15 ms·

Kelly Criterion

jmount 6y ago
I'd like to share a video I prepared on Kelly betting: https://youtu.be/6xhjbgREGDA https://youtu.be/6xhjbgREGDA
one-more-minute 6y ago
Some interesting psychology here: > In one study, each participant was given $25 and asked to place even-money bets 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. > 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.
- blowski 6y ago
  > two-thirds gambled on tails at some stage in the experiment I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not irrational to bet on tails is it?
  - jfk13 6y ago
    No, the next toss still has a 60% chance of being heads. The coin doesn't remember how it landed last time.
    - blowski 6y ago
      If I'm expecting 60% of my flips to be heads, and I've already had 60%, isn't it more likely that the next one will be tails? I'm sure you can probably tell I know next to nothing about either maths or probability, so feel free to explain why I'm wrong.
      - ska 6y ago
        Yes, you are wrong, but your confusion is very common. It's so common it even has a name "The Gamblers Fallacy". Over the long run, you expect 40% tails, but if you run the experiment an infinite amount of time there will be sequences of all-heads or all-tails. Because the events are independent, the previous flips don't change anything about what happens next.
        FredPret 6y ago
        https://dilbert.com/strip/2001-10-25 https://dilbert.com/strip/2001-10-25
        blowski 6y ago
        So the probablity of 60% assumes infinite flips. Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads. I'd also need to know the probability that I'm in one of the cases where I get 60% of heads. Is that right?
        FredPret 6y ago
        With a 60%-heads coin you can still get 10 straight heads. It’s just that over many many flips, the average will gradually tend towards 60%. You can still have streaks of hundreds, thousands, millions of either heads or tails in a row.
        ska 6y ago
        This gets into interesting stuff! So the situation described in that paper is that you are given the true odds of the coin, 60% heads. In this case it's just as I described - knowing previous results doesn't tell you anything useful. > Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads. This is true. In fact there is only about 25% chance of getting exactly 6 of the 10 to be heads (but nearly 70% chance of >= 6 heads). You can work this out with something called the binomial distribution. Chance of getting 10 heads in a row is .6% A more interesting aspect is when you don't know the odds (or don't trust what you've been told). In this case it's definitely important what the history is. So given your 10 flips, we can ask questions like "how likely is it that this coin is fair (50/50) given the 10 flips I just saw". It turns out the best estimation of the true probability is, pretty intuitively, (h+t)/h; this will jump aroudn for small N . In practice you are more often looking at something like P(0.55 < p < 0.65 | samples) , i.e. the probability that the true value lies between 0.55 and 0.65 heads, given the 10 flips I've seen). Obviously in these cases, the more samples you have seen the tighter the estimate get. You can also ask questions like how many flips do I need to see to be confident at a certain the coin is really 0.6 heads.
  - afranchuk 6y ago
    Those are independent variables. The fact you've had X heads has no bearing on the future flips. It is irrational to bet on tails statistically speaking, though psychologically that line of reasoning is common.
  - theschwa 6y ago
    No, the coin doesn't have a memory, so the chance of tails is still 40% making it still optimal to choose heads.
  - skeeter2020 6y ago
    you're not betting on the number of heads/tails per 10 trials though, each trial is independent with a 60% of heads. In a striaght-up prediction you should always choose heads, it the how much to wager that is the question.
  - baobabKoodaa 6y ago
    > I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not an add thing to bet on tails is it? I realize you're probably joking, but since this argument is intuitively appealing to many people, I will answer as if it was serious: if you have a weighted coin that is 60% likely to land on heads, that means it's 60% likely to land on heads on any given toss. On the first toss. On the second toss. Any given toss. Even after you have tossed it 6 times and seen 6 heads in a row, the coin is still 60% likely to land on heads. The coin has no "memory". Previous results have no effect on future results.
    - 6gvONxR4sf7o 6y ago
      Moreover, the important feature of coin flips isn’t randomness, it’s independence (from previous coin flips and from everything else). Independence is in fact a useful mental model for randomness.
    - vgeek 6y ago
      I quickly searched but couldn't find the exact study, but I've read that by adding the past numbers digital signage to roulette tables, casinos experience a significant (I'm thinking it was like 100%+) increase in wagers when people believe that a color is "due" simply from not understanding independent vs dependent events. Humans love to look for patterns, even when there isn't any real _meaning_ behind them.
    - dehrmann 6y ago
      There's a corollary to the gambler's fallacy that says is P(heads) is 60% and you get 6 heads in a row, the people running the experiment probably lied to you.
      - lupire 6y ago
        but that means you should bet into the bias, not against it.
        dehrmann 6y ago
  - prometheus76 6y ago
    You've just discovered the Gambler's Fallacy.
  - JacobLinney 6y ago
    Yes it is irrational. That's a common statistical misconception, the key thing here is that every flip has a 60% chance of being heads. The result of each flip is completely independent of what came before it. In your example the 7th flip is just as likely to be heads as the first flip, or any of the other 5 flips that landed on heads.
    - blowski 6y ago
      It says "a coin that would land heads 60% of the time". If it's already landed heads 60% of the time, I'd expect the remaining 40% for it to land on tails.
      - antasvara 6y ago
        The key here is that it's expected to land heads 60% of the time. Take a normal coin, which is expected to land heads 50% of the time. If you flip a heads, do you instantly expect it to be tails next time? By your logic it would be impossible to ever flip heads twice in a row. Coins as a general rule aren't impacted by previous flips.
      - bscphil 6y ago
        That's not a guarantee for any number of flips. For example, if you only flipped the coin one time, what does "60% of the time" even mean in that context? As your other replies have indicated, this is getting at the long-run frequency, meaning as you flip the coin more and more times, approaching infinity, the number of heads approaches 60%.
      - sokoloff 6y ago
        Thought experiment: in what way has it landed heads 60% of the time? It landed heads 100% of the trials so far, but the coin has no way of keeping track of that.
  - justinpowers 6y ago
    Each toss is independent of prior (and subsequent) tosses, so no matter what, a given tosshas 60% chance of landing heads. Rationally, one should bet heads on any given toss. But most people would agree with the irrational bet. This tendency is known as the Gambler’s fallacy (https://en.wikipedia.org/wiki/Gambler's_fallacy) https://en.wikipedia.org/wiki/Gambler's_fallacy).
  - Jtsummers 6y ago
    That's the gambler's fallacy in action. So long as each event is independent, the prior ones have no impact on the likelihood of future events. If you've flipped the coin 60 times and they've all been heads, there's no reason to expect the next 40 will be tails. They still have better odds of being heads.
    - travisjungroth 6y ago
      If you see 60 heads in a row in the real world you've got a trick coin. The odds of that are 1/10^17.
      - Jtsummers 6y ago
        It's certainly low odds, but it's not impossible nor does it require a trick coin. I've seen people roll a 20 on a d20 10 times in a row, and then not a single 20 the rest of the session on the same die. Shit happens, it's probability and it may be improbable but it isn't impossible.
        lupire 6y ago
        I don't believe you.
        Jtsummers 6y ago
        I mean, that's fine, it's an anecdote. If you'd like, take a few dice and set up cameras and an automatic rolling mechanism and see if there are any improbable sequences like alternation between two or three number or a long run of a single number, or a long run without a particular number appearing. Over enough trials you are likely to encounter these kinds of events.
        deleted 6y ago
        [deleted]
        travisjungroth 6y ago
  - mytherin 6y ago
    The probability of a coin flip being heads or tails is completely independent from the previous flips. If the coin lands 6 heads in a row, the next coin flip still has a 60% chance of being heads, hence it is always unwise to bet on tails in this experiment. This is an example of the Gambler's fallacy [1]. [1] https://en.wikipedia.org/wiki/Gambler%27s_fallacy https://en.wikipedia.org/wiki/Gambler%27s_fallacy
  - ska 6y ago
    > If you've just had 6 heads in a row the next 4 "should" be tails That's not how this works. Each toss is independent, so you should never pay attention to previous results if you know the true odds.
  - deeg 6y ago
    This wiki page can explain why better than me: https://en.wikipedia.org/wiki/Gambler%27s_fallacy https://en.wikipedia.org/wiki/Gambler%27s_fallacy
  - stocknoob 6y ago
    Your friend walks up while you're playing. They haven't seen the game, so think heads is coming up. Your other friend has been playing longer, before you even started. They saw 13 tails and then your 6 heads. The next throw should be heads to even it out for them. Why is your history more of an influence than theirs?
  - sorokod 6y ago
    While this is irrational in this experiment, but it is likely that the biological systems in which humans evolved, tend to not have truly independent events - hence our intuition.
- frabjoused 6y ago
  I made a little playground for this, you can fiddle with the numbers. https://parsebox.io/dthree/lnumtuenmskr https://parsebox.io/dthree/lnumtuenmskr
- tialaramex 6y ago
  To be fair, as a participant in psychology experiments I go in aware that it's plausible, even likely that I am being misled about what's really going on. That's even necessary in some experiments. Maybe I'm not technically lied to but if deliberately engineering a false impression is the goal, psychologists are the people to do it in a controlled experiment. The experimenters aren't (ethically) allowed to cause you harm, and they'll probably tell you exactly what was really going on afterwards at least if you ask, but during the experiment everything is potentially suspect. Maybe the task you're focused on was just a distraction and they really care whether you notice the clocks in the room are running too fast so that "five minutes" to do the task is really only 250 seconds - but equally maybe the apparent "time pressure" to complete the task is the distraction and they really care whether you lie about completing it properly given an opportunity to cheat. So if the experimenter in a psych experiment tells me the coin is biased 60% heads, I don't consider that the same way I would if the friend I play board games with says it. As a result chances are my first few dozen bets are confirming this unusual claim about the world. Biased coins are hard to make, is this coin really biased? Maybe I try fifty bets in rapid succession, $1 on heads each time. Apparently that's expected to take about five minutes of my half an hour, and before that's done I won't feel comfortable even assuming it's really 60% heads. And at the end of those five minutes on average I turn $25 into $35 and feel comfortable it's really 60% heads or that I can't tell what's wrong. Now, why gamble on tails? Well like I said, Psychologists mislead you intentionally during experimentation. Maybe the experimenter tells you it's 60% likely to be Heads. If the gamer told me that, I believe it's 40% likely to be Tails because that's logical, but when an experimenter tells me that, I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check.
  - lupire 6y ago
    Biased coins are *impossible" to make if the coin is flipped not spun. I doubt any story about a biased coins in the real world.
    - function_seven 6y ago
      If the coin was made from a thin magnet, and being flipped onto a weak magnetic plate, couldn't you bias the result? If the landing pad was a strong magnet, then you could trivially make it a "100% heads" coin. Just weaken the magnetic field so it's not strong enough to flip a coin flat at rest, but has enough oomph to take a coin landing near its edge to the preferred result.
- jfengel 6y ago
  Did they know that it was biased towards heads? With only a 60-40 split I probably wouldn't notice it unless I was actually keeping track, which could take a while. A 6-4 split on 10 tosses doesn't tell you anything. If you told me it was a fair coin and I thought the experiment was about something else, it might take a very long time before it occurred to me to test the hypothesis that the coin wasn't fair. If they knew it was biased... I'm sure there's an optimal strategy, but a simple strategy would be "bet half of what you have on heads every time". Any idea how much worse that is than the optimal strategy?
  - kqr 6y ago
    You can plot g = 0.6 log (1 + 2f) + 0.4 log (1 - f) And locate f=0.5 and compare to the maximum g. Edit: I wanted to check my intuition so I did: https://www.wolframalpha.com/input/?i=plot++0.6+log+%281+%2B+2f%29+%2B+0.4+log+%281+-+f%29%2C+f%3D0..1 https://www.wolframalpha.com/input/?i=plot++0.6+log+%281+%2B... Looks like 0.5 is a slight overbet, but still very, very good.
  - tedunangst 6y ago
    If only there was a link to the study so we could see how it was setup.
  - deleted 6y ago
    [deleted]
  - seoaeu 6y ago
    > Did they know that it was biased towards heads? "Prior to starting the game, participants read a detailed description of the game, which included a clear statement, in bold, indicating that the simulated coin had a 60% chance of coming up heads and a 40% chance of coming up tails."
- ISL 6y ago
  The paper is pretty awesome and accessibly-written: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963 The PDF is free-to-read.
syassami 6y ago
If you enjoyed this, I highly recommend reading Fortune's Formula.
- kqr 6y ago
  And when you're done with that, the Kelly Capital Growth Investment Criterion is one of the better books I've read. But it's a much more advanced read.
  - _e 6y ago
    And when you are done with that, I highly suggest reading Edward Thorp's autobiography "Man for all markets" where he employs the Kelly Criterion in adventure after adventure. He not only developed the first card counting system for blackjack but he also created the first wearable computer to beat roulette (with Claude Shannon).
anonu 6y ago
This has been posted 5 other times on HN with no real discussion [1]. I'll add my 2 cents: I used to use the principles of kelly betting back when I designed systematic HFT strategies. It gives you a good framework to think about how much to bet based on the batting average of a particular pattern you recognize in the market... [1] https://hn.algolia.com/?q=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FKelly_criterion https://hn.algolia.com/?q=https%3A%2F%2Fen.wikipedia.org%2Fw...
- jared_buckner 6y ago
  There's a bit of a discussion here: https://news.ycombinator.com/item?id=18484631 https://news.ycombinator.com/item?id=18484631
- fighterpilot 6y ago
  How did you apply Kelly to a HFT strategy? Usually those strats don't have a binary outcome so standard Kelly wouldn't fit.
  - dcolkitt 6y ago
    For continuous payoffs, Kelly sizing reduces to the square of Sharpe ratio.
    - howlin 6y ago
      Kind of. Most simple models for continuous payoffs will assign a nonzero probability to losing all your wealth or your wealth going negative. The Kelly bet size for any thing with a nonzero chance of "ruin" is zero.
      - dcolkitt 6y ago
        Sharpe is typically calculated on log returns. Price going to zero would weigh as negative infinity in log return space. Therefore Sharpe would also prescribe zero bet on finite chance of ruin.
tediousdemise 6y ago
Kelley betting could probably be applied with some success to momentum trading strategies. Momentum trading is more deterministic than purely speculative strategies since it is based on observed/historical behavior.
- objektif 6y ago
  Momentum itself is as speculative as it gets.
  - tediousdemise 6y ago
    I would say pure speculation is not based on tangible data. Pure speculation: I think consumer space travel will be popular in the future, let me buy some SpaceX shares. Momentum: SpaceX seems to be trading higher in pre-market, let me buy some SpaceX shares at market open. Edit: I know SpaceX is not public, this is just an example.
ttty 6y ago
Kelly criterion (or Kelly strategy or Kelly bet) is a formula for bet sizing that leads almost surely to higher wealth. It was described by J. L. Kelly Jr, a researcher at Bell Labs, in 1956. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive. In one study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. 18 of the participants bet everything on one toss, while two-thirds gambled on tails. The average payout was just $91.36. Kelly bet: Bet one-nineteenth of the bankroll that red will not come up. If gamble has 60% chance of winning, gambler should bet 20% of bankroll at each opportunity. In American roulette, the bettor is offered an even money payoff (even money payoff, with a 95% probability of reaching the cap) If edge is negative, then the formula gives a negative result, indicating that the gambler shouldn't bet. The top of the first fraction is the expected net winnings from a $1 bet, since the two outcomes are that you either win with probability or lose the $1 wagered, i.e. win $ + $1, with probability $+.1. For even-money bets, the formula can be simplified to the first simplified formula: b=p-q-q/p, b=1, and b=0.20. Probability of success is: If you succeed, the value of your investment increases. If you fail (for which the probability is): In that case, the price of the investment decreases. Kelly criterion maximizes the expected value of the logarithm of wealth. Using too much margin is not a good investment strategy when the cost of capital is high, even when the opportunity appears promising. The general result clarifies why leveraging (taking out a loan that requires paying interest in order to raise investment capital) decreases the optimal fraction to be invested, as in that case the odds of winning are less than 0.1. The expected profit must exceed the expected loss for the investment to make any sense. The Kelly criterion can be used to determine the optimal amount of money to invest. Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the Kth win, that the rest of the bets will lose), one will end up with the most money if one bets. In the long run, final wealth is maximized by setting the derivative to zero, which means following the Kelly strategy. The function is maximised when this derivative is equal to zero. For a rigorous and general proof, see Kelly's original paper[1] or some of the other references listed below. Some corrections have been published[12] . In the long run, Kelly always wins. This is true whether N is small or large. In practice, this is a matter of playing the same game over and over, where the probability of winning and the payoff odds are always the same. Betting each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and the odds of winning are the same for each trial. The heuristic proof for the general case proceeds as follows. In a single trial, if you invest the fraction f of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor 1-f+f(1+b) Exponential mechanism (differential privacy)
xiphias2 6y ago
A simple description of the Kelly criterion is that if you want to grow wealth over a long period time, at each decision point take the one that maximizes your average expected log wealth. I'm trying to use it in real life, though sometimes the decisions are quite scary, as it's hard to estimate the probability of outcomes. Also my wealth is much more volatile than most people can stomach, but I look at it like a game.
- joosters 6y ago
  Betting with 'full' Kelly-calculated stakes is highly volatile. If I'm remembering correctly, if you get your probabilities/edge exactly right, you will still have a 50/50 chance of losing half of your bank at some point in the future (i.e. after some number of future bets) It's very common to bet just some fraction of the Kelly stakes in order to smooth out the roller coaster ride.
  - xiphias2 6y ago
    Sure, I've gone through losing more than 80% of my wealth multiple times by being 100% in BTC, so I got used to that already. At the same time it stresses my friends out a lot. I'm expecting to lose more than 50% of my wealth, but at this point it doesn't really change my life style.
- barbazoo 6y ago
  Do you have examples of how that would be used in real life decisionmaking?
  - CodesInChaos 6y ago
    For people who earn a wage and don't just make money by investing, the Kelly Criterion can't be applied in its basic form, since it means your capital gain has both constant and linear components, instead of just being linear as the formula assumes, which complicates matters a lot. Plus for low probability high reward bets you have the additional complication that you probably can't make them often enough to get a decent chance of hitting the jackpot.
minitoar 6y ago
I used this to win AI Rock Paper Scissors competition in undergrad. I just played random symbols, but used Kelly criterion to compute my bid. This worked well because the game wouldn’t allow your bankroll to go to 0 — the floor was 1.
- glacials 6y ago
  Can you explain why the Kelly criterion wouldn't have you bet 0 every time? The chance of winning a round of rock-paper-scissors when throwing a random symbol seems to be 50% (if ties cause re-dos), so wouldn't that work out to 50 * 2 - 100 = 0?
  - minitoar 6y ago
    It’s a good question. You’re right, if I followed it strictly it wouldn’t work. I suppose I rationalized offsetting it because I couldn’t actually go bankrupt. If I was better at math there’s probably some other criterion that takes into account how hard it would be to get back to where you are, given that you couldn’t go below 1.
- dplavery92 6y ago
  That's strange, given that the Kelly criterion maximizes the expectation of the log of wealth--that is, it's maximizing over multiplicative percent gains in a scenario where you can go bankrupt.
  - minitoar 6y ago
    I don’t get why it’s strange? What I learned from that competition was that bid sizing was way more important than the symbol selection strategy. Trying to beat the other students at iocaine powder wasn’t really a winning proposition.
bko 6y ago
If anyone wants to see Kelly in action, I made an app where you define an edge and a wager (% of pot or absolute amount) and see how you fare compared to the optimal bet strategy. https://kelly-criterion.netlify.app/ https://kelly-criterion.netlify.app/ https://github.com/breeko/kelly-criterion https://github.com/breeko/kelly-criterion
sl8r 6y ago
I made a streamlit app about Kelly last year, showing how to bet when you have an "edge" over a toy market of coin flippers: https://kelly-streamlit.herokuapp.com/ https://kelly-streamlit.herokuapp.com/ Other references I found interesting: - Cover and Thomas's "Elements of Information Theory" shows some interesting connections between Kelly betting and optimal message encoding. - Ed Thorp, the inventor of card counting, has a nice compendium of papers on this in "The Kelly Capital Growth Investment Criterion".
a11r 6y ago
For anyone actively managing investment portfolios, a deep understanding of the Kelley criterion is very important. For example, it is common practice to use "Half Kelly" to size positions, but most sources only provide a hand-wavey intuitive explanation. Thorp's paper[2] quantifies the benefits for any fraction of the "full Kelly" bet and its implications. In addition to Poundstone's book [1] I strongly recommend Ed Thorp's highly readable paper[2]. [1] Poundstone, William (2005), Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street, [2]https://wayback.archive-it.org/all/20090320125959/http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/KellyCriterion2007.pdf https://wayback.archive-it.org/all/20090320125959/http://www...
- User23 6y ago
  For any kind of trading activity, the most important skill to have is risk management. You can get everything else completely wrong, but if you have your risk management down you're still in the game and can learn from your mistakes. If you don't you're liable to be ruined and be out of the game until you can build back a bankroll some other way.
- hodder 6y ago
  Understanding Kelly criterion is almost useless in practical investment management. I’m a professional trader and former quant and I don’t know a single actual pro who uses anything like Kelly to size bets. I’m not saying understanding the methodology isn’t commonplace and useful, I’m saying this isn’t how portfolios are structured in the real world. Securities are not like a deck of cards. This seems to be discussed at greater length among retail traders who have no way of even knowing their odds than any professional.
  - anotheranon631 6y ago
    https://blog.alphatheory.com/2013/01/kelly-criterion-in-practice-grizzlyrock-capital-part-1.html https://blog.alphatheory.com/2013/01/kelly-criterion-in-prac...
    - 6y ago
cyberlab 6y ago
Naval Ravikant has a small post about this here: https://nav.al/kelly-criterion https://nav.al/kelly-criterion I first heard about it from him. He summarizes it as follows: > Naval: The Kelly criterion is a popularized mathematical formulation of a simple concept. The simple concept is: Don’t risk everything. Stay out of jail. Don’t bet everything on one big gamble. Be careful how much you bet each time, so you don’t lose the whole kitty.
- hogFeast 6y ago
  Lol. He must have never met anyone who has bet full Kelly.
  - auntienomen 6y ago
    Seriously. Full Kelly betting involves the use of significant leverage. The correct Kelly bet on the S&P index would be long 2.5x your total wealth.
    - kqr 6y ago
      This result depends on assumptions about the future that would not sit easy with me.
      - xiphias2 6y ago
        I can't tell you if democrats or republicans will win, but I'm quite confident that QE won't stop.
        kqr 6y ago
        And this is what makes SPY mispriced? Why do you think other investors don't share your view?
    - sigstoat 6y ago
      it literally can't tell you to bet more than your bankroll. if you include margin in your bankroll, well, that's on your head.
nwsm 6y ago
I have used the Kelly Criterion successfully in automated sports gambling. It's relevant anywhere you are doing confidence-based arbitrage.
- waynecochran 6y ago
  Did you determine the probability of winning based merely on the sport-book odds are do something more sophisticated? The sport-book odds, as I understand, are merely trying to divide the bets on each side evenly (i.e., they don't necessarily represent a probability).
  - nwsm 6y ago
    The bookie odds go into the formula as b- the net fractional odds received on the wager. We have our own models for our confidence, and the Kelly criterion decides our wager size (though we don't use a full Kelly bet). Yes, the sportsbook minimizes their own risk by setting a spread or odds with respect to how patrons are wagering. This actually makes it easier to make money if your model is much better than the average bettor. There will be games where public opinion and the majority of bets are on the wrong side of a matchup, and the bookie adjusts the odds accordingly, so the correct bet's payout is bigger than it should be. In high school I tried to do more what you are asking- use one bookie's odds (which I deemed the most accurate) as the "true probability", and another as the payout. This was not successful, but theoretically could be if the two bookies' clientele were consistently better or worse than each other, therefore influencing their odds consistently.
    - kqr 6y ago
      I can attest to this. Successful sports betting is about betting on gamblers, not games. As for your last points: bookies often book bets with each other in order to even out the odds. Otherwise you would be able to arbitrage bookies against each other. (Which would also result in their odds evening out, of course, but then the bookies wouldn't get the proceeds so they prefer to do it themselves.)
senthil_rajasek 6y ago
Also worth looking at is this previous discussion on HN, https://news.ycombinator.com/item?id=13143821 https://news.ycombinator.com/item?id=13143821
nicholast 6y ago
Hi, I wrote an essay about Kelly Criterion a while back based on a review of paper by Edward Thorpe. Cheers. https://medium.com/from-the-diaries-of-john-henry/an-optimal-trade-a374064fda91 https://medium.com/from-the-diaries-of-john-henry/an-optimal...
kqr 6y ago
For anyone interested in practising your Kelly estimation, I made a game inspired by Bernoulli's original paper on the subject for a lunch and learn at my job: https://static.loop54.com/ship-investor.html https://static.loop54.com/ship-investor.html There's also a sequel for the case of continuous outcomes: https://static.loop54.com/ship-investor-2.html https://static.loop54.com/ship-investor-2.html Before my parental leave is over, I hope to make two more sequels, one with futures and one with options. Maybe also a fixed-income version, but I'd have to learn more about that myself first.
- dshacker 6y ago
  Hey, this is really cool! What's the optimal strategy? Would love to learn more
  - tome 6y ago
    I found good success with going for the smallest investment in Bering and the second smallest investment in the other two straits. The optimal strategy would be to estimate which investment maximises your log returns :) but I don't have time for that.
- dorgo 6y ago
  Edit: First investment always fails. The next 5 always succeed. Predetermined randomness?
  - kqr 6y ago
    Not intentional, but I expect there to be some bugs -- didn't want to spend too much time on it.
deleted 6y ago
[deleted]
TameAntelope 6y ago
Kelly himself ended up using 1/n for his own personal portfolio management. Gerd Gigerenzer has a lot to say about how harmful this model has been to finance and the world, because it creates "false certainty". https://news.ycombinator.com/item?id=26325425 https://news.ycombinator.com/item?id=26325425 has further discussion.
- kqr 6y ago
  I'm not sure what you mean by using 1/n, but the Kelly criterion optimised on past returns for common portfolios of thickly traded assets does suggest something very close to 1/n very often. I've always attributed this to market efficiency (if it suggested anything else, that's what investors would do until the mispricing went away) but maybe there's a deeper reason it happens.
  - TameAntelope 6y ago
    This random person's thesis describes 1/N in a way I think is understandable: > In circa 400 A.D. Jewish Rabbi Issac Bar Aha recommended always to invest a third into land, a third into merchandise and to keep a third at hand. This method later became well-known under the name “1/n asset allocation strategy”, “equal asset allocation strategy” or “naïve strategy” and is further defined by DeMiguel et al.(2009) as ”the one in which a segment 1/n of wealth is allocated to each of N assets available for investment at each rebalancing data.” The strategy requires investing an equal part of the capital in the different present assets. Nowadays this rule is often labelled as naïve and too simple, by McClatchy and VandenHul (2005) for example. http://arno.uvt.nl/show.cgi?fid=129399 http://arno.uvt.nl/show.cgi?fid=129399 Gerd Gigerenzer has a number of books, the one I recently read was, "Risk Savvy" and he goes into some detail about the topic. All I'd do here is write a terrible book review, so if you're curious, I definitely recommend taking a look at the book. I'm not sure I totally agree with his arguments (I had a hard time understanding how he would suggest accounting for human bias), but they're definitely interesting.
    - kqr 6y ago
      Ah, that's what I thought, and what I'd expect Kelly optimisation to come up with too. So they're not really different approaches, 1/n is a special case of Kelly in somewhat efficient markets. I will look up your references though!
larrydag 6y ago
Edward Thorp used the Kelly Criterion for success in blackjack strategies and later the stock market. He has articles on his statistical methods. http://www.edwardothorp.com/articles/ http://www.edwardothorp.com/articles/
jasebell 6y ago
Fortune’s Forumla by William Poundstone is an excellent book. Edward Thorpe has most of his papers published too, they’re all good for a read [though I cannot be held responsible if you’re going to beat the dealer at Blackjack, I don’t need Kelly to know how that will turn out :)]
fny 6y ago
Many here are correct that the Kelly criterion is relatively useless compared to standard portfolio management techniques for a basket of assets. However... I will say that it's incredible useful when deciding on more high risk bets based on binary outcomes which is not something portfolio managers would dream of doing for their clients. Consider a long dated call spread on the SPY that goes out to 12/2023. Say you think the SPY will be over $600. Today, for $140 of risk, you stand to make $1,860 if you're right if you buy a $570 call and sell a $590. This is exactly what Kelly was made for. The proper strategy, IMO, is to find a comfortable allocation for trades of this sort as a portion of an overall portfolio (Say 1-2%), then of that percentage use Kelly to allocate capital to different bets of this nature to lower the variance. So sure, Kelly isn't useful for portfolio management writ large, but for managing a portfolio of binary trades, it's a useful metric.
keithalewis 6y ago
A Nobel winning economist was not impressed by the Kelly criterion. http://www-stat.wharton.upenn.edu/~steele/Courses/434F2005/Context/Kelly%20Resources/Samuelson1979.pdf http://www-stat.wharton.upenn.edu/~steele/Courses/434F2005/C...
aborsy 6y ago
It’s worth mentioning that Kelly was an associate of Claude Shannon (the father of information theory) at Bell Labs. Kelly’s criterion is in fact based on Shannon’s theory. It seems they developed the approach together. Shannon, his wife and Ed Thorp later went to Las Vegas gambling using this method, and apparently made some money.
- crubier 6y ago
  If they made some money gambling in Vegas it was clearly not thanks to Kelly’s criterion, because Kelly’s criterion clearly (and correctly) states that the optimal bet in Vegas is zero dollar.
  - kqr 6y ago
    They played, among other things, blackjack. The rules of blackjack are such that the edge varies around zero. It's mostly a tiny bit negative, but sometimes creeps over zero and that's when you apply bigger than minimum bets, following the Kelly criterion.
- deepnotderp 5y ago
  And after Vegas Ed Thorp started a hedge fund and made even more money ;)
treesrule 6y ago
brb updating the stardew valley wiki https://stardewvalleywiki.com/Stardew_Valley_Fair https://stardewvalleywiki.com/Stardew_Valley_Fair Edit: whoops they already had used this strategy
demircancelebi 6y ago
Hey, I made a game recently for people to gain an intuition for Kelly: https://beatkelly.celebi.me/ https://beatkelly.celebi.me/ Let me know what you think :)
- xiphias2 6y ago
  It sounds great, but I like instant results, can you write out after each round how much Kelly put up and has?
bxrxdx 6y ago
Its really fun to learn something new and realize how incredibly naive you've been your whole life.
aidenn0 6y ago
Note that the martingale, a common betting strategy, does exactly the opposite of the Kelly criterion. If you have a small edge and bet with the martingale against a very wealthy house, you have a fairly large chance of going bankrupt!
jedberg 6y ago
Sadly this only works in games with a positive expected outcome, so it's not actually useful in a casino unless you're a card counter.
- hansvm 6y ago
  Nah, you just use the technique to conclude that the optimal percentage of your bankroll to bet to maximize expected logarithmic wealth is 0% ;)
  - jedberg 6y ago
    Ok, fair point. :)
forinti 6y ago
I remember the first time I read about this. I put in the numbers for the lottery and a negative number came out. Of course! Your expected winnings are negative and you shouldn't play the lottery.
- jmharvey 6y ago
  Well, most of the time, anyway. If you do find a lottery game where the odds are in your favor, something resembling the Kelly criterion is a reasonable starting point for a bankroll-management strategy.
- deleted 6y ago
  [deleted]

I'd like to share a video I prepared on Kelly betting: https://youtu.be/6xhjbgREGDA https://youtu.be/6xhjbgREGDA

Some interesting psychology here: > In one study, each participant was given $25 and asked to place even-money bets 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. > 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.

blowski 6y ago
> two-thirds gambled on tails at some stage in the experiment I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not irrational to bet on tails is it?
- jfk13 6y ago
  No, the next toss still has a 60% chance of being heads. The coin doesn't remember how it landed last time.
  - blowski 6y ago
    If I'm expecting 60% of my flips to be heads, and I've already had 60%, isn't it more likely that the next one will be tails? I'm sure you can probably tell I know next to nothing about either maths or probability, so feel free to explain why I'm wrong.
    - ska 6y ago
      Yes, you are wrong, but your confusion is very common. It's so common it even has a name "The Gamblers Fallacy". Over the long run, you expect 40% tails, but if you run the experiment an infinite amount of time there will be sequences of all-heads or all-tails. Because the events are independent, the previous flips don't change anything about what happens next.
      - FredPret 6y ago
        https://dilbert.com/strip/2001-10-25 https://dilbert.com/strip/2001-10-25
      - blowski 6y ago
        So the probablity of 60% assumes infinite flips. Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads. I'd also need to know the probability that I'm in one of the cases where I get 60% of heads. Is that right?
        FredPret 6y ago
        With a 60%-heads coin you can still get 10 straight heads. It’s just that over many many flips, the average will gradually tend towards 60%. You can still have streaks of hundreds, thousands, millions of either heads or tails in a row.
        ska 6y ago
        This gets into interesting stuff! So the situation described in that paper is that you are given the true odds of the coin, 60% heads. In this case it's just as I described - knowing previous results doesn't tell you anything useful. > Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads. This is true. In fact there is only about 25% chance of getting exactly 6 of the 10 to be heads (but nearly 70% chance of >= 6 heads). You can work this out with something called the binomial distribution. Chance of getting 10 heads in a row is .6% A more interesting aspect is when you don't know the odds (or don't trust what you've been told). In this case it's definitely important what the history is. So given your 10 flips, we can ask questions like "how likely is it that this coin is fair (50/50) given the 10 flips I just saw". It turns out the best estimation of the true probability is, pretty intuitively, (h+t)/h; this will jump aroudn for small N . In practice you are more often looking at something like P(0.55 < p < 0.65 | samples) , i.e. the probability that the true value lies between 0.55 and 0.65 heads, given the 10 flips I've seen). Obviously in these cases, the more samples you have seen the tighter the estimate get. You can also ask questions like how many flips do I need to see to be confident at a certain the coin is really 0.6 heads.
- afranchuk 6y ago
  Those are independent variables. The fact you've had X heads has no bearing on the future flips. It is irrational to bet on tails statistically speaking, though psychologically that line of reasoning is common.
- theschwa 6y ago
  No, the coin doesn't have a memory, so the chance of tails is still 40% making it still optimal to choose heads.
- skeeter2020 6y ago
  you're not betting on the number of heads/tails per 10 trials though, each trial is independent with a 60% of heads. In a striaght-up prediction you should always choose heads, it the how much to wager that is the question.
- baobabKoodaa 6y ago
  > I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not an add thing to bet on tails is it? I realize you're probably joking, but since this argument is intuitively appealing to many people, I will answer as if it was serious: if you have a weighted coin that is 60% likely to land on heads, that means it's 60% likely to land on heads on any given toss. On the first toss. On the second toss. Any given toss. Even after you have tossed it 6 times and seen 6 heads in a row, the coin is still 60% likely to land on heads. The coin has no "memory". Previous results have no effect on future results.
  - 6gvONxR4sf7o 6y ago
    Moreover, the important feature of coin flips isn’t randomness, it’s independence (from previous coin flips and from everything else). Independence is in fact a useful mental model for randomness.
  - vgeek 6y ago
    I quickly searched but couldn't find the exact study, but I've read that by adding the past numbers digital signage to roulette tables, casinos experience a significant (I'm thinking it was like 100%+) increase in wagers when people believe that a color is "due" simply from not understanding independent vs dependent events. Humans love to look for patterns, even when there isn't any real _meaning_ behind them.
  - dehrmann 6y ago
    There's a corollary to the gambler's fallacy that says is P(heads) is 60% and you get 6 heads in a row, the people running the experiment probably lied to you.
    - lupire 6y ago
      but that means you should bet into the bias, not against it.
      - dehrmann 6y ago
- prometheus76 6y ago
  You've just discovered the Gambler's Fallacy.
- JacobLinney 6y ago
  Yes it is irrational. That's a common statistical misconception, the key thing here is that every flip has a 60% chance of being heads. The result of each flip is completely independent of what came before it. In your example the 7th flip is just as likely to be heads as the first flip, or any of the other 5 flips that landed on heads.
  - blowski 6y ago
    It says "a coin that would land heads 60% of the time". If it's already landed heads 60% of the time, I'd expect the remaining 40% for it to land on tails.
    - antasvara 6y ago
      The key here is that it's expected to land heads 60% of the time. Take a normal coin, which is expected to land heads 50% of the time. If you flip a heads, do you instantly expect it to be tails next time? By your logic it would be impossible to ever flip heads twice in a row. Coins as a general rule aren't impacted by previous flips.
    - bscphil 6y ago
      That's not a guarantee for any number of flips. For example, if you only flipped the coin one time, what does "60% of the time" even mean in that context? As your other replies have indicated, this is getting at the long-run frequency, meaning as you flip the coin more and more times, approaching infinity, the number of heads approaches 60%.
    - sokoloff 6y ago
      Thought experiment: in what way has it landed heads 60% of the time? It landed heads 100% of the trials so far, but the coin has no way of keeping track of that.
- justinpowers 6y ago
  Each toss is independent of prior (and subsequent) tosses, so no matter what, a given tosshas 60% chance of landing heads. Rationally, one should bet heads on any given toss. But most people would agree with the irrational bet. This tendency is known as the Gambler’s fallacy (https://en.wikipedia.org/wiki/Gambler's_fallacy) https://en.wikipedia.org/wiki/Gambler's_fallacy).
- Jtsummers 6y ago
  That's the gambler's fallacy in action. So long as each event is independent, the prior ones have no impact on the likelihood of future events. If you've flipped the coin 60 times and they've all been heads, there's no reason to expect the next 40 will be tails. They still have better odds of being heads.
  - travisjungroth 6y ago
    If you see 60 heads in a row in the real world you've got a trick coin. The odds of that are 1/10^17.
    - Jtsummers 6y ago
      It's certainly low odds, but it's not impossible nor does it require a trick coin. I've seen people roll a 20 on a d20 10 times in a row, and then not a single 20 the rest of the session on the same die. Shit happens, it's probability and it may be improbable but it isn't impossible.
      - lupire 6y ago
        I don't believe you.
        Jtsummers 6y ago
        I mean, that's fine, it's an anecdote. If you'd like, take a few dice and set up cameras and an automatic rolling mechanism and see if there are any improbable sequences like alternation between two or three number or a long run of a single number, or a long run without a particular number appearing. Over enough trials you are likely to encounter these kinds of events.
        deleted 6y ago
        [deleted]
        travisjungroth 6y ago
- mytherin 6y ago
  The probability of a coin flip being heads or tails is completely independent from the previous flips. If the coin lands 6 heads in a row, the next coin flip still has a 60% chance of being heads, hence it is always unwise to bet on tails in this experiment. This is an example of the Gambler's fallacy [1]. [1] https://en.wikipedia.org/wiki/Gambler%27s_fallacy https://en.wikipedia.org/wiki/Gambler%27s_fallacy
- ska 6y ago
  > If you've just had 6 heads in a row the next 4 "should" be tails That's not how this works. Each toss is independent, so you should never pay attention to previous results if you know the true odds.
- deeg 6y ago
  This wiki page can explain why better than me: https://en.wikipedia.org/wiki/Gambler%27s_fallacy https://en.wikipedia.org/wiki/Gambler%27s_fallacy
- stocknoob 6y ago
  Your friend walks up while you're playing. They haven't seen the game, so think heads is coming up. Your other friend has been playing longer, before you even started. They saw 13 tails and then your 6 heads. The next throw should be heads to even it out for them. Why is your history more of an influence than theirs?
- sorokod 6y ago
  While this is irrational in this experiment, but it is likely that the biological systems in which humans evolved, tend to not have truly independent events - hence our intuition.
frabjoused 6y ago
I made a little playground for this, you can fiddle with the numbers. https://parsebox.io/dthree/lnumtuenmskr https://parsebox.io/dthree/lnumtuenmskr
tialaramex 6y ago
To be fair, as a participant in psychology experiments I go in aware that it's plausible, even likely that I am being misled about what's really going on. That's even necessary in some experiments. Maybe I'm not technically lied to but if deliberately engineering a false impression is the goal, psychologists are the people to do it in a controlled experiment. The experimenters aren't (ethically) allowed to cause you harm, and they'll probably tell you exactly what was really going on afterwards at least if you ask, but during the experiment everything is potentially suspect. Maybe the task you're focused on was just a distraction and they really care whether you notice the clocks in the room are running too fast so that "five minutes" to do the task is really only 250 seconds - but equally maybe the apparent "time pressure" to complete the task is the distraction and they really care whether you lie about completing it properly given an opportunity to cheat. So if the experimenter in a psych experiment tells me the coin is biased 60% heads, I don't consider that the same way I would if the friend I play board games with says it. As a result chances are my first few dozen bets are confirming this unusual claim about the world. Biased coins are hard to make, is this coin really biased? Maybe I try fifty bets in rapid succession, $1 on heads each time. Apparently that's expected to take about five minutes of my half an hour, and before that's done I won't feel comfortable even assuming it's really 60% heads. And at the end of those five minutes on average I turn $25 into $35 and feel comfortable it's really 60% heads or that I can't tell what's wrong. Now, why gamble on tails? Well like I said, Psychologists mislead you intentionally during experimentation. Maybe the experimenter tells you it's 60% likely to be Heads. If the gamer told me that, I believe it's 40% likely to be Tails because that's logical, but when an experimenter tells me that, I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check.
- lupire 6y ago
  Biased coins are *impossible" to make if the coin is flipped not spun. I doubt any story about a biased coins in the real world.
  - function_seven 6y ago
    If the coin was made from a thin magnet, and being flipped onto a weak magnetic plate, couldn't you bias the result? If the landing pad was a strong magnet, then you could trivially make it a "100% heads" coin. Just weaken the magnetic field so it's not strong enough to flip a coin flat at rest, but has enough oomph to take a coin landing near its edge to the preferred result.
jfengel 6y ago
Did they know that it was biased towards heads? With only a 60-40 split I probably wouldn't notice it unless I was actually keeping track, which could take a while. A 6-4 split on 10 tosses doesn't tell you anything. If you told me it was a fair coin and I thought the experiment was about something else, it might take a very long time before it occurred to me to test the hypothesis that the coin wasn't fair. If they knew it was biased... I'm sure there's an optimal strategy, but a simple strategy would be "bet half of what you have on heads every time". Any idea how much worse that is than the optimal strategy?
- kqr 6y ago
  You can plot g = 0.6 log (1 + 2f) + 0.4 log (1 - f) And locate f=0.5 and compare to the maximum g. Edit: I wanted to check my intuition so I did: https://www.wolframalpha.com/input/?i=plot++0.6+log+%281+%2B+2f%29+%2B+0.4+log+%281+-+f%29%2C+f%3D0..1 https://www.wolframalpha.com/input/?i=plot++0.6+log+%281+%2B... Looks like 0.5 is a slight overbet, but still very, very good.
- tedunangst 6y ago
  If only there was a link to the study so we could see how it was setup.
- deleted 6y ago
  [deleted]
- seoaeu 6y ago
  > Did they know that it was biased towards heads? "Prior to starting the game, participants read a detailed description of the game, which included a clear statement, in bold, indicating that the simulated coin had a 60% chance of coming up heads and a 40% chance of coming up tails."
ISL 6y ago
The paper is pretty awesome and accessibly-written: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963 The PDF is free-to-read.

syassami 6y ago

If you enjoyed this, I highly recommend reading Fortune's Formula.

kqr 6y ago
And when you're done with that, the Kelly Capital Growth Investment Criterion is one of the better books I've read. But it's a much more advanced read.
- _e 6y ago
  And when you are done with that, I highly suggest reading Edward Thorp's autobiography "Man for all markets" where he employs the Kelly Criterion in adventure after adventure. He not only developed the first card counting system for blackjack but he also created the first wearable computer to beat roulette (with Claude Shannon).

anonu 6y ago

This has been posted 5 other times on HN with no real discussion [1]. I'll add my 2 cents: I used to use the principles of kelly betting back when I designed systematic HFT strategies. It gives you a good framework to think about how much to bet based on the batting average of a particular pattern you recognize in the market... [1] https://hn.algolia.com/?q=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FKelly_criterion https://hn.algolia.com/?q=https%3A%2F%2Fen.wikipedia.org%2Fw...

jared_buckner 6y ago
There's a bit of a discussion here: https://news.ycombinator.com/item?id=18484631 https://news.ycombinator.com/item?id=18484631
fighterpilot 6y ago
How did you apply Kelly to a HFT strategy? Usually those strats don't have a binary outcome so standard Kelly wouldn't fit.
- dcolkitt 6y ago
  For continuous payoffs, Kelly sizing reduces to the square of Sharpe ratio.
  - howlin 6y ago
    Kind of. Most simple models for continuous payoffs will assign a nonzero probability to losing all your wealth or your wealth going negative. The Kelly bet size for any thing with a nonzero chance of "ruin" is zero.
    - dcolkitt 6y ago
      Sharpe is typically calculated on log returns. Price going to zero would weigh as negative infinity in log return space. Therefore Sharpe would also prescribe zero bet on finite chance of ruin.

tediousdemise 6y ago

Kelley betting could probably be applied with some success to momentum trading strategies. Momentum trading is more deterministic than purely speculative strategies since it is based on observed/historical behavior.

objektif 6y ago
Momentum itself is as speculative as it gets.
- tediousdemise 6y ago
  I would say pure speculation is not based on tangible data. Pure speculation: I think consumer space travel will be popular in the future, let me buy some SpaceX shares. Momentum: SpaceX seems to be trading higher in pre-market, let me buy some SpaceX shares at market open. Edit: I know SpaceX is not public, this is just an example.

ttty 6y ago

Kelly criterion (or Kelly strategy or Kelly bet) is a formula for bet sizing that leads almost surely to higher wealth. It was described by J. L. Kelly Jr, a researcher at Bell Labs, in 1956. The Kelly Criterion is to bet a predetermined fraction of assets, and it can seem counterintuitive. In one study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. 18 of the participants bet everything on one toss, while two-thirds gambled on tails. The average payout was just $91.36. Kelly bet: Bet one-nineteenth of the bankroll that red will not come up. If gamble has 60% chance of winning, gambler should bet 20% of bankroll at each opportunity. In American roulette, the bettor is offered an even money payoff (even money payoff, with a 95% probability of reaching the cap) If edge is negative, then the formula gives a negative result, indicating that the gambler shouldn't bet. The top of the first fraction is the expected net winnings from a $1 bet, since the two outcomes are that you either win with probability or lose the $1 wagered, i.e. win $ + $1, with probability $+.1. For even-money bets, the formula can be simplified to the first simplified formula: b=p-q-q/p, b=1, and b=0.20. Probability of success is: If you succeed, the value of your investment increases. If you fail (for which the probability is): In that case, the price of the investment decreases. Kelly criterion maximizes the expected value of the logarithm of wealth. Using too much margin is not a good investment strategy when the cost of capital is high, even when the opportunity appears promising. The general result clarifies why leveraging (taking out a loan that requires paying interest in order to raise investment capital) decreases the optimal fraction to be invested, as in that case the odds of winning are less than 0.1. The expected profit must exceed the expected loss for the investment to make any sense. The Kelly criterion can be used to determine the optimal amount of money to invest. Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the Kth win, that the rest of the bets will lose), one will end up with the most money if one bets. In the long run, final wealth is maximized by setting the derivative to zero, which means following the Kelly strategy. The function is maximised when this derivative is equal to zero. For a rigorous and general proof, see Kelly's original paper[1] or some of the other references listed below. Some corrections have been published[12] . In the long run, Kelly always wins. This is true whether N is small or large. In practice, this is a matter of playing the same game over and over, where the probability of winning and the payoff odds are always the same. Betting each time will likely maximize the wealth growth rate only in the case where the number of trials is very large, and the odds of winning are the same for each trial. The heuristic proof for the general case proceeds as follows. In a single trial, if you invest the fraction f of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor 1-f+f(1+b) Exponential mechanism (differential privacy)

xiphias2 6y ago

A simple description of the Kelly criterion is that if you want to grow wealth over a long period time, at each decision point take the one that maximizes your average expected log wealth. I'm trying to use it in real life, though sometimes the decisions are quite scary, as it's hard to estimate the probability of outcomes. Also my wealth is much more volatile than most people can stomach, but I look at it like a game.

joosters 6y ago
Betting with 'full' Kelly-calculated stakes is highly volatile. If I'm remembering correctly, if you get your probabilities/edge exactly right, you will still have a 50/50 chance of losing half of your bank at some point in the future (i.e. after some number of future bets) It's very common to bet just some fraction of the Kelly stakes in order to smooth out the roller coaster ride.
- xiphias2 6y ago
  Sure, I've gone through losing more than 80% of my wealth multiple times by being 100% in BTC, so I got used to that already. At the same time it stresses my friends out a lot. I'm expecting to lose more than 50% of my wealth, but at this point it doesn't really change my life style.
barbazoo 6y ago
Do you have examples of how that would be used in real life decisionmaking?
- CodesInChaos 6y ago
  For people who earn a wage and don't just make money by investing, the Kelly Criterion can't be applied in its basic form, since it means your capital gain has both constant and linear components, instead of just being linear as the formula assumes, which complicates matters a lot. Plus for low probability high reward bets you have the additional complication that you probably can't make them often enough to get a decent chance of hitting the jackpot.

minitoar 6y ago

I used this to win AI Rock Paper Scissors competition in undergrad. I just played random symbols, but used Kelly criterion to compute my bid. This worked well because the game wouldn’t allow your bankroll to go to 0 — the floor was 1.

glacials 6y ago
Can you explain why the Kelly criterion wouldn't have you bet 0 every time? The chance of winning a round of rock-paper-scissors when throwing a random symbol seems to be 50% (if ties cause re-dos), so wouldn't that work out to 50 * 2 - 100 = 0?
- minitoar 6y ago
  It’s a good question. You’re right, if I followed it strictly it wouldn’t work. I suppose I rationalized offsetting it because I couldn’t actually go bankrupt. If I was better at math there’s probably some other criterion that takes into account how hard it would be to get back to where you are, given that you couldn’t go below 1.
dplavery92 6y ago
That's strange, given that the Kelly criterion maximizes the expectation of the log of wealth--that is, it's maximizing over multiplicative percent gains in a scenario where you can go bankrupt.
- minitoar 6y ago
  I don’t get why it’s strange? What I learned from that competition was that bid sizing was way more important than the symbol selection strategy. Trying to beat the other students at iocaine powder wasn’t really a winning proposition.

bko 6y ago

If anyone wants to see Kelly in action, I made an app where you define an edge and a wager (% of pot or absolute amount) and see how you fare compared to the optimal bet strategy. https://kelly-criterion.netlify.app/ https://kelly-criterion.netlify.app/ https://github.com/breeko/kelly-criterion https://github.com/breeko/kelly-criterion

sl8r 6y ago

I made a streamlit app about Kelly last year, showing how to bet when you have an "edge" over a toy market of coin flippers: https://kelly-streamlit.herokuapp.com/ https://kelly-streamlit.herokuapp.com/ Other references I found interesting: - Cover and Thomas's "Elements of Information Theory" shows some interesting connections between Kelly betting and optimal message encoding. - Ed Thorp, the inventor of card counting, has a nice compendium of papers on this in "The Kelly Capital Growth Investment Criterion".

a11r 6y ago

For anyone actively managing investment portfolios, a deep understanding of the Kelley criterion is very important. For example, it is common practice to use "Half Kelly" to size positions, but most sources only provide a hand-wavey intuitive explanation. Thorp's paper[2] quantifies the benefits for any fraction of the "full Kelly" bet and its implications. In addition to Poundstone's book [1] I strongly recommend Ed Thorp's highly readable paper[2]. [1] Poundstone, William (2005), Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street, [2]https://wayback.archive-it.org/all/20090320125959/http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/KellyCriterion2007.pdf https://wayback.archive-it.org/all/20090320125959/http://www...

User23 6y ago
For any kind of trading activity, the most important skill to have is risk management. You can get everything else completely wrong, but if you have your risk management down you're still in the game and can learn from your mistakes. If you don't you're liable to be ruined and be out of the game until you can build back a bankroll some other way.
hodder 6y ago
Understanding Kelly criterion is almost useless in practical investment management. I’m a professional trader and former quant and I don’t know a single actual pro who uses anything like Kelly to size bets. I’m not saying understanding the methodology isn’t commonplace and useful, I’m saying this isn’t how portfolios are structured in the real world. Securities are not like a deck of cards. This seems to be discussed at greater length among retail traders who have no way of even knowing their odds than any professional.
- anotheranon631 6y ago
  https://blog.alphatheory.com/2013/01/kelly-criterion-in-practice-grizzlyrock-capital-part-1.html https://blog.alphatheory.com/2013/01/kelly-criterion-in-prac...
  - 6y ago

cyberlab 6y ago

Naval Ravikant has a small post about this here: https://nav.al/kelly-criterion https://nav.al/kelly-criterion I first heard about it from him. He summarizes it as follows: > Naval: The Kelly criterion is a popularized mathematical formulation of a simple concept. The simple concept is: Don’t risk everything. Stay out of jail. Don’t bet everything on one big gamble. Be careful how much you bet each time, so you don’t lose the whole kitty.

hogFeast 6y ago
Lol. He must have never met anyone who has bet full Kelly.
- auntienomen 6y ago
  Seriously. Full Kelly betting involves the use of significant leverage. The correct Kelly bet on the S&P index would be long 2.5x your total wealth.
  - kqr 6y ago
    This result depends on assumptions about the future that would not sit easy with me.
    - xiphias2 6y ago
      I can't tell you if democrats or republicans will win, but I'm quite confident that QE won't stop.
      - kqr 6y ago
        And this is what makes SPY mispriced? Why do you think other investors don't share your view?
  - sigstoat 6y ago
    it literally can't tell you to bet more than your bankroll. if you include margin in your bankroll, well, that's on your head.

nwsm 6y ago

I have used the Kelly Criterion successfully in automated sports gambling. It's relevant anywhere you are doing confidence-based arbitrage.

waynecochran 6y ago
Did you determine the probability of winning based merely on the sport-book odds are do something more sophisticated? The sport-book odds, as I understand, are merely trying to divide the bets on each side evenly (i.e., they don't necessarily represent a probability).
- nwsm 6y ago
  The bookie odds go into the formula as b- the net fractional odds received on the wager. We have our own models for our confidence, and the Kelly criterion decides our wager size (though we don't use a full Kelly bet). Yes, the sportsbook minimizes their own risk by setting a spread or odds with respect to how patrons are wagering. This actually makes it easier to make money if your model is much better than the average bettor. There will be games where public opinion and the majority of bets are on the wrong side of a matchup, and the bookie adjusts the odds accordingly, so the correct bet's payout is bigger than it should be. In high school I tried to do more what you are asking- use one bookie's odds (which I deemed the most accurate) as the "true probability", and another as the payout. This was not successful, but theoretically could be if the two bookies' clientele were consistently better or worse than each other, therefore influencing their odds consistently.
  - kqr 6y ago
    I can attest to this. Successful sports betting is about betting on gamblers, not games. As for your last points: bookies often book bets with each other in order to even out the odds. Otherwise you would be able to arbitrage bookies against each other. (Which would also result in their odds evening out, of course, but then the bookies wouldn't get the proceeds so they prefer to do it themselves.)

senthil_rajasek 6y ago

Also worth looking at is this previous discussion on HN, https://news.ycombinator.com/item?id=13143821 https://news.ycombinator.com/item?id=13143821

nicholast 6y ago

Hi, I wrote an essay about Kelly Criterion a while back based on a review of paper by Edward Thorpe. Cheers. https://medium.com/from-the-diaries-of-john-henry/an-optimal-trade-a374064fda91 https://medium.com/from-the-diaries-of-john-henry/an-optimal...

kqr 6y ago

For anyone interested in practising your Kelly estimation, I made a game inspired by Bernoulli's original paper on the subject for a lunch and learn at my job: https://static.loop54.com/ship-investor.html https://static.loop54.com/ship-investor.html There's also a sequel for the case of continuous outcomes: https://static.loop54.com/ship-investor-2.html https://static.loop54.com/ship-investor-2.html Before my parental leave is over, I hope to make two more sequels, one with futures and one with options. Maybe also a fixed-income version, but I'd have to learn more about that myself first.

dshacker 6y ago
Hey, this is really cool! What's the optimal strategy? Would love to learn more
- tome 6y ago
  I found good success with going for the smallest investment in Bering and the second smallest investment in the other two straits. The optimal strategy would be to estimate which investment maximises your log returns :) but I don't have time for that.
dorgo 6y ago
Edit: First investment always fails. The next 5 always succeed. Predetermined randomness?
- kqr 6y ago
  Not intentional, but I expect there to be some bugs -- didn't want to spend too much time on it.

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TameAntelope 6y ago

Kelly himself ended up using 1/n for his own personal portfolio management. Gerd Gigerenzer has a lot to say about how harmful this model has been to finance and the world, because it creates "false certainty". https://news.ycombinator.com/item?id=26325425 https://news.ycombinator.com/item?id=26325425 has further discussion.

kqr 6y ago
I'm not sure what you mean by using 1/n, but the Kelly criterion optimised on past returns for common portfolios of thickly traded assets does suggest something very close to 1/n very often. I've always attributed this to market efficiency (if it suggested anything else, that's what investors would do until the mispricing went away) but maybe there's a deeper reason it happens.
- TameAntelope 6y ago
  This random person's thesis describes 1/N in a way I think is understandable: > In circa 400 A.D. Jewish Rabbi Issac Bar Aha recommended always to invest a third into land, a third into merchandise and to keep a third at hand. This method later became well-known under the name “1/n asset allocation strategy”, “equal asset allocation strategy” or “naïve strategy” and is further defined by DeMiguel et al.(2009) as ”the one in which a segment 1/n of wealth is allocated to each of N assets available for investment at each rebalancing data.” The strategy requires investing an equal part of the capital in the different present assets. Nowadays this rule is often labelled as naïve and too simple, by McClatchy and VandenHul (2005) for example. http://arno.uvt.nl/show.cgi?fid=129399 http://arno.uvt.nl/show.cgi?fid=129399 Gerd Gigerenzer has a number of books, the one I recently read was, "Risk Savvy" and he goes into some detail about the topic. All I'd do here is write a terrible book review, so if you're curious, I definitely recommend taking a look at the book. I'm not sure I totally agree with his arguments (I had a hard time understanding how he would suggest accounting for human bias), but they're definitely interesting.
  - kqr 6y ago
    Ah, that's what I thought, and what I'd expect Kelly optimisation to come up with too. So they're not really different approaches, 1/n is a special case of Kelly in somewhat efficient markets. I will look up your references though!

larrydag 6y ago

Edward Thorp used the Kelly Criterion for success in blackjack strategies and later the stock market. He has articles on his statistical methods. http://www.edwardothorp.com/articles/ http://www.edwardothorp.com/articles/

jasebell 6y ago

Fortune’s Forumla by William Poundstone is an excellent book. Edward Thorpe has most of his papers published too, they’re all good for a read [though I cannot be held responsible if you’re going to beat the dealer at Blackjack, I don’t need Kelly to know how that will turn out :)]

fny 6y ago

Many here are correct that the Kelly criterion is relatively useless compared to standard portfolio management techniques for a basket of assets. However... I will say that it's incredible useful when deciding on more high risk bets based on binary outcomes which is not something portfolio managers would dream of doing for their clients. Consider a long dated call spread on the SPY that goes out to 12/2023. Say you think the SPY will be over $600. Today, for $140 of risk, you stand to make $1,860 if you're right if you buy a $570 call and sell a $590. This is exactly what Kelly was made for. The proper strategy, IMO, is to find a comfortable allocation for trades of this sort as a portion of an overall portfolio (Say 1-2%), then of that percentage use Kelly to allocate capital to different bets of this nature to lower the variance. So sure, Kelly isn't useful for portfolio management writ large, but for managing a portfolio of binary trades, it's a useful metric.

keithalewis 6y ago

A Nobel winning economist was not impressed by the Kelly criterion. http://www-stat.wharton.upenn.edu/~steele/Courses/434F2005/Context/Kelly%20Resources/Samuelson1979.pdf http://www-stat.wharton.upenn.edu/~steele/Courses/434F2005/C...

aborsy 6y ago

It’s worth mentioning that Kelly was an associate of Claude Shannon (the father of information theory) at Bell Labs. Kelly’s criterion is in fact based on Shannon’s theory. It seems they developed the approach together. Shannon, his wife and Ed Thorp later went to Las Vegas gambling using this method, and apparently made some money.

crubier 6y ago
If they made some money gambling in Vegas it was clearly not thanks to Kelly’s criterion, because Kelly’s criterion clearly (and correctly) states that the optimal bet in Vegas is zero dollar.
- kqr 6y ago
  They played, among other things, blackjack. The rules of blackjack are such that the edge varies around zero. It's mostly a tiny bit negative, but sometimes creeps over zero and that's when you apply bigger than minimum bets, following the Kelly criterion.
deepnotderp 5y ago
And after Vegas Ed Thorp started a hedge fund and made even more money ;)

treesrule 6y ago

brb updating the stardew valley wiki https://stardewvalleywiki.com/Stardew_Valley_Fair https://stardewvalleywiki.com/Stardew_Valley_Fair Edit: whoops they already had used this strategy

demircancelebi 6y ago

Hey, I made a game recently for people to gain an intuition for Kelly: https://beatkelly.celebi.me/ https://beatkelly.celebi.me/ Let me know what you think :)

xiphias2 6y ago
It sounds great, but I like instant results, can you write out after each round how much Kelly put up and has?

bxrxdx 6y ago

Its really fun to learn something new and realize how incredibly naive you've been your whole life.

aidenn0 6y ago

Note that the martingale, a common betting strategy, does exactly the opposite of the Kelly criterion. If you have a small edge and bet with the martingale against a very wealthy house, you have a fairly large chance of going bankrupt!

jedberg 6y ago

Sadly this only works in games with a positive expected outcome, so it's not actually useful in a casino unless you're a card counter.

hansvm 6y ago
Nah, you just use the technique to conclude that the optimal percentage of your bankroll to bet to maximize expected logarithmic wealth is 0% ;)
- jedberg 6y ago
  Ok, fair point. :)

forinti 6y ago

I remember the first time I read about this. I put in the numbers for the lottery and a negative number came out. Of course! Your expected winnings are negative and you shouldn't play the lottery.

jmharvey 6y ago
Well, most of the time, anyway. If you do find a lottery game where the odds are in your favor, something resembling the Kelly criterion is a reasonable starting point for a bankroll-management strategy.
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