# Kelly criterion

## How much to bet: formula for determining the optimal wager

Searching for long-term profit is mathematically equivalent to that of value betting.

Now that we know what a value bet is, and we have ideas for detecting them, the question remains:

**how much should you bet ?**

The safer the bet seems to us, the more we should engage from our bankroll. This is a 1st point.

Furthermore, even if the bet does not seem very certain to us, but the odds are high, there too it is in our interest to bet high stakes.

Finally, all this is obviously conditioned by our betting capacity, our bankroll.

How then can be estimated the fair share of our bankroll to invest ? depending on the odds and probability estimated for the event outcome.

The exact answer to all of this is given by the mathematical formula, also known as the Kelly criterion.

## Kelly mathématical formula

John Larry Kelly, Jr. was an American scientist (1923 - 1965) who worked in particular on information theory and game theory. In particular, he established the criterion, now known as the Kelly criterion, which relates the share of wealth to be invested, in a risky investment, so as to maximize the rate of return.The use of the Kelly criterion leads to a capital, or bankroll, higher than with any other long-term strategy: it therefore gives the theoretical optimal efficiency among all bets.

This formula mathematically reads

*f*

^{*}=

*p*−

*q*

*c*− 1

*f*^{*}is the optimal fraction of one's bankroll to bet*p*is the estimated probability of victory*q*= 1 −*p*is the estimated loss probability- odds of
*c*

See also the mathematical details and proof of this formula.

## Kelly criterion calculator

Bankroll fraction to wager: |
f ^{*} = % |

stakes | m^{*} = € |

## Special cases

Some cases- if
*p*= 100%: we consider that the event on which we are betting is 100% sure. In this case we logically have*f*^{*}= 100 %.

In other words, if you are categorically sure of winning, you should**wager everything you can, everything you have**. - if
*p*= 1/*c*, then we find that*f*^{*}= 0.

If the probability you estimate of winning is that which corresponds to the odds offered, then you should not bet.

We find here that this case corresponds to the very definition of odds and yields a zero expected profit: in the long term no gain (nor loss) can be expected here, and there is therefore no point in betting. - Kelly's formula tells us that we must bet a fraction
*f*^{*}> 0 when*p*>1/*c*, that is to say when the estimated probability is greater than that associated with the odds (the implies probability given by the bookmaker therefore) : This is called a value betting situation.

We find here that we must**only bet in a value bet situation**...

Kelly's formula is therefore a quantitative complement to the value bet notion: the determination of the value bet situation indicates when we must bet, and the formula of Kelly then specifies the amount of the bet in this case. - If we calculate that
*f*^{*}< 0, it means that not only should we not bet, but above all we should bet on the contrary event outcome which is more probable.

We then exchange the values of the probabilities*p*and*q*as well as the value of the odds*c*.

## Limits of the formula, disadvantages

The weak point, an undeniable drawback, of this formula lies in the estimation of the probability*p*. The difficulty is exactly the same as detecting a potential value bet: how do you know that a team or a player has an actual 82% chance of winning?

Without a precise mathematical and numerical evaluation of this probability, the result given by this Kelly formula is just as imprecise... However, although an imprecise tool, it nonetheless remains a tool which makes it possible to guide the decision of the bettor in a good direction: should we wager small stakes, or on the contrary is this a very favorable situation ?

Moreover, to arrive at this formula, we assume that the probability

*p*of victory for each bet remains the same and constant, which is open to criticism, see the mathematical details leading to the Kelly formula.

## Real interest in Kelly's formula?

Using the Kelly formula leads to obtain the theoretical maximum yield. This can be observed, for example, by comparing its long-term use with other strategies.On the other hand, from a practical point of view, and given its limitations and the difficulty of applying it (in particular truly estimating the probability of winning), Kelly's formula may seem somewhat vain, ineffective, unusable, etc.

It is not !

- Firstly, it clearly shows the sensitive point of any form of betting: the real estimation of the probability of winning. This is where every bettor should focus their energy. Without this estimate, Kelly's formula is ineffective, of course, but without this estimate a bettor only in fact bets
*a priori*, according to his feelings and not mathematically. - Secondly, this formula must be seen as very useful, for any investor or bettor, in order to frame his bets: avoid excessive optimism through excess enthusiasm and an excessively high bet, just as well, conversely, as do not fall into pessimism which would limit excessively low bets even in a truly advantageous situation.

**See also:**