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"Optimal Gambling System for Favorable Games" is by Breiman, not Thorpe. I haven't read the Breiman paper so I don't know if the correct citation is that paper by Breiman, or some other paper by Thorpe. TimothyFreeman ( talk) 08:25, 14 April 2023 (UTC)
"This is done by maximizing the logarithm of wealth, which is mathematically simpler to do, and also maximizes wealth."
Wealth is a random variable (not a scalar), and you can't maximize wealth or the logarithm of wealth. You can maximize expected wealth or the expected logarithm of wealth, but those are NOT equivalent! is NOT equivalent to . The former maximizes the expected return, the latter maximizes the expected geometric growth rate of wealth. The former leads to going broke with probability 1 (and measure 0 probability of infinite wealth), while the latter leads to the Kelly criteria.
The distinction between maximizing expected wealth and the expected logarithm of wealth is absolutely critical. Someone reading this introduction may come away with serious misconceptions.
"... is a formula for bet sizing that leads almost surely to higher wealth compared to any other strategy in the long run. This bet size is found by maximizing the expected geometric growth rate (which is equivalent to maximizing the expected logarithm of wealth).
Mgunn ( talk) 15:53, 9 January 2019 (UTC)
I went ahead and changed it. Mgunn ( talk) 01:09, 10 January 2019 (UTC)
I disagree. Mgunn's statement above does correct the vague statement about "maximizing wealth", but the conclusion is incorrect. The first proof of the Kelly criterion given in the article maximizes the expected value of wealth after n bets of size x, which is (1+bx)^(pn) * (1-x)^(qn). It maximizes this by maximizing the log of the expected value of wealth, NOT the expected value of the logarithm of wealth. log(E[W]) = log((1+bx)^(np) * (1-x)^(nq)) = np*log(1+bx) + n(1-p)*log(1-x). We divide by n before maximizing, to maximize the log of the geometric mean of E[W]. So it does maximize the expected geometric growth rate, but maximizes the logarithm of expected wealth, not the expected logarithm of wealth. Reply with counterargument by Sept. 15, or I'll change it. Philgoetz ( talk) 03:21, 3 September 2019 (UTC)
The formula f = p/a-q/b is wrong. It should be f = p-q*a/b. It simplifies to f = (pb-qb)/b = expected profit / gains if success.
The way it's stated, f is usually bigger than 1, which is incorrect. This should be corrected. — Preceding unsigned comment added by 189.112.213.146 ( talk) 23:04, 21 March 2013 (UTC) Agreed the formula given gives ridiculous numbers (despite the apparent "proof"). E.g. p=0.8, q=0.2, a=0.25, b=0.10, so four out of five times we invest we make a 10% profit, the fifth time we make a 25% loss. The f = p/a-q/b forumula tells us to invest 120% of our capital (0.8/0.25 - 0.2/0.1 = 3.2 - 2.0 = 1.2). Not only is this incorrect but it's dangerous as users may take this formula and invest/bet amounts which are detrimental to them, and possibly bankrupting. There are no references to the wrong formula, so I intend to correct it to the formula you have mentioned, which is backed up my many sources elsewhere on the internet and reference them. In the event I am interrupted before I can put the edit in, please can somebody else do the correction? Here's an example of a reference with the correct formula (p-q*a/b): http://www.investopedia.com/terms/k/kellycriterion.asp — Preceding unsigned comment added by 85.255.232.117 ( talk) 18:43, 7 May 2016 (UTC)
You guys are going off the rails here. The formula p-qa/b gives the Kelly fraction to RISK. The formula p/a-q/b gives the Kelly fraction to BET. You want the second one, and these are not the same in your example. You are risking the amount a, but you are betting the different amount 1 dollar ("the value of your investment"). The 2 formulas differ by a factor of a, and they are only the same when a=1, that is, when you can lose your entire bet. In the example given above, the answer of 120% is neither ridiculous nor incorrect - it is correct. You need leverage to bet the Kelly optimal in this case, but betting 120% of the bankroll means that we are only placing 30% of the bankroll at risk since we can only lose 25% of our bet. To see that this is optimal, simply note that 120% indeed maximizes the expected value of the log of the bankroll 0.8log(1+0.1f)+0.2log(1-0.25f). This should be clear from the derivation of p/a-q/b under PROOF. This last revision should be reversed. Brucezas ( talk) 19:46, 3 September 2019 (UTC)
There should probably be a statement pointing out that this formula can advise betting more than 100% of the bankroll, but that the amount at risk will be less than 100%. The amount at risk can be obtained by scaling the result by a. Otherwise this is likely to be a common source of confusion. Brucezas ( talk) 19:53, 4 September 2019 (UTC)
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.
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)
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)
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)
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.
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)
A lot of this section is factually incorrect and needs to be revised. (I have no clue how to sign this, nor do I just wanna lop the whole section out of the page. But the 2nd and 4th paragraphs in this section are either misleading or factually incorrect)
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-proportion rebalanced portfolio", and has been studied by economists Mark Rubenstein, Eugene Fama, and Thomas Cover.
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)
There's a huge probleme with the objectif of this quadratic problem. At first, Q is not positive definite and Dim(Ker(Q)) = N-1, so under these constraints, in the best case, we can find N-3 different solutions which make the result unstable. Just try do find Xi+1 = Xi + dXi and look at the norm of dXi.
If you use a "regular" solver with an iterative method, you'll not converge in dXi but in dU (utility). That implies the problem is ill-posed. We cannot use it that way.
The Bad Boy (
talk) 14:33, 9 October 2013 (UTC)
The Application's to Stock Market section is quite poorly written and should, in my opinion, be removed. — Preceding unsigned comment added by 104.162.109.38 ( talk) 09:12, 12 September 2016 (UTC)
I am still puzzled what the objective function is that we are trying to maximise and what the constraints are. In my view, the article would benefit from a precise mathematical expression. I take it it's the limit of the expected wealth as the number of periods goes to infinity? For any finite number of periods T, I suppose I could beat Kelly, for instance, by betting Kelly until time t and put all my money on one side in the last round. — Preceding unsigned comment added by Derfugu ( talk • contribs) 11:25, 26 March 2011 (UTC)
Derfugu ( talk) 11:26, 26 March 2011 (UTC)
In the article it seems to be missing the fact that we are looking to maximise the expected value of the log of the wealth. Since I find it very confusing (I only found out the answer by checking this talk page), I'm going to add a line in the intro to make it explicit. Student73 ( talk) 13:29, 18 November 2017 (UTC)
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.
Before I derive it, let me list some characteristics I expect the "correct" formula to have:
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:
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)
-- User:derfugu
Hi, I deleted some stuff from the proof section - it appeared to me that the section on how to underbet and beat kelly was original research and I don't think it works - here's the section: suppose you bet a small amount less than Kelly on the first bet, and double the amount less than Kelly every bet until you finally lose a bet. At that point, you will be a small amount ahead of Kelly. If you follow Kelly thereafter, you will end up better than Kelly. This will happen unless you start with such a long string of wins that you can no longer double the amount you underbet Kelly. You can make the probability of this as low as you want by making the initial underbet very small. Thus you can come up with a strategy that will always outperform Kelly.. This strategy cannot be better than Kelly - the reason is simple, if it were better than Kelly, I would use the strategy on itself. Then I would have a strategy that was better than the better strategy. Ad infinitum. -- 165.222.184.132 ( talk) 09:01, 16 March 2009 (UTC)
You are correct that the argument can be used on itself to form an infinite series of better strategies. But that does not invalidate its point that it is possible to find a strategy better than Kelly. Whether this doubling strategy actually is better than Kelly, or even different from Kelly is a matter of dispute. Certainly it doesn't point to a practical way to improve on Kelly.
Anyway, the doubling argument is often made, and it was referenced. Whether you agree with it or not is not the point in Wikipedia. I do not think it should have been deleted. On the other hand, I don't have any strong feeling that it needs to be included. AaCBrown ( talk) 21:04, 10 October 2013 (UTC)
This scheme is like the argument that is known as Leib's paradox, and a proof concerning it appears in the 2008 article Understanding The Kelly Criterion by Edward O. Thorp--- the man. The article is a revised reprint from two columns from the series A Mathematician on Wall Street in Wilmott Magazine, May and September 2008. Thorp explains, contrary to the commentary here to the effect that nothing like this could work, that Lieb's trick works. Here's Thorp's summary of it:
"To prove the first part, we show how to get ahead of Kelly with probability 1 − ε within a finite number of trials. The idea is to begin by betting less than Kelly by a very small amount. If the first outcome is a loss, then we have more than Kelly and use the strategy from the proof of the second part to stay ahead. If the first outcome is a win, we’re behind Kelly and now underbet on the second trial by enough so that a loss on the second trial will put us ahead of Kelly. We continue this strategy until either there is a loss and we are ahead of Kelly or until even betting 0 is not enough to surpass Kelly after a loss. Given any N , if our initial underbet is small enough, we can continue this strategy for up to N trials. The probability of the strategy failing is p N , 12 < p < 1. Hence, given ε > 0, we can choose N such that p N < ε and the strategy therefore succeeds on or before trial N with probability 1 − p N > 1 − ε."
My first thoughts--- I have to study the paper more--- are that the resolution may be to understand that the stated and mathematically proven Kelly "dominance", as Thorp sometimes calls it, the superiority, is established in a gross statistical sense and only with respect to what he calls (and mathematically defines) "essentially different" strategies. A strategy that is a close shadow of Kelley that is generally implemented for only a few trials is not essentially different. The basic meaning of that could be that the Leib strategy could in the long run amount to small potatoes compared to the difference between Kelley and an essentially different strategy. Mihael O'Connor ( talk) 12:04, 6 April 2014 (UTC)
There are two sentences in this article I disagree with:
"However, as is evident in the derivation below, the Kelly strategy is applicable only when the same proportion of one's bankroll is invested in the same investment vehicle ad infinitum, or at least a huge number of times, e.g., thousands or even millions of times.[citation needed]"
There is no assumption in Kelly of either the same proportion of bankroll or the same investment vehicle. It does assume a large number of independent bets, but this is discussed later in the article. And "large" need not mean thousands or millions.
"In conclusion, the Kelly strategy is the best strategy for beating a casino, because it is (upper limit) invariant."
Kelly is irrelevant for most casino games, because they have negative expectation and do not allow negative bets. The last clause requires a lot more discussion to be meaningful, and it doesn't support the rest of the sentence.
I'm going to take them out.
AaCBrown ( talk) 02:42, 21 June 2010 (UTC)
The article does not clearly state what goal is being maximized and where the logarithm comes from.
So my first attempt to make sense of the article was to maximize the expected profit after n bets. This leads to f being 1 under favourable conditions (p(b + 1) > 1) and 0 otherwise. Clearly not the Kelly criterion.
Then I realized that, under favourable conditions, a skilled gambler will be able to break the bank with probability 1. (Or to reach a target wealth). That lead me to second goal function, namely to minimize the expected time to break the bank. With this goal, he will never bet all his money, because then there is a positive probability that he will never break the bank leading (making the expected time infinite). With f* < 1, the logarithm of the capital follows a one dimensional Random walk and the gambler wants to maximize the rate at which he is moving towards the target. Now the Proof section makes sense. -- Nic Roets ( talk) 10:54, 4 January 2012 (UTC)
I believe having a link to an interactive kelly calculator adds value to this page for the reader. I have a kelly calculator on my site and I posted a link to it from here under a section titled External Links at the bottom of the page. Over time, people saw it, and posted links to their own calculators and it became spammy. An editor recently removed them all of them and asked that anyone who wants to have their link restored has to justify it here first. I think the value it brings to the reader justifies the link itself and the record of the page will show that the link to the calculator on the website RLD Investments was the first one posted. — Preceding unsigned comment added by RyanRLD ( talk • contribs) 17:21, 12 December 2013 (UTC)
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The Kelly criterion assumes that only one bet is available during each time period, and maximizes the growth over time by using that one opportunity to its fullest. If however multiple bets are available at any given time, that run in parallel and whose outcomes are not completely correlated, then it makes sense to distribute a smaller amount of money over each bet.
For example if p=0.5 and B=2, the Kelly bet is 25%. If three of these bets are available at the same time and are uncorrelated, then the optimum bet (maximizing EV of log) is 21% each -- a total of 63% betted. If ten are available, then optimum bet is 9.96% each (total 99.6%). As a rough guide, if N equivalent and uncorrelated bets are available and Kelly says to bet much less than 1/N, then the N bets should be made with values slightly less than Kelly. On the other hand if Kelly says to bet more than 1/N then one should (and can only) do N bets of a bit less than 1/N. [these guides change drastically if there are correlations]
In other words, another reason to de-rate the Kelly bet is its opportunity cost, that the money betted could be spread out into other avenues. Moreover, if the opportunities have partially correlated outcomes, then this gives additional push in EV(log) to strongly de-rate the Kelly bet on each. The combination of these effects thus offers a naturally emerging reason for fractional Kelly betting. I am sure these reasons are mentioned in some finance book somewhere and would be worth mentioning in the introduction, besides the "practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations".
(In, general with correlated probabilities one sees the emergence of hedged strategy like long/short equity. This is similar to the section "Many Assets" but without the Taylor approximation.) -- Nanite ( talk) 20:18, 22 November 2016 (UTC)
In paragraph 2 I think it's worth noting that the sample of 61 people was made up of a combination of college-age students in finance and economics and some young professionals at finance firms (including 14 who worked for fund managers)[1], as a random sample of the general population would probably have fared even worse.
[1] https://www.economist.com/blogs/buttonwood/2016/11/investing
111.220.125.72 ( talk) 12:42, 4 March 2018 (UTC)
... and can be counterintuitive. In one study, each participant was given $25 and asked to bet on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. Behavior was far from optimal. "Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment." Using the Kelly criterion and based on the odds in the experiment, the right approach would be to bet 20% of the pot on each throw (see first example below). If losing, the size of the bet gets cut; if winning, the stake increases.
This lead is very heavy for the length of the article to begin with. I'd personally consider moving the anecdotal material out of the lead. — MaxEnt 00:55, 11 August 2018 (UTC)
The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations.
Okay, semicolon inside a parenthetical—I do that all the time (but not on Wikipedia). And then it veers off into camp dissection (even Alex Jones concedes that Hillary stands by her man).
Couldn't we just say that Kelly theory is equivalent to expected utility theory under a logarithmic utility function?
This seems too prolix for what it conveys. — MaxEnt 01:04, 11 August 2018 (UTC)
@ Stdazi:
No need to add legs for the snakes on a painting, don't you think so?
— Wikipedian Right ( talk) 13:10, 16 February 2020 (UTC)
If you read the cited paper for this section, the steps taken in the paper are quite different to those given here.
Step 1 (calculating earning rate) is given in the paper (formula 2.3) as:
In step 4, you need to recalculate the reserve rate. The paper cited suggest this is the correct formula (6.8 in the paper):
The paper then suggests (on page 9) that the optimal Kelly fractions are then calculated using formula 6.5:
I would like to propose these changes, but am concerned I'm misinterpreting the paper.
- PaulRobinson ( talk) 18:11, 19 May 2020 (UTC)
A year ago, I used this page in the past to create a software implementation of this algorithm, and it worked. Today, I tried creating another software implementation, again using this page, but I was getting some very strange results. I can say almost definitively that the algorithm now described in this section as a consequence of these changes is incorrect. If you try the case of a single outcome, the algorithm will always recommend you bet nothing.
Meowxr ( talk) 11:22, 19 February 2021 (UTC)
I removed this paragraph. The video (questionable source on its own) was not backing the claim. The video shows that bets that are too large (way larger than the Kelly criterion) will ruin you. That's correct, but that's not betting according to the Kelly criterion. The rest of that pararaph (completely unsourced) was just wrong or applies to fair bets only, but for fair bets the Kelly criterion says not to bet. -- mfb ( talk) 02:44, 3 October 2020 (UTC)
Shouldn't in ″(e.g. betting $10, on win, rewards $4 plus wager; then )" be instead? — Preceding unsigned comment added by Convolutional Network ( talk • contribs) 05:00, 4 February 2021 (UTC)
I’m planning on switching the (b-1)s back to b. Like just about every other article, including Kelly’s paper. I’m also thinking the article is a bit proof heavy… seems like the various authors of the article are expanding it to justify their points, rather than make it readable… thoughts? -- Zojj t c 02:20, 10 August 2021 (UTC)
So I had initially added a growth rate formula to the article. It came from the 60/40 coin flip study that listed a 4% expected growth rate, and a $3 million expected value. I assume this came from the simple formula . But notice that the growth rate according to this formula increases with a higher bet fraction, which conflicts with the Kelly criterion. So something isn't right... thoughts? -- Zojj t c 18:13, 14 August 2021 (UTC)
The paper is correct that the average expected winnings should be $3,220,637 if the payout is left uncapped. The paper has the math listed in their footnotes. You are showing the expected utility which is the median result while expected value is the mean result. This is discussed in footnotes [13] and [14] of the paper. This is also easy to verify with a quick Monte Carlo sim. I can share mine with you if you'd like.
Yes you're correct, the paper has it right. Thanks for pointing out the footnote. -- Zojj t c 08:13, 8 January 2022 (UTC)
The article is titled "Kelly criterion", and says that the Kelly criterion is "a formula". However, the article gives a bunch of different formulas and never actually says which one is the "Kelly criterion". Is the Kelly criterion one of the formulas in the article, or something else?
In Kelly's paper the thing he refers to as a "criterion" is the idea that at every bet the gambler should maximise the expected value of the logarithm of their capital. I think this is probably what is meant by "Kelly criterion" -- which means that the Kelly criterion is not in fact a formula at all. If that's right then the article should be corrected. Otherwise, it should be explicitly stated which formula is the one called "Kelly criterion."
Nathaniel Virgo ( talk) 01:37, 27 February 2023 (UTC)
This was anonymously added a few days ago:
with the edit summary
If your utility function is monotone, its precise shape doesn't matter much: more is better. It seems to me that Kelly optimization assumes that your wager is not the only wager you'll ever make; you want to grow your pot over the long run, to have more to bet with next time, and this means multiplying your wealth by an optimum factor rather than adding to it.
If you optimize linearly, won't you bet your shirt every time the odds are in your favor? — Tamfang ( talk) 04:46, 29 March 2023 (UTC)
William Poundstone's book Fortune's Formula says somewhere that sophisticated gamblers often bet half the Kelly amount, because of uncertainties. Perhaps this could be mentioned somewhere. (I'll dig out the book one of these days.) — Tamfang ( talk) 06:58, 13 April 2023 (UTC)
re: a is the fraction that is lost in a negative outcome. If the security price falls 10%, then
b is the fraction that is gained in a positive outcome. If the security price rises 10%, then .
We're saying "negative outcome" and "positive outcome" but the example is an odd ratio. Which is correct? Dforootan ( talk) 04:12, 26 January 2024 (UTC)
"One claim was that one can only lose the amount bet so there was no reason to consider the (simple) generalization of this formula to the situation where a unit wager wins b with probability p > 0 and loses a with probability q."Dforootan ( talk) 07:25, 26 January 2024 (UTC)
The second sentence claims that maximising expected log wealth and maximising the expected CAGR are equivalent. But this is false. Most obviously, if running the bet for only one round, CAGR is maximised by better 100% of wealth. What is true is only that in the limit of infinitely many repeated bets, nearly all the probability is in the typical set where the fraction of Heads is very nearly np, and that maximising CAGR for that set is equivalent to maximising log wealth. Echidna44 ( talk) 09:16, 16 March 2024 (UTC)
The real problem with Kelly Criterion in the Real World is that it expects that you know the TRUE probability of winning. But you do not. You can only estimate the probability of winning based on the data you have. So your estimate of the TRUE probability of winning has a degree of uncertainty. See the wikipedia article of estimating the probability of a coin toss.
Another problem is that the TRUE probability of winning could be changing over time in the real world as the real world environment changes. Ohanian ( talk) 06:02, 18 May 2024 (UTC)
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"Optimal Gambling System for Favorable Games" is by Breiman, not Thorpe. I haven't read the Breiman paper so I don't know if the correct citation is that paper by Breiman, or some other paper by Thorpe. TimothyFreeman ( talk) 08:25, 14 April 2023 (UTC)
"This is done by maximizing the logarithm of wealth, which is mathematically simpler to do, and also maximizes wealth."
Wealth is a random variable (not a scalar), and you can't maximize wealth or the logarithm of wealth. You can maximize expected wealth or the expected logarithm of wealth, but those are NOT equivalent! is NOT equivalent to . The former maximizes the expected return, the latter maximizes the expected geometric growth rate of wealth. The former leads to going broke with probability 1 (and measure 0 probability of infinite wealth), while the latter leads to the Kelly criteria.
The distinction between maximizing expected wealth and the expected logarithm of wealth is absolutely critical. Someone reading this introduction may come away with serious misconceptions.
"... is a formula for bet sizing that leads almost surely to higher wealth compared to any other strategy in the long run. This bet size is found by maximizing the expected geometric growth rate (which is equivalent to maximizing the expected logarithm of wealth).
Mgunn ( talk) 15:53, 9 January 2019 (UTC)
I went ahead and changed it. Mgunn ( talk) 01:09, 10 January 2019 (UTC)
I disagree. Mgunn's statement above does correct the vague statement about "maximizing wealth", but the conclusion is incorrect. The first proof of the Kelly criterion given in the article maximizes the expected value of wealth after n bets of size x, which is (1+bx)^(pn) * (1-x)^(qn). It maximizes this by maximizing the log of the expected value of wealth, NOT the expected value of the logarithm of wealth. log(E[W]) = log((1+bx)^(np) * (1-x)^(nq)) = np*log(1+bx) + n(1-p)*log(1-x). We divide by n before maximizing, to maximize the log of the geometric mean of E[W]. So it does maximize the expected geometric growth rate, but maximizes the logarithm of expected wealth, not the expected logarithm of wealth. Reply with counterargument by Sept. 15, or I'll change it. Philgoetz ( talk) 03:21, 3 September 2019 (UTC)
The formula f = p/a-q/b is wrong. It should be f = p-q*a/b. It simplifies to f = (pb-qb)/b = expected profit / gains if success.
The way it's stated, f is usually bigger than 1, which is incorrect. This should be corrected. — Preceding unsigned comment added by 189.112.213.146 ( talk) 23:04, 21 March 2013 (UTC) Agreed the formula given gives ridiculous numbers (despite the apparent "proof"). E.g. p=0.8, q=0.2, a=0.25, b=0.10, so four out of five times we invest we make a 10% profit, the fifth time we make a 25% loss. The f = p/a-q/b forumula tells us to invest 120% of our capital (0.8/0.25 - 0.2/0.1 = 3.2 - 2.0 = 1.2). Not only is this incorrect but it's dangerous as users may take this formula and invest/bet amounts which are detrimental to them, and possibly bankrupting. There are no references to the wrong formula, so I intend to correct it to the formula you have mentioned, which is backed up my many sources elsewhere on the internet and reference them. In the event I am interrupted before I can put the edit in, please can somebody else do the correction? Here's an example of a reference with the correct formula (p-q*a/b): http://www.investopedia.com/terms/k/kellycriterion.asp — Preceding unsigned comment added by 85.255.232.117 ( talk) 18:43, 7 May 2016 (UTC)
You guys are going off the rails here. The formula p-qa/b gives the Kelly fraction to RISK. The formula p/a-q/b gives the Kelly fraction to BET. You want the second one, and these are not the same in your example. You are risking the amount a, but you are betting the different amount 1 dollar ("the value of your investment"). The 2 formulas differ by a factor of a, and they are only the same when a=1, that is, when you can lose your entire bet. In the example given above, the answer of 120% is neither ridiculous nor incorrect - it is correct. You need leverage to bet the Kelly optimal in this case, but betting 120% of the bankroll means that we are only placing 30% of the bankroll at risk since we can only lose 25% of our bet. To see that this is optimal, simply note that 120% indeed maximizes the expected value of the log of the bankroll 0.8log(1+0.1f)+0.2log(1-0.25f). This should be clear from the derivation of p/a-q/b under PROOF. This last revision should be reversed. Brucezas ( talk) 19:46, 3 September 2019 (UTC)
There should probably be a statement pointing out that this formula can advise betting more than 100% of the bankroll, but that the amount at risk will be less than 100%. The amount at risk can be obtained by scaling the result by a. Otherwise this is likely to be a common source of confusion. Brucezas ( talk) 19:53, 4 September 2019 (UTC)
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.
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)
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)
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)
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.
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)
A lot of this section is factually incorrect and needs to be revised. (I have no clue how to sign this, nor do I just wanna lop the whole section out of the page. But the 2nd and 4th paragraphs in this section are either misleading or factually incorrect)
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-proportion rebalanced portfolio", and has been studied by economists Mark Rubenstein, Eugene Fama, and Thomas Cover.
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)
There's a huge probleme with the objectif of this quadratic problem. At first, Q is not positive definite and Dim(Ker(Q)) = N-1, so under these constraints, in the best case, we can find N-3 different solutions which make the result unstable. Just try do find Xi+1 = Xi + dXi and look at the norm of dXi.
If you use a "regular" solver with an iterative method, you'll not converge in dXi but in dU (utility). That implies the problem is ill-posed. We cannot use it that way.
The Bad Boy (
talk) 14:33, 9 October 2013 (UTC)
The Application's to Stock Market section is quite poorly written and should, in my opinion, be removed. — Preceding unsigned comment added by 104.162.109.38 ( talk) 09:12, 12 September 2016 (UTC)
I am still puzzled what the objective function is that we are trying to maximise and what the constraints are. In my view, the article would benefit from a precise mathematical expression. I take it it's the limit of the expected wealth as the number of periods goes to infinity? For any finite number of periods T, I suppose I could beat Kelly, for instance, by betting Kelly until time t and put all my money on one side in the last round. — Preceding unsigned comment added by Derfugu ( talk • contribs) 11:25, 26 March 2011 (UTC)
Derfugu ( talk) 11:26, 26 March 2011 (UTC)
In the article it seems to be missing the fact that we are looking to maximise the expected value of the log of the wealth. Since I find it very confusing (I only found out the answer by checking this talk page), I'm going to add a line in the intro to make it explicit. Student73 ( talk) 13:29, 18 November 2017 (UTC)
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.
Before I derive it, let me list some characteristics I expect the "correct" formula to have:
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:
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)
-- User:derfugu
Hi, I deleted some stuff from the proof section - it appeared to me that the section on how to underbet and beat kelly was original research and I don't think it works - here's the section: suppose you bet a small amount less than Kelly on the first bet, and double the amount less than Kelly every bet until you finally lose a bet. At that point, you will be a small amount ahead of Kelly. If you follow Kelly thereafter, you will end up better than Kelly. This will happen unless you start with such a long string of wins that you can no longer double the amount you underbet Kelly. You can make the probability of this as low as you want by making the initial underbet very small. Thus you can come up with a strategy that will always outperform Kelly.. This strategy cannot be better than Kelly - the reason is simple, if it were better than Kelly, I would use the strategy on itself. Then I would have a strategy that was better than the better strategy. Ad infinitum. -- 165.222.184.132 ( talk) 09:01, 16 March 2009 (UTC)
You are correct that the argument can be used on itself to form an infinite series of better strategies. But that does not invalidate its point that it is possible to find a strategy better than Kelly. Whether this doubling strategy actually is better than Kelly, or even different from Kelly is a matter of dispute. Certainly it doesn't point to a practical way to improve on Kelly.
Anyway, the doubling argument is often made, and it was referenced. Whether you agree with it or not is not the point in Wikipedia. I do not think it should have been deleted. On the other hand, I don't have any strong feeling that it needs to be included. AaCBrown ( talk) 21:04, 10 October 2013 (UTC)
This scheme is like the argument that is known as Leib's paradox, and a proof concerning it appears in the 2008 article Understanding The Kelly Criterion by Edward O. Thorp--- the man. The article is a revised reprint from two columns from the series A Mathematician on Wall Street in Wilmott Magazine, May and September 2008. Thorp explains, contrary to the commentary here to the effect that nothing like this could work, that Lieb's trick works. Here's Thorp's summary of it:
"To prove the first part, we show how to get ahead of Kelly with probability 1 − ε within a finite number of trials. The idea is to begin by betting less than Kelly by a very small amount. If the first outcome is a loss, then we have more than Kelly and use the strategy from the proof of the second part to stay ahead. If the first outcome is a win, we’re behind Kelly and now underbet on the second trial by enough so that a loss on the second trial will put us ahead of Kelly. We continue this strategy until either there is a loss and we are ahead of Kelly or until even betting 0 is not enough to surpass Kelly after a loss. Given any N , if our initial underbet is small enough, we can continue this strategy for up to N trials. The probability of the strategy failing is p N , 12 < p < 1. Hence, given ε > 0, we can choose N such that p N < ε and the strategy therefore succeeds on or before trial N with probability 1 − p N > 1 − ε."
My first thoughts--- I have to study the paper more--- are that the resolution may be to understand that the stated and mathematically proven Kelly "dominance", as Thorp sometimes calls it, the superiority, is established in a gross statistical sense and only with respect to what he calls (and mathematically defines) "essentially different" strategies. A strategy that is a close shadow of Kelley that is generally implemented for only a few trials is not essentially different. The basic meaning of that could be that the Leib strategy could in the long run amount to small potatoes compared to the difference between Kelley and an essentially different strategy. Mihael O'Connor ( talk) 12:04, 6 April 2014 (UTC)
There are two sentences in this article I disagree with:
"However, as is evident in the derivation below, the Kelly strategy is applicable only when the same proportion of one's bankroll is invested in the same investment vehicle ad infinitum, or at least a huge number of times, e.g., thousands or even millions of times.[citation needed]"
There is no assumption in Kelly of either the same proportion of bankroll or the same investment vehicle. It does assume a large number of independent bets, but this is discussed later in the article. And "large" need not mean thousands or millions.
"In conclusion, the Kelly strategy is the best strategy for beating a casino, because it is (upper limit) invariant."
Kelly is irrelevant for most casino games, because they have negative expectation and do not allow negative bets. The last clause requires a lot more discussion to be meaningful, and it doesn't support the rest of the sentence.
I'm going to take them out.
AaCBrown ( talk) 02:42, 21 June 2010 (UTC)
The article does not clearly state what goal is being maximized and where the logarithm comes from.
So my first attempt to make sense of the article was to maximize the expected profit after n bets. This leads to f being 1 under favourable conditions (p(b + 1) > 1) and 0 otherwise. Clearly not the Kelly criterion.
Then I realized that, under favourable conditions, a skilled gambler will be able to break the bank with probability 1. (Or to reach a target wealth). That lead me to second goal function, namely to minimize the expected time to break the bank. With this goal, he will never bet all his money, because then there is a positive probability that he will never break the bank leading (making the expected time infinite). With f* < 1, the logarithm of the capital follows a one dimensional Random walk and the gambler wants to maximize the rate at which he is moving towards the target. Now the Proof section makes sense. -- Nic Roets ( talk) 10:54, 4 January 2012 (UTC)
I believe having a link to an interactive kelly calculator adds value to this page for the reader. I have a kelly calculator on my site and I posted a link to it from here under a section titled External Links at the bottom of the page. Over time, people saw it, and posted links to their own calculators and it became spammy. An editor recently removed them all of them and asked that anyone who wants to have their link restored has to justify it here first. I think the value it brings to the reader justifies the link itself and the record of the page will show that the link to the calculator on the website RLD Investments was the first one posted. — Preceding unsigned comment added by RyanRLD ( talk • contribs) 17:21, 12 December 2013 (UTC)
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The Kelly criterion assumes that only one bet is available during each time period, and maximizes the growth over time by using that one opportunity to its fullest. If however multiple bets are available at any given time, that run in parallel and whose outcomes are not completely correlated, then it makes sense to distribute a smaller amount of money over each bet.
For example if p=0.5 and B=2, the Kelly bet is 25%. If three of these bets are available at the same time and are uncorrelated, then the optimum bet (maximizing EV of log) is 21% each -- a total of 63% betted. If ten are available, then optimum bet is 9.96% each (total 99.6%). As a rough guide, if N equivalent and uncorrelated bets are available and Kelly says to bet much less than 1/N, then the N bets should be made with values slightly less than Kelly. On the other hand if Kelly says to bet more than 1/N then one should (and can only) do N bets of a bit less than 1/N. [these guides change drastically if there are correlations]
In other words, another reason to de-rate the Kelly bet is its opportunity cost, that the money betted could be spread out into other avenues. Moreover, if the opportunities have partially correlated outcomes, then this gives additional push in EV(log) to strongly de-rate the Kelly bet on each. The combination of these effects thus offers a naturally emerging reason for fractional Kelly betting. I am sure these reasons are mentioned in some finance book somewhere and would be worth mentioning in the introduction, besides the "practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations".
(In, general with correlated probabilities one sees the emergence of hedged strategy like long/short equity. This is similar to the section "Many Assets" but without the Taylor approximation.) -- Nanite ( talk) 20:18, 22 November 2016 (UTC)
In paragraph 2 I think it's worth noting that the sample of 61 people was made up of a combination of college-age students in finance and economics and some young professionals at finance firms (including 14 who worked for fund managers)[1], as a random sample of the general population would probably have fared even worse.
[1] https://www.economist.com/blogs/buttonwood/2016/11/investing
111.220.125.72 ( talk) 12:42, 4 March 2018 (UTC)
... and can be counterintuitive. In one study, each participant was given $25 and asked to bet on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250. Behavior was far from optimal. "Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment." Using the Kelly criterion and based on the odds in the experiment, the right approach would be to bet 20% of the pot on each throw (see first example below). If losing, the size of the bet gets cut; if winning, the stake increases.
This lead is very heavy for the length of the article to begin with. I'd personally consider moving the anecdotal material out of the lead. — MaxEnt 00:55, 11 August 2018 (UTC)
The conventional alternative is expected utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes expected utility, so there is no conflict; moreover, Kelly's original paper clearly states the need for a utility function in the case of gambling games which are played finitely many times). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations.
Okay, semicolon inside a parenthetical—I do that all the time (but not on Wikipedia). And then it veers off into camp dissection (even Alex Jones concedes that Hillary stands by her man).
Couldn't we just say that Kelly theory is equivalent to expected utility theory under a logarithmic utility function?
This seems too prolix for what it conveys. — MaxEnt 01:04, 11 August 2018 (UTC)
@ Stdazi:
No need to add legs for the snakes on a painting, don't you think so?
— Wikipedian Right ( talk) 13:10, 16 February 2020 (UTC)
If you read the cited paper for this section, the steps taken in the paper are quite different to those given here.
Step 1 (calculating earning rate) is given in the paper (formula 2.3) as:
In step 4, you need to recalculate the reserve rate. The paper cited suggest this is the correct formula (6.8 in the paper):
The paper then suggests (on page 9) that the optimal Kelly fractions are then calculated using formula 6.5:
I would like to propose these changes, but am concerned I'm misinterpreting the paper.
- PaulRobinson ( talk) 18:11, 19 May 2020 (UTC)
A year ago, I used this page in the past to create a software implementation of this algorithm, and it worked. Today, I tried creating another software implementation, again using this page, but I was getting some very strange results. I can say almost definitively that the algorithm now described in this section as a consequence of these changes is incorrect. If you try the case of a single outcome, the algorithm will always recommend you bet nothing.
Meowxr ( talk) 11:22, 19 February 2021 (UTC)
I removed this paragraph. The video (questionable source on its own) was not backing the claim. The video shows that bets that are too large (way larger than the Kelly criterion) will ruin you. That's correct, but that's not betting according to the Kelly criterion. The rest of that pararaph (completely unsourced) was just wrong or applies to fair bets only, but for fair bets the Kelly criterion says not to bet. -- mfb ( talk) 02:44, 3 October 2020 (UTC)
Shouldn't in ″(e.g. betting $10, on win, rewards $4 plus wager; then )" be instead? — Preceding unsigned comment added by Convolutional Network ( talk • contribs) 05:00, 4 February 2021 (UTC)
I’m planning on switching the (b-1)s back to b. Like just about every other article, including Kelly’s paper. I’m also thinking the article is a bit proof heavy… seems like the various authors of the article are expanding it to justify their points, rather than make it readable… thoughts? -- Zojj t c 02:20, 10 August 2021 (UTC)
So I had initially added a growth rate formula to the article. It came from the 60/40 coin flip study that listed a 4% expected growth rate, and a $3 million expected value. I assume this came from the simple formula . But notice that the growth rate according to this formula increases with a higher bet fraction, which conflicts with the Kelly criterion. So something isn't right... thoughts? -- Zojj t c 18:13, 14 August 2021 (UTC)
The paper is correct that the average expected winnings should be $3,220,637 if the payout is left uncapped. The paper has the math listed in their footnotes. You are showing the expected utility which is the median result while expected value is the mean result. This is discussed in footnotes [13] and [14] of the paper. This is also easy to verify with a quick Monte Carlo sim. I can share mine with you if you'd like.
Yes you're correct, the paper has it right. Thanks for pointing out the footnote. -- Zojj t c 08:13, 8 January 2022 (UTC)
The article is titled "Kelly criterion", and says that the Kelly criterion is "a formula". However, the article gives a bunch of different formulas and never actually says which one is the "Kelly criterion". Is the Kelly criterion one of the formulas in the article, or something else?
In Kelly's paper the thing he refers to as a "criterion" is the idea that at every bet the gambler should maximise the expected value of the logarithm of their capital. I think this is probably what is meant by "Kelly criterion" -- which means that the Kelly criterion is not in fact a formula at all. If that's right then the article should be corrected. Otherwise, it should be explicitly stated which formula is the one called "Kelly criterion."
Nathaniel Virgo ( talk) 01:37, 27 February 2023 (UTC)
This was anonymously added a few days ago:
with the edit summary
If your utility function is monotone, its precise shape doesn't matter much: more is better. It seems to me that Kelly optimization assumes that your wager is not the only wager you'll ever make; you want to grow your pot over the long run, to have more to bet with next time, and this means multiplying your wealth by an optimum factor rather than adding to it.
If you optimize linearly, won't you bet your shirt every time the odds are in your favor? — Tamfang ( talk) 04:46, 29 March 2023 (UTC)
William Poundstone's book Fortune's Formula says somewhere that sophisticated gamblers often bet half the Kelly amount, because of uncertainties. Perhaps this could be mentioned somewhere. (I'll dig out the book one of these days.) — Tamfang ( talk) 06:58, 13 April 2023 (UTC)
re: a is the fraction that is lost in a negative outcome. If the security price falls 10%, then
b is the fraction that is gained in a positive outcome. If the security price rises 10%, then .
We're saying "negative outcome" and "positive outcome" but the example is an odd ratio. Which is correct? Dforootan ( talk) 04:12, 26 January 2024 (UTC)
"One claim was that one can only lose the amount bet so there was no reason to consider the (simple) generalization of this formula to the situation where a unit wager wins b with probability p > 0 and loses a with probability q."Dforootan ( talk) 07:25, 26 January 2024 (UTC)
The second sentence claims that maximising expected log wealth and maximising the expected CAGR are equivalent. But this is false. Most obviously, if running the bet for only one round, CAGR is maximised by better 100% of wealth. What is true is only that in the limit of infinitely many repeated bets, nearly all the probability is in the typical set where the fraction of Heads is very nearly np, and that maximising CAGR for that set is equivalent to maximising log wealth. Echidna44 ( talk) 09:16, 16 March 2024 (UTC)
The real problem with Kelly Criterion in the Real World is that it expects that you know the TRUE probability of winning. But you do not. You can only estimate the probability of winning based on the data you have. So your estimate of the TRUE probability of winning has a degree of uncertainty. See the wikipedia article of estimating the probability of a coin toss.
Another problem is that the TRUE probability of winning could be changing over time in the real world as the real world environment changes. Ohanian ( talk) 06:02, 18 May 2024 (UTC)