The Kelly Criterion

Yeah, I know.  The title of this article sounds like the title of a spy novel.  John Kelly was an interesting fellow, but as far as I know he was never an undercover agent.  John Kelly worked at AT&T’s Bell Labs in the fifties.  In 1956 he developed what has become known as the Kelly Criterion. His original formulas dealt with long-distance telephone transmission signal to noise ratios. Exciting stuff, right? But the gambling community quickly grasped that Kelly’s approach could be used to help calculate the optimal amount to bet on a horse race.

How does that work?  Essentially, the “Kelly”  is an “up as you win” wagering system designed to maximize the expected growth of the player’s bankroll. This is accomplished by adjusting the size of the bet to an optimum level based on the player’s advantage or positive expectation. It assumes that the player only places action when he has a positive expectation.

In horse racing, this is based on the player’s own handicapping of the race. Of course, the Kelly has also been adapted for other “bets,” including blackjack, sports betting, and playing the stock market. Does it have any application in casino craps? Well, the answer to that is – depends. It depends on your advantage – and how consistent it is.

Let’s walk through a simple example first, then I’ll show you how to determine just what fraction of your bankroll should be wagered based on the Kelly.

Let’s say you have $100 bankroll and are skilled enough at tossing the dice to win on the ten fifty percent of the time. For the sake of this example we’ll bet $20 – or 20% of your bankroll on the ten. Winning bets will be paid at 2-to-1. We’ll ignore the commission for now just to keep it simple. If you win this first $20 bet, you win $40, increasing your bankroll to $140. How much is your next bet?

Based on Kelly’s principles, you would again bet 20% of your bankroll. Remember, that percentage is dictated by your advantage on the ten – not by your bankroll. But since your bankroll increased to $140, you’re next bet is larger than your initial bet. The next bet is $28. Now imagine you win again employing the same strategy. Your wager of $28 would win $56, increasing your total bankroll to $196. That’s a very nice two-win increase over your initial $100 bankroll – almost doubling your buy-in.

But what do you do on a loss? Simply put – the same thing. Let’s say you continue this strategy and bet 20% of $196 – or $39 on your next hand. If you lose that bet your total winnings slip back to $157. That’s still $17 more than you had after your first win. If you continue to play, your next wager will still be 20% of your bankroll. But since your bankroll is now $157, your bet size is reduced to $31. Now let’s assume you lose that wager as well. You still have $126 in the rack – a net $26 win for the series. Note that you had two wins and two losses for the series, but are still dollars ahead. That is the power of the Kelly.

Most people who play the Kelly base their progressions on the saying, “Bet the fraction of your bankroll that equals your percentage advantage.” However, this saying assumes you are playing at an advantage on even-money wagers only. A better choice would be to bet a fraction of your bankroll equal your percentage advantage divided by the “to” odds on that wager.

For example, if you have a 20% advantage on the nine and you are getting 7-to-5 odds, then the fraction of your bankroll to bet is (0.20)/1.4 = 0.14 =14%. Of course, if you toss out 14% of your bankroll on a single number, lose, then post about it on the craps forum – a certain fat guy we all know and love will probably say you over-bet your bankroll. But that’s a topic for another day.

The Kelly Criterion is a recognized money management system used extensively in horse and sports betting, and in the stock market. Whenever the question of optimal betting size pops up in handicapping or money management books, you usually see Kelly formula mentioned. It’s a great “up as you win” way to play – provided you put the tracking time in to know where your advantage is.