Kelly criterion

The Kelly formula, also called Kelly criterion of profit maximization of bets with a positive expectation of profit serves. It goes back to the scientist John Larry Kelly, Jr., who published them in 1956.

With a bet is in this context means risking a sum of money (insert), the (fixed rate ) will be rewarded in case of winning with a fixed multiple of the bet. In case of loss of use is issued.

The formula

The Kelly formula is the calculation rule of the Kelly fraction, the fraction of the bankroll, which is used in the bet to maximize profits taking into account the repayment rate and the probability of winning. The calculation rule is

Here are

The counter can be interpreted as the chance, more than the expected value of the return on a bet of 1

The denominator is the profit rate, ie the repayment rate minus the insert.

Thus reads a memorable version of the Kelly formula:

Bets with positive profit expectations are betting with advantage (value) for the player, so-called " value -bets " ( value = value ). A bet with a winning percentage and a repayment rate is a value bet or has a positive expectation when

Applies. In this case, is always positive.

An idealized example

If we consecutively play many similar bets each with the same amount as the operation, we expect to about the times the total money invested.

Suppose we play 1000 similar bets with a winning percentage of 40 %, that and the odds are, ie the insert in the case of winning triples ( net income is twice the insert ). If we each put the same amount in all of our 1000 betting, say, 1 €, we will win about 400 of those bets and lose 600.

We get so

Back and have it

Expended. We have won a total of 200 €, ie 0.2 times our entire mission:

It is around the expected value. In reality, can come out a little more or a little less than 0.2 times of the insert.

When using the Kelly formula, we would each have risked the times the existing bankroll. So each

One tenth of each existing bankroll.

With a starting budget of 1000 € which would be € 100 for the first bet. If we win the first bet, then we would have a total of 1200 €, would thus risking € 120 at the second bet. If we, however, lose the first bet, we had only 900 €, would thus in the second bet just 90 € risking etc.

We will win by betting back in 1000 about 400 and losing 600. In the event of a win is a credit balance after the use of a credit balance of

In the event of a loss is from a credit, a credit of

At a gain of 400 bets and a loss of 600 bets so our seed capital as a whole multiplied 400 times with 1.2 and 600 times with 0.9. The results after 1000 bets a capital of

The order in which the gains or losses occur, it does not matter.

In reality, however, we shall hardly find such a series of bets.

Larger or smaller bets

What are the consequences if we always put a larger or smaller share instead of the Kelly stake?

Let us first in our example. Suppose we set twice, so we put instead of 0.1 from the existing credit 0.2. We would therefore at a starting balance of € 1,000 with an insert of 200 start, etc. For each bet we would have obtained from a credit, a credit of

Achieved at each losing bet would from a credit, a credit of

Be. After 1000 betting we would therefore

Although we had a lot more risk, would come out significantly less gain than the simple Kelly - use. Even more clearly, it is the Triple Kelly insert ( 0.3). We would bet after 1000

Since you can not share the penny, which would be a total loss.

Had we used smaller bets, a profit would still come out. Although this would not be so high as the Kelly - use, but we would have less risk. For example, when you half- Kelly - use (0.05 ) would be after 1000 bets a credit of

For our example, the following figure shows the result after 1000 bets on each various multiples of the Kelly - use dar. It is assumed here that the number of winning bets the winning probability corresponds.

The maximum profit is achieved when always exactly the Kelly - use ( 1 on the X -axis) is set. Too small tips may cause less profit is made, however, to big bets carry the risk of total loss in itself. The following figure shows the significant cut-outs are shown enlarged in each case.

Even with different values ​​for the probability of winning and the repayment rate as in the above example, the curve will have a shape such that the maximum is reached at the Kelly use, after which the gain curve falls relatively quickly. The prerequisite is that the following holds.

For a similar example see also.

Incorrect probability information

In reality, you do not know the probability often, but estimates only. In the worst case, it is not a value bet, so no use would be at all appropriate.

Sticking to the example considered above and assume that the probability of winning would not be 40 % but only 36 %, that is. At a rate of nevertheless it would still be a bet with positive expected profits, because

The Kelly - share would be

If we had previously applied with the wrong probability of 0.4 out expecting Kelly - use, that would be more than double the correct Kelly bet. When starting budget of 1000 € we had after 1000 bets a credit of

That would be a loss. If we had only half Kelly bet that we had calculated with the little high -assessed probability risks, our operations would have been just a bit too high. That would not have had such a devastating impact. The credit in this case would after 1000 bets

In comparison, the result would be the use of the actual use of Kelly

Because of possible errors in the estimation of probabilities, it is advisable to play only those bets that would also have a slightly smaller probability nor a positive expectation, and then only a part of the Kelly - use, eg insert half.

Fluctuations

Even if we know the probability of winning a bet, and thus the correct proportion Kelly sure fluctuations in the balance when setting the appropriate bets are enormous and take with growing credit to. In the following figure, which is illustrated. The values ​​of the idealized example used, ie a probability and a repayment rate of. The course of 1000 bets may be as follows.

The development credit only the first 200 bets for this example looks like this:

An idea to mitigate the fluctuations would be to divide the balance of the paper into multiple virtual accounts and play separately with each account. This can be aptly described with the word diversification.