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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.
*****************
For a related
article concerning money management and the Kelly criterion, please visit: www.professionalgambler.com
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