The Kelly Criterion

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The Kelly Criterion provides the optimal strategy when betting on random outcomes with known probabilities.
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TIMESTAMPS
0:00 Intro/The Game
1:16 Growth Rates
3:43 People Play Terribly
4:46 The Best Strategy Possible
7:10 The Kelly Formula Is Dangerous
CODE
github.com/Duane321/mutual_in...
SOURCES
I first came across the Kelly Criterion in [1], which features it as a powerful example of the relevance and usefulness of information theory. In chapter 6, you'll find an exact and more general discussion of this idea.
Wikipedia, [2], was also useful, as it is frequently. That's where I learned about [3], which was an interesting case study about how bad we are at this game. As an aside, I actually ran the experiment on a friend and didn't find similar results - I guess she was pretty smart! She bet about 3-5 dollars on each flip, with the wagers moving to the larger side of this range as the wealth increased.
[4] is the original paper. It's a nice explanation, but relies on some background knowledge of information theory. If you're curious about the connection to information theory, this is the place to look.
--------------------------------
[1] T. M. Cover and J. A. Thomas. Elements of Information Theory. 2nd edition. John Wiley, 2006.
[2] Kelly Criterion, Wikipedia, en.wikipedia.org/wiki/Kelly_c...
[3] V. Haghani, R Dewey, "Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin", Journal of Portfolio Management, 2016
[4] J. L. Kelly Jr, "A New Interpretation of Information Rate", Bell System Technical Journal, 1956

[@robmarks6800]

Kelly smoking six packs a day and died at 41 really knew the ins and outs of risk lmao

[@Boringpenguin]

P(death) = 1💀

[@michaelmellinger2324]

Top comment by someone who understands little about history. In the year 1965 when Kelly died, the first warning on cigarettes was required by law. It simply stated: “Caution: Cigarette Smoking May Be Hazardous to Your Health”. You could smoke on airlines into the 1980’s and 1990’s. Office buildings, …

[@RenePatrique]

People have misleadingly been brainwashed by the tabacco industry propaganda. - Kelly was a victim of exactly that. - Blame the lobbyists that a genius mind killed himself. - Watch the movie "Thank you for smoking" for deeper insights.

[@zebulon220]

​@@michaelmellinger2324 you have to admit as a punchline its pretty funny.

[@michaelmellinger2324]

@@zebulon220 What’s so funny about millions of people dying from cigarettes? 60 years later people are still dying from it. Almost half of Americans smoked in 1960.

[@MaxPicAxe]

"travel evenly through logspace" wow that was an amazing way of explaining it.

[@Mutual_Information]

ha yea I didn't even realize that until a made the log-plot. Very on-the-fly comment

[@antopolskiy]

For the last part: I guess a more sensible thing would be to collect data continuously and update the probabilities after each observation and recalculate the percentage. Now I wonder what would be the results of that :)

[@jamesfulford]

Hey who let Bayes in here?

[@coolshoos]

BRING IN THE MULTI ARMED BANDITS!!!

[@imreolah6077]

The answer no, not sensible, it is in the video ( ~7.10)

[@greg.ocallaghan]

@@imreolah6077 in the video the probability and odds data is only gathered for the first 10 flips, not recalculated after each flip from then onwards as @antopolskiy suggests

[@VincentKun]

Maybe one can change the probabilities of the outcome each time in real scenarios

[@Sam-qi2gw]

just fantastic video, thanks for all the effort - currently reading fortune's formula and fascinated !

[@threepointonefour607]

Great to see you back! Looking forward to your new videos, all your content is very interesting and well-presented

[@Mutual_Information]

I've been working on something pretty big, hence no recent uploads. But I'm not going anywhere :)

[@justynsweeting]

This is an extremely high quality vid that is well explained, you have a new sub!

[@PowerhouseCell]

I can't believe I didn't come across this channel until now! You make amazing content- as a creator myself, I understand how much work must go into each of these videos. Liked and subscribed- keep it up!!

[@Mutual_Information]

Thank you brother! Yea, it's a lot of work - hence so few videos. Glad to meet someone who understands the pain. Just subbed to your channel. We have a long journey in front of us 😅

[@pedrocolangelo5844]

This video is incredible. I'm really looking forward in watching other videos of yours

[@fhdusjxjdjxjjdj]

Great video, well explained!

[@Elemblue2]

You held my attention even though I didnt mean to watch this entire video. Your content is to the point, interesting, well presented, and the code thing is 10/10.

[@Mutual_Information]

Aw thank you my friend - It's a work in progress :)

[@Arsenal2045]

one of the best videos ive seen explaining the kelly criterion

[@EzBz982]

Great video. Subscribed!

[@jb_makesgames2264]

Good video - Kelly is like the "great white whale" of portfolio management - great to read about but hard to implement. Your description was very good. The harder part is how if at all can you make it work. Several authors such as Vince have proposed an optimal F approach. I've also seen instances of applying VaR techniques to limit downside risk of using full Kelly.

[@Mutual_Information]

Thanks JB - yea I noticed that connecting this technique to real world investing strategies involves fairly heavy hole-patching..

[@markluka7356]

I Love finding undiscovered gems on youtube, like this channel.

[@angusadams]

Great vid! If you continue making content like this the channel will grow like crazy 🙏🙏🙏

[@Mutual_Information]

Lol I will keep my expectations nice and chill for the time being

[@FightingTorque411]

Recycling 37.5% of content into each new video is the surest way of achieving this

[@goesi1]

Kelly can be used in reality card counting.

[@yunusd8167]

Really nice video, well explained, good work... Thank you,

[@Mutual_Information]

Glad it was helpful!

[@harry8]

This content is amazing. I subscribed before the end of the video!

[@Mutual_Information]

Thank you Harry - I'll do my best to keep this stuff coming

[@milostean8615]

Brilliant video. Deserves a lot more views!

[@Mutual_Information]

Thanks! Help me get there! Tell Everyone!! Just kidding.. just tell like a few close friends :)

[@milostean8615]

@@Mutual_Information 🤝

[@robertsmentkowski312]

Great insights! When I was trading bonds, our economist gave his estimates of economic numbers AND the likelihood of each to occur. From there, the desk (all traders) would estimate how far and the direction the market would move. As a market maker for a primary dealer, I realized I only had to be right about +- half the time. The professional risk manager knows when to cut losses - unlike retail investors - and when to let profits run. Nearly anyone can model risk. Deciding what to do is risk management.

[@Mutual_Information]

That sounds like a quality separation. I worked at a hedge fund for a few years, but never had the luxury of such a separation of work. We had to both model expected returns/risk *and also* make investment decisions with them. That's not too unusual, but it's gut wrenching and makes you question the model constantly.

[@PeteZhu-zw9do]

@@Mutual_Information I have been using the Kelly formula to calculate short-term options trading on Bitcoin, and I am doing very well. Thanks to the blogger for sharing the formula video, which is very helpful to me.

[@JC-db6oy]

Really underrated channel. Please keep posting videos!

[@Mutual_Information]

No plans to stop :)

[@MathVisualProofs]

This is an excellent one. Really nicely done. I think I missed it the first time around.

[@Mutual_Information]

Thanks Micro - that's at least one good reason for the re-upload ;)

[@murilopalomosebilla2999]

I bet that this channel will grow to 1M+ subscribers! Nice work!!

[@Mutual_Information]

Thank you very much! That would be absolutely wild but I have no expectations of that lol

[@BreadAndGatorade]

-_-

[@williamlethco7208]

Thanks for making this video. The topic of risk management and knowing how much to risk per bet is complicated.

[@Mutual_Information]

Indeed it is.. hopefully no one thinks that's not the case after this vid.

[@MrSilviut]

What an amazing video and channel!!!!

[@Mutual_Information]

Thank you Silviu! Stick around, it'll get better :)

[@keauna1893]

Hey DJ your videos are like consistently well paced while also being informative. It's clear that you're putting in the time before uploading, so keep up the good work! Also, maybe you should consider making a discord community for your channel. It might be one of the easiest ways to get a community going while also marketing your channel at no cost.

[@Mutual_Information]

Hey Keaun, I'm happy you're noticing the effort - it's not easy :) Discord is something I actually haven't thought about yet. I figured I wasn't at the size where people will show up. But maybe that's a bad assumption. I'm also not sure what my personal time commitment would have to be. I'm already at capacity between work and these vids, so I'd have to make room for it. But if it's the wise investment you point out, I should probably do that. I'll start thinking about it - thanks for the suggestion

[@keauna1893]

@@Mutual_Information No problem :D And yeah just keep up the great work and if you do end up deciding to make a server I'll be more than happy to join at that time.

[@gaming4fun419]

@@Mutual_Information Make it a place to discuss the upcoming video for input and help with resources. Could be a great way to be able to get to know what questions to answer in the video before it is made.

[@vinzanity68]

Interesting. Sounds applicable to farming or any business venture.

[@abraarz2971]

Thank you!

[@escapefelicity2913]

well done!

[@fitybux4664]

Holy crap. AI was just telling me about this strategy. 😲 Great explainer video!

[@denesdolor975]

been looking for this for a very long time.....

[@jerryware1970]

Fortune’s Formula…great book on the subject

[@southalpha9569]

Wow that was totally fun man.

[@Ehoenix]

Loved the easy-to-follow explanation and the fact about Kelly's lifestyle choices...Thanks, DJ!

[@Mutual_Information]

Thank you Rain!

[@omegasigma4500]

I remember Ed Thorp once recommended to use something like a quarter of the Kelly fraction if probabilities are uncertain. What do you think about that? By the way, thanks for the video and keep up the great work. Subscribed to your channel immediately after this video. ;-)

[@Mutual_Information]

I haven't heard of that rule before but it makes sense in terms of magnitude. You really need to cut back to the bets with uncertain probabilities.. And thank you :)

[@honor9lite1337]

@@Mutual_Information It in his book.

[@mgostIH]

On Gwern's blog there's a study on the actual coin flipping game when the total amount of flips is bounded. You could discuss this in your new RL series! Regarding the strategy under uncertainty, I wonder how a bayesian approach would look and perform like.

[@Mutual_Information]

Interesting - Funny enough, if you go to the public notebook, someone actually did a Bayesian approach. Must be a common curiosity.

[@huckleberryfinn8795]

To be fair, back in Kelly's time, there was significant misinformation on the dangers of smoking. It didnt really become common knowledge of how bad it is for us until decades after he passed away.

[@Houshalter]

Yeah experiments at the time put rats in cages with much higher cigarette smoke than any human would ever breathe. And didn't find any higher rate of cancer. Among other experiments. It was basically considered debunked pseudoscience at the time.

[@kennethcrandall8131]

Don't forget taxes too! Great video.

[@HappyMan36]

If your looking for the holy grail in trading there you go… many people will still never do it even when it’s giving for to them for free….

[@ShadowMcSneaky]

why did the algorithm take so long to recommend me this XD

[@geraltofrivia9424]

Maybe using the Central Limit Theorem to estimate the value of mean and since result of the flip of a coin is a Bernoulli random variable, this will give us P(H). Of course ti use rhe CLT, it's better to make your calculations after having more than 30 values so it makes sense to make the first 30 bets with the minimum amount, calculate the expected value of our Bernoulli variable, infer P(H) and finally apply Kelly's criterion. Maybe.

[@Mutual_Information]

Yea, that's the instinct we all have. But it only works after a lot of bets. 30 might be good, but it depends on the value of P(H).

[@tauhidkhan4849]

I came to this channel yesterday and for some reason this guy is looking alot like Professor David kipping from 'cool worlds' KZbin channel

[@christophgouws8311]

So by how much must you shrink them?

[@sidework1]

Thinking about applying this to sports betting, specifically a sport i'm familiar with and can make estimations more confidently. Any advice?

[@JoeTaber]

Where is bayes when you need him?

[@WilliamDye-willdye]

Hmm. This video might do well in the form of an interactive web game or free Steam game. Yeah the code is open-source (thanks for that), but installing Jupyter Notebook may scare off most people.

[@Mutual_Information]

Yea you're right. It's only there b/c a few people have requested code, but it's not something I expect to invest a lot in at the point. Maybe in the future though

[@b0nce]

I guess that the target audience already knows how to use jupyter notebooks :)

[@yoloswaggins2161]

What if you estimate the probability by continuously updating a beta distribution over time? You can take conservative bets over the belief distribution on p(H).

[@Mutual_Information]

Yea, that's the Bayesian way to do is. But if you use the distribution over p(H) to inform your optimal bet.. you *don't* get the Kelly Formula.

[@xorenpetrosyan2879]

in the example of estimated P(H) what if instead of observing for 10 flips than playing according to that estimate you update your estimate along the way. So after 20 bets you'll have a estimate based on 10 initial + 20 new outcomes? Also can you elaborate on why is optimal betting strategy always a fraction of your current money?

[@orlandomoreno6168]

Would it work if we were scaling the bet by dividing by the entropy of the Bayesian probability of our estimate being correct? Or log of the entropy?

[@Mutual_Information]

That sounds reasonable, but I'm not familiar with that as the result of a derivation for an optimal strategy. I'm not actually sure what the optimal bayesian strat looks like..

[@nihilsson]

The histogram of observed growth rates using Kelly criterion (7:04) looks funny. It is reasonably bell-shaped but has weirds "horns" more or less ±2 STDs from the mean. Are you sure that's not a bug?

[@Mutual_Information]

Yes, but i forget exactly what the issue is. It's something to do with how the clipping is done. But I distinctly remember trying to get rid of them. It's something to do with rounding and keep the ticks at what I wanted them at. But you're right. The truth underneath doesn't have those horns.

[@aarontamaddon9417]

Thank you. Fun and not pedantic

[@EthanCowlbeck]

How would you apply this to a wager with multiple outcomes? Say you wager $10 You have a 31% chance of losing your $10 You have a 35% chance of only losing $6 You have a 26% chance of profiting $10 You have an 8% chance of winning $90 What percentage of your wealth should you wager?

[@Mutual_Information]

There's no longer an analytic expression for that solution, but there is an algorithm that produces the answers. I forget where I saw it exactly, but it's in one of the sources.

[@roncahlon]

What does it mean if you have a negative fixed percent (e.g betting a color on roulette) where you get paid out 1-to-1 but your odds of hitting are .473. Does that mean you can expect a negative growth rate? Similarly, it could be zero fixed percent (e.g getting paid o 1-to-1 on a real coin flip).

[@Mutual_Information]

Then the answer is to never bet. Or always bet zero. Or always bet as close to zero as is allowed by whatever the rules are.

[@SamuelLiJ]

Intuitively, the optimal betting strategy should be scale-free; i.e. since the unit of money is arbitrary, the only meaningful parameter is the proportion of your current wealth to bet. The Kelly criterion then gives you this optimum.

[@adrianriosmares1727]

If you could do a video with examples on how to use that formula in sports betting lol but anyway great information thank you

[@Mutual_Information]

Maybe I will do sports betting one day.. I used to dabble it in back in the day 🤔

[@gaming4fun419]

What if you keep plugging in the average after every bet? Perhaps adjusting our bet depending on the probability of us knowing the correct probability. After 10 flips, how close are we of knowing the correct probability? Or how far off can we reasonably expect to be? Take that and adjust the bet down accordingly. Then do the same with 11 flips, 12 flips 13 flips etc, all the way up to 100.

[@Mutual_Information]

Yea that's a very natural idea. That's what I do in the final little experiment of the video and I show the kelly criteria really falls apart (except I wait for 10 flips before making any bets). The Kelly Criteria **assumes** we know the probability of heads.. which is a luxury we rarely have in the real world. But I can say this, if we do your strategy (which in the long run, will make money), the optimal bet involves merely shrinking the bet that the Kelly Criteria would recommend.

[@paulk756]

​@@Mutual_Informationalso known as 'fractional Kelly', right?

[@jimjackson4256]

The youtube algorithm did good with this one .

[@seva9375]

Does Kellys formula maximizing the median? Or is the average growth rate something different. I think in the real world you would try to maximize the outcome of for example the badest 5%. I limitate the example on wealth because if you consider relationships, friends, health etc. its getting to complicated. So my reason would be because a wealth over 3 million does not really makes you life better. So if the median is for example 10 million, you would like to decrease the median if it increases the probability to get 3 million (without making the outcomes under 3 million worse). What is your thought which badest precent you should try to optimize? Median? 25%? 5%? 1%? Or do you have a better idea for which outcome should be optimized

[@NikolajKuntner]

My conclusions is, if you design the game, make b worse than 1/P(H)-1, so that f* < 0 and the be strategy for my opponent would be not to play at all.

[@Mutual_Information]

For sure - if f* < 0, your expected return on a flip is negative so you don't bet. Or, if the game allows it, you bet on tails :)

[@amg11901]

Let me ask you a question, let's say that using the Kelly formula you know that you can risk 5% of your bankroll in a game of flipping a coin. But instead you use 2.5% on 50% of your bets and 7.5% on the other 50% of your bets. This gives you 5% on average. Would you get the same results as using 5% all the time?

[@Mutual_Information]

Interesting idea.. It turns out that strategy would be sub-optimal. On each flip, your bet amount implies an expected growth rate over that flip. 2.5% and 7.5% would imply two growth rates both less than that of 5%. Combining them will yield a strategy which will have some average of their growth rate (maybe a geometric average?).. and averaging two sub-optimal things, in this case, won't give you something optimal. Intuitively, when it comes to betting, variance is not your friend. So if you randomly switch between 2.5 and 7.5%, you're injecting variance with no benefit, so going to the fixed 5% is "free" variance reduction.

[@matiasvergara6826]

Missing your videos man.....you are not going to upload any more? :(

[@Mutual_Information]

I'm coming back! I've just been working on this massive 6 part series on Reinforcement Learning. The videos all depend on each other, so it's hard to release them in sequence. I expect to post the first one in end of this month/early next month

[@matiasvergara6826]

@@Mutual_Information That's great man, thank you so much! Please don't forget whenever you can do a primer on Dirichlet Processes. Your videos are incredibly good!

[@AlisonStuff]

yaaaaaaaaaaahhh!!!

[@Mutual_Information]

Lol yah indeed

[@aljazbrilj1698]

Wooooooooo

[@scarfo441]

sooo kelly really undestood risk huh?..

[@Feds_the_Freds]

Wow, what. Is there no formula to describe how much we should wager if uncertainty is given? Like an "applied kelly criterion" or smth.

[@Mutual_Information]

The approach would be to be Bayesian about this.. but then you have to bring your own prior to the table, and that's why there is no single, simple formula for the bet amount. That formula would need to depend on the prior.. and that prior can be anything, so a simple formula is not going to be able to handle that. The natural thing is to model p(H) with a beta distribution. I am not aware of an optimal bet amount formula that follows from that.

[@IOverlord]

Thanks. Now I can lose money slower on my trades.

[@Mutual_Information]

lol you're not alone

[@notdan995]

I have a question for the mathematically inclined. I'm developing a trading system using Kelly for day trading the S&P 500 futures market. I plan to only take trades if I have a Kelly number over 25%, but I am unsure how many data points is the minimum I should require before trusting a particular trade set up. My question is related to the appropriate sample size for back testing particular set ups. Specifically, does anyone know if the Central Limit Theory applies to calculating the Kelly formula? In other words as long as I have a sample size of 30 or more for a particular set up, can I trust the Kelly number?

[@Mutual_Information]

I'd be careful. As mentioned, the KC is notoriously aggressive b/c it assumes you know the true probabilities. In reality, you should treat those probabilities as unknown variables themselves, and then model you're uncertainty around those parameters. That's where sample-sizes and maybe the CLT apply (CLT requires IID samples, which you probably don't have when trading). If you were going down this route, I'd suggest a bayesian approach to these probabilities and then use decision theory.

[@notdan995]

@@Mutual_Information Thanks for the reply. I'm not a statistics expert, but I see great value in using it to analyze trading outcomes. I'm curious why you think day trading samples would not be IID? Isn't every futures trade essentially independent and mutually exclusive from any other?

[@Mutual_Information]

@@notdan995 Unfortunately not at all. Just imagine if there's a whale making a strong directional bet. That will correlated many trades - all those they participate in. Also, if there's a news source that all traders are reading, that'll correlate their trades. Nonstationarity and these correlations make trading much harder than the toy stat exercises you'll see (like this video). Personally, I think to do trading effectively (better than just market exposure), you should be some kind of statistics expert. It's really a statistics game! You can become one :) just takes time.

[@notdan995]

@@Mutual_Information I've always really liked math, maybe I should look into taking some statistics courses... 🤔 I'm definitely going to keep trading so it would probably help. Thanks again!

[@nickmartin3647]

Genetic algorithms

[@Diego17511]

🥏✨

[@StevenSiew2]

Why can't you change the estimate of the probability as you get new data about the bet? More data equals better estimate. Continuus improvement.