The Kelly Betting Criterion

 

Imagine you are gambling and find out that you have an edge on the game. For instance, say you are betting on the outcome of a coin flip which is usually a 50% heads / 50% tails proposition, but in this case, you have knowledge that the coin being flip will show heads 60% of the time, and tails just 40%. If given these facts, one might want to put everything that have on heads. The downside of such a strategy, is there is still a 40% chance you might go bankrupt on the first flip.

Given the known advantage you have, if you take the opposite extreme where you wager just a small portion of your bankroll to be safe, you’ll avoid bankruptcy, but you’ll also fall short of the optimal profit you might make over the long run.

This problem of optimizing profit was recognized going back to beginning of probability theory in the 18^th century when Daniel Bernoulli suggested that when one has a choice of a series of bets or investments, one should choose that with the highest geometric mean of outcomes.

Move forward to the 20^th century, Bell Labs researcher, J. L. Kelly, Jr., formalize this strategy in establishing the Kelly Criterion (also known as the Kelly Bet or Kelly Strategy).

The strategy to optimize the long-term outcome, is with each bet to a certain fraction of your total bankroll. This fraction (f) is given by the formula:

f = p – q/b,

Where f is the fraction of the current bankroll to wager,
p is the probability of winning
q is probability of losing, and
b is proportion of the bet gained with a win (the odds being offered)

In our example with the dishonest coin flipping, p=0.6, q=0.4, b=1

f = 0.6 – (0.4/1) = 0.2

So, our first bet would be wagering 20% of our bankroll. If we won the first bet, our next wager would be adjusted higher to reflect the new, higher bankroll. Or, if we lose the first bet, our next wager would be lowered because our lower bankroll.

If you enter a casino, where with very few exceptions, you don't have an advantage, the fraction, f, is zero or negative - meaning you shouldn't wager anything. 

Interestingly, there are applications beyond gambling where this strategy is used. Warren Buffett and other well-known investors used this strategy in allocating their investments.

There is a good video by Adam Kucharski that also explains some applications of the Kelly Bet: (1584) How Science is Taking the Luck out of Gambling - with Adam Kucharski - YouTube

The website, Wizard of Odds, has a page on the Kelly Criterion and has the results of some simulations based on the criterion. https://wizardofodds.com/gambling/kelly-criterion/