+x_{i}} \quad \text { subject to } \sum_{i=1}^{n} x_{i}=b, \quad x_{1}, \ldots, x_{n} \geq 0$$

Assume that $p_{1} / s_{1} \geq p_{2} / s_{2} \geq \ldots \geq p_{n} / s_{n}$. Applying the Lagrangian sufficiency theorem, prove that the optimal solution to (1) satisfies

$$\frac{p_{1} s_{1}}{\left(s_{1}+x_{1}\right)^{2}}=\ldots=\frac{p_{k} s_{k}}{\left(s_{k}+x_{k}\right)^{2}}, \quad x_{k+1}=\ldots=x_{n}=0$$

with maximal possible $k \in\{1,2, \ldots, n\}$.

[You may use the fact that for all $\lambda<0$, the minimum of the function $x \mapsto \frac{p s}{s+x}-\lambda x$ on the non-negative axis $0 \leq x<\infty$ is attained at

$$\left.x(\lambda)=\left(\sqrt{\frac{p s}{-\lambda}}-s\right)^{+} \equiv \max \left(\sqrt{\frac{p s}{-\lambda}}-s, 0\right) .\right]$$

Deduce that if $b$ is small enough, the gambler's optimal strategy is to bet on the horses for which the ratio $p_{i} / s_{i}$ is maximal. What is his expected gain in this case?