# Betting simulation with and without the Kelly criterion

## Optimal bet for maximum gain

## Kelly criterion

The Kelly criterion gives, depending on the odds and the estimated real probability of winning, the fraction of one's bankroll to bet (or of one's capital to invest) to obtain the**theoretical maximum return**.

This theoretical return is of course probabilistic in nature and, as with the fundamental notion of mathematical expectation, it must be understood as follows: if we bet a large number of times on this event, it is by betting each time this "Kelly proportion" of our bankroll that in the long term we will have the highest bankroll, higher than with any other bet (a fixed bet, or another proportion of the bankroll).

These elements are of course found in the mathematical proof of this Kelly formula .

To fully realize, and therefore fully understand, the benefit of using this Kelly formula, we compare in the following simulator the long-term effect of using the Kelly formula compared to two other types of bets:

**Fixed bet:**we always bet the same amount, regardless of our bankroll**Fixed percentage:**we always bet the same percentage of our bankroll, again independently of our bankroll.

This method is often recommended for frequent bettors, in particular to avoid bankruptcy. In general, we advise not to bet more than a few percent of your bankroll with each bet (2% by default in the following simulator). This allows you to be able to withstand bad beats and other dark series of losses.

The Kelly criterion is a fixed percentage strategy, with the particularity that this percentage has been calculated exactly in order to obtain a maximum overall return.

## Simulation and comparison of the Kelly method and the two other strategies

Vary the different quantities of the simulation: initial bankroll (or capital), odds of the event and estimated real probability, as well as the characteristics of the two other strategies that we want to compare: fixed percentage or fixed bet.The resulting graph shows the evolution of the bankroll according to the number of successive bets. The long term is considered in Kelly's formula: do not hesitate to simulate a "larger number" of games.

## Discussion and application of the Kelly method

Quite clearly in the long term, using the Kelly formula to exactly proportion your bet according to your bankroll leads to a higher bankroll compared to the other two strategies. Eventually, increase the number of games to realize that the other two methods are quickly and quite far behind.We remember that, in mathematical terms, and in the demonstration of Kelly's formula, this large number means exactly "at the limit where this number tends towards infinity"…

Nevertheless, on a small number of bets, the use of the Kelly proportion is not as convincing.

We must add to this that the "real" probability of winning, the central parameter in Kelly's formula, is difficult to estimate.

Thus, for a large number of bets (hundreds, thousands, more, etc.) the knowledge and use of the Kelly formula is essential, for example, for those who want to place bets or investments in very large quantities, computationally, high-frequency trading is a typical example.

On the other hand, for the more modest bettor…

This formula remains fundamentally interesting and should be known by any bettor wanting to

**win in the long term**.

- the first lesson of this formula is the benefit of betting a percentage of your bankroll, a low percentage (a few small percentages), and fixed why not.

The Kelly formula is then an aid in estimating what percentage to bet. - secondly, this formula allows you to
**dimension your bets**:- avoid excessive optimism due to excess enthusiasm which leads to an excessively high bet: never bet your entire bankroll! (or even half).

The gains are made**in the long term**. - avoid, conversely, falling into pessimism which would border on excessively low stakes even in a truly advantageous situation.

- avoid excessive optimism due to excess enthusiasm which leads to an excessively high bet: never bet your entire bankroll! (or even half).

## Mathematical formula of the Kelly criterion

Finally, we recall the mathematical formula which allows the calculation of the Kelly proportion*f*

^{*}, calculated automatically in the simulator on this page.

Using the Kelly criterion leads to the

**theoretical maximum return**over all bets.

The mathematical expression of this formula is

*f*

^{*}=

*p*−

*q*

*c*− 1

*p*is the estimated probability of winning*q*= 1 −*p*is the estimated loss probability*c*is the odds of winning (given for example by the bookmaker)

This calculator allows you to automatically calculate the excate Kelly proportion based on the characteristics of the bet.

See also the mathematical details and proof of this formula.

**See also:**