Kelly Criterion - Many Horses

Many Horses

Kelly's criterion may be generalized on gambling on many mutually exclusive outcomes, like in horse races. Suppose there are several mutually exclusive outcomes. The probability that the k-th horse wins the race is, the total of bets placed on k-th horse is (in dollars), and

where are the pay-off odds., is the dividend rate where is the track take or tax, is the revenue rate after deduction of the track take when k-th horse wins. The fraction of the bettor's funds to bet on k-th horse is . Kelly's criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions of bettor's wealth to be bet on the outcomes included in the optimal set . The algorithm for the optimal set of outcomes consists of four steps.

Step 1 Calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes:

Step 2 Reorder the outcomes so that the new sequence is non-increasing. Thus will be the best bet.

Step 3 Set (the empty set), . Thus the best bet will be considered first.

Step 4 Repeat:

If then insert k-th outcome into the set:, recalculate according to the formula: and then set ,

Else set and then stop the repetition.

If the optimal set is empty then do not bet at all. If the set of optimal outcomes is not empty then the optimal fraction to bet on k-th outcome may be calculated from this formula: .

One may prove that

is the reserve rate. Therefore the requirement may be interpreted as follows: k-th outcome is included in the set of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction may be interpreted as the excess of the expected revenue rate of k-th horse over the reserve rate divided by the revenue after deduction of the track take when k-th horse wins or as the excess of the probability of k-th horse winning over the reserve rate divided by revenue after deduction of the track take when k-th horse wins. The binary growth exponent is

and the doubling time is

This method of selection of optimal bets may be applied also when probabilities are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that and .