The Kelly Criterion

[stags]

Can someone explain to me how you would use the below Kelly Criterion in sports betting?


The Kelly Criterion is a math formula that tells you what fraction of your bankroll to wager on a given bet to attain the most rapid possible growth of your bankroll. It was developed in 1956 by physicist and Bell Labs research scientist John L. Kelly.

The formula is quite simple from a math standpoint:
f = (bp-q)/b

where

* f is the portion of the current bankroll to wager
* b is the ratio of profit to amount risked on the bet when you win
* p is the probability of winning the bet
* q is the probability of losing the bet (1-p)

If the bet size is 0 or negative, you should not take the bet.
You will be losing money in the long run.

A simple illustration of this would be a situation where you are rolling a single die. If the number rolled in a 6, you win 7 times the amount wagered. If the number rolled is 1,2,3, or 5, you lose your bet. What is the optimal amount to bet on this game?

* b = 7
* p = .1667
* q = .8333

f = [(7)(.1667) - .8333]/7
f = 0.0477

Your Optimum bet size in this situation would be 4.77% of your bankroll.
As your bankroll grows the dollar amount you bet would also grow.
The 4.77% would remain constant.

This formula can be applied to sports betting with great effect, but only if the bettor can accurately estimate the probability that his bet will win. This tends to be very difficult, so as a precaution against over betting many bettors will bet half the Kelly recommended amount. This Half Kelly method produces about 75% of the rate of return of full Kelly, while also muting volatility and preventing inadvertent over betting from win probability estimates that are too high.

[uva3021]

A more urgent expression of Kelly stakes from the point of view of a sports bettor who tries to quantify an edge versus a market, compared to other gambling enterprises where the odds are a static, is thus:

Stake = Edge / (Decimal Odds -1)

Stake represents percentage of bankroll, edge is expected win probability minus market win probability.

For example. I calculate win probability of 55% for team A, while the market price is even.

5% / (2.00 - 1) = 5%

The optimal wager in this scenario is 5% using a full Kelly philosophy.

[Kramer]

INTERESTING ARTICLE :

HERE'S WHAT YOU'LL NEED WITH ALONG WITH
A ½ HOUR TIME


1- A HAND CALCULATOR
2- TWO DECKS OF CARDS
3 – SHEETS OF LINED PAPER
4 – PEN
5 – A THANK YOU NOTE


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,’


ACES ARE TIES

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.


....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.

Like religionists, 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

[stags]

thanks for info

[jarhead60]

kramer, I disagree with your analysis.

Kelly assumes that the win percentage is constant and the the games are independent events. Your deack of cards analogy is not independent events. Kelly maximizes profits regardless of the order of wins and losses, so long as the win percentage is accurate. Run a spread sheet with an assumed win percentage. Run one spread sheet with all the expected wins coming at the front end followed by all the losses in a row. Use a 60% win rate (14.5 % bet size), odds of bet 11 to win 10, and 5 games. The final bankroll is same if the three wins come in the first 3 games or in the last 3 games.

The weakness in Kelly is that win actual win rate must be the same as the estimated win rate to maximize profits, which rarely happens in pro football because the annual sample is too small. Multiple bets at the same time also is a problem in practice.

[Kramer]

Fair enough , back to NFL

The Kelly Criterion is a bet-sizing technique, which balances both risk
and reward for the advantage gambler.
The same principle would work for any investment with an expectation
of being profitable.


For the gambler/investor with average luck bankroll and a fixed bet size,
bankroll growth is defined as:
Producti [(1+wixi)^(n*pi)] - 1, where
wi is the net payout for the ith outcome
xi the stake for the ith outcome
pi the probability of the ith outcome.

This product is maximized by Kelly betting.


Kelly betting also minimizes the expected number of bets
required to double the bankroll, when bet sizing is always
in proportion to the current bankroll.


The Kelly bet amount is the optimal amount for maximizing

the expected bankroll growth,
for the gambler with average luck.

While betting more than Kelly will produce greater expected
gains on a per-bet basis,
the greater volatility causes long-term bankroll growth to decline
compared to exact Kelly
bet sizing.


Betting double Kelly results in zero expected growth.


Anything greater than double Kelly results in expected bankroll decline.


What is more commonly seen is betting less than the full Kelly amount.
While this does lower expected growth, it also reduces bankroll volatility.
Betting half the Kelly amount, for example, reduces bankroll volatility by 50%, but growth by only 25%.


For simple bets that have only two outcomes,
the optimal Kelly bet is the advantage
divided by what the bet pays on a "to one" basis.


For bets with more than one possible outcome, the optimal Kelly wager
is that which maximizes the log of the bankroll after the wager
. However, for bets with more than one outcome, that can be hard to determine .

A sports wager has a 20% chance of winning, and pays 9 to 2.
The advantage is 0.2×4.5 + 0.8×-1 = 0.1.

The optimal Kelly wager is 0.1/4.5 = 2.22%.

Following is the exact math of example .


Let x be optimal Kelly bet, with a bankroll of 1 before the bet.
The expected log of the bankroll after the bet is...
f(x) = 0.2 × log(1+4.5x) + 0.8 × log(1-x)
To maximize f(x), take the derivative and set equal to zero.
f'(x) = 0.2 × 4.5 / (1+4.5x) - 0.8 / (1-x) = 0
0.9 / (1+4.5x) = 0.8/(1-x)
0.9 - 0.9x = 0.8 + 3.6 x
4.5x = 0.1
x = .1/4.5 = 1/45 = 2.22%


The math gets much messier when there is more than one possible outcome
(E.G, - A PUSH)

CHEERS