Optimization | Part IB, 2002

A gambler at a horse race has an amount b>0b>0 to bet. The gambler assesses pip_{i}, the probability that horse ii will win, and knows that si0s_{i} \geq 0 has been bet on horse ii by others, for i=1,2,,ni=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 (x1,x2,,xn)\left(x_{1}, x_{2}, \ldots, x_{n}\right) so that it maximizes

i=1npixisi+xi subject to i=1nxi=b,x1,,xn0\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 xix_{i} is the amount the gambler bets on horse ii. Show that the optimal solution to (1) also solves the following problem:

 minimize i=1npisisi+xi subject to i=1nxi=b,x1,,xn0\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 p1/s1p2/s2pn/snp_{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

p1s1(s1+x1)2==pksk(sk+xk)2,xk+1==xn=0\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{1,2,,n}k \in\{1,2, \ldots, n\}.

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

x(λ)=(psλs)+max(psλs,0).]\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 bb is small enough, the gambler's optimal strategy is to bet on the horses for which the ratio pi/sip_{i} / s_{i} is maximal. What is his expected gain in this case?

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