Kelly Criterion

Kelly Betting Explained

Bankroll management is one of the most important aspects of successful gambling. If you don't stake intelligiently using a suitable method of bankroll management you will lose even if you are betting with an edge.

Too often people bet more than they can afford, decide to bet their entire bank on the last race of the day or are simply not consistent enough with their staking. Unfortunately the problem with many gamblers is that they have no control over how much they bet when really one should be keeping an accurate record of every bet they make.

Whist there are many staking systems (martingale, fixed bets, percentage bets, star system and so on), only one has been proven mathematically to optimize the rate of bankroll growth. That method is the Kelly Criterion.

The Kelly Betting formula was originally written by John Kelly in 1 956 for information rates, but has been adapted to Sports Betting recently. The original paper was released in a journal article in the Bell System Technical Journal. The formulas have been simplified so the criteria can be adapted to betting on sports.

The Kelly criterion assumes you know the probability of the result of an event. We have mentioned in previous articles that the odds can be converted into the probability and vice versa, but what we are after here is the 'true probability' of a team to win a match.

What’s the true probability you ask? Well whenever you place a bet on a team, you are betting because you believe that they have greater chance to win than what the odds suggest, as you’re looking for value in the odds. Take for example, a match in which both sides are at $ 1 .90, and you decide to bet on the team that are playing at home. $ 1 .90 represents a probability of , but because you’ve bet on the home side you probably think that the probability of them winning is more like 55%, 60% or even more. Therefore, that is your accessed probability of winning.

Of course punters will have varying opinions as to what the true probability is and that is what separates a profitable gambler from a losing gambler. The Kelly criteria works on betting large amounts for favourites and smaller amounts for outsiders. It also suggests larger bets when you have a greater advantage over the bookmaker’s odds. So in the above example, if you thought that the home side were a 70% chance of winning, then according to the Kelly criteria you would bet more than if you thought they were a 60% chance of winning.

The Kelly criteria also works off a moving bank, so this means that the more money you have the more money you bet, however there are several variants of the Kelly method of bankroll management which we shall look at.

The original Kelly Criteria which is often referred to as 'Full Kelly'. The formula to use to work out how much to bet is:

The overlay is the advantage you have over the bookmaker and is equal to the following formula:

So if we decide to combine the two formulas above we come up with the well recognised formula for the Kelly criteria:

So let’s assume that we have a bank worth $ 1 ,000 and we want to bet on the home side that are paying $ 1 .70. We believe that they are a 65% chance to win. We plug these numbers into the formula above to work out how much we should bet:

If we thought that they were only a 60% chance to win, then we should bet:

Which shows that the greater the overlay (the advantage that you have over the bookmaker) then the more you bet, which makes sense. But note here that we are betting more than a quarter of our bank! Most professional punters will suggest that an average bet size of 1 % to 2% of your bank is most preferable, and up to 5% maximum.

This leads us on to the variants of the Kelly Method. Obviously if we were 100 % sure of the probability of the match being correct, then the full Kelly method would be most advantageous, however in reality we are never going to know the exact true probability of a team winning. We may be able to estimate an approximate probability but never the exact one. Hence the method of Full Kelly Betting can be seen as too volatile.

Hence many people suggest the use of a Half Kelly, Quarter Kelly or some other fractional variation of the Kelly criteria. The Half Kelly method for example simply bets half the amount as suggested, and the quarter Kelly method bets a quarter of the amount suggested.

These variants are much preferred amongst gamblers and are used widespread. The final variant is the constant Kelly variant as opposed to the moving bank.

Basically after each bet your bank changes depending if you won or lost the bet. As your bank increases and decreases so does the amount that you bet as the variable “Bank” is in the formula for calculating how much to bet. This essentially means that it is impossible to go bankrupt, however with several losses in a row you might find yourself betting very small amounts, so realistically there is a smaller chance of losing all your money but it is still possible.

If someone decides to keep the variable 'Bank' constant then we are using the constant Kelly variant. In other words, irrelevant to how much you have won or lost in the past, you decide to keep the Bank variable set at $ 1,000. So now, unlike normal Kelly betting, you are now betting fixed amounts and not a percentage of your current bank. Of course they are not fixed exactly as the amount that you bet is still dependant on the odds and the advantage that you have over the bookmaker’s odds.

So which variant is the best? Well that is entirely up to the individual, whilst it is generally understood that full Kelly betting is far too risk adverse, fractional Kelly’s and constant Kelly are often used by professional punters as they hold a mathematical advantage over all other forms of bankroll management.