Kelly criterion

How much to bet: formula for determining the optimal wager

Searching for long-term profit is mathematically equivalent to that of .
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 * = pq/c − 1
See also .

Kelly criterion calculator

Bankroll fraction to wager:  f * = %
stakes m* =

Special cases

Some cases

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 .

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 !

See also: