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Debunking The Kelly Criterium
by J.R. Miller
I apologize up front, but this article isn't for everyone. It's aimed specifically at sports bettors that are using - or that are planning to use - the so-called Kelly criterion to size their bets.
The Kelly criterion is essentially a progressive betting system wherein the higher your probability of winning, the more you're supposed to risk; the less your probability of winning, the less you're supposed to risk. (Sounds reasonable, all right.) We won't describe the Kelly system in detail here because it's boring and it takes too long. Those of you who are using it already know how it works. Besides, by now there are so many variations that the one you might be using could be a lot different from any particular one we might describe. I'll cut right to the chase. None of the variations work; - at least, not against sports betting. I'm going to explain to you right here, right now, once and for all why the Kelly criterion as applied to sports betting would be better called the Kamikaze criterion. You can prove it for yourself, and here's how:
Here's what you'll need, along with at least a half-hour of time:
1. A hand calculator
2. Two decks of ordinary playing cards
3. Lined paper
4. Pen or pencil
5. A 'Thank You' note (to send to me after you complete this exercise).
Pick any size fantasy bankroll to use as your total bankroll. Why not $10,000? Thoroughly shuffle the 2 decks together and place them face down in front of you. We're going to turn one card at a time and count it as a win, loss, or tie. Everything 7 through King will be a 'winner,' everything 2 through 6 will be a 'loser,' the Aces will be ties. With those rules the double deck contains 56 'winners,' 40 'losers,' and 8 'ties.' That makes an overall winning expectation of 58.3 percent, but that expectation will vary widely - just as it does in sports betting - as you turn the cards and the deck turns 'positive' and 'negative.' Figure the sizes of your Kelly bets accordingly. If the first card is a 'loser,' there are only 39 losers left in the deck, but still 56 winners. Your winning expectation for the second draw ('bet') increases to 56 out of 95, or 58.9 percent. If the first card is a 'winner,' your winning expectation for the second draw drops to 55 of 95 or 57.9 percent. This, of course, is where the hand calculator comes in.
Be sure to record whether you won or lost the first bet, and how much you won or lost. Go ahead and do this 50-or-so more times before reshuffling the deck and starting over. (Don't do it more than 50 or 60 times without reshuffling.) Remember, according to the Kelly criterion if the deck goes 'negative' and you do not have a positive expectation don't bet anything. Just flip the next card and the next until you do have a positive expectation.
To make the exercise more realistic, as when actually betting against sports, flip several cards at once. After all, NFL, NBA, MLB and NHL games often go off several at a time and cannot be bet sequentially. You have to lay several bets at once. Try flipping 3 or 4 or more cards at once. After doing another 50-60-or-so observations with the reshuffled deck, it's time to compare your results using the Kelly criterion against so-called 'flat' bets. You cannot compare results as you go along because there's no telling what the size of a comparable flat bet should be. The only way to fairly compare the Kelly system (or the so-called "star" system or any other progressive betting scheme) to flat betting is to use a flat bet the same size as the AVERAGE size of all your Kelly bets. That way you're risking the same total amount against the same overall 58.3 percent expectation. No fair risking more money overall with one system than the other. That would obviously skewer the results. In fact, that is precisely where most proponents of the Kelly criterion get bushwhacked. Without noticing, their average bet with the Kelly criterion is bigger than their flat bets. It's only natural that you'll win more money (when you're winning) if your bets are bigger. It is critical to your results that the total amount risked is the same for both systems. Comparing flat betting against a "one-star, two-star, three-star" system, if all your flat bets are only the same size as your "one-star" bets, you will naturally get an erroneous comparison. All right, time to check the profits from flat betting against the record of the Kelly criterion, and ta-daa! There's your proof. The Kelly loses, and it loses every time. In fact, using most forms of the Kelly criterion, I would be surprised if after 70 or 80 'bets' you are not - for all intents and purposes - broke.
You can use the same results to compare the old "one-star, two-star, three-star" system. You don't have to flip the cards again, you can use the same won-lost progression you got while testing the Kelly criterion. Set your own parameters concerning when to use a "one-star" bet, a "two-star" bet or a "three-star" bet. Perhaps between 55 and 58 percent you could use a "one-star" bet, etc. (Of course, when your winning expectation is less than 53 or 54 percent, there is no reason to bet at all.) The cold hard fact is that all progressive betting systems are nothing more than modified versions of the Martingale system. In the Martingale, you risk one unit, and if you win you keep risking one unit. If you lose, you double your bet, and if you lose again you re-double and keep re-doubling until you finally do win. Then you go back to risking one unit. As any fool can plainly see, the Martingale can't miss, so long as you win one more bet before you die you're going to be a winner. Well, yeah, if you lose10 bets in a row you'd have to risk $1,024 to win $1, but how often is that gonna happen?
As it turns out, plenty. Modifications of the Martingale have been devised to be more "forgiving." One of the ways to soften the Martingale is to double your bet after two losses instead of after every loss. Or how about increasing the bet by only 50% instead of doubling?
With progressive betting schemes the ratio of risk rises or falls in direct proportion to the ratio of "guaranteed" profit. This is true of all progressive betting systems, including the Kelly criterion. With the Martingale, the promise of profit is essentially absolute, so the potential for disaster is also essentially absolute. With the Kelly criterion the promise of profit is not so absolute, so the potential for disaster is not so absolute. Nevertheless, you can be sure the potential for disaster is increased by the use of the Kelly criterion, and the potential for disaster is increased dramatically. The Kelly criterion is sometimes touted as the best strategy against casino 21, but during my years as a card counter I finally learned otherwise. You will be urged by self-proclaimed blackjack experts to use graduated bet sizes, depending on whether your expectation of winning the next hand is 51 percent or 53 percent or 55 percent. Frankly, that's a lot of hooey, and I've played an awful lot of blackjack. My strategy finally evolved into trying to risk my maximum bet whenever the deck was in my favor and whenever I felt no heat from the floor people, and trying to risk nothing at all when the deck was negative (and I could get away with passing the hand). Any other bet size, including whatever size bet is made after the dealer shuffles, is nothing but camouflage in order to hide from the pit boss. The key phrase, of course, is "maximum bet size." Supporters of the Kelly criterion are apparently espousing that when you have a 55% winning expectation you're supposed to use a bet so large that it would bankrupt you if you had only a 52% winning expectation. I can pretty much guarantee that if your bets are big enough to break you with a 52% winning expectation, they will sooner or later break you with a 55% winning expectation. Blackjack players using the Kelly system are kidding themselves. If they could go back and average all their Kelly bets and simply bet that average amount every time the deck was in their favor they would end up with a lot more profit.
At least, good card counters can know relatively clearly what the expectation of winning might be. The numbers on the cards speak for themselves. Against sports, however, you can never know that. There are countless variables involved in deciding which team is going to win a football, basketball, baseball or hockey game. These subjective and abstractual factors are not so precisely calculable. In other words, you can never know what your winning expectation might be against a sports event. It is futile to try to handicap your own handicapping.
In my books and articles at our website,
www.professionalgambler.com, and in our PROFESSIONAL GAMBLER Newsletter I repeatedly warn bettors that the size of their bets cannot be used as a pry-bar to win more than they deserve. If you'll do the exercise above you will prove it for yourself. …But y'know what? I don't expect a big upsurge in my mail due to Thank You notes. Hardly anyone will do the test. People tend to believe what they wish to be true; they tend to disbelieve what they wish to be untrue. Using a flat bet is the most boring of all betting "systems" because it does not offer more profits than you deserve. Nevertheless, it is the most effective "system" over the long haul largely because it is certainly the safest. We recommend using an absolute maximum of 2 percent of your working capital for individual bets, and not changing your bet size until your bankroll has increased or decreased by about 50 percent. Many non-professionals think 2 percent is too conservative, but most professional bettors use less than 2 percent. The consensus among pros I've known is 1 percent.
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