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].
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.
I don't have the source at hand but by looking at what data we have from successful investors, many of them have returns that statistically seem like what you'd expect from E log X strategies.
In fact, it's not even a point of debate. If you target growth, you are using the Kelly criterion whether you know it or not. It's just the name for the thing you do when you optimise for growth.
1. Investment returns are multiplicative and should be looked at as a geometric series. To optimize the portfolio, optimize for geometric mean not arithmetic mean.
2. To optimize the geometric mean of some specific games, apply some specific mathematical rules that Kelly derived.
Then 2nd part is not applicable to general market investing. The 1st part is.
I think it’s more a difference in what people think the term “Kelly criterion” means, which is somewhat fair. The concept of optimizing for geometric mean came before Kelly, as well as the math showing that optimizing for log utility is a way to do that.
Kelly can work if you can properly model your uncertainty over the probability of outcomes and take this into account. You can either do some sort of Bayesian averaging over your posterior belief of the risk, or you can use the pessimistic side of the confidence interval of the actual risk probability.
The key understanding of the Kelly Criterion is that you need to scale your investment size with risk; riskier investments require smaller investments. How you estimate risk and how that informs your investments is rather fluid, but understanding it is the cornerstone of professional investing.
If you don't understand that, then you are going to go eventually go bust.
From the second article[2] that follows: “This makes sense because the problem with the Kelly Formula for portfolio management is that it looks at each bet individually” i.e. the Kelly Criterion bets your whole portfolio on a single position. I presume any strategy that has multiple positions (a portfolio) cannot use the Kelly Criterion by definition.
From what I can tell “Kelly Criterion” originally applied to one bankroll and a single repeated bet.
It seems the choosing the optimal strategy for allocating a portfolio to maximise growth is often called “Kelly style”, “Kelly strategies”, “Kelly methods”, and also “Kelly criterion” by some people (which is why I was confused).
The details of an optimal strategy are completely different depending upon your assumptions (how reallocation is performed as new information is received, accounting for error in predicted outcomes, blah blah blah) so there cannot be a single definition for the Kelly Criterion for a portfolio, instead there are a variety of strategies (each with different assumptions and constraints).
For example the “many assets” model you refer to looks like it models a single market correlation (alpha), and not the multiple correlations within a real market.
Disclaimer: I am not an investment professional, but a small amount of software experience with hedge fund NAV calculations.
A tiny firm managing under $100m uses Kelly...and...
The post you replied to is right. The fundamental principle of Kelly is that you know your edge, in the markets that is mostly untrue. Funds will volatility-weight their portfolio but this isn't the same as Kelly in practice. Most fund managers will also weight their portfolio towards their "best" position but that is not necessarily based on return. Indeed, picking high return assets is only half the battle.
I also bet a lot, so I am familiar with Kelly. It is totally unusable in finance, no-one uses it in finance, and retail investors have an obsession with it.
In particular, if you Kelly-weight a value portfolio (which the firm linked to in your post is) then you are setting cash on fire. And if you Kelly-weight a long/short portfolio (again, the firm linked to appears to be doing this) then you are setting cash on fire. It is important to understand how a tool works at a practical level.
“ The fundamental principle of Kelly is that you know your edge, in the markets that is mostly untrue.” Most every professional investor I’ve met attempts to quantify the risk-reward of each trade and size accordingly. I agree that naieve investors engage in false precision eg assuming backtest sharpe for position sizing, or ignoring correlated risks. That makes them size too aggressively. But that doesn’t mean pros don’t try their best to estimate risk reward and size accordingly. Indeed many of the best traders ever (Buffett, Soros) put on massive bets when the risk reward were highly in their favor.
When I place a bet, I can estimate my edge because the outcome is binary. When the outcome is continuous, it is far more tricky. It is like saying a kid who learns to ride his trike is ready for MotoGP...they are just totally different.
And yes Soros put on big bets, but what you are missing with Soros is the fact that his hit rate was still 30%. Most of the stuff he did didn't work out, macro is largely bets on skewness not returns. Buffett had a higher hit rate but trying to suggest someone optimise a strategy based on what literally the best investor of all time did is...not smart. Even if you were better than Buffett, you might not be lucky.
The reason why Kelly doesn't work with value investing in particular is because your returns are largely random, you know that your portfolio has an edge but you don't usually know which position is going to revalue.
The reason why Kelly doesn't work with long-short in particular is because you aren't only betting on return but correlation. Anyone who runs Kelly will eventually get a correlation spike and blow up (this is also roughly true of macro, again why Soros isn't a good example, he largely bet on skewness).
I was a "pro" so I am also aware of what most pros do. Again, investors don't only look at return, they have to look at correlation, volatility (note that if you are betting on sports, you don't have to worry about things like correlation).
It sounds like we might mostly agree in substance and a debate over semantics isn’t productive.
I agree that “the inputs to the Kelly formula are imprecise and therefore we should not mindlessly implement its recommendations.”
I agree that retail investors should not model their 401k allocations like Soros and Buffett.
Having run a factor neutral long short book I’m extremely familiar with the role of correlation and volatility in portfolio management and position sizing. As others have noted, there are extensions of Kelly (and related portfolio construction formulas) that account for correlations.
I disagree that risk-reward (broadly defined) shouldn’t be the primary bet sizing metric. I think many investors ignore risk reward calculations in their sizing and they would be better off if they paid attention to it. Many of the smartest investors I know have their entire sizing strategy based on risk reward.
To suggest that active investors should ignore risk reward / odds / whatever you want to call it, is wrong, in my opinion.
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.
Ed Thorp AND Claude Shannon! One of the best nontechnical finance books ever written.
In practice though, positioning doesn’t work like that in modern times because a lot of your entries and exits happen around liquidity events. However, it is very pertinent for biotech stocks and special situations where you are dealing with discrete outcomes.
The problem with anything that isn't 1/n is the large estimator variance of the mean of asset returns. There's such little signal there that Markowitz et al invariably fit to mostly noise, which reduces diversification, increases transaction costs, among other problems.
A similar phenomenon occurs in ensemble methods in statistics. It's often better to equal weight many estimates than try to fit weights to them, since that fitting process introduces lots of variance.
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.
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.
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.
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 am confused so hope you will clarify. I thought the article argues that Markowitz mean variance has problems and 1/n is a reasonable estimator. You seem to be arguing for the opposite? Or perhaps you mean 1/n vs. Kelly but that article does not talk about Kelly.
Sorry, I didn't mean to argue any point really, just expose folks to Gerd Gigerenzer's work, as it seems relevant to this topic. He makes the arguments much more strongly than I ever could.
Any confusion or inconsistency I'm presenting is my fault, and I apologize!
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.
> 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.
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.
I kinda feel sorry for psychology and related social science fields. They have an immense hurdle to clear when designing experiments. Both protocol and statistical analysis.
50 or 100 years ago, a study participant might have gone in oblivious to the possibility of subterfuge. Totally unaware that the "taste test" they're participating in for the "marketing majors" was really a study on how political party affiliation affects choices between lemon cake and chocolate chip cookies. Or whatever.
But I have a feeling that college students are much more aware of how these things go today. The experiment is tainted from the get-go by all the participants looking for the "real" data being collected.
I know for damn sure that if I'm recruited for an experiment where I'm taking some sort of test, when a "fellow student" suggests we cheat, that this is an honesty test. Or maybe if the clock runs out before I'm done, I'm being watched for how I handle stress. Wait, is it kind of cold in here? Ah, they must be gauging performance as a function of comfort.
And of course, study participants are way too often 18-24 year olds who happen to go to college. Such a tiny slice of the general population.
So I could see myself placing bets on the "40%" outcome. I wonder if the coordinators straight up told the participants, "Look, we're really testing your betting decisions. This coin really has a 60/40 bias. This isn't a ruse. Please treat this info as true; we're not doing deception testing here" if that would eliminate the kind of second-guessing we're talking about. (I guess we need to study that:) But if that became a norm, then it would further highlight the deceptive tests when that statement is missing.
And of course, study participants are way too often 18-24 year olds who happen to go to college. Such a tiny slice of the general population.
It gets worse. Typically 18-24 year olds who happen to go to the same college as the researcher is working at. So, for example, if this is a large state school then it is a population selected for having SAT scores in a range. Namely above the cutoff to get into the school, but below the cutoff for more desirable schools.
Now suppose that you're doing ability testing. You should expect that any pair of unrelated abilities that help you on SATs will be inversely correlated, because being good at the one thing but landing in that range means you have to be worse at something else. And sometimes that will be the other thing you're looking at.
Several years ago I remember running into a bunch of popular science articles that I found dubious. I tracked down the paper and decided that their analysis suffered from exactly that flaw.
"I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check."
Only if you were clueless, or perhaps if the experimenter said "if you bet on heads it has a 60% chance of winning". Being unstated what would happen if you bet on tails, you might forget that the coin has know knowledge of how you bet, thus making it impossible for there to be any different outcome than a 60% chance of loss by betting on tails.
Even worse, the experimenters didn't actually provide real coins. They just sent around links to a website that they said was simulating a biased coin. Participants presumably had no actual way to know whether the flips were actually 60% biased towards heads, whether the results were truly independent from one flip to the next, or even whether their bet might impact the outcome.
All those sources of uncertainty of the actual probabilities are, while in some cases not typical of a real coin (although uncertainty about actual bias one has been informed of certainly is), fairly typical all of real-world situations in which people face, so I’m not at all certain that that invalidates any application of the results to real-world situations.
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.
If you don't flip the coin within any reasonable definition of flip, sure.
But if you flip a coin and it turns about N times, you can't make the sum (over all k) of the probability of N+2k turns substantially more likely than thr sum of probability of N+2k+1 turns.
If the mat that my coins are landing on is a strong magnet, I know I can make every single flip land heads. Even when the coin would otherwise land tails, it will instantly flip to align with the strong magnet beneath.
So what if I dial the magnetic field back just a bit? So that only when the coin is oriented flat as it lands will it maintain that orientation in spite of the opposing magnetic forces. But if the coin's orientation is near vertical, then the forces are directed to nudge it "headwards" instead of "tailwards".
Your math applies to weighting the coin. It makes sense in that context. I'm talking about a system of magnetic coin and matched magnetic landing pad.
If you bend a coin, one side has larger area than the other and will prefer to land on that side accordingly. The turn-based argument depends on the fact that both sides of the coin are the same size, which is not true if you bend the coin.
Sometimes an experiment to see if you can go five minutes without eating the marshmallow is just an experiment to see if you can go five minutes without eating the marshmallow, and not a trick to see what happens if they give you three marshmallows after eating the first one.
Yes, this is what every very smart person who underperforms or behaves illogically in a study says. Well, actually, I didn't choose wrong, I was testing the experiment. I chose to eat the marshmallow because I wanted to force them to reveal what would happen next, and then they told me the experiment was over, exactly as I predicted. I win again.
Here's a related yet totally different take: your comment demonstrates flawlessly the reason why sufficiently intelligent people must be weeded out of these experiments (or at least the results). And that in turn helps explain why we end up with people who bet tails.
(Note that the thrill of gambling is another explanation; I'm not claiming "those people are less intelligent, it's the only explanation" but rather "a bias against a certain kind of intelligence could lead to an increase in the observed outcome".)
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?
"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."
> 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.
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.
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.
If they said P(heads) is 60% and you get 4 tails in a row, you also might think the people running the experiment lied to you, especially if it happens near the beginning. But there’s a 13% chance in any sequence of four tosses.
True; my point was that the person falling for the gambler's fallacy was wrong, but in a sense, so were all the people explaining the gambler's fallacy.
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.
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?
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.
Let's say you're throwing a piece of paper into the trash can from a short distance. Suppose you can successfully throw the paper in the trash 100% of the time. You move your hand. Your hand moves the paper. Gravity pulls the paper down. It collides with the trashcan. It's just a bunch of physical objects exerting forces upon one another.
Now, suppose you keep throwing, but somebody has opened a window, so now there's an occasionally gust of wind, which moves the paper in unexpected ways while the paper is in the air. Now you no longer hit 100% of your throws. Sometimes the paper lands in the trashcan, sometimes you miss. Regardless, the paper is still only affected by physical forces: your hand, gravity, wind.
Now, suppose you've been really unlucky the past few throws: you have missed 5 throws in a row because of the darn wind. Does it make you more likely to win the next throw, because you are "due" a win? Of course not, because the wind doesn't know or care about your paper throwing hobby. The wind does what it does, regardless of how many of your throws landed in the trashcan. If anything, missing 5 throws in a row makes it _less_ likely to land the next shot, because it may indicate conditions unfavorable to throwing (strong wind, loss of confidence, etc.)
Now, the coin flipping experiment with the weighted coin obeys the same physical laws as the paper tossing experiment. It's just a physical object that's affected by forces from your hand, gravity, air, etc. If you throw 6 heads in a row, there's no magic that somehow alters the coin's path in the air on the 7th toss to make it come down tails. The universe doesn't care about our little games.
There are a few other nice answers here, but I think it's important to attack it from as many angles as possible.
The intuition that you're going for is that if the true rate is 60% heads and you've seen more than that then to hit 60% odds you _must_ have some extra tails _eventually_. Interestingly, that isn't actually required to make the odds work out to 60% eventually. I'll try for an intuitive explanation:
Say you've gotten 10 heads in a row but that the coin really only has a 60% chance of coming up heads.
- After 1000 extra flips you'll have 610 heads and 400 tails total on average for a 60.4% chance of heads so far.
- After 10k extra flips you'll have 6010 heads and 4000 tails for a 60.04% chance of heads so far.
- After 1M extra flips you'll have 600010 heads and 400k tails for a 60.0004% chance of heads so far.
Notice how the average percentage of heads is getting closer and closer to 60% even though the extra flips don't have _any_ bias toward tails. A temporary bias toward tails would _also_ suffice, and in much less time (some games like WoW use this for their loot tables I think), but it isn't necessary, and in the example of independent coin flips it does not happen.
> 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?
Nope.
> 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.
Lots of people have explained in terms of independence, which is correct. Another way of looking at it (definitely not more correct, but maybe more compatible with the “a series should eventually match the quoted probability” thinking) is in terms of infinity:
If you are expecting 60% of results to be heads, you expect that to hold over an infinite series of flips.
If you see any finite number of heads in a row, the probability for each of the remaining flips in the infinite series to get the total to 60% is…still 60%.
No finite series of results can change the probabilities necessary to get the infinite series to turn out as expected.
You’re talking about reversion to the mean, which is a phenomenon that’s related to the law of large numbers.
Law of Large Numbers says that, over an arbitrarily large random sampling size, you will eventually end up with a sample that perfectly fits the probability distribution.
But the probability of each individual sample is random. This means that, if each sample is randomly-selected and independent, your history of N samples does not affect your N+1th sample.
The regression to mean curve is only predictable in the big picture, each bump is 50/50 (or 60/40 in this case).
It's not that you're expecting 60% of your flips to be heads, but the coin has a 60% probability of being heads.
The former implies that previous flips have an effect on future flips. Or that, if you land on heads 6 times in a row, then the probability of it landing on tails goes up.
How would a coin that's weighted to increase the odds of it landing on heads, somehow start landing on tails more frequently?
If you flip a normal coin and it lands on heads 10 times, you still have a 50% chance of getting heads the 11th time. The odds of it landing on heads 10 times in a row in the first place is vanishingly small (0.5^10 or 0.097%). But if it Does, the 11th flip still has a 50% chance. The first 10 flips don't affect the 11th. Physically, how Would the first 10 flips affect the 11th?
This is all assuming that the coin flips aren't somehow magically linked or casually dependent on each other. The math changes if the previous coin flip could somehow affect the next one. But in a situation where every single roll of the dice is purely independent, then by definition (Because they are Independent ) a previous roll doesn't have an impact on future rolls
You expect 60% of your flips to be heads at the outset. Let's say you flipped it a bunch and you're running at some rate.
How could the past flips of the coin possibly influence the flips you get in the future? The coin hasn't changed, the surrounding area hasn't changed, why would the coin suddenly have a different chance of turning up heads on your next flip? There's no probability god that mucks with random chance to make sure 'runs' are balanced overall. Every coin flip is independent, which means all the coin flips are also independent of the past coin flips.
If you've "had 60%", that means you've had an unlikely run of heads. Let's say the last 6 flips were 5 heads and a tail, a slightly unlikely outcome (3 in 16, I think). What physical force is acting on the coin to make it less likely to be heads, in the future? Why wouldn't it still have a 60% chance of coming up heads on the next flip?
It's always exactly 60%, no matter how many heads you have already had. That is pretty much by definition, since the problem states that the chances of heads are 60%.
In fact, in the real world getting an unlikely string of heads (or tails, or sixes, or whatever) outside of a casino setting probably means that the coin/dice/whatever are unfairly loaded and you should adjust your expectation for the next coin toss even further towards heads.
I upvoted you because, while you aren't correct in your assessment, I think this is a good "teachable moment". Human intuition about statistics is really, really bad.
I think people who have a better-than-average understanding of statistics forget how bad their intuition is. I suspect it leads to a lot of incorrect assumptions about what a "rational" behavior for someone working from only their statistical intuition would be.
The fact that your previous 6 flips were all heads was an unlikely outcome, but the coin has no recollection of what just happened and doesn't "care" about the past when you flip it again. The maths term for this is to say that each coin toss is "independent." I would not bet that you'd get another 6 heads in a row, but I would bet that the next coin flip will be heads.
I understand, once 60% chance is established - then - that's what it is.
However, until such probability is established, if I see heads in a row - my intuition would tell me that the physics is skewed towards heads. I don't think that it would be unreasonable to think that in such circumstances until one gets a larger sample of throws.
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.
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.
If you see 60 heads in a row from a coin you’ve been informed is biased to produce heads on average 60% of the time, you'd need a pretty strong bases for trust in your information to not conclude that the most likely explanation is that the bias was underreported. Yes, its possible with the reported bias (or even if the bias was overreported), but that's not the most likely conclusion absent some pretty firm external evidence of the accuracy of the bias estimate you were provided with.
> 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.
People rolling dice aren’t, even when they try to be, perfect randomizers, and with a maximally favorable result and an action which demonstrably repeats it, there’s a strong incentive to repeat the action as accurately as possible rather than even trying to be a perfect randomizer.
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.
There will always be improbable sequences; with a fair coin, every possible sequence of length N is equally improbable, after all; if you flip a fair coin 64 times, the sequence is guaranteed to be a 1 in 2^64 event.
OTOH, the probability of some other explanation besides a fair coin isn’t consistent among all other possible sequences, so what the actual result does to your estimate of the likelihood of a fair coin depends on the actual sequence, and your basis for believing the coin was fair going in.
Things are only slightly different with, say, a coin you’ve been told has a 60% bias.
EDIT: For instance, if there is a 1:1,000,000 chance that you would be given an underestimate of bias and a 1:1,000,000,000 chance of the outcome you actually receive being true if the coin had only the bias you were informed of, its a lot more likely that you were lied to than that you just got an unusually consistent set of results.
If you had a camera pointing at a thousand coins that flipped once every second since the beginning of the universe, you still would probably not see 60 heads in a row.
If you had flipped one coin 4.35e17 times and never saw 60 heads in a row, on a biased coin, I'd be rather surprised. (took 13.8 billion years as the age of the universe). Do that 1000 more times and still don't see 60 heads in a row it would be even more surprising.
It doesn't change the point of my original comment, regardless of the improbability of 60 heads in a row, you aren't "due" 40 tails in a row because the events are independent. That's all I was getting at before you took us on a weird tangent.
I did some miscalculations. 2^60 is 1.15E18. So you couldn't do a thousand times per second. But it probably wouldn't happen at 1 per second.
The original point of your comment is correct, at least from a probability standpoint. You don't get "owed" tails. I guess my hint was that there are sometimes other factors at play that mean the theory goes out the window. Like if someone shuffles a deck in front of you and it ends up new deck order, it's more likely they're a magician than lucky.
I used to drive a fellow RPG-er crazy with this. Whenever I would roll a few times low numbers, I would say "Alright, next time will be high, that is obvious, it's pure statistics!". At first he would object, but still even after he knew that I knew, that statement would still drive him mad.
I remember one time when I rolled really low numbers on a D20, and then there was this really important roll, where I had to get a 20. I confidently said "No problem, I rolled a few really low numbers in a row, so this is definitely going to be a 20, it's pure statistics". Also throwing some calculation in there: "I rolled a 2 and a 1, so in 3 rolls I should get a total of 30 on average, so that means I actually still need 27 to reach the average. That results in more than 100% chance of rolling a 20 right now". And then I actually rolled a 20, was able to keep my cool and a straight face "see, it's just theory". Pure gold! LOL :D
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?
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.
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.
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.
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%.
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.
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.
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].
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.
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.
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.
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
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.
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...
You may be interested to know that Kelly's work was instrumental in a company called Axcom in the 60s. Elwyn Berlekamp, previously an assistant to Kelly at Bell Labs, implemented Kelly et al's work in early financial trading at Axcom, which was later turned into the Medallion Fund at Renaissance Technologies. Wikipedia [1] has some info on this, but I also highly recommend "The Man Who Solved The Market" (Zuckerman, 2019) for more history.
You may be interested to know that Ed Thorps - Princeton Newport Partners/ the Santa Fe school work lives on at an even better performing fund called TGS Management based in Irvine.
> I used to use the principles of kelly betting back when I designed systematic HFT strategies.
possibly a dumb question, but how did this work exactly? the kelly criterion assumes you know the amount by which the coin is weighted, how would you know the equivalent for the stock market in the very near term?
Kelly goes beyond binary outcomes. The underlying principle is the same, though: you maximise expected logarithmic wealth.
To do that you need the joint distribution of outcomes (what are the possible future scenarios and how likely are they?) Estimating this well is the trick to successful application of the Kelly criterion.
You wouldn't unless you could vary your exposure to such a sequential bet.
Suppose you can though. For simplicity, suppose you can expose yourself to 0.4U(-1, 1.1), 40U(-1, 1.1), or any other fractional amount F U(-1, 1.1) you might like. Kelly is a technique for choosing F (maybe you had some other idea in mind like that you have to buy into a bet on U(0, 2.1) -- if so, that's nearly equivalent other than putting bounds on F -- the idea of maximizing expected logarithm will carry through to other bet structures).
Going through the motions, suppose you're starting with a bankroll B then you want to choose some ratio F=rB maximizing the expected logarithm of the bet. The distribution of your outcome is another uniform distribution U(B-rB, B+1.1rB), and you want to choose r maximizing the expected logarithm of that distribution. The details of that are probably beyond the scope of a HN comment, but you wind up with r approximately equal to 0.13624.
If you'd like you could plot the result of many instances of 100 such sequential bets with r varying. You'll find that those with r around 0.13624 will usually be much larger than for other choices of r.
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.
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.
the binary outcome formulation you see everywhere is just "real" kelly boiled down. the real thing, which is contained fully in the first paragraph ("The Kelly bet size is found by maximizing the expected value of the logarithm of wealth"), has no such restrictions.
How do you maximize the E(log(wealth)) when applied to a HFT strategy? In such a strategy we have N sequential bets, each bet has a roughly normal distribution outcome with mean just above zero.
The example on Wikipedia supposes we are investing in a geometric Brownian motion and a risk free asset.
in the U(-1.0, 1.1) case you mentioned, kelly says not to bet.
optimize the value of the bet size over the expected value of the log of bankroll + betsize*outcome. you can do that for any probability distribution of outcomes.
if you can't write that in 5 minutes, then i already did half your homework for you.
> each bet has a roughly normal distribution outcome
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.
It doesn't seem obvious that this is a good strategy for personal wealth management because besides maximizing expected wealth, there's another very important criterion: minimizing probability of going broke. I only get to play one game, after all. Obviously you can't go entirely broke if you always bet a fraction of your portfolio, but are there results of how these strategies compare in, say, the probability of dipping below 10%, or 1%, of the starting value?
I can't tell you about the 1% version, but when it dipped to 15%, it was a strange feeling that I made a bad decision with the thinking that I'm making a great decision (or more trying not to think about it and trust the decision that I made earlier). It's a mental game at that point that you have to wait through. At least with investing it's just about waiting through those periods, being a CEO of a company and making decisions in that state would have been much harder.
I love the quote “the money in investing isn’t in the buying and selling but in the waiting”.
I know for me I had the moment where things had gone down to roughly 15% and I questioned my decision making. Learning to wait through those periods is super important. Years ago I made the repeated mistakes of not waiting through those periods and missed out on log gains in favor of linear gains.
Agree that it’s easier as an investor and not a CEO to manage that experience day to day.
For me most of my BTC is in a multisig contract between 3 physical trezors in another continent, so actually I am not able to change my decision just because the price dips. Still, as I'm planning to change my portfolio, I'm afraid more of the execution risk than the volatility.
One thing I can tell you is that banks hate people executing the Kelly strategy, as they expect wealth of people to be predictable so that they can issue loans against it to other people.
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.
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.
E log X strategies are known for Being very volatile.
However, there are two things that take the scariness out of estimating probabilities for me:
- You're often maximising something that looks like a quadratic function. This means you're aiming at a plateau more than a peak: if you make small errors in either direction it doesn't affect growth that much.
- You always have the safe option of underestimating. The E log X strategy forms an "efficient frontier" (to borrow terminology from MPT) of linear combinations from the risk-free rate to the full Kelly bet (and even past it into leveraged Kelly strategies.) You can always mix in more of the risk-free rate and get lower growth but at higher safety.
These two properties makes the Kelly criterion very forgiving to estimation. (In contrast to MPT style mean--variance estimations, and other less principled strategies.)
I find both mean variance and Kelly to be very poor in practice due to the dependence on the expected return term. Like, if I knew that, I wouldn't be wasting my time with all this math! (half joking)
One simple example is buying 2X S&P index ETF instead of 1x. There was a great article about the Kelly optimal S&P allocation, and with all the fees included it's about 2x. Of course there's increased execution risk for the ETF itself, which needs to be estimated.
Another thing where I may look stupid from outside is that I started to take some loan against my BTC and use that to finance my lifestyle, as currently (under $100k BTC price) my estimate of the Kelly optimal BTC allocation is more than 1. This is of course a personal estimate, I don't suggest other people to do the same thing, and again there's a lot of execution risk, so I do this only with a part of my portfolio.
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.
For people who expect to have stable earnings with the current interest rates being below the real inflation the Kelly optimal strategy is to be in debt use it to finance investments (of course this works only if the future earnings are really stable).
As a business example startups are starting to apply for loans against their future subscription earnings to reinvest in their companies. Debt against your salary is the personal version of the same strategy.
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.
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.)
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.
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?
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.
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.
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.
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.
You are right, but with execution risk / slippage it gets closer to 2x (2x and 3x are both close to 2.5x, but 2x has been performing better in the past).
Yes it can. Kelly can be applied to determine optimal leverage ratios. Assuming a risk free rate of zero, that formula is expected return divided by expected variance.
so 10% expected return and 10% expected volatility, optimal Kelly is 10x leverage.
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.
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).
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.
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 :)]
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/
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".
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.
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.
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.
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.
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.
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.
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!
Please stop doing this (with any account). HN threads are supposed to be conversations. Copying a mass of content from someplace else is not conversation.
[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...