Given a set of probabilities for the runners in a multi-player competition, how much should you bet on each runner to maximise your expected profits?

The standard reference paper for this topic was published by Isaacs over 60 years ago[1]. Unfortunately the presentation is not always easy to follow, so this article presents an alternative approach.

Suppose that a pool with runners has amounts

bet by oneself, and amounts

bet by the rest of the market. The true probabilities of the runners winning the race are

calculated from the previous form of the runners.

Let  Q  be the total proportion of the pool that is paid out. Then the optimal bet proportions are given by

where

where the summation notation on the top means ‘add all values of for which the expected return from runneris negative’, and the corresponding sum on the bottom means ‘add all values offor which the expected return from runneris positive’.

 

Worked example

Consider a race with 8 dogs. The win pool sizes and probabilities are as shown in the following table:

1	$  148.00	0.1204
2	$  151.00	0.1741
3	$    36.00	0.0668
4	$  105.00	0.2048
5	$  135.00	0.2474
6	$    39.00	0.0500
7	$    22.00	0.0393
8	$    75.00	0.0974

The total pool size is $711, the track take-out is 19%, and the rebate is 9%. We assume that the pool is will grow by 15% from the time of placing any bets to when final dividends are calculated, and we decide to use 40% of the recommended Isaacs stake to reduce volatility. What bet sizes give the highest return?

The first step is to work out which dogs have a positive expectation, once take-out and rebate are taken into account. These are dogs for whichThe total pool size is $711, the track take-out is 19%, and the rebate is 9%. We assume that the pool is will grow by 15% from the time of placing any bets to when final dividends are calculated, and we decide to use 40% of the recommended Isaacs stake to reduce volatility. What bet sizes give the highest return?

where, for dog i,

is the probability of the dog winning;

is the amount staked in the public pool on this dog winning;

is the total amount staked in the pool (i.e., the pool size);

T is the track take-out;

and is the track rebate. The results are:

Dog	LHS	RHS	Bet
1	0.578239	1.132631	0
2	0.819606	1.132631	0
3	1.319096	1.132631	1
4	1.386482	1.132631	1
5	1.302859	1.132631	1
6	0.910893	1.132631	0
7	1.27093	1.132631	1
8	0.922943	1.132631	0

In this instance, dogs 3, 4, 5 and 7 have positive expectations.

The next step is to form the expression

 

This is straightforward using the results of the earlier calculation. Note that the factor  Q  refers to the track take-out and does not include any rebate.

Dog	sumSi	sumPi
1	148.00	0
2	151.00	0
3	0.00	0.06679
4	0.00	0.204755
5	0.00	0.247378
6	39.00	0
7	0.00	0.039326
8	75.00	0
TOTAL	413.00	0.56

In this example,  has a value of sqrt (0.9*$413/ 1-(0.9*0.56)) = 23.77.

The last step is to calculate the size of the actual bets using

The value of this expression will only be positive for the dogs with positive expectation. Again using the values above, the amounts to be staked on dogs 3, 4, 5, 7 are ($6.38, $21.73, $22.95, $3.42).

The last step is to take the increase in the pool size and the reduction factor into account. We expect the pool size to increase in size by 15% before the race starts, so all bets are multiplied by 1.15. Also, we have decided to bet 40% of the recommended Isaacs stake, so all bets are also multiplied by 0.40. The resulting bet sizes are ($2.94, $10.00, $10.56, $1.57).

 

[1] Isaacs, R. (1953)“Optimal Horse Race Bets” American Mathematical Monthly Vol. 60, No.5, (May, 1953), pp. 310-315, (May, 1953), pp. 310-315