Talk:Kelly criterion

message: "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)
replies: text: The citation is correct. There is a paper by Thorpe  of that name. Limit-theorem (talk) 11:47, 14 April 2023 (UTC)
thread_title: Citation 14 is wrong

message: "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!  \max_{W \in \mathcal{A}} \operatorname{E}[W]  is NOT equivalent to  \max_{W \in \mathcal{A}} \operatorname{E}[\log W] . 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.
===I suggest instead===
"... 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)
replies: text: I believe it is computing the expected log of wealth, not the log of expected wealth. Wouldn't maximizing the log of expected wealth give simply the same solution as maximizing expected wealth directly? (An expected value is a deterministic number, not a random variable, and a logarithm is a simple monotonic function, so whatever maximizes the expected value of something will also maximize the log of that expected value. Wealth, however, is a random variable, so the log of wealth is also a random variable, and maximizing the expected value of wealth might provide a different solution from maximizing the expected value of the log of wealth.)  —BarrelProof (talk) 00:47, 4 September 2019 (UTC)
thread_title: A sentence in the introduction is wrong / confusing / deeply problematic

message: "... 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)
replies: text: I believe it is computing the expected log of wealth, not the log of expected wealth. Wouldn't maximizing the log of expected wealth give simply the same solution as maximizing expected wealth directly? (An expected value is a deterministic number, not a random variable, and a logarithm is a simple monotonic function, so whatever maximizes the expected value of something will also maximize the log of that expected value. Wealth, however, is a random variable, so the log of wealth is also a random variable, and maximizing the expected value of wealth might provide a different solution from maximizing the expected value of the log of wealth.)  —BarrelProof (talk) 00:47, 4 September 2019 (UTC)
thread_title: I suggest instead

message: 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)
replies: text: You're right. Sorry it took so long to fix. -- Derek Ross | Talk 17:37, 31 August 2019 (UTC)
thread_title: General Formula Wrong

message: 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)
replies: text:  To quote from Kelly's original paper "The reason (for the log rule) has nothing to do with the value which (the gambler) attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets" 158.143.86.159 (talk) 16:14, 25 January 2010 (UTC)
replies: text: Reading some of his paper, the math is about maximizing the log of the expected value, but he seems to justify it on the basis of maximizing the mode. Someone who uses the Kelly Criterion will probably do better than someone who does not, even though it's easy to get a higher expected value than someone who uses it. — DanielLC 22:09, 10 December 2014 (UTC)
thread_title: Factual basis of Kelly Criterion is in dispute

message: 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)
thread_title:  Generalized Kelly Strategy 

message: 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)
thread_title:  Formula presentation 

message: 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)
replies: text: The clue how to sign this is just above the edit panel. – Do you mean part of the article is wrong, or part of this section of the Talk page? The Talk page is not part of the article, it's for discussing what belongs in the article, so errors and misunderstandings are to be expected; we do not "correct" what others post here. —Tamfang (talk) 03:22, 8 December 2009 (UTC)
thread_title:  Added disadvantages section 

message: 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.
:Mark Rubinstein was an economist (or "financial engineer") and Eugene Fama is an economist, but Thomas Cover was an information theorist rather than an economist. Proportional rebalancing is a very common strategy in portfolio management. It is advocated in a very large amount of the personal finance literature, there are "balanced" mutual funds based on it, etc. —BarrelProof (talk) 01:09, 1 September 2019 (UTC)
thread_title:  Shannons stock system 

message: Kelly Criterion For Stock Market should be merged into this article. (Nuggetboy) (talk) (contribs) 19:11, 25 January 2007 (UTC)
thread_title:  [[Kelly Criterion For Stock Market]] 

message: 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)
replies: text: It's maximizing the log of the wealth. The reason for this doesn't seem to be very justified. It mentions that using a different probability consistently will result in less wealth with a probability that approaches unity, but the example you gave will do better most of the time (assuming you're putting all your money on the most likely side, even if this reduces the expected value). — DanielLC 22:14, 10 December 2014 (UTC)
thread_title:  Rigorous statement of the objective function 

message: 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)
replies: text: A minor historical worry: In Kelly's paper, he uses the word "odds" but not to mean what you mean here. He had his odds such that the (fair) odds are the reciprocal of the probability. This is the situation where eg. odds on a fair coin landing heads are 2 to 1, not 1 to 1. See p. 921 of his paper. This doesn't fit with odds nor with your "b" above. I'm not questioning the derivation, I'm wondering how (or whether) to present the divergence of terminology... 158.143.86.159 (talk) 16:18, 25 January 2010 (UTC)

text: I thought this is rather clear in the article. We speak of "to" odds, when we mean the fraction of the wager that we receive in addition to the original wager and of "for" odds when we mean including the wager. Hence the 'fair' odds (fair in a sense that the implied probabilities sum up to one) in coin tossing are 1 to 1 and 2 for 1.  — Preceding unsigned comment added by Derfugu (talk • contribs) 11:05, 26 March 2011 (UTC)
thread_title:  formula derivation 

message: 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)
thread_title:  Proof section 

message: 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)
thread_title: Two disagreements

message: 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)
thread_title: Definition of a goal function that is to maximized

message: 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)
thread_title:  Link to Kelly Calculator 

message: Cyberbot II has detected links on Kelly criterion which have been added to the blacklist, either globally or locally. Links tend to be blacklisted because they have a history of being spammed or are highly inappropriate for Wikipedia. The addition will be logged at one of these locations: local or global
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If you would like me to provide more information on the talk page, contact User:Cyberpower678 and ask him to program me with more info.
From your friendly hard working bot.—cyberbot IITalk to my owner:Online 16:56, 11 August 2015 (UTC)
replies: text: What link are you talking about? I replaced the link months ago with this one from Ed Thorp's own site: http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/beatthemarket.pdf AndyBloch (talk) 23:45, 11 November 2015 (UTC)
thread_title:  Blacklisted Links Found on [[Kelly criterion]] 

message: 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)
replies: text: I was searching for the Kelly optimum at parallel bets, what you write intuitively seems correct. Can you give a link to a paper or similar? Thanks. Richard  — Preceding unsigned comment added by 141.69.58.57 (talk) 16:29, 2 November 2017 (UTC)
thread_title:  Parallel bets 

message: 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)
thread_title:  Irrational coin tossers 

message: 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)
replies: text:  Yes, one may shorten, but while stressing that it is coincidental. It so happened that maximizing the rate of growth maps to log utility. Limit-theorem (talk) 11:02, 11 August 2018 (UTC)
thread_title:  Lead lacks concision 

message: 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)
thread_title:  Grammar: "probability of win" vs. "probability of a win" ([[Special:Diff/940400173|940400173]]) 

message: 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:
er_i = \frac{Dp_i}{\beta_i} = p_i(Q_i + 1)
In step 4, you need to recalculate the reserve rate. The paper cited suggest this is the correct formula (6.8 in the paper):
R(S) = \frac{D\sum_{k \in S}p_k}{D-\sum_{k \in S}\beta_k}
The paper then suggests (on page 9) that the optimal Kelly fractions are then calculated using formula 6.5:
f_i = p_i - \beta_i \frac{\sum_{k \in S}p_k}{(D - \sum_{k \in S}\beta_k)}
I would like to propose these changes, but am concerned I'm misinterpreting the paper. 
- PaulRobinson (talk) 18:11, 19 May 2020 (UTC)
thread_title:  "Multiple outcomes" formulas are wrong 

message: 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)
replies: text: If you want to optimize the expected linear wealth then you should bet everything every time, indeed. This is clearly not how people behave in practice. There are more options than just linear and logarithmic, however. Different options lead to different strategies. --mfb (talk) 08:45, 29 March 2023 (UTC)
thread_title:  logarithms and values 

message: 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)
thread_title:  Maximising log wealth and maximising CAGR are *not* equivalent 

message: 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)
thread_title:  The real problem with using Kelly Criterion in the Real World