Tuesday, 16 December 2008

Kelly Criterion

There is currently some debate on the Betfair as to the merits of the Kelly Criterion, and I thought I would post an excerpt from a piece I wrote some time ago on the subject.

In Kelly's analysis, the smart gambler should be interested in "compound return" on capital. He showed that the same math a colleague (Claude Shannon) had used in his theory of noisy communications channels applies to the gambler. The gambler's optimal policy is to maximize the expected logarithm of wealth.

Though an aggressive policy, this offers important downside protection. Since log(0) is negative infinity, the ideal Kelly gambler never accepts even a small risk of losing everything.

Fortunately for non-mathematical people, you don't even have to know what a logarithm is to use the so-called Kelly Criterion. You should wager this fraction of your bankroll on a favourable bet:

Edge / Odds
Edge is how much you expect to win, on the average, assuming you could make this wager over and over with the same probabilities. It is a fraction because the profit is always in proportion to how much you wager. The edge is usually diminished by tax or commission. When your edge is zero or negative, the Kelly Criterion says not to bet.

Odds is a measure of the profit if you win.

In the Kelly Criterion, odds is not necessarily a good measure of probability. Odds are determined by market forces, by everyone else's opinions about the chance of winning. These opinions may be wrong, and in fact MUST be wrong for the Kelly gambler to have an edge.

For example: The odds on Red Rum are 4 to 1 (i.e. the market estimates that Red Rum has a 1 in 5 chance of winning, or a 20% probability), but by your calculations, Red Rum has a 1 in 4 chance of winning (i.e. odds of 3 to 1, or a 25% probability).

Assuming your estimation is correct, then by betting £100 on Red Rum you stand a 1/3 chance of ending up with £500. On the average, that is worth £166.67, a net profit of £66.67. The edge is the £66.67 profit divided by the £100 wager, in this case 0.67.

The Kelly formula of edge/odds, is therefore 0.67 / 4, or .1675. This means that you should bet 16.75% of your bankroll on Red Rum.

A quick search of the Internet will provide you with links to a number of easy to use Kelly calculators.

Unlike some mathematical formulas, the Kelly formula does have the virtue of being easy to remember.

By always making the Kelly bet, your bankroll will increase faster than with any system.

However, gamblers need to understand that their progress and bank balance will not be a smooth upward slope, but will be interrupted by frequent drawbacks. For this reason, a common practice among investors and gamblers is to use the Half-Kelly bet. This greatly reduces the volatility of the Kelly bet, but returns 3/4 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.

For sports betting, there is the added complication that the true odds on an outcome are not known. When calculating your Kelly bet, your estimate may well differ significantly from the true odds.

Both under-betting and over-betting will give you a reduced rate of return. Under-betting, which the Half Kelly is, will provide steadier growth, but with reduced returns, whereas over-betting can be fatal, as betting twice the optimal Kelly bet results in almost no long-term growth at all.

Most gamblers are probably best served by using a flat 2% of their bank per bet, since figuring edges in sports is, as mentioned earlier, very difficult. 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.

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 maximize their returns.

3 comments:

Anonymous said...

Hi,

Very interesting article, but 1 point i dispute (and it links back to your value debate).

"the market estimates that Red Rum has a 1 in 5 chance of winning, or a 20% probability), BUT BY YOUR CALCULATIONS, Red Rum has a 1 in 4 chance of winning"

Now, what makes the bettor's opinion more accurate / important / relevant than the market's?? Example - Man U v Accrington stanley. Market says 50/1 away win. I believe this should be evens. Does that make my bet value or helpful in the kelly factor?? Obviously not cos my opinion is wrong.

It's just this massive grey area that it's all opinions at the end of the day, no matter what formulas etc are used

Thanks
Simon
ps i really do enjoy reading your blog by the way!! :-)

saperduper said...

one more insightful post!
thank you!

Anonymous said...

Simon,

Your quite right that this is an assumption of the Kelly Criterion. It is its one biggest drawbacks when used in a real situation (usually anyway.)

If the size of his edge (or the existance of any edge) is over-estimated by the bettor, then betting the suggested Kelly stake will be "overstaking" leading to an increased likelihood of going broke at some point.

However, this doesnt dispute any of the mathematics/principles behind the Kelly staking system. It merely reinforces & highlights the assumptions that must be made or catered for when using it.

The Kelly staking system (or any other for that matter) will not turn a losing/poor approach to betting into a profitable one in the long term. The skill of the bettors judgment is still being put to the test. Just that if this skill/judgment is good enough, then from a mathematical point of view, it is the optimum/most efficient stake to use.

JPG