The "Just One More" Paradox
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- เผยแพร่เมื่อ 23 ก.ย. 2022
- Code I wrote for this video: github.com/manforowicz/Manim-...
Great ergodicity economics blog post by Jason Collins: www.jasoncollins.blog/posts/e...
Related article:
Myers, J. K. (2021). Multiplicative Gains, Non-ergodic Utility, and the Just One More Paradox (with Supplemental Information). www.researchsquare.com/articl...
Another nice video about the Kelly criterion: • The Kelly Criterion
It seems some viewers were confused by how I calculated 15% in the first equation.
To clarify, 15% is the expected return for only one bet, in isolation from the others.
Here's the same equation written in a more conventional way:
1/2 (0.5) + 1/2 (1.8) = 1.15
And indeed, 15% is the growth rate of the population average (but not median and mode).
If you'd like to learn how to calculate expected return: en.wikipedia.org/wiki/Expected_return
And while this isn't strictly mathematically a paradox, I used the term in the colloquial sense, as it appeared in this article:
Myers, J. K. (2021). Multiplicative Gains, Non-ergodic Utility, and the Just One More Paradox (with Supplemental Information). www.researchsquare.com/article/rs-237495/latest.pdf
Given how much the comments are disagreeing, I think it is a (veridical) mathematical paradox, just like the Monty Hall Paradox. Otherwise, I‘d say the latter also wouldn’t be a paradox :D
After spending some time in the comments, it seems to me not using the geometric mean, which would predict the mode here, confused a lot of people as well and that lead them to believe the whole premise of the paradox is faulty.
But even then the paradox remains, only that it now is paradoxical how one can expect to gain money despite the geometric mean predicting loss. In fact, I think this is what drives the paradox: Arithmetic and geometric mean seem to disagree on what should happen, and so loss and gain are both possible, depending on how one plays their cards.
th-cam.com/video/_FuuYSM7yOo/w-d-xo.html - .15% average in compound interest does indeed grow exponentially but its a paraodox because returns dont always go up every year, sometimes they go down or less. If you watch the video above at 15% growth for compound interest the average is over 100,000 for their example but the mean ( which most people can expect - is a measly 7 dollars ). The market will go up but every year on average but its not compounding year over year on continuous growth, it fluxuates. In order to properly calculate the rate of return MOST people will get you would need to run it through a simulation of the market - which is likely much less impressive but honestly i havent done it so i could be totally wrong. So ya you call out average vs mean/mode here but your leaving out that the average is insanely different from the mean. cant just gloss over it. I mean if it really was this easy everyone would be rich. By all means save and im sure you will become a millionaire... but multi millionaire? Youd have to get lucky. Just look at anyones 401k. if its like mine its less than it was 5 years ago. It might average out over time - but thats 5 years of consistent lost compound growth that this entire model is based on.
You are underrated
Does using the word paradox in a "colloquial" sense just mean something is unexpected? That's a pretty weak excuse for click bait and using a word so poorly that it doesn't even make sense.
I don't understand the purpose of this video, this is just an elementary thing that a 5th grader would know.
To have the same number after a division by 2 you need to multiply by 2, but what you're doing is multiplying by 1.8, therefore there can be no expectation of certain profit over multiple tries.
Anyone who can multiply and divide by 2 should be able to understand this almost immediately.
The problem is that -% are not the same as +%. +100% is the opposite of -50%, not 80%. Treat all negative %s as a division.
This was my exact thought. A more accurate game would have either +100% or minutes whatever 1/1.8 is as a percentage
yep, +80% is *1.8, -50% is *0.5. the reciprocal of 0.5 is 2, which is greater than 1.8. 1 * 1.8 * 0.5 = 0.9. the expected value should be a loss of 10% each round.
@@Joffrerap this wouldn’t work, the same 10% percent reduction would apply over an infinite number of tosses regardless of your betting strategy
Yes, the issue is that 80% up sounds as if it's greater than 50% down, but it isn't.
@@felepos surprisingly, you're wrong, and the video is about how you're wrong.
I learned this as a child in MMOs, when I calculated through why debuffs are so much stronger than similar buffs.
That's a nice comparison
exactly
Amen, I learned basic probability and statistics via videogames lol
How many allies, and how many enemies does it take before Buffing allies is better than debuffing enemies and vice versa.
@@Aurongroove For me it was more like: "Why have I less health than before, despite me compensating for the -20% Debuff with a +20% potion? Oh......" 😂
If this is still confusing you, here's a more stark example of the same setup: Instead of -50% and +80%, the two options are -100% and +300%. Going all in, the number of players who have any money at all is cut in half, but the total amount of each winner is quadrupled and the total pool of money between players doubles. That means that in that game it only takes two rounds for the median and mode to go to zero, but the dwindling population of consistent winners becomes so incredibly rich that the average goes up every round. By round 10 only one player in about a thousand isn't broke but those players have about a hundred million dollars.
"Now let's substitute those variables in." f* (not sure what the * means) = 0.5/1 - 0.5/4 = 0.375, again! Just a coincidence.
(Splitting comment because TH-cam is filtering it)
thank you
I was confused for a few minutes at first, but then I realized the video was the same as economics. If you take the average of a nation, then you don't get the real economics of a nation, because the numbers are weighted by the rich. Thus to get the real economics of a nation you need to look at the median and modes.
@@earthenscience Exactly. There are cases where the average is the most useful statistic, but those are generally cases where samples are contributing to something important rather than cases where what is typical for a random single sample matters.
great addition to the context
When I was getting my bachelor's degree in math, I had to take one "Applied" math course. I picked Probability, but I wasn't too happy about it. It turned out to be one of my favorite upper-level math courses in college (it was one of the easier ones, too, which helped.) There are probability/game theory puzzles everywhere in my life and they're so much fun to think about. Plus... for someone who no longer practices math regularly, they're very approachable!
wait so u werent happy about ur choice when u made it but u changed ur mind after u made it? were the other options already full or something?
Haha, no, I didn't want to take *any* of the applied math courses! I'm sure statistics was another choice, I don't remember what the others were. @@molrat
What are some interesting every day things regarding probabilities can you tell us? Like specific examples from your life maybe?
wow its really strange that probability its not mandatory, i wouldnt even consider it as applied math
What are the chances the one course you picked was one you disliked?
The fact that you could change your strategy to additive and have your mode slope upwards was fascinating!
This is very similar to dollar cost averaging strategy in stock market trading. However, the problem there is nowhere this clean.
Dollar cost averaging doesn't do anything for you in an efficient market.
@ or a rigged market hey oh
@@Shellll amen
I have a problem with the additive formula and graph at 4:50 . Once at zero the curve never goes up, so many of the paths shown need to be trimmed. Catastrophic loss has no recovery. I would expect it to change the result, but haven't checked it.
Very cool! Basically the point is that an 80% gain followed by a 50% loss is a net loss (because the 50% applies to the entire 180%, not just the original 100%). You’d need a 100% gain to cancel out a 50% loss
"the 50% applies to the entire 180%" No it doesn't, you bet 100 and lose, you end up with 50, not 60.
"You’d need a 100% gain to cancel out a 50% loss" That part is right.
@@Derzull2468 you bet 100 and win, you end up with 180, you bet again and you loose, you then end up with 60. That's the order he meant to go by.
@h33p where are you guys getting $60 from?
@@derrickfoster644 On which planet? Not in this universe!
@@derrickfoster644 what is half of what now?
2:32 the yellow line highlighting each possible tree of 3H,3T was just a minor detail, but I wanted to express how well done I think it was. Very simple, and really cool visualization of branches.
That "the problem is that each tails reduces the amount we can gain in the future" hit hard. Realization point for me. The expected value problem is too shallow.
That's because the losses only seem less than the gains. They are substantially greater. The opposite of -50% is x2, not x1.8.
I try to explain this to my financial guy who wants to be happy one year of making 20% gains after losing 20% the last year. We're not back to even yet, buddy.
@@creefer well, it was a good year anyway.
@creefer my stocks are great! I went up 120% last year! only this year's been bad, down 90%, but in average of 2 years im a good investor right?
@@user-wq9mw2xz3jIt is a rather good start, I would only give a 2 cents of advice to not get too euphoric about your small term gains, because the market is always cyclic.
I don't know about your region and how it works on a regular basis, but here in Brazil it is "obvious" that the market operates on cycles, some long ones and some short ones, dictating profit and losses on a general scale.
Why do I say that? 2023 was a year of growth cycle for us, and many investors who gained money might think and get ahead of themselves "oh I'm so good at investiment in stocks, look how much I profited", but inevitably the market always reajusts itself, has its downs, etc.
We are living a growth cycle and maybe we have 1 or a few years of growth still, but always should one be balanced.
This paradox worked in reverse for me. My intuition was that the opposite of halving isn’t +50%, but doubling, and therefore I thought that the gambling would just be bad. I was surprised that the outliers on the top of the curve were SO successful that they were able to pull the average up.
I sorta feel like there is a lesson in here about the psychology of the lottery. Obviously the lottery is a significantly worse game, since the lottery is inherently designed to extract a profit, but there is still an element of “The best case is incredibly good, and the worst case is that I only lose this small amount I had at the start, therefore I should gamble”, regardless of how bad the average or median expected return are.
Same! No need to write that comment now that you wrote it :D
This was a big takeaway from the video for me. Even tho the odds were not fair, I’m surprised there is an actual strategy to make money on (geometric) average
@@ToriKo_ That's because the expected value on any bet is indeed positive.
Given that the "average" winnings is 114k, you *should* actually play this game when presented the opportunity.
For the lottery, *BOTH* the average AND median are lower than what you start with. Here, only the Median is less.
@@iCarus_A Yeah, I tried to point out in my comment that the upside to the lottery is obviously incredibly lower, since it’s a business and obviously isn’t going to give out more than it receives. I was just trying to say that for me, it helped give me an insight into the psychology, because even if the vast majority of people lose money, the thought that a very small number of people benefit so hugely while a massive number of people suffer so smally.
I was in no way encouraging playing the lotto, as a mathematician I get no joy out of throwing money away, just saying that for me personally I found this video to give me an insight into the rational of what other people think when they decide to do it.
My first thought was that it wasn't fair since to reverse a halting requires doubling, not adding just 80%.
@@jach2513How does that make it pointless?
@@jach2513 That assumes you are the median person who wins half loses half. But if you are signifigantly lucky, the payout is potentially thousands of times larger than the initial sum. And if you are sufficiently unlucky, you just lose, well, the sum.
The whole point of interest is that the average return is very positive which is why its not pointless to consider
halving*
@Dafatpiranha no, not fair as in the information was completely wrong or misleading
I think the point of the video is that the multiplicative nature of gains and losses isn't very intuitive unless you already know the secret. For me, I understood the nature of gains and losses, but it was nice to see it finally put into a mathematical definition. Just knowing that losses are twice as much doesn't quite get you the understanding of how to calculate the optimal bet.
This was one hell of an interesting video, the graphical representations were just amazing. Great work!!
This reminds me of the "coin game" from the Game Boy game Wario Land: Super Mario Land 3. It's an optional bonus gambling game you can play at the end of each level. You pick one of two buckets. One contains a moneybag which doubles the coins you got in the level, and the other contains a 10 ton weight which will cut your coins in half (rounding down.) You can pick up to three times. So even though you're basically getting 2 to 1 odds on a coin flip, you have to wager all the coins you have for the level each time, so one moneybag and one 10 ton weight would have you break even (or maybe lose a coin with rounding down.) It's still worth playing, though, since one time in 8 you'll octuple your money (that is, if you don't hit the cap of 999 coins), while one time in 8 you'll lose 7/8 (or a little more, rounding down) of the money.
The main point is, a 50% loss cancels a 100% gain if it's all your chips at stake every time. So with the game in this video, the mode and median go down with more plays, while a few lucky people win jackpots and drag up the average result.
If you lose the coin flip your wealth is divided by 2, but if you win the coin flip is multiplied by less than 2. This means a lost flip loses you more than what a win would gain you.
Changing it to betting a fixed amount under these rules reverses the trend - a win gains you more than what a lose takes from you.
That's why it's counter-intuitive though. If you only look at "Should I flip *this* coin", the answer is always yes, as the expected value is positive - playing one more flip is always advantageous, since you have the opportunity to win more value than you lose, by 15% of your bet. Assuming the flips are actually fair and independent, there is no internal reason not to go all in on the next flip, as that gives you maximum current return on *this* next flip. It's only when you look at the strategy that results from doing that repeatedly *on the whole* that it becomes clear that it is a losing strategy.
In a sense, the reasons not to go all in on the next flip are *external* to the single flip, because the flip is part of a larger game, and it is not entirely obvious when playing the best play for the current decision affects other decisions in the game positively or negatively when you're 'zoomed in'. It's similar to why 'greedy algorithms' don't give optimal results on difficult problems - they only account for the 'next step' result, not how that step interacts with the larger structure.
@@HeavyMetalMouse Is it a losing strategy thou? The average is up, so on average you are better off. It's just that you either win huge or lose relatively small. If you value your "odds" so to speak, you go full send. If you want to be more likely to win, but win less, you act differently.
I don't think it's actually counter-intuitive. If I charge you $10 to roll a die, and on 1 you gain $1000 but on 2-6 I keep your $10 would you take your chance? On average you win big but the mode is that you lose $10. Is taking your chance here a losing strategy as well?
@@randomnobody660 in your scenario, there is a net positive of 950 every 6 rolls on average. In the video's scenario, betting all in every time leads to a 10% loss every 2 flips on average. Therefore, on average, you will lose out, so you have to get lucky to profit, which isn't the case in your scenario. That's why it's better to be careful as the average you're referring to is the mean, which is violently skewed here. A better average would be the median, as it won't be skewed.
@@spaceship7007 I think you misunderstand. In my game you get exactly 1 go and that's it. There is no "every time". The median and mode are both -10, and you have to be lucky to profit exactly like the scenario in this video. The mean is ~156 ish however and your expected outcome is positive.
Conventional wisdom and common sense (at least mine) seems to suggest I offered you a pretty favorable game, yet the "better average" of median is negative. What do you mean by "skewed"? Skewed from what?
@@randomnobody660 “Common sense” ideas of what is or isn’t a favorable game is tainted by utility calculations - $10 is an insignificant amount of money so you don’t care about losing it, while $1000 is high enough that it’s worth trying for the low chance of winning. Plus, 1/6 is still a reasonable chance. Change the bet to your entire life savings and lower the odds of winning to 2% and suddenly the bet becomes a lot less attractive, even if the expected value is still positive.
Since this is multiplicative, taking the arithmetic mean at the beginning leads to the confusion. Taking the geometric mean gives a mean of ~.94, which is less than 1. If you then take ~.94^50 to calculate the 50-game outcome that you simulated, you will get roughly .072 - which corresponds to the median/mode of $7.20 you saw in the simulation.
There's no confusion, the mean really is extremely high. The median is low. Do you not understand why?
@@coscanoe I’m really not sure how to respond to your comment. For one, there most certainly _is_ confusion, which is one of the reasons this video was made, and the author of the video specifically references this confusion and attempts to explain it. It’s not that _I’m_ personally confused by it; I’m just responding to the confusion that was being discussed in the video.
And for two, you say “the mean really is extremely high, but the median is low. Do you not understand why?” I’m not sure why you’re asking me this question, given that my comment specifically explains why. So yes, I do understand why. That’s why I wrote the comment explaining it. I’m not trying to be rude, but perhaps you need to read what I wrote a second time. Maybe you misunderstood something the first time you read it.
Also, the critical point I’m referring to is arithmetic versus geometric mean, which is the primary source of the confusion for someone looking at this for the first time and perhaps feeling that it’s counterintuitive. The mean versus median distinction is not what I was discussing, and merely saying “the mean is high but the median is low” contains no explanatory power; it is a descriptive statement about the end result, rather than an explanation of how or why it is the way that it is, and how to arrive at the correct answer.
@@therainman7777 so you cleared up the "confusion" by introducing the idea of a "geometric mean" without further explanation, when "mean" means the arithmetic mean to 100% of people? You think that cleared up the "confusion" when the video did a perfectly good job of explaining the situation already? I wasn't being rude, I literally was forced to assume you didn't understand the video because you believed you were "clearing up confusion" somehow, like you thought people would be saying "Ohhhh, I was confused after watching this video but now I'm not confused --this guy says the geometric mean is about 7!"
@@coscanoe Again, I really don’t know how to respond to your comment. I’m not sure why you seem aggravated by a comment written in good faith and intended to offer a bit of supplementary information that some people might find useful. Why would that bother you? It’d be one thing if I was wrong and spreading misinformation, but I’m not wrong, and you’re not claiming that I am. So why does it bother you? The video didn’t discuss the specific point that I raised, and I felt it was an important part of the explanation. If you disagree that it’s important, that’s fine. I feel it’s important to know which type of mean is appropriate for this type of problem, and that using it can avoid some confusion. If you don’t think that’s important, no problem. Not everyone will come into the video with your exact level of knowledge and background, and maybe others will find it useful. I expect they will, as it’s a core part of what’s being discussed here.
You also criticize me by pointing out that the word “mean” means arithmetic mean to 99% of people.” Yes, I know that. That is why I clearly distinguished arithmetic mean from geometric mean in my comment, using the phrases “geometric mean” and “arithmetic mean” rather than just saying “mean.” So what exactly is your point? I said one was right for this situation and the other wasn’t. And you said I “didn’t offer further explanation,” but I’m not sure what you want from me in a TH-cam comment-should I write LaTeX to show the full formula for calculating a geometric mean? All someone has to do is google “geometric mean” if they’re interested, and they’ll get to the Wikipedia article which explains it far better and in far more detail than I could possibly do in TH-cam comment.
I’m not sure what your problem is, but I’m fairly certain it’s not me.
@@therainman7777 Geometric mean is not "right" for this situation. It's completely irrelevant. The video concerns expected value (which relates to arithmetic mean) and risk. In gambling we typically try to maximize expected value, but with this game if you try to maximize expected value by betting the maximum each time you will almost certainly lose almost all your money. The video then solves the dilemma by introducing the Kelly Criterion, which is the proven best way to maximize your bankroll at any given point in time.
I didn't think you understood the basic premise of the video, which is why I commented on your initial post. And honestly, I still don't think you understand it.
The math went over my head at some point, but this was super interesting! Great video!
Fact: 90% of gambling addicts quit right before they are about to hit it big
Yeah because they went broke for some reason
99.99%!! 😲😲
this basically says, if you want a good median outcome, even if you have an advantage, don't go all in at once. make sure to protect your odds of staying in the game.
That's why you have a bankroll in poker.
I'm aware of the Kelley criterion but it was really cool to see it derived and visualized like this. Good stuff. 👍
Kelly, actually
yeah, i learned it from stardew valley out of all things lol
Tired of "I knew this but cool video" ass comments smh
@@arztschwanzfurz1631 lol no one cares buddy, don't read the comments if you're that sensitive
@@arztschwanzfurz1631 cool
Excellent description of the Kelly Criterion! Wonderful video, and captures the intuition perfectly.
I think it's more impresive that you manage to create this video with code. Amazing
Man, this video is so underrated. I never expected that it could be possible for most people to make a loss even if the expected value of money gained for the game was still positive. And then showing that a simple change of strategy (from a multiplicative to an additive approach) could actually make the median rise? simply mind blowing!! Really interesting phenomenon, well-explained and well-animated. Congrats.
Incidentally, this is also one of the reasons billionaires who are also morons exist. When you start with millions of dollars in backing, it doesn't matter if most of your investments are boneheaded. If even one turns out well, you become obscenely wealthy, and then the stupidity of future bets matters even less. Us ordinary folks, though, can't afford to make so many idiotic bets that one of them is sure to pay off. So a system arises where it looks like anyone can make it with enough skill, but really it's all luck and having enough resources to start with that matters. Capitalism is not a meritocracy.
I don't know if you should be surprised. Imagine I offer you a game for 10 dollars. You get to roll a dice and on a 1 you get $1000, but if you get 2 thru 6 you get nothing. Here your average $156 or so, but both medium and mode are -10. Isn't this a much simpler scenario with the same situation?
I think this is because for it to balance out multiplicatively, you would need to double all your money on heads, and half it all on tails. That way the heads and tails directly cancel each other out.
@@Frommerman agree, and I would add that : "the belief that billionaire are smart" help them, so even if actually you are stupid, the belief in a specific society that "rich people = smart people" will make people trust you, which will help you make more money, as people trusting you is a big advantage in term of economic opportunities (like : you are more likely to invest in a country/venture/etc that you trust, and you are more likely to loan money to an entity that you trust with low or even negative interest rates. This is why countries can have loans with negative interest rate, as it's a "safer" way to store money as country are so "sure" to loan money, compared to le'ts say companies or individuals, who are much much more likely to default payment.
the belief that "rich people = smart people" explains also why a billionaire can become president (hello orange man) or scam people with stupid product (hello exploding rocket man) like electric cars in a tunnel (which is just metro but worse) and people will STILL support you and pay for your "products" despite that they objectively suck, because they have so much trust into you.
This aggravate the phenomenon you are describing, that expalins even more why trust fund babies can fail upwards and despite being bad, still succeed.
The simple belief that "billionaire = smart and therefore they will succeed more in the future" is a somewhat self fullfilling procecy that guarantee further success for the elite.
As economy tend to grow, I, for example, would also be able to generate a profit on by investing if y have enough money. I would simply invest in many things and the think that work bring money, it's a high risk high reward situation that is beneficial if you have enough capital to invest, and if the dominant ideology make that people actually trust me more, i can even sell them worse product than the competition and therefore make MORE MONEY, it's the Elon Musk way of becoming rich, just have people trust you because "billionaire = good" is efficient money glitch.
Agree on the end sentiment :
Capitalism is not meritocratic by nature, Free market capitalsim as an economic model may have some qualities (for example regarding regarding how people can set value, which can increase economic efficiency) but clearly, rewarding people for their skill is not one of them, because of the "accumulation of capital" that is antithetical to any functionning meritocracy.
There is also others problem with such an economic system (free market capitalism) :
If this economic system is efficient to give value to things with positive value (compared to let's say central planned economies, feudalism or old mercantilism), free market capitalism suck for stuff that have "negative value" : like, any kind of waste or pollutant : which end up usually being left behind by capitalist economies.
Which is why (free market) capitalism is pretty bad at... anything related to protecting the environment. And this is why a capitalist economy can't function on the long term (at least without proper extensive government regulation) without destroying it's environment.
(I have heard some anticapitalists that say "capitalism need growth, and infinite growth is not possible so capitalism will collapse", and it's extremely stupid : first : capitalism don't need growth, Japan is a good example of non growing capitalist economy, since now a decade at least, and it's far from looking like Madmax. Capitalism will probably need extensive government regulation to survive as an economic system, but clearly, it's not impossible to imagine a post growth capitalist economy. But to be clear this don't mean capitalist economy is good, this simply mean the anticapitalist prophecy of "capitalism is bond to collapse" is simply... a myth, a marxist myth that is simply not proven. While i think capitalism have some inherent problems, i don't think we should believe that it's downfall is inevitable (especially if the way this downfall is going to come is also with the downfall of modern civilization, which is probably not a good thing if we want to stay alive (i think it's wrong to cheer on the "end of capitalism" if the way this "end" is gonna be is "civilization collapsing")) especially when it's not the most likely scenario (an authoritarian economically improverished oligarchy is much more likely than Madmax in a post growth world, which is NOT good by any means, but very different from "capitalism collapse")).
@@QajjTube Yes.... author evidently doesn't have anything better than to put his viewers to false beliefs by converting percentage gain to the exact value so that it looks astounding that this "paradox work" while it is not surprising at all. I don't believe he can have any other intentions considering that he starts the video by saying you get 80 dollars on win and 50 dollars on lose, which is obviously manipulative.
In finance, the basic Kelly's Criterion is not particularly useful as it only applies to binary outcomes, however the principle behind it is very important. Higher volatility results in a greater difference between arithmetic and geometric returns which is known as volatility decay. It's vital to consider multi-period geometric (compounded) returns.
You can work out pseudo- Kelly's criteria by approximating it as a series of binary outcomes though, and it works OK.
The issue more is absolute maximisation of median wealth isn't always what people want; often they want to reduce risk further than that.
Does it have any relationship to option pricing? As soon as the binary tree came out I was thinking of a very rudimentary, fixed time step, binary model for European options. I don’t remember the exact formula for the arbitrage-free price but it was certainly dependent on the expected value
@@casperguo7177
For option, in-the money ones are more expensive with volatile stocks whereas out-of-the money ones are cheaper.
It's about the probability of them flipping in- or out- of the money.
Also, if you don't already understand stuff like this, I would highly discourage buying options.
Tbh I'd probably discourage it in any case (it's easy to make miatakes), but definitely if you don't understand them.
@@IamGrimalkin I just remember the models from a financial math course that practically taught me no real-world know-how lol. Thanks for the answer
Kelly criterion is useful and it used in real world situation. A good blog to check out is breaking the market which goes over the usefulness of Kelly.
I really need more videos like this one. Good job!
I just watched the entire thing about a topic I resented in school.
Well done.
I love the way you generalized this problem, and how the solution ties back to the real world. Excellent video!
Can explain how this can be implemented to the real world
@@proayachi9262 gambling
This was an extremely good watch and you narrated so nicely. Really kept me as interested as my intro to statistics courses.
honestly, that is very interesting! There are some aspects to math that are very cool and engagingm I wish it was taught like that in schools. Thank you so much for that!
Thanks for your time in Illustration.
And adding knowledge to TH-cam.
Cheers❤
It always makes me happy to see math videos like this made using the software from 3 blue 1 brown. If only all math could be taught in such an elegant way.
Also, congratulations on having this video go crazy with TH-cam's algorithm
"If only all math could be taught in such an elegant way." - I don't want to sound like an TH-cam add but check out Brilliant (that famous add when they show you how you could learn anything by interacting with graphs, code and so on).
What I find most interesting are the applications of the "Just One More" paradox in human behavior and psychology.
The results are quite shocking, goood job editing!
Wow I'm just seeing this now! An excellent video - well done!
And of course, a big thank you for the shout out :)
There is a little carnival game in Stardew Valley where you bet some currency double-or-nothing on a weighted coin flip. I was always curious what the optimal strategy is for reaching the maximum currency. The more you have, the more you can wager, but you lose it all on a lost flip. This video helped to contextualize that problem, as it seems very closely related!
I was thinking about that game too!
My intuitive strategy was apparently pretty close to optimal: always bet 1/3 to 1/2 (depending on mood). 😂
Wasn't one of the colors significantly more likely to win or sth? I think I wagered half the points and managed to get max points fairly easily
@@akiraigarashi2874 Yeah I think it was like way way unbalanced and that was sort of the joke
Is it weighted in your favor?
It isn‘t weighed in your favour by just randomly betting, but one colour (green) has a higher chance (3/4) of winning. So it is rigged indeed and I wasn‘t going crazy while palying it haha
this is a perfect explanation why salary should never be compared to the average of a given group, but rather the median.
It's always good to use both and even that might not be good enough depending on the distribution. I've seen software companies advertise job positions saying that their median salary is whatever, but that tells absolutely nothing about how much the bottom 49% (and thus every junior position) earn.
@@pajander This example does not seem to be related to my statement...Comparing your own salary to "all people 20-30 years old" or "all people in this same position" was what I had in mind. and yes, the median is perfect for this,the average doesn't say anythig about it. The median answers "Am Idoing okay compared to others of this group" perfectly: half of the people are better, haalf are worse. if you're above the median you are doing okay, if you are below the median you might want to improve.
@@Todbrecher what Pajander said is still true, you need both to make accurate assessments. With median you can't differentiate between a situation where 100% get around $100k and one where 51% get $100k and 49% that hold say a degree get $200k. In situation 1 you are good, in situation 2 it depends on what "good" is for you. But I'd say being in bottom half is not great at all.
@@pajander Yeah, preferably one would want mean, median and standard deviation as well to find out the spread.
Okej. Mam to teraz na matematyce. Rachunek prawdopodobieństwa i kombinatorykę. Szczerzę szanuję, za takie podejście do tego. Trzymaj się.
I think an easier way to visualise this is consider getting 1 head and 1 tails. As you said in the video, $100 x 1.8 = $180, $180 * 0.5 = $90. So getting the same number of heads and tails leaves you with LESS than what you started with.
Now obviously you aren't going to get an tequal number of heads and tails. However, as the number of tosses tends to infinity, the number of heads/tails equalise.
Therefore with enough play you're going to tend towards equal numbers of heads/tails, which we established is a loss.
That is exactly what I thought, just from seeing a small example we can extrapolate that this is going to go bad
That's pretty mind blowing to think about. The expected value is still positive so if you played the game infinite times would you end up broke or with infinite money? The law of large numbers would say heads and tails equalizes and median tends to zero but expected value goes to infinity. Kind of polar stretch
As a sports bettor, as soon as you mentioned should you bet a fraction of wealth, kelly criterion instantly came to mind. There is also 1/4 kelly for more variable markets (like sports betting) which can be followed to maybe not maximize wealth, but minimize losses WHILE still making money.
Far less than the kelly criterion makes a lot of sense when you're guessing the odds
Fractional Kelly is really important in terms of risk of ruin. Typically you want 1% or less risk of ruin - sub 0.1% is really nice - and while each individual will have a different definition of what 'ruin' is for themselves, the fact is that if you're as aggressive as KC sometimes dictates, that RoR will be above 1%.
Awesome video! Optimizations like this are always a fun way to put some math behind intuition and get something concrete out
I remember reading "Greed" by Marc Elsberg and there was a mathematics professor that showed this exact game with coins in a bar. Let's say it dissolved into a brawl quickly :D
@@Maximilian-SchmidtThe reason you shouldn't play is that individually you will most likely end up losing money.
It is only if you can make many plays, whether one person woth many bets or many people who then share the prize, that you can enjoy the profits.
The first proposition of the game results in the casino losing money, most people losing money and a lucky few raking in some serious money.
you know the video gonna be good whenever it starts with manim animations
It really refreshed my memories of probability and average i used to love those subjects
This is just a random walk on the log scale skewed towards loss. You go down by log 2 with 50% prob. and up by log 1.8 with 50% prob.
The 15% gain per attempt is irrelevant because you never repeat any attempt with the same amount of money. You need to keep in mind what you're calculating, expectation value assumes repetition of the same experiment and taking the arithmetic mean. But in this random walk the final outcome is a product, not a sum so you would have to take the geometric mean, so you need the expectation value on logscale.
Now the expectation value on logscale is 50% log 1.8 + 50% log 0.5 < 0, so you're losing on average, as you would expect.
Missed the point
And expected value is Always an additive mean , thats the very definition of it
amazingly well explained and animated video, glad I was recommended this.
Given the amount of resources available online, one must wonder why schools fail to meet expectations (hint: the human element).
Wow, 1100 lines of code for this whole video is indeed quite impressive! Thanks for sharing this fact!
This also applies to warfare. Multiple battles won could sometimes lead to the war being overall lost, especially when, despite winning large plots of land, you lose large amounts of men. Or even if you lose small amount of men, but lose large amount of supplies due to suffering logistics, you're bound to bog down, or even lack crucial supplies for a crucial battle.
if you win 9 mid boss battles but lose the final boss battle you still lose the game
Yeeeah, I think it happened to us in the post-Soviet space. We won WWII, but we lost an extremely huge number of people to win it, and obviously we were extremely short of manpower to work, besides, debts were paid to the end only in this century
I had dealt with this problem directly when I was making a twitch minigame which is basically a simplified stock exchange. I made the stock prices go up and down randomly a certain percent every 10 mins. however, I noticed something very weird about how this all scales up. it didn't matter how much the "expected average increase" was, even when it was positive, the price would still go very low at times. in the game I made it so, if a company had its stock price below 0.5, it'd bankrupt and another company spawned in its place
fun story, I actually had to learn math (or at least ask the help of a math guy) to make a formula on "the expected price cycles" a company could survive, and the average increase
The intuition behind this is that multiplicative processes tend to amplify small differences and create extreme outcomes. For example, if you invest $1000 in the stock market and you multiply it by a random number between 0 and 2, you will get a different outcome every time. However, if you repeat this process many times, you will notice that your money will either grow very fast or shrink very fast, depending on whether the random numbers are mostly above or below 1. This is because each multiplication changes your money by a percentage, not by a fixed amount.
Respect for the 1000+ lines of code you provided for free, Much appreciated
I guessed 1/3, so I was pretty close!
Rule of thumb :)
What was your thought process? Did you base it on mathematics, or was it just a random guess?
When I saw the curve on the graph, I thought pi might be involved, but my usual strategy for guessing percentages is to say "About 38%". That would have been about right for this one.
Oh, a wild Lindybeige! Howdy!
@@jackpeters2884 You can recognize them by their sweaters.
After just 40 seconds of video I knew he was operating in multiplicative space.
Btw, similar trick managers use when reporting profit expectations to their superiors:
They dont report the correct geometric average for growth of the investment, but average out using the arithmetic average. This effectively raises the profit implied to expecting investors. I was doing my computer science internship at some major global corporation and once asked why this error was systematically done... in the end I got a reaction mix out of people saying it was irrelevant, others saying it's stupid and others trying to hush me up into silence. Sad to see in a corporations which puts a lot of importance of everyone doing everything to improve the company and not abuse their positions or lie to improve their personal profile in the company.
I don’t think it’s sad, but it is revealing. Thanks for your comment
That's quite telling, indeed. Seems like many establishments love to mess with the numbers, and because *MOST* people don't know enough math to spot such discrepancies / sleight-of-hand (not exactly the correct term, but you get the idea), they often get tricked into investing / gambling more than they should.
But then again, maybe it's proof that those managers are appealing to an old fact about us human beings: we make decisions based on emotions, and justify them with logic.
Even if said logic is flawed because we don't have enough information to make a properly-informed decision.
Nice having the visualisation. Thanks man.
holy, the graphics here are very good! Such a smooth visualization
I love mathematical "paradoxes" and this is one that I hadn't met yet. The most surprising thing for me is that I didn't know that 3b1b had published manim. Thank you for both pieces of information.
dude its not a paradox its bad mathematics, The dude is not using the proper average, he is using it on a linear scale. but since the values are multiplied and divided it should be used on a logarithmic scale, since if you multiply 3 by itself you aren't going to get 6 you are going to get 9 same with multiplying 0.5 by itself its not going to be 0 its going to be 0.25. so the proper average would be that you would lose 5.13167% of your total money per coin toss
This is not a paradox
@@ares4130 i agree that ';paradox' is being misappropriated here. it's like a mental fumble. or like the monty hall problem. where common sense or a surface level understanding can lead to an unexpected and wrong/bad outcome.
@@binz2056 That is a paradox
@@ares4130 That is a paradox
Okay this makes more sense to me now. Investing has always been really confusing and just weird to me. But thank you for showing us the math behind basic healthy investing. I know the real world investing is more complicated than +80% or -50%. But knowing the basics is key.
I needed this for something I'm working with on game theory. Thank you
Wow, thanks for the video man, it is amazing!
This is the best explainer for the Kelly Criteria that I could ever imagine. Nicely done.
Amazing video! Love the increase in well visualized math videos recently
You should definitely make more of these type of videos
Perfect explanation of the stock market, especially when compared to the downward slope of volatility funds over time
Awesome video. Investors don't use the Kelly criterion though, because the win and loss probabilities in stock markets are not fixed. They are unknown and they vary with time. To handle this additional uncertainty, you have to be even more risk-averse, which means you have to bet even less. So a heuristic way to handle this is to bet 1/2 of the Kelly fraction, or 1/4 of the Kelly fraction. If you have a more complicated model for the evolution of the win and loss probabilities over time, then you can derive the sequence of optimal fractions using some very sophisticated mathematical techniques from a field of probability theory called stochastic calculus.
This is the comment I was looking for. As an extention: Even if the win and loss-probabilities are fixed, but only slightly skewed against your favor, like e.g. in roulette, e.g. playing black/red (18/37 win chance, 19/37 loss chance, i.e. 48.65% win, 51.35% loss-chance), there is *no* strategy at all that even results in an *average* win (and certainly none for the median).
I don't use it for investing myself, but I believe you can use it in investing if you create an array of outcomes and calculate based on that (ie. 25% of +20% gain, 25% of +5% gain, 25% of 0% gain, 25% of 50% loss).
Bettors do also use fractional Kelly; typically we do it not because we're afraid of the large stake, but rather because we've wanted to reduce our risk of ruin (usually we aim to get 1% or less; sub-0.1% is the goal but may not be attainable without sacrificing too much growth).
You be surprised that investors does use Kelly criterion. I use it my self using the average win and average loss % formula and trying to adapt it to investing by using geometric rebalancing.
This is mind-blowing stuff! Thank you for all the trouble you went to to make it visual. Excellent and easy to follow. I only wish High School mathematics would have been taught with pragmatic purposes such as this. I would have been useful.
I did think this was a 3blue1brown video until the end, when you mentioned the library. Great content!
Instead of averaging 0.8 and -0.5 arithmetically, you should be averaging 1.8 and 0.5 geometrically. Then it's clear that the median result is approximately a negative exponential with base sqrt(0.9) = 3/sqrt(10) < 1.
yeah i didn't bother watching, the starting conclusion of "average gain 0.15" was wrong, there's no paradox here
@@Firelucid it’s crazy, this guy got a formula wrong and made a 10 min video about how it’s a “paradox”
The geometric mean alone cannot explain how a change of strategy makes it possible to achieve a gain on average. This is possible since the average outcome of any one bet is +15%.
If you find the geometric mean more natural, then there is indeed no paradox in the outcome when betting all the money every time - there is however one in how it’s possible to gain money nonetheless.
lmao you guys heven't even gotten to the paradox at all bruh
@@BlastingAgents The geometric mean is not useful in the majority of probability scenarios, but this one is multiplicative, and thus naturally the results will be exponential. I never said the geometric mean of the variables explains the arithmetic mean of the result after a certain amount of trials. That one is explained by the +15% calculated earlier. I said that the median was explained by the geometric mean and its exponential function.
I think this effect is much easier to see with the same game but with +200% and -100%. Every time you play the game, your EV is +50%, but if you EVER hit tails, your total money will be 0 for the rest of time.
That's an effect of x2 vs x0. I could see some gamblers trying to win big and losing everything this way tho
@@wabc2336 +200% is x3, not x2
@@wabc2336 But in casinos, your EV is negative, so if you want to win big, you should bet a lot.
It amazes me that someone has the confidence to make a video about maths without knowing how to do the most basic maths, hats of to you my friend
This is Mind Blasting. Well Done Sir!
So intriguing to see that you basically get the "worst punishment" from only one tails, but require multiple heads in a row just to cover that one tails flip.
Indeed. You can relate it to real life too. It's really hard to end up rich if you are born into difficult circumstances (your first coinflip was tails) but the privileged classes (starting with heads) can be way out in front before you get started.
@@AutPen38I like that idea. Someone in poverty might only be able to survive a few tails while a rich person can take the hit to make it to a run of heads
This concept also emphasizes the importance of not putting all your eggs in 1 basket; if you dump 100% of your money into each investment you make, then your investment trajectory follows the downward trajectory portrayed in this video.
Years ago I poured 100% of my money into 1 property and lost everything, which was a lesson that cost me much more than my entire college education. Now my real estate portfolio is diversified among a handful of different properties thanks to that hard earned lesson.
Unless you are buying fractions of properties or you have enough starting capital to buy multiple properties all at once then at some point you will have 1 property, your first property. You were unlucky that your first property failed before you could buy a 2nd but luckily your 2nd property lasted long enough for you to buy more.
Being successful long enough to be able to buy multiple properties requires 'putting all your eggs in one basket' at least for a little while.
It is the same with businesses. A lot of the time people will have many failed business ventures until one of them becomes successful enough to allow them to 'diversify' into multiple ventures.
@@Subjagator I have to disagree somewhat here. I could have kept my money in a diversified stock market position (such as an ETF) for a few more years and then I could have purchased 2 or 3 small properties at the same time.
@@scottfrayn
I was talking specifically about property. There are of course other investment options besides property.
I'm a semi-pro sports bettor, and I lurk on some of the betting subreddits for various reasons (usually I'm trying to be helpful, ironically sometimes with stuff about the Kelly Criterion :P).
It is crazy and saddening how many people try a strategy of "bet my whole bankroll on a heavy favourite" strategy, particularly when they're doing $50 to $1000 challenges or something like that. Not only do these people usually not have any sort of algorithm or system that can, with confidence, say that their picks are +value (most bettors bet based on intuition, and most people's intuition isn't good enough to cover the bookie's ~5% edge), but even if they were +value that's terrible bankroll strategy.
As an example of just how much they're overbetting, lets say someone offers you a bet - for $100, you roll a d100, and if anything but 100 comes up, you win $103. That's an extremely likely scenario - 99% - and it's definitively +value, at +1.97%. So the people on the subs with their challenges, would place their whole bankroll on it. You know what KC says you should bet on this 99% likely, 2% +value wager?
65.67%. Basically two thirds of your bankroll. It's still huge, but a lot lower than you might expect for a 99% likely market.
This is super cool video! It would be perfect to see the model updated a little bit. In real life, all of 'not invested' money is a guaranteed loss (due to inflation that is higher than interest rate) it is very small compared to gains but still loss. And maybe if we adjust gains and losses to more realistic in terms of stocks. I'm curious what will be the result.
Thanks for the video, pretty nice work !
This can lead to some insane situations. For example, the expected value of this game is in fact infinity. Which creates the uncomfortable situation of it *always* giving below average results.
This actually shed some light to me because if you use a calculator when you win you only multiply by 1,8 however if you lose you divide by 2. 2>1,8 and that's how i understanded this video
It's so counterintuitive I love it.
So counterintuitive. Thanks for pointing it out!
That one I immediately got it... you're multiplying on each interation, not adding. So the average path is 1.8*0.5=0.9, so on average it decreases 10% on every interation
That’s the easy part. The interesting thing is that you can still make money by investing a small part
Wow, I really find it counterintuitive that there can be both losing and winning strategies even though the payouts of the game are always fixed, and it’s a game of perfect chance.
It’s understandable that betting strategies can improve your odds in a game like blackjack, like when to double down when a favorable scenario arises… but this is just flipping a coin… no scenario is more or less favorable than any other. Interesting concept.
Good video my dude gonna save it for when I have to explain this stuff to others!
Great video on a surprisingly interesting subject!
It's quite funny how just by intuition,
let x = current balance;
if heads: x = x * 1.8
if tails: x = x / 2
we quickly see that for this to be at all favorable for us, we must multiply (when heads occurs) by at least the same amount as by which we divide (when tails occurs).
e. g. if we would double our balance upon receiving heads, it would be a completely fair system
Which is why people--or businesses--that are trying to talk people into playing these games always use percentages. They make people feel smart and in control while also making it really easy to trick them into a losing game.
You've missed the paradox, or that this is actually a profitable game. You should play this game because it has a positive expectation of 15%, but the paradox is that most people will lose. By pure luck you could be one of the people that starts with a couple of heads and ends up with much more than 100, but if you start with tails, you're immediately down to 50 and it will take a good run of luck to get even again. Half the players will start with heads and be above average with a chance to get way above the pack. After two flips a quarter of the players have 324 dollars. You just need luck to be in that quarter that made a big profit and not in the three quarters that lost.
@@AutPen38 you could say the same for the lottery, that there is just one fraction of players that will end up millionaires, even though games like this are specifically engineered for you to lose.
@@steviousmusic Yes, a small proportion of lottery players (including the jackpot winners) are actually profitable over the course of their lifetimes due to random good luck. But the lottery is different to this game in that the "house" (the company running the lottery) takes money out of the prize pool in order to pay its shareholders or donate money to charities. The player pool as a whole in a lottery loses money. But no company would run a coinflipping game that paid +80 for heads and collected 50 for tails because it would immediately go bankrupt when the coin came up heads. (Imagine offering this wager: "I'll give you 80 quid for heads, and you only have to give me 50 for tails". Only a lunatic would offer such a wager, since they would be offering free money to the players). This +80/-50 game is weighted in favour of the players, who - as a whole - literally get more money back than they paid in. But that's the paradox. The players as a whole in this coinflip game make a profit (+15% per game), but - as with the lottery - individual profit is concentrated on the luckiest few (who start with a win and get more heads than tails). Half of the players get tails at the start and have their "bankroll" halved right at the start, meaning it's really hard to get back in profit, but *the winners are making more than the losers lose*. It's just that there are a lot more losers than winners. In this game, a quarter of players are in profit after two flips, but that fraction gets smaller and smaller. The profit of the luckiest people gets much bigger than the relatively small losses of the majority.
People really fail this one harder than the Monty Hall problem.
For one throw 80% gain and a 50% loss means that for one throw half the time we end with $1.8 and half the time with $0.5, this means on average we end with (1.8+0.5)/2= $1.15. This is 15% more than we started with.
For 2 throws We expect:
25%: 2 wins -> 1*1.8*1.8 = 3,24
50% 1 win 1 loss -> 1* 1.8 * 0.5 = 0.9
25% 2 loss -> 0.5 * 0.5 = 0.25
Expected value = (3.24 + 0.9 * 2 + 0.25) / 4 = 1,3225 (same as 1.15^2)
The expected value is always positive, the mean doesn't matter. The intuitive trick would be to see that if we win we win way more than we would lose were we to lose with every throw.
It's absolutely insane how many people are writing comments along the lines of "+80% is less than -50%". As if these values somehow "cancel" each other out. Completely ignoring the reality of coin toss distributions (the very large profit of flipping heads more than tails makes up for the fact that getting heads as many times as tails is slightly losing). There is no "paradox" here at all, only the fact that you cannot use the expected value alone to determine whether you should gamble with finite amount of money (which should be obvious). But you wonder if the people here understand what "expected value" even means.
@@casimir4101 There are lots of people in the comments that seem to think this is a losing game with a negative expected value. Those people are wrong. There are also a lot of people in the comments that have failed to see the paradox, perhaps because the video-maker didn't quite hammer the point home. The paradox is that the total bankroll of all the players added together keeps growing, because the person running the game is giving away free money away at $15 per coinflip, but the number of people that have less money than they started with is also growing. After one flip, half the players are in the red. After two flips, three-quarters of players have less money than they started with. After three flips, even more people have less than $100. But paradoxically, when added together with the minority of players that are in profit, the total population has more money than they started with. The paradox is that an ever-shrinking minority of profitable players makes more profit than the growing number of losers have racked up as losses. In short, total profit keeps rising, but so does the number of losers. More and more people lose, even though the total amount of money in the system keeps rising. That's quite paradoxical, don't you think?
I've just scimmed the video so far, but this seems like incredibly valuable information.
Excellent. I knew about Kelley, but didn't know the derivation. Thanks!
This is a lovely explanation of the difference between single and iterated plays in game theory.
The 80% 50% set up is interesting. It would be a terrible game for a casino to run and they would lose a lot of money but it's also a terrible game for the average punter that uses the wrong strategy as most lose money. Highlights the importance of proper strategy nicely, great video
Awesome vid, mildly wondered about this and now I know
Awesome work bro
the issue is you're not only losing 50 dollars on the first flip, you are also losing the ability to gain 144 dollars on the next flip.
My immediate reaction on seeing the initial game was that x0.5 is equivalent to dividing by 2 and that taking each outcome once would therefore give 1.8/2 or O.9 times the initial value, therefore the graph should overall trend down as an average. It didn't occur to me that this could be perceived as favourable initially
I think this problem is much simpler if you think of it in terms of regular gambling odds instead of a strange dependence on bank-roll. I believe you’re giving odds of $2.60 for the coinflip (bet $25, win $40). This means we can just plug this straight into the Kelly criterion to get the 31%
very cool video, this is my first time here on your YT.
I love how Pascal's triangle just casually jumps out at you from a logarithmic view of the possible scenarios
I love when my math students ask me "why do I need this?" Well, some people make lots of money understanding this.
And when they ask you "and why don't you own a lot of money, and have to work as a professor?"
"I actually do have a lot of money, and I actually don't have to work a s a professor. I just chose this profession because I am a sadist. Fear my next exam."
Ad lib mad laugh if you need.
This is a common problem with trading. This is why people loose a crap ton of money in the market when they start out...especially when you include broker fees. The Just One More paradox is almost like FOMO, fear of missing out. This problem also becomes compounded, as you are not dealing with 50/50 probability.
Thank you for sharing!
Interesting graphics and thanks. I've heard of Kelly before, on a different clip. Back when computers were very expensive, and I attended University, vectors were largely the point. Maths were largely done with the CRC Handbook of Mathematics, and you looked it up. And done graphs. Then came relatively cheap ($200.00) hand held calculators, if memory serves, in the mid to late 70s and things changed.
The original bet is not necessarily bad. It all depends on context.
For instance, suppose you had the option of gambling your weekly paycheck with the 50 flips multiplicive method. Taking that bet every time will cause your net earnings over time (ignoring interest/inflation/opportunity costs) to increase, because occasionally you'll get a lucky streak that more than makes up for all your previous losses.
(Of course, that all depends on you having an unlimited steady stream of money, that you can only bet a limited amount of. Unlike in real life, where it's easy to double-down on gambles until you go bust.)
7:38 I love that he basically says “if you’re too dumb to understand what I’m about to say, just be quiet for the next 20 seconds” bc as someone who doesn’t know much past basic geometry ,but loves math problems, this kind of video pops up often and am regularly met with not understanding what’s being said. Funny he made a point to say that is all
Really nice animations and explanations
You earned a sub. Thanks for sharing!