Suppose m horses run in a race, where horse i wins with probability p(i), i = 1,...,m.
For every dollar you bet on horse i you get o(i) dollars if that horse wins. You divide
your total wealth in the horse race according to b(i), i = 1,..., m, where b(i) represents
the proportion of the money you bet on horse i. Hence $\sum_{i=1}^{m} b(i) = 1$.
For example, if horse 1 wins, your wealth is multiplied by b(1)o(1). Note that the
money you bet on any other horse will be lost.
We assume that the horses run in n races, and that outcome of race j, denoted by
$X_j \in \{1,..., m\}$, is iid according to p = (p(1),...,p(m)). You use the same betting
strategy b = (b(1),...,b(m)) at each race to reinvest your wealth and the return
o = (0(1),..., o(m)) remains the same for each race. Then your wealth (relative to
your initial investment) after n races is
$S_n = \prod_{j=1}^{n} b(X_j)o(X_j)$.
(a) Show that
$\lim_{n \to \infty} \frac{1}{n} \log(S_n) = W(b)$,
where
$W(b) = \sum_{i=1}^{m} p(i)\log(b(i)o(i))$,
the convergence is in probability and log is base 2. The term W(b) is called the
doubling rate. Clearly explain your steps.
Note that the above result suggests that for large n,
$S_n \approx 2^{nW(b)}$.
(b) Suppose p(1) > 0, and you set b(1) = 0, while b(i) > 0 for i = 2,3,..., m.
i. Let N be the first race after which you lose all your money. Of course for
all $j \ge N, S_j = 0$. Find the probability distribution of N assuming you bet
indefinitely ($n \to \infty$).
ii. Find $P(S_n = 0)$. What happens as $n \to \infty$?
iii. Compute $W(b)$. Is your answer from part 1(b)ii consistent with equation
(1)? Comment.
(c) For a race with two horses where p(1) = p, find the best betting strategy b =
(b(1), b(2)) that maximizes your doubling rate W(b). Show your work.
