Outer space: Racing certainties

A few months ago I saw a TV crime drama that involved a plan to defraud bookmakers by nobbling the favourite for a race. The drama centred around other events and the basis for the betting fraud was never explained. What might have been going on?

Suppose that you have a race where there are published odds on the competitors of $a_1$ to 1, $a_2$ to 1, $a_3$ to 1, and so on for any number, $N$, of runners. For example, if the odds are 5 to 4 then we express that as an $a_i$ of 5/4 to 1. Now if we lay bets on all of the $N$ runners in proportion to the odds so that we bet a fraction $1/(a_i+1)$ of the total stake money on the runner with odds of $a_i$ to 1 then we will always show a profit so long as the odds satisfy the inequality $$Q=\sum _{i=1}^N\frac{1}{a_i+1}<1,$$ and if $Q$ is less than $1$ then our winnings will be at least equal to $$W=[\frac{1}{Q}-1]\times \text{our total stake}.$$

Let's look at some examples. Suppose there are four runners and the odds for each are 6 to 1, 7 to 2, 2 to 1 and 8 to 1. Then we have $a_1=6,a_2=7/2,a_3=2$ and $a_4=8$ and $$Q=\frac{1}{7}+\frac{2}{9}+\frac{1}{3}+\frac{1}{9}=\frac{51}{63}<1$$ and so by betting our stake money with 1/7 on runner 1, 2/9 on runner 2, 1/3 on runner 3, and 1/9 on runner 4 we will win at least 51/63 of the money we staked (and of course we get our stake money back as well).

However, suppose that in the next race the odds on the four runners are 3 to 1, 7 to 1, 3 to 2 and 1 to 1 (ie evens). Now we see that we have $$Q=\frac{1}{4}+\frac{1}{8}+\frac{2}{5}+\frac{1}{2}=\frac{51}{40}>1$$ and there is no way that we can guarantee a positive return. Generally, we can see that if there is a large field of runners ($N$ is big) there is is likely to be a better chance of $Q$ being greater than 1. But large $N$ doesn't guarantee $Q>1$. Pick each of the odds by the formula $a_i=i(i+2)-1$ and you can get $Q=3/4$ and a healthy 30\% return even when $N$ is infinite!

But let's return to the TV programme. How is the situation changed if we know ahead of the race that the favourite in our $Q>1$ example will not be a contender because he has been doped?