5. Negative horse race.
Consider a horse race with $m$ horses with win probabilities $p_1, p_2, \dots, p_m$. Here the
gambler hopes a given horse will lose. He places bets $(b_1, b_2, \dots, b_m)$, $\sum_{i=1}^m b_i = 1$, on
the horses, loses his bet $b_i$ if horse $i$ wins, and retains the rest of his bets. (No odds.)
Thus $S = \sum_{j \neq i} b_j$, with probability $p_i$, and one wishes to maximize $\sum p_i \ln(1 - b_i)$
subject to the constraint $\sum b_i = 1$.
(a) Find the growth rate optimal investment strategy $b^*$. Do not constrain the bets
to be positive, but do constrain the bets to sum to 1. (This effectively allows
short selling and margin.)
(b) What is the optimal growth rate?
