# The Kelly Criterion : Finding An Edge

Jul 16th, 2012 - Posted by in Betting Theory

Musings on Backing, Laying, Trading, Punting, In-Running and more on the Betting Exchanges and related items of interest in the wide world of sports investing

What is the Kelly Criterion and how can applying it help us find an edge against the oddsmakers? Today on the blog Cassini tells us.

That some punters win consistently is not in doubt. Bookmakers close, or severely limit, accounts and even the world’s largest betting exchange has found it necessary to introduce a Premium Charge, implicitly acknowledging that a number of people have the ability to consistently identify favourable bets and the discipline to bet only, or perhaps mostly,when it is value for them to do so.

If the goal of these punters is to grow their banks as optimally as possible, how should they stake?

It’s the subject of much debate, but the answer to this is that stakes should be calculated according to the Kelly Criterion, but unfortunately when this solution is applied in the field of sports investing, there are a number of challenges.

## Kelly Drawbacks

As John Larry Kelly Junior’s original 1956 paper makes clear, the criterion is only valid “when the investment or 'game' is played many times over, with the same probability of winning or losing each time, and the same payout ratio”.

This is unfortunate. In the world of sports, no two events are ever exactly the same. Bet on red on the roulette wheel and you know exactly what the probability is, but since the edge in casinos is in favour of the house, Kelly isn’t going to help you here.

For Kelly to work, you must have a positive edge. If the edge is precisely zero, Kelly recommends no bet, and of course if the edge is negative, again there is no bet.

So how can Kelly help in the real world of sports, where precise probabilities are not known?

To help answer this, let’s take a simple example of a game where the true probability is 50%, but the payout exceeds this. You have a value bet, but it should be obvious that betting the full bank each time is not the best way to approach this. Your chances of going bust are extremely high.

## Kelly In Action

At the other end of the risk spectrum is the ultra-cautious approach of staking a small fraction of the bank each time. While the chances of going bust are now very small, your bank isn’t going to grow very fast.

Clearly the optimal strategy lies between these two extremes, and Kelly calculated that the fraction of the bank to be staked equals the size of your edge.

For example, if the chance of a win is 51%, and the price available is evens, you should bet your edge of 2% (51% -49%), 49% being the probability of losing.

If you have a bigger edge, for example your chance of a win is 53%, your stake should be 6% (53% - 47%).

When the odds offered are other than evens, the calculation is a little more complicated, but a quick search of the Internet for ‘Kelly calculator’ will help you out.

## Real Examples

To see the formula in action, let’s take an example of a football match where the odds available on the draw are 3.50 (or 5/2 with an implied probability of 28.6%) but your estimate of the ‘true’ probability is 30%.

The formula for calculating the Kelly stake is:

[(Probability multiplied by odds) – 1] divided by (odds-1).

Thus Kelly stake:

= [(0.3 * 3.50) – 1] / (3.50 - 1)

=[(0.3 * 3.50) – 1] / 2.5

=[1.05 – 1] / 2.5

=0.05 / 2.5

=0.02, i.e. the stake should be 2% of the bank.

## Fractional Kelly

When applying Kelly, the consequences of over-estimating your edge are serious, and as I mentioned earlier, in sports the probability of an outcome is imprecise. It is for this reason that most punters err on the side of caution, and use the more cautious strategy of ‘fractional Kelly’. This means that rather than bet the suggested percentage, you use a fraction of it, commonly a half (Half-Kelly) but it can be any fraction.

This is a sensible way of handling the inevitable losing runs which occur, even if you have a favourable bet. Your bank will increase in the long run, only more slowly, but the risk of blowing the bank are reduced.

Progress and bank balance will not be a smooth upward slope, and will be interrupted by frequent drawbacks (losing runs) but by using the Half-Kelly bet, volatility is greatly reduced, yet returns 3/4 of the compound return. For many gamblers, that is a price worth paying.

It can be shown that a Kelly bettor has a 1/2 chance of halving a bankroll before doubling it, and that you have a 1/n chance or reducing your bankroll to 1/n at some point in the future. For comparison, a“Half Kelly” bettor only has a 1/9 chance of halving their bankroll before doubling it.

## Kelly Conclusion

Because of the problems in calculating the precise edge in sports, most gamblers are probably best served by using a flat 2% of their bank per bet. For a season-long win rate of 55% (on a bet paying at evens), a good target for most bettors, this represents a little more than 1/3 Kelly, which is a conservative compromise between risk and return. This figure of 2% is coincidentally, or perhaps not so coincidentally, often suggested as an appropriate bet size. Without access to inside information, it's unlikely that your edge will ever be much bigger than this anyway, especially in the liquid markets.

Increasing this to 3%, or occasionally 4% on an especially good play, is reasonable. More experienced gamblers, with a good understanding of the downsides of Kelly and an above average ability in estimating betting advantages, may wish to adopt the more aggressive Kelly approach to maximise their returns, but for the mere mortals among us, 2% should suffice.