A common question in betting is determining how much to bet on a given opportunity. The Kelly criterion aims to answer that question in an objective way.

If you find a bet with positive expected value, then the more you can bet at these odds, the more you expect to win. This can make it hard to figure out exactly how much to bet - the more you bet, the higher your expected value, but also the more you have at risk. If you routinely risk a large percentage of your bankroll, even if you are following profitable bets, a bad run of luck will eventually wipe out your bankroll.

Scientist John Kelly found that by following a specific bet sizing strategy, you can maximize your expected bankroll growth. His Kelly strategy optimally balances risk and reward. This strategy prefers opportunities where you are likely to win, rather than longshots, and also prefers opportunities with higher model edges.

Although there are some issues with using this in practice, it is a very good starting point, and some of the issues can be accounted for. Let's take a look at the formula. 

Kelly Bet Size = Bankroll * [ Model Edge / (Bookmaker Odds - 1) ]
Model Edge = Model Probability * Bookmaker Odds - 1

As an example, let's say we have a bankroll of $1000, and we are looking at a bet with decimal odds of 2.20, that we feel has a 50% chance to win.

Model Edge = 50% * 2.20 - 1 = 10%
Kelly Bet Size = $1000 * 10% / (2.2 - 1)  = $83.33

The Kelly strategy recommends a bet of $83.33.

Kelly Fraction

In practice, most bettors find the Kelly strategy to be overly aggressive, and most bettors will apply a fraction when using Kelly bet sizing. The fraction simply multiplies the final bet size - so using a fraction of 1/2 would result in half the bet as using the 'standard' Kelly bet size.

There are several good reasons to apply such a fraction. The strategy assumes that your model is perfectly accurate - in reality, there's always some error in your model's probabilities, no matter how accurate they are.

Another assumption going into the strategy is that all bets are independent. If you bet on multiple maps of the same series, these bets are dependent, and having a more conservative fraction is wise to account for this. To properly apply Kelly bet sizing to dependent bets is considerably more challenging.

Professional bettors that we have talked to recommend Kelly fractions anywhere from 1/4 to 1/40.

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