Talk:Kelly criterion
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[edit] Factual basis of Kelly Criterion is in dispute
The Kelly Criterion is not universally accepted in the mathematical community. For example, see http://www.bjmath.com/bjmath/kelly/mandk.htm. The dispute seems to hinge on the fact that the choice of utility function is arbitrary. There is no reason to prefer the log utility function in the current version of the article over others a priori, as any monotonically increasing utility function will result in infinite predicted wealth with time.
- Sorry but this has nothing to do with utility theory. Kelly's theory that you should choose the result with the highest expected log result. This is the same as Bernoulli's theory with a choice of bets you choose that with the highest geometric mean of outcomes. Perhaps utility theory can be derived from Kelly, but it's not necessary for Kelly. This source isn't great it's from a blackjack forum, and the writer doesn't appear to have academic credentials. Please don't forget to sign your posts. --Dilaudid (talk) 12:13, 24 January 2008 (UTC)
The formula can be simplified:
(bp-q)/b => b(p-q/b)/b => q=1-p so k=p-(1-p)/b
--Geremy78 09:49, 28 January 2006 (UTC)
[edit] Generalized Kelly Strategy
The "generalized form of the formula" given in the article isn't really the most general. The most general expression of the Kelly criterion is to find the fraction f of the bankroll that maximizes the expectation of the logarithm of the results. For simple bets with two outcomes, one of which involves losing the entire amount bet, the formula given in the article is correct and is easily derived from the general form. For bets with many possible outcomes (such as betting on the stock market), the calculation is naturally more complicated.
One statement in the article,
In addition to maximizing the long-run growth rate, the formula has the added benefit of having zero risk of ruin, as the formula will never allow 100% of the bankroll to be wagered on any gamble having less than 100% chance of winning.
isn't strictly true from a theoretical standpoint. It is always true that Kelly strategy has zero risk of ruin, but in the general case it is not true that a bet of 100% of the bankroll is not allowed. If the probability of losing the entire amount of the bet is zero, then bets of 100% and even larger (buying stocks on margin, for example) are allowed. Investing in a stock index (as opposed to a single stock or small number of stocks) could allow such percentages, if we assume the index can never go broke (even though individual stocks might), and that the index has a positive expectation of outcome (adjusted for inflation, since we are dealing with money invested over time). Of course, any real-world investment will have a non-zero chance of going bust, and therefore Kelly strategy will indicate a bet of less than 100% of bankroll.
Rsmoore 07:55, 4 February 2006 (UTC)
[edit] Formula presentation
While the Kelly Criterion formula can be "simplified" to remove the q term, it actually becomes longer, less intuitive, and harder to remember. As a result, it is generally presented as (bp-q)/b.
Whoever "corrected" the formula to (bp-1)/(b-1), this is incorrect. I have changed it back to the correct formula. I cite as my source William Poundstone's book Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. Additional sources can also probably be found all over the internet.
(bp-1)/(b-1) is not incorrect, it depends on the definition of odds, i.e. how "b" is defined. (bp-1)/(b-1) is for European/decimal odds. —Preceding unsigned comment added by Cwberg (talk • contribs) 13:16, 11 January 2008 (UTC)
[edit] Added disadvantages section
The information in that section is all from reading Poundstone's book. The book is very very verbose and completely non-mathematical - designed for bedtime reading I expect. It could have been condensced to 1/10th. of the length without losing much. I have seen more concise explainations of Kelly in other popular books about chance, although I do not remember reading before about the volatility problem or over-betting. Something only mentioned in one sentance is that its easy to overestimate the true odds and unconciously overbet, leading to ruin. It is suggested this happened to Long Term Capital Management.
There are no explainations of the maths behind the information stated in the book - you have to take it on trust. For those with busy lives you can find all the relevant info by looking up Kelly criterion and geometric average in the index. The pages in the 2005 hardback edition I thought were most relevant were pgs. 73, 191, 194-201,229,231,232, 297, 298.
It does have an extensive bibliography, and there is a reference for: Bernoulli, Daniel (1954) "Exposition of a New Theory on the Measurement of Risk" Trans. Louise Sommer, Econometrica 22:23-36. Wonder if its available online? Henry A Latane/ did some academic papers about the geometric mean criterion. The books (academic or popular) of William T Ziemba also seem of interest, including Beat the Racetrack.
Poundstone describes the Kelly criterion in his own way (pg.73): he says you should gamble the fraction edge/odds of your bankroll. Edge is how much you expect to win on average. Odds are the public or 'tote-board' odds. Example: the tote board odds for a horse are 5 to 1. You think the horse has a 1 in 3 chance of winning. So by betting on the horse you on average get $200 back for a $100 stake, giving a net profit of $100. The edge is the $100 profit divided by the $100 stake, giving 1. So in this case the edge is 1. The odds are 5 to 1 - you only need the 5. So edge/odds is 1/5 - you should bet one fifth of your bankroll.
As someone who has never gambled on races, I find "odds" confusing. I wish someone would also provide a formula in the article where only p is used, that is more suitable for use with investments.
Where the book really falls down is in describing multiple bets. Poundstone just baldly says you can bet more of your bankroll with simultaneous bets - but he dosnt give any how or why, although this would be useful to know. Perhaps he doesnt understand this himself.
As someone who is currently making heavily geared real-estate investments, I think the encyc. article should go into much more detail than currently, including practical applications. I wish I had some guidance on how much I should optimally invest. I find the idea of choosing the greatest geometric mean much easier to understand than the Kelly criterion. In business investments I suppose you would take the geometric mean of the expected net present values - or would you?
As Poundstone points out (I think), the geometric average rule does have a flaw. For example, if you had a bet for a $10 stake where you had a 99% chance of winning $1000000 and a 1% chance of winning $0, then the geometric mean criteria would tell you to ignore this bet completely! (Please tell me if I've got this wrong.)
The book is about 90% chat about financial things only tenuously linked to Kellys criteria - about various imprisoned and/or ruined Wall St. multi-millionaires, about the links one large well known entertainment company is said to have/had with the Mafia. It says nothing about Shannon's communication theory, and zilch about the links between this and Kelly's criterion, which was my reason for ordering the book. It does describe Thorp quite a lot though.
Perhaps someone could add some references to some more concise popular expositions of the criterion.
- For example, if you had a bet for a $10 stake where you had a 99% chance of winning $1 000 000 and a 1% chance of winning $0, then the geometric mean criteria would tell you to ignore this bet completely! (Please tell me if I've got this wrong.)
- Let's calculate:
- f == (p*b - q)/b
- in this case, p = 0.99, q = 1-p = 0.01, and b = 999 990, so
- f = ( 0.99*(999 990) - 0.01) / ( 999 990 ) = 0.99000
- . So the geometric mean criteria recommends you keep 1% in your pocket, and bet everything else.
- --68.0.120.35 19:55, 3 March 2007 (UTC)
Hi I think RE the 1% chance of winning $0, I think you meant a 1% chance of losing all your money. Then the Kelly criterion would say do not bet. —Preceding unsigned comment added by 82.26.92.226 (talk) 09:49, 4 January 2008 (UTC)
[edit] Shannons stock system
Continuing from the above, Poundstones book also mentions an interesting (theorectical) investment system devised by Shannon.
Shannons actual stock investments (the book says) were buy and hold. He selected stocks by extrapolating earnings growth (using human judgement). Two or three of Shannons stocks accounted for nearly all the value of his portfolio.
He also devised an interesting theorectical system for investing in stock with a lot of volatility but no trend (pg. 202). Put half your capital into stock and half into cash. Each day rebalance by shifting from stock to cash or vice versa to keep these proportions. Surprisingly, the total value grows. In practice the dealing commissions would remove any profit.
This system is now known as a "constant-proprtion rebalanced portfolio", and has been studied by economists Mark Rubenstein, Eugene Fama, and Thomas Cover.
[edit] Kelly Criterion For Stock Market
Kelly Criterion For Stock Market should be merged into this article. (Nuggetboy) (talk) (contribs) 19:11, 25 January 2007 (UTC)
I think merging will create a lot of confusion. For practicality they should be separated because Kelly Criterion For Stock Market requires the reader to have some math and finance background. (User:Zfang)
I would support merging. The article here is much more encyclopedic, the one to merge has some tone/content issues - the approach is more instructional and doesn't sound appropriate here.--Gregalton 22:44, 15 February 2007 (UTC)
[edit] formula derivation
Rather "take it on trust" that the formula is correct, (or, worse, refuse to believe and repeatedly substitute random variations), I would much rather people check my calculation and fix any flaws I introduce.
- f is the fraction of the current bankroll to wager;
- b is the odds received on the wager;
- p is the probability of winning;
- q is the probability of losing, which is 1 − p.
- m0 is your initial bankroll.
- m1 is your bankroll after 1 round of the game.
- When you lose, you lose the entire amount of your wager.
- when you lose, the amount in-the-bankroll becomes m1=m0*(1-f).
- When you win, you win back your wager, plus the amount bet times b, the payoff odds.
- when you win, the amount in-the-bankroll becomes m1=m0*(1+b*f).
Before I derive it, let me list some characteristics I expect the "correct" formula to have:
- as long as p is strictly less than 1.0, f is less than 1.0 : "never bet the whole wad".
- If p is 1.0 and b is greater than zero, f should be exactly 1.0 : "bet the whole wad on a sure win"
- If p is 0, f should be ... less than 0 ?
- as b increases from q/p ("expect to break even") up, f should increase monotonically ("non-decreasing") from 0 towards 1.0. "Even if there is only a 1 in 10 chance of a horse winning, you can still make money if the odds are 11 to 1"
- As p increases from 1/(1+b) ("expect to break even") up to 1.0, f should increase monotonically from 0 to 1.0 .
We want to maximize the geometric mean of the ... (fill in here ...). To do that, we pick f to maximize the expected value of the log of the final amount in-the-bankroll m1.
pick f to maximize g(f), where
g(f) = expectation( log( m1 ) ) = g(f) = p*log( m1_when_we_win ) + q*log( m1_when_we_lose ) = g(f) = p*log( m0*(1+b*f) ) + q*log( m0*(1-f) ) = g(f) = log(m0) + p*log(1+b*f) + q*log(1-f).
For a smooth function like this, the maximum is either at the endpoints (f=0 or f=1.0) or where the derivative of the function is zero: ( k1 depends on whether we use log10(), log2(), loge(), etc. -- but it turns out to be irrelevant. )
(d/df)g(f)= 0 + p*k1*( 1/(1+b*f) )*b + q*k1( 1/(1-f) )*(-1) = (d/df)g(f)= p*k1*b/(1+b*f) - q*k1/(1-f) find f where 0 == (d/df)g(f). 0 == p*b/(1+b*f) - q/(1-f) 0*(1+b*f)*(1-f) == p*b*(1-f) - q(1+b*f) 0 == p*b - p*b*f - q - q*b*f 0 == p*b - q - ( p*b + q*b )*f 0 == p*b - q - ( b )*f f == (p*b - q)/b
And there we have it.
(Should I cut-and-paste this derivation into the article, like Kelly Criterion For Stock Market includes the derivation in the article?)
new term:
- n is the "know-nothing" estimate of the probability given only the posted odds b, n = 1/(b+1), b=(1-n)/n
- If the true probability of winning exactly matches the posted odds (p=n), I expect f=0.
Other ways of expressing the value of f:
f == (p*b - q)/b f == p - (1-p)/b f == p - (q/b) f == (p(b+1) - 1)/b f == (p-n)/(1-n) f == 1 - q/(1-n)
Special cases:
for even-money bets (b=1, so n=0.5), f=p-q. for "huge payoff" bets, where 1 << b but p << 1, we can approximate f ≅ p - 1/b ≅ p - n.
-- User:DavidCary --68.0.120.35 19:55, 3 March 2007 (UTC)
