How to Compute Optimal Betting Sets

Let’s say that you want to learn how to compute optimal betting sets on horse racing; in that case, let’s say that in a ten horse race you think that three of the horses are undervalued and thus profitable to bet on.  Should you just bet on the most profitable?  Should you bet on all three?  How should you divide your bets up?  Are there situations where you might hedge your bets further by betting on a horse with a negative expected value?

It’s obvious that your overall expected value is maximized when you bet everything on the one horse with the highest expected value.  However, as we’ve discussed previously “expected return over time” is a much more important concept than expected value.  “Expected return over time” tells you how to safely bet the largest possible amount of money while still protecting your bankroll.  It’s this figure that you’ll want to compute to figure out your optimal betting combination.

The Wikipedia article on Kelly Betting has an excellent overview of the math involved when you want to bet on multiple horses. Unfortunately, the article is not very readable for casual mathematicians, so in this article we’ll explore the key concepts in a friendlier way.

In a nutshell, there are two important equations that we care about:

Reserve Rate = (1 – (sum of each probability bet on)) / (1 – (sum of each 1/payoff))

Fraction to Bet = Probability – Reserve Rate/Payoff

In Kelly Betting reserve rate simply refers to the fraction of your bankroll that you are not betting.  For example, let’s say a horse’s true probability of winning is 10% and its payoff is $20 (meaning 19 to 1, including the return of your $1 bet.)  If you only bet on this one horse, then your reserve rate would be (1 – .1)/(1 – (1/20)) = 94.7%.  As we’ve discussed previously, this is a hugely important number.  It means that to maximize your profit over time, you should only bet up to 5.3% of your money on this horse.

Now let’s say there are two other horses in the same race, which also have a 10% true probability of winning, but are paying 14 to 1.  Using the reserve rate formula, we learn that you can safely bet a much larger percentage of your bankroll if you are betting on all three horses.  Here the reserve rate is (1 – (.1 + .1 + .1)) / (1 – (1/20 + 1/15 + 1/15)) = .7/.817 = 85.7%.  So, even though you are betting on less profitable horses, you can safely average a larger profit because you can now safely wager up to 14.3% of your total bankroll.

Finally, let’s look at a third horse has a 30% chance of winning and a payoff of $3 (2 to 1 plus the return of your $1 bet.)  Note that you would normally not bet on this horse.  Its expected value is only .9, meaning that you lose 10 cents on every dollar that you bet.  However, if we look at the reserve rate, we discover that by betting on this horse we can again make a much larger much safer overall wager.  Here the reserve rate is (1 – (.1 + .1 + .1 + .3)) / (1 – (1/20 + 1/15 + 1/15 + 1/3)) = .4/.484 = 82.6%.  So even though this horse is a slightly bad bet by itself, it’s still useful for hedging purposes because we can safely place a larger bet on our other, better horses if we also place a small bet on this horse as well.

Once you have used the reserve rate formula to select the set of horses you will bet on, the final step is to compute how much to bet on each horse.  Using the “Fraction to Bet” formula, we can compute our bets as follows:

Fraction to Bet (Horse 1) = .1 – .826/20 = 5.9%

Fraction to Bet (Horse 2) = .1 – .826/15 = 4.5%

Fraction to Bet (Horse 3) = .1 – .826/15 = 4.5%

Fraction to Bet (Horse 4) = .3 – .826/3 = 2.5%

So ultimately, on this race you should bet 17.4% of your money, and you should divide it up using the percentages above, betting 5.9% of your bankroll on Horse 1.  We’ve now computed our best possible set of wagers on these four horses.  By using the Kelly Betting formulas and by hedging, we have the best of all worlds.  We are able to place a larger, safe bet on the most profitable horse, and can also place a set of bets that will win 60% of the time; hence, you now know how to compute optimal betting set for horse racing, and probably for any other type of sports betting.