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The Kelly criterion: How to size bets (2019)
- fouc 5y agoThe first time I encountered Kelly Criterion was when I was scouring books on trading. It got mentioned in one chapter in a book that happened to be a collection of interviews of traders, out of about 30 books on trading that I had read. Bet sizing can be used for money management and risk control for professional traders.
- jdmoreira 5y agoI highly recommend the book "Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street"
- gardenfelder 5y agohttp://www.fortunesformula.com/ http://www.fortunesformula.com/
- clove 5y agoTLDR?
- kqr 5y agoAnd if you want to dig deeper, I'd follow up with The Kelly Capital Growth Investment Criterion. It's just a collection of peer-reviewed science on the Kelly criterion and various applications. Very dense reading, but also the best overview of the subject I've seen. https://www.amazon.com/Kelly-Capital-Growth-Investment-Criterion-ebook/dp/B00CFG09WG/ref=sr_1_1 https://www.amazon.com/Kelly-Capital-Growth-Investment-Crite...
- te 5y agoBig book. Any of the chapters/papers within that you'd particularly recommend?
- fredgrott 5y agoIt's 1.58 dollar heads double your money at 100 bets and 220 bets. How did I do that? I use to win money playing poker to pay for college books at beg of Semester using the same math strategy as it always has to be between I/15th and 1/20 if you have 60% odds
- alisonkisk 5y ago
- gfd 5y agoHow should you bet if you're not comfortable with variance? I tried the "autobetter" simulation at the bottom of the page, always clicking heads, and it took 150 tries to max out. For most of the time it was way below the initial $25 investment: https://imgur.com/a/HKKHMRL https://imgur.com/a/HKKHMRL
- krajzeg 5y agoIf you're not comfortable with variance, you should bet 0$ each time. This is only partially tongue-in-cheek, since the only way to reduce variance is to bet lower amounts, but this will also reduce your expected return, all the way to zero at 0$ bets.
- alisonkisk 5y ago
- contravariant 5y agoA simple way would be to pick a strategy with a slightly different utility function from Kelly's (which implicitly uses log as the utility function). The downside of this is that Kelly's utility function has some very nice properties. In particular the total utility of a sequence of bets will turn into a sum which means you can ignore dependencies between events. Note for instance that the argument in this article still holds even if you've got very strong dependencies between coin flips, it could repeat the pattern HTHTH forever and the Kelly criterion would still give the same optimal proportion, though obviously there are better strategies if you don't restrict yourself to blindly betting on heads.
- ojbyrne 5y agoNot to mention that in real life, there is probably a minimum bet.
- cevi 5y agoIf you bet with the -1/x utility function, then the optimal strategy is to always bet about 10.1% of your money. I find this to give a good tradeoff between the variance and the rate of exponential growth.
- 5y ago
- stevespang 5y ago
- im3w1l 5y agoTime is money, so my definition of the best strategy is the one that gets you your $250 asap. Potentially giving up if your bankroll gets so small that it would take unreasonably long to climb back. With this idea we can assign a dollar value to every possible bankroll. Those values should satisfy. V(250) = 250 V(x) = max(x, max over 0<y<=x pV(x+y) + (1-p)V(x-y) - a) I can't immediately see how to solve it though.
- kqr 5y agoThe Kelly criterion, which maximises long-term growth, also has the nice property that it gets to any particular desired level faster than any other strategy, on average.
- im3w1l 5y agoI don't think this can be true. Consider if you start with x money and want to reach x+epsilon. Then the first bet in an optimal strategy should be to bet epsilon (if you win you reach your goal, and if you dont the setback is smaller than kelly).
- kqr 5y agoBut what do you do if you don't win? Then the target level is two epsilon away -- do you bet that amount? Maybe that works if the probability of success is high enough and the initial epsilon is small enough. If that is not the case, it starts to sound like a martingale type strategy.
- im3w1l 5y agoIf you start super close to the goal, then yes you would double for every loss... for a while. When it starts actually cutting into your bankroll the bets would grow slower and slower and then even start shrinking. It should not surprise me if, far from the goal, you would bet a fraction of your bankroll every time, like Kelly does.
- EMM_386 5y agoI ended up like this by simple reducing my win wager to $10 and doubling my loss wager every time I lost. https://i.imgur.com/QUbOopz.jpg https://i.imgur.com/QUbOopz.jpg Essentially using the Martingale system.
- altcognito 5y agoI went with betting half my assets every time, and for whatever reason, the first go around I managed to cash out the first time at 7 flips. (I did better than 79% of folks) I was so spooked I ran the simulation again, this time just maxing out. After a couple dozen tries I cashed out at 5 attempts. I still only did better than 79% of participants in that study. (I always went with 60% no matter what, in both extreme cases I never lost)
- kqr 5y agoFor the first 50 throws, your Martingale-like strategy actually performs fairly competitively with Kelly – if we ignore the very serious consequence that it can lead to complete ruin. Kelly is red, and your system is black in this plot: https://i.xkqr.org/kelly-vs-martingale-50.png https://i.xkqr.org/kelly-vs-martingale-50.png Of course, given a longer opportunity to invest, Kelly wins every time by growing exponentially: https://i.xkqr.org/kelly-vs-martingale-100.png https://i.xkqr.org/kelly-vs-martingale-100.png
- danielmarkbruce 5y agoIt's a nice piece of math, but it just confirms common sense. If you have a small edge and the payoff is very likely, you bet a decent chunk of cash. If you have a big edge and payoff is unlikely, you bet a half decent chunk of cash. If you have a big edge and the payoff is very likely, you bet a double decent sized of cash. If you have a small edge and payoff is unlikely, you bet a little bit. If you don't have an edge, you don't bet.
- ultra_nick 5y agoYou're describing the anti-martingale which is a different strategy with a statistically expected profit of $0 for random coin flip type games. The Kelly criterion is different and has a positive expected profit.
- danielmarkbruce 5y agoNo, that's not what I'm describing at all. Re-read it, or re-read what martingale/anti-martingale is. I don't say anything about betting size based on a previous bet.
- modeless 5y agoYou'd think the optimal strategy is common sense but for small probabilities and large payoffs it really isn't. Many people simply ignore very risky investments on the theory that they're just too risky, when the actual optimal thing is to invest small amounts if the payoff is big enough.
- danielmarkbruce 5y agoDo people really ignore this? The entire VC industry seems dedicated to these exact bets. The guys at Sequoia for example are approximating the kelly criterion and likely don't ever think about it explicitly.
- SkyMarshal 5y ago> If you have a big edge and payoff is unlikely Why would the payoff be unlikely if you have a big edge?
- dang 5y agoPast related threads: Kelly Criterion – how to calculate optimal bet sizes - https://news.ycombinator.com/item?id=27431627 https://news.ycombinator.com/item?id=27431627 - June 2021 (129 comments) Kelly Criterion - https://news.ycombinator.com/item?id=26834333 https://news.ycombinator.com/item?id=26834333 - April 2021 (194 comments) Log Optimal Betting – An interactive demo of the Kelly criterion - https://news.ycombinator.com/item?id=23894425 https://news.ycombinator.com/item?id=23894425 - July 2020 (1 comment) Kelly Criterion - https://news.ycombinator.com/item?id=21559782 https://news.ycombinator.com/item?id=21559782 - Nov 2019 (1 comment) Kelly Criterion (2007) - https://news.ycombinator.com/item?id=18484631 https://news.ycombinator.com/item?id=18484631 - Nov 2018 (96 comments - great top subthread!) Kelly Criterion in detail - https://news.ycombinator.com/item?id=13143821 https://news.ycombinator.com/item?id=13143821 - Dec 2016 (8 comments) Kelly Criterion in detail - https://news.ycombinator.com/item?id=9401821 https://news.ycombinator.com/item?id=9401821 - April 2015 (1 comment) Kelly Criterion For Sport Betting - https://news.ycombinator.com/item?id=2504222 https://news.ycombinator.com/item?id=2504222 - May 2011 (23 comments) Ed Thorp: It is realistic to multiply your capital by 1000 using Kelly Criterion - https://news.ycombinator.com/item?id=1119318 https://news.ycombinator.com/item?id=1119318 - Feb 2010 (1 comment)
- hamiltonians 5y agosorta an obcession, huh
- bluetwo 5y agoTrying to avoid site amnesia, like Reddit has, I assume.
- lifeisstillgood 5y agoThink of it more as spaced repetitive learning for the HN crowd. I rremembered what Kelly Criterion was, that it was an ideal betting method, but could not remember the formula. Now I might remember both the next time it comes up. And maybe use it in my life! Ie should I play the lottery? 1/45m - (1-1/45m)/1M 0.00000002 - 0.000001 -0.00000098 so, no.
- 5y ago
- tunesmith 5y agoUsing the Kelly criterion is brutal though - the volatility is crazy.
- shapefrog 5y agoFractional kelly is the answer.
- ultra_nick 5y agoWould this work for calculating the odds of startup success? Runway of 500k, a 5% chance, 100:1 odds, so don't bet more than ~$22k on each bet/pivot.
- shakezula 5y agoI think it would be a good heuristic if you were able to be that precise about expenses, but I think most startups aren’t.
- paulgb 5y ago(Author here) I agree, but I think there is some applicability of the Kelly Criterion to startups (at least qualitativey) in explaining why an entrepreneur who is all-in on a company may be more inclined to take an exit than their investors, who have spread their bets.
- kqr 5y agoThe Kelly criterion is a U-shaped optimisation problem, so it is rather tolerant of imprecise estimations!
- rohitb91 5y agoIt is definitely one way to approach startup investment and why VC funds spread their bets out on so many startups. Almost all go to 0, some go 100x.
- cortesoft 5y agoThe only number you know there is the 500k… everything else is just a guess.
- demopathos 5y agoI took a similar approach where I defined "ruin" as losing more than half my money and set a target "probability of ruin" for 5%. With a specific set of parameters I think this ends up being very similar to the maximize the median approach.
- donquixote25 5y agoIf you knew you had a 60% chance of winning, wouldn't you want to bet a small amount to reduce variance? I guess on the down side that is A LOT of clicking. It is similar to how when playing a better team in basketball, you want to slow down the pace of the game to limit possessions.
- modeless 5y agoThe assumption is that you want the maximum return for any given number of bets. If you have unlimited bets then sure, maximizing the return per bet is not really important. But where can you find unlimited bets with positive expected return? There are other assumptions in there, like that there's no minimum bet and you can divide your bet to arbitrary precision. Always check your assumptions! I'm sure someone has modified the Kelly criterion for minimum bets and quantized bets, but a cursory search didn't turn it up immediately.
- cortesoft 5y agoThat assumption was never stated in the question. It didn’t say there was any cost to placing a bet, and I learned quite quickly that I could mash the bet button very fast… why would I take any risk at all when I can get a guaranteed win betting small and mashing the button?
- thehappypm 5y agoYep just bet a dollar every time. Over enough presses you’ll see the law of large numbers guarantee you getting something that looks like exponential growth.
- istjohn 5y agoIf you bet a dollar every time, you'll average $0.60 of growth every click no matter how much money you accumulate. That is linear growth, not exponential.
- thehappypm 5y ago
- chickenpotpie 5y agoKelly criterion is how I explain the importance of insurance. Insurance is a net loss and you’re better off on average not having it, but you’re betting your entire wealth on that working out.
- dcl 5y agoThis paper discusses that exact issue: Insurance makes wealth grow faster Ole Peters, Alexander Adamou https://arxiv.org/abs/1507.04655 https://arxiv.org/abs/1507.04655
- misja111 5y agoIt is not the same. The Kelly criterion is for when your capital is required to make you money; without capital you can't make bets so you'll be ruined forever. Without insurance you risk losing all your money, but if you have a day job, you'll eventually make it back again.
- kqr 5y agoIt is the same. It takes the same time for your savings to go from $1 k to $10 k as it does from $10 k to $100 k. In other words, if you have $10 k in savings, and then skip the insurance and go down to $1 k, you might end up retiring with $10 k. Whereas if you do take the insurance and your savings stay at roughly $10 k, you can retire with $100 k – given the exact same day job, etc. Your savings compound. That's the only requirement of the Kelly criterion, regardless of everything else.
- misja111 5y ago> Your savings compound. That's the only requirement of the Kelly criterion, regardless of everything else. The use case of the Kelly criterion is to determine the optimal size of your capital to put at risk, where the profit you're expecting to make is linearly related to that size. In your example you have savings which will bring in some percentage no matter what. And you have the choice of taking an insurance or not. Taking the insurance would cut your profit, but most likely not in the linear way that the Kelly criterion assumes when it calculates the optimal bet size.
- merek 5y agoHow do I cash out? Do you support ACH transfers?
- adiM 5y agoAn alternative derivation using dynamic programming for multi-round bets: https://adityam.github.io/stochastic-control/mdp/optimal-gambling/ https://adityam.github.io/stochastic-control/mdp/optimal-gam...
- OscarCunningham 5y agoThe dynamic programming and multi-round bets are a distraction there. Since it uses a logarithmic utility function the Kelly criterion is optimal for even a single bet.
- bumbledraven 5y agoNo, the two betting strategies diverge. The Kelly criterion assumes an unlimited steam of future bets, while the dynamic programming approach assumes a fixed number of remaining bets.
- OscarCunningham 5y agoThe logarithmic utility function makes the Kelly criterion optimal even when there is only one remaining bet. For example say you have $x, the probability of winning is p, and you bet $y. Then you want to maximise plog(x+y) + (1-p)log(x-y). Setting the derivative with respect to y equal to 0 yields p/(x+y) - (1-p)/(x-y) = 0. This rearranges to give y = (2p - 1)x, which is precisely the Kelly criterion.
- adiM 5y agoThis is precise the argument at the penultimate time-step in the dynamic programming solution of the multi-round case. The other interesting aspect is that the expected returns are logarithmic, i.e, with y = (2p -1) x p log(x+y) + (1-p) log(x-y) = log(x) + C where C is the Shannon capacity of the binary symmetric channel with cross-over probability p. By the same argument, the expected wealth after T rounds will be log(x) + T C So, in addition to the optimal strategy, we have also derived the rate of growth of wealth. This is also in tune with the motivation of Kelly's paper where he was showing a relationship between Shannon capacity and optimal gambling (without using a dynamic programming argument)
- throwaway_Aef8 5y agoI did in 23 flips always betting heads, but raising my bet if I lose twice in a row. Also the first couple times I all-in for a streak of 2.
- OscarCunningham 5y agoThe Kelly criterion is only optimal if you have logarithmic utility. People like to pretend that it somehow becomes optimal in other cases if you have multiple rounds, but when you do the calculations it just doesn't. Having the highest probability to win among the strategies that bet a constant amount is not a useful property, because you're not in a competition and don't have to bet a constant amount. In particular, in that Haghani and Dewey study they mention, the optimal strategy actually involved playing a lot safer that the Kelly bet. (Although the participants weren't told about the cap at the start of the game, so they would be reasonable to take more risk. Incidentally, it strikes me as slightly unethical to mislead the participants in a way that makes it more likely that they'll risk losing more money.)
- abm53 5y agoWhat about in the limit where the number of rounds tends to infinity?
- OscarCunningham 5y agoThe longer you have to reach the cap, the smaller your initial bet becomes, because you have more time to catch up. It doesn't converge to Kelly, it converges to 0.
- bluetwo 5y agoThe way it is used in movies and tv comes to mind.
- r-zip 5y agoCan you elaborate on these "other cases" with multiple rounds where logarithmic utility doesn't make sense?
- kqr 5y agoThis is a common misconception. The Kelly criterion maximises growth when the wealth compounds multiplicatively. It makes no assumption about the shape of utility. In other words, it's prescriptive, not descriptive. The Kelly criterion says that if you want to maximise growth, you should adopt log utility. If you don't have log utility, you won't maximise growth. (Though, as you point out, maximising growth isn't the only goal possible -- limiting drawdown is another useful goal. The linear combination of "no bet" and "full Kelly" is an optimal frontier that maximises growth given a particular limit to drawdown.)
- MaxMoney 5y ago
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- bottled_poe 5y agoI was betting 20% before reading the article and still went almost straight to zero. That’s gambling I guess.
- kqr 5y agoHere's a similar, but slightly more advanced, game I made for a lunch-and-learn session at work: https://static.loop54.com/ship-investor.html https://static.loop54.com/ship-investor.html Use the "link to this scenario" button if you want to compete with friends on the same seed.
- hnkain 5y agoI have a blog-post on the Kelly Criterion, and some (I think interesting) realistic examples of its application in decision-making: https://blog.paulhankin.net/kellycriterion/ https://blog.paulhankin.net/kellycriterion/
- phs318u 5y ago“When the probability of heads is 50%, there is no benefit from betting.” :)
- bluecalm 5y agoKelly criterion is the solution for optimal bet size with the assumption that utility is a log function. While it's often modelled as such we know it's not perfect (utility of money is bounded or even goes downwards after certain point while log is unbounded) and more importantly utility of money is different for different people. That means that, while being an important illustration of the concept of utility of money, Kelly is most likely not a good practical guideline for most people. Many professional gamblers for example feel that something closer to 1/2 Kelly or 1/4 Kelly is better but again it all depends on your personal situation and what money means to you.
- kqr 5y agoAnother comment already posted this misconception. The Kelly criterion assumes only two things: compounding and that the objective is maximised growth. It makes no assumptions about utility functions. There are several good reasons to use fractional Kelly strategies, but shapes of utility functions has nothing to do with it.
- bluecalm 5y agoHow do you define maximized growth and why is it important for anything if utility is not log? Shapes of utility function have everything to do with determining your bet sizes. Can you provide one reason to use Kelly that is not connected to the shape of the utility function? Not to escalate but calling a very well known fact about Kelly a misconception doesn't make for a good start. I mean it's math you can verify or open Wikipedia and read the very first paragraph there.
- Gravityloss 5y agoI think it misses the explanation that you can play multiple times...
- anonymoushn 5y agooptimizing your log wealth at a single step seems pretty good, even if you don't get multiple plays of the same game.