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The Best Way to Gamble
R. J. Miller on Money Management
DEBUNKING THE KELLY CRITERION
Here's how to compare betting systems
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:
Pick any size fantasy bankroll to use as your total bankroll. Why not $10,000?
Thoroughly shuffle the 2 decks of playing cards 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, and that would be a great long term winning percentage against sports betting.
But that expectation will vary widely as you remove cards from the deck. As you turn the cards and remove them from the remaining deck the deck will turn ‘positive’ or ‘negative'; - that is, if you remove more 'losers' (2's through 6's) from the deck than 'winners' (7's through K's), the remaining deck will offer a higher expectation of 'winning' on the next draw, and vice-versa.
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. As a Kelly bettor, of course, your bet sizes will vary up and down as your winning expectation goes up and down. Go ahead and do this 50-or-so times before reshuffling the deck and starting over. (Don’t do it more than 50 or 60 times without reshuffling. Always reshuffle when you've been through about half the double deck. Don't go through the entire double deck.)
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. Size your Kelly bets exactly as you do against sports, and 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 - maybe even a dozen or so - just like when you're betting on sports.
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.
Right here is precisely where Kelly promoters always screw up. Let's say they have 100 actual bets wherein they win, say, 58 and lose 42. That's a great winning percentage of 58%, of course. Now, they'll explain that their basic bet is, say, $100, but if their expectation is higher than such-and-such percentage they risk $120, or $130, or whatever, and if their expectation is even higher than such-and-such they might risk $200 or more.
Then they compare what they won by using the Kelly system to what they would have won had they been risking only $100 on each of the 100 bets.
.....Duhhh!.....They risked more money with the Kelly system and they made more money after going 58-42. I hate to burst their balloon, but when you go 58-42, the more money you risk the more money you figure to make. Sorry, boys, but that is not news.
The only way to fairly compare the Kelly 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 the Kelly bets. That way, you’re risking the same total amount against the same overall won-lost results. No fair risking more money overall with one system than the other. That obviously skewers the results.
Or, another way to fairly compare betting systems is to keep track of the winnings as a percent of the total amount risked. If Betting System A wins 8 percent of all monies risked while Betting System B wins 12 percent of all the monies risked, Betting System B is obviously better than Betting System A.
This is precisely what Kelly-promoters choose to ignore. They want so very hard to believe their fantasy they won't be made to face facts, no matter what. Comparing flat betting against a "1-star, 2-star, 3-star" system, for example, and going 58-42, if all your flat bets are only as big as your "1-star" bets, of course you will win more with the star system. You're risking more and you're winning 58% of your bets.
All right, back to our double-deck of cards. Time to check the profits from flat betting against the record of the Kelly criterion, and ta-daa! There’s your proof. Using the average size of your Kelly bets as your flat bet, 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 "1-star, 2-star, 3-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 "1-star" bet, a "2-star" bet or a "3-star" bet. Perhaps between 55 and 58 percent you could use a "1-star" bet, etc. Of course, when your winning expectation is less than 53 or 54 percent, there is no reason to bet at all. Bet any system - including flat betting - only when you have an acceptable winning expectation.
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 lose 12 bets in a row you’d have to risk $409,600 to win $100, but how often is that going to happen?
As it turns out, plenty.
Modifications of the Martingale can make it 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? You see, with all progressive betting schemes the ratio of risk rises or falls in direct proportion to the ratio of "guaranteed" profit. This fact includes 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 short-term profit is not so absolute, so the potential for short-term disaster is not so absolute. Nevertheless, you can be sure the potential for disaster is increased dramatically. Over a relatively short period of time you will invariably go broke due to the vagaries of binomial distribution concerning your winners and losers. (See our article, "Binomial Distribution and You.")
NOTE: According to expert researcher Dr. Nigel E. Turner, Ph.D., Scientist, Centre for Addiction and Mental Health ( email@example.com ), incremental betting is one of the telltale signs of someone with a gambling problem.
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. Against casino blackjack you will be urged by a lot of "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, I think 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 - my "real" bet - whenever the deck was in my favor and whenever I felt no heat from the floor people, and trying to risk as little as possible or nothing at all when the deck was negative. Any other bet size, including whatever size bet 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 simply 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 and a lot less risk.
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 abstract factors are not so precisely calculable. In other words, you can never know what your winning expectation might be against a sports event. The best you can hope for is to believe you have at least a 55%-60% chance of winning. It is futile to try to handicap your own handicapping.
In our books and articles and in 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 any believer in progressive betting systems will actually do the test. People want very hard to believe what they wish to be true, and they simply choose to disbelieve what they wish to be untrue. Once people denied the earth was not the center of the universe, or denied it was round. Users of progressive betting schemes want very much to believe they can earn more than they deserve. They refuse to be confused by facts.
- J.V. Miller
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