A gambler at a horse race has an amount to bet. The gambler assesses , the probability that horse will win, and knows that has been bet on horse by others, for . 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 so that it maximizes
where is the amount the gambler bets on horse . Show that the optimal solution to (1) also solves the following problem:
Assume that . Applying the Lagrangian sufficiency theorem, prove that the optimal solution to (1) satisfies
with maximal possible .
[You may use the fact that for all , the minimum of the function on the non-negative axis is attained at
Deduce that if is small enough, the gambler's optimal strategy is to bet on the horses for which the ratio is maximal. What is his expected gain in this case?