[HN Gopher] Kelly Criterion
___________________________________________________________________
Kelly Criterion
Author : niklasbuschmann
Score : 300 points
Date : 2021-04-16 14:30 UTC (7 hours ago)
(HTM) web link (en.wikipedia.org)
(TXT) w3m dump (en.wikipedia.org)
| jasebell wrote:
| 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 :)]
| demircancelebi wrote:
| Hey, I made a game recently for people to gain an intuition for
| Kelly: https://beatkelly.celebi.me/ Let me know what you think :)
| xiphias2 wrote:
| It sounds great, but I like instant results, can you write out
| after each round how much Kelly put up and has?
| minitoar wrote:
| I used this to win AI Rock Paper Scissors competition in
| undergrad. I just played random symbols, but used Kelly criterion
| to compute my bid. This worked well because the game wouldn't
| allow your bankroll to go to 0 -- the floor was 1.
| glacials wrote:
| Can you explain why the Kelly criterion wouldn't have you bet 0
| every time? The chance of winning a round of rock-paper-
| scissors when throwing a random symbol seems to be 50% (if ties
| cause re-dos), so wouldn't that work out to 50 * 2 - 100 = 0?
| minitoar wrote:
| It's a good question. You're right, if I followed it strictly
| it wouldn't work. I suppose I rationalized offsetting it
| because I couldn't actually go bankrupt. If I was better at
| math there's probably some other criterion that takes into
| account how hard it would be to get back to where you are,
| given that you couldn't go below 1.
| dplavery92 wrote:
| That's strange, given that the Kelly criterion maximizes the
| expectation of the log of wealth--that is, it's maximizing over
| multiplicative percent gains in a scenario where you _can_ go
| bankrupt.
| minitoar wrote:
| 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.
| forinti wrote:
| I remember the first time I read about this. I put in the numbers
| for the lottery and a negative number came out. Of course! Your
| expected winnings are negative and you shouldn't play the
| lottery.
| jmharvey wrote:
| 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.
| kqr wrote:
| 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
|
| There's also a sequel for the case of continuous outcomes:
| https://static.loop54.com/ship-investor-2.html
|
| Before my parental leave is over, I hope to make two more
| sequels, one with futures and one with options. Maybe also a
| fixed-income version, but I'd have to learn more about that
| myself first.
| dshacker wrote:
| Hey, this is really cool! What's the optimal strategy? Would
| love to learn more
| tome wrote:
| I found good success with going for the smallest investment
| in Bering and the second smallest investment in the other two
| straits.
|
| The optimal strategy would be to estimate which investment
| maximises your log returns :) but I don't have time for that.
| bko wrote:
| If anyone wants to see Kelly in action, I made an app where you
| define an edge and a wager (% of pot or absolute amount) and see
| how you fare compared to the optimal bet strategy.
|
| https://kelly-criterion.netlify.app/
|
| https://github.com/breeko/kelly-criterion
| cyberlab wrote:
| Naval Ravikant has a small post about this here:
| https://nav.al/kelly-criterion
|
| I first heard about it from him. He summarizes it as follows:
|
| > Naval: The Kelly criterion is a popularized mathematical
| formulation of a simple concept. The simple concept is: Don't
| risk everything. Stay out of jail. Don't bet everything on one
| big gamble. Be careful how much you bet each time, so you don't
| lose the whole kitty.
| hogFeast wrote:
| Lol. He must have never met anyone who has bet full Kelly.
| auntienomen wrote:
| Seriously. Full Kelly betting involves the use of significant
| leverage. The correct Kelly bet on the S&P index would be
| long 2.5x your total wealth.
| xiphias2 wrote:
| 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).
| sigstoat wrote:
| it literally can't tell you to bet more than your bankroll.
|
| if you include margin in your bankroll, well, that's on
| your head.
| smabie wrote:
| 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.
| kqr wrote:
| This result depends on assumptions about the future that
| would not sit easy with me.
| xiphias2 wrote:
| I can't tell you if democrats or republicans will win,
| but I'm quite confident that QE won't stop.
| fny wrote:
| 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.
| aborsy wrote:
| 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.
| [deleted]
| larrydag wrote:
| Edward Thorp used the Kelly Criterion for success in blackjack
| strategies and later the stock market. He has articles on his
| statistical methods.
|
| http://www.edwardothorp.com/articles/
| tediousdemise wrote:
| Kelley betting could probably be applied with some success to
| momentum trading strategies. Momentum trading is more
| deterministic than purely speculative strategies since it is
| based on observed/historical behavior.
| objektif wrote:
| Momentum itself is as speculative as it gets.
| tediousdemise wrote:
| I would say pure speculation is not based on tangible data.
|
| Pure speculation: _I think consumer space travel will be
| popular in the future, let me buy some SpaceX shares_.
|
| Momentum: _SpaceX seems to be trading higher in pre-market,
| let me buy some SpaceX shares at market open_.
|
| Edit: I know SpaceX is not public, this is just an example.
| jmount wrote:
| I'd like to share a video I prepared on Kelly betting:
| https://youtu.be/6xhjbgREGDA
| nwsm wrote:
| I have used the Kelly Criterion successfully in automated sports
| gambling. It's relevant anywhere you are doing confidence-based
| arbitrage.
| waynecochran wrote:
| Did you determine the probability of winning based merely on
| the sport-book odds are do something more sophisticated?
|
| The sport-book odds, as I understand, are merely trying to
| divide the bets on each side evenly (i.e., they don't
| necessarily represent a probability).
| nwsm wrote:
| The bookie odds go into the formula as b- the net fractional
| odds received on the wager.
|
| We have our own models for our confidence, and the Kelly
| criterion decides our wager size (though we don't use a full
| Kelly bet).
|
| Yes, the sportsbook minimizes their own risk by setting a
| spread or odds with respect to how patrons are wagering. This
| actually makes it easier to make money if your model is much
| better than the average bettor. There will be games where
| public opinion and the majority of bets are on the wrong side
| of a matchup, and the bookie adjusts the odds accordingly, so
| the correct bet's payout is bigger than it should be.
|
| In high school I tried to do more what you are asking- use
| one bookie's odds (which I deemed the most accurate) as the
| "true probability", and another as the payout. This was not
| successful, but theoretically could be if the two bookies'
| clientele were consistently better or worse than each other,
| therefore influencing their odds consistently.
| kqr wrote:
| I can attest to this. Successful sports betting is about
| betting on gamblers, not games.
|
| As for your last points: bookies often book bets with each
| other in order to even out the odds. Otherwise you would be
| able to arbitrage bookies against each other. (Which would
| also result in their odds evening out, of course, but then
| the bookies wouldn't get the proceeds so they prefer to do
| it themselves.)
| senthil_rajasek wrote:
| Also worth looking at is this previous discussion on HN,
|
| https://news.ycombinator.com/item?id=13143821
| bxrxdx wrote:
| Its really fun to learn something new and realize how incredibly
| naive you've been your whole life.
| nicholast wrote:
| Hi, I wrote an essay about Kelly Criterion a while back based on
| a review of paper by Edward Thorpe. Cheers.
|
| https://medium.com/from-the-diaries-of-john-henry/an-optimal...
| aidenn0 wrote:
| 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!
| one-more-minute wrote:
| Some interesting psychology here:
|
| > In one study, each participant was given $25 and asked to place
| even-money bets on a coin that would land heads 60% of the time.
| Participants had 30 minutes to play, so could place about 300
| bets, and the prizes were capped at $250.
|
| > Remarkably, 28% of the participants went bust, and the average
| payout was just $91. Only 21% of the participants reached the
| maximum. 18 of the 61 participants bet everything on one toss,
| while two-thirds gambled on tails at some stage in the
| experiment.
| jfengel wrote:
| Did they know that it was biased towards heads? With only a
| 60-40 split I probably wouldn't notice it unless I was actually
| keeping track, which could take a while. A 6-4 split on 10
| tosses doesn't tell you anything. If you told me it was a fair
| coin and I thought the experiment was about something else, it
| might take a very long time before it occurred to me to test
| the hypothesis that the coin wasn't fair.
|
| If they knew it was biased... I'm sure there's an optimal
| strategy, but a simple strategy would be "bet half of what you
| have on heads every time". Any idea how much worse that is than
| the optimal strategy?
| kqr wrote:
| You can plot
|
| g = 0.6 log (1 + 2f) + 0.4 log (1 - f)
|
| And locate f=0.5 and compare to the maximum g.
|
| Edit: I wanted to check my intuition so I did: https://www.wo
| lframalpha.com/input/?i=plot++0.6+log+%281+%2B...
|
| Looks like 0.5 is a slight overbet, but still very, very
| good.
| tedunangst wrote:
| If only there was a link to the study so we could see how it
| was setup.
| [deleted]
| seoaeu wrote:
| > Did they know that it was biased towards heads?
|
| "Prior to starting the game, participants read a detailed
| description of the game, which included a clear statement, in
| bold, indicating that the simulated coin had a 60% chance of
| coming up heads and a 40% chance of coming up tails."
| frabjoused wrote:
| I made a little playground for this, you can fiddle with the
| numbers. https://parsebox.io/dthree/lnumtuenmskr
| ISL wrote:
| The paper is pretty awesome and accessibly-written:
| https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963
|
| The PDF is free-to-read.
| blowski wrote:
| > two-thirds gambled on tails at some stage in the experiment
|
| I'm not sure why that's called out. If you've just had 6 heads
| in a row the next 4 "should" be tails, so it's not irrational
| to bet on tails is it?
| afranchuk wrote:
| Those are independent variables. The fact you've had X heads
| has no bearing on the future flips. It is irrational to bet
| on tails statistically speaking, though psychologically that
| line of reasoning is common.
| theschwa wrote:
| No, the coin doesn't have a memory, so the chance of tails is
| still 40% making it still optimal to choose heads.
| skeeter2020 wrote:
| you're not betting on the number of heads/tails per 10 trials
| though, each trial is independent with a 60% of heads. In a
| striaght-up prediction you should always choose heads, it the
| how much to wager that is the question.
| baobabKoodaa wrote:
| > 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.
| vgeek wrote:
| I quickly searched but couldn't find the exact study, but
| I've read that by adding the past numbers digital signage
| to roulette tables, casinos experience a significant (I'm
| thinking it was like 100%+) increase in wagers when people
| believe that a color is "due" simply from not understanding
| independent vs dependent events. Humans love to look for
| patterns, even when there isn't any real _meaning_ behind
| them.
| dehrmann wrote:
| There's a corollary to the gambler's fallacy that says is
| P(heads) is 60% and you get 6 heads in a row, the people
| running the experiment probably lied to you.
| lupire wrote:
| but that means you should bet into the bias, not against
| it.
| dehrmann wrote:
| 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.
| intuitionist wrote:
| 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.
| 6gvONxR4sf7o wrote:
| 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.
| prometheus76 wrote:
| You've just discovered the Gambler's Fallacy.
| stocknoob wrote:
| 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?
| JacobLinney wrote:
| Yes it is irrational. That's a common statistical
| misconception, the key thing here is that _every_ flip has a
| 60% chance of being heads.
|
| The result of each flip is completely independent of what
| came before it. In your example the 7th flip is just as
| likely to be heads as the first flip, or any of the other 5
| flips that landed on heads.
| blowski wrote:
| 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.
| sokoloff wrote:
| 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.
| antasvara wrote:
| The key here is that it's expected to land heads 60% of
| the time. Take a normal coin, which is expected to land
| heads 50% of the time. If you flip a heads, do you
| instantly expect it to be tails next time? By your logic
| it would be impossible to ever flip heads twice in a row.
| Coins as a general rule aren't impacted by previous
| flips.
| bscphil wrote:
| 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%.
| justinpowers wrote:
| Each toss is independent of prior (and subsequent) tosses, so
| no matter what, a given tosshas 60% chance of landing heads.
| Rationally, one should bet heads on any given toss.
|
| But most people would agree with the irrational bet. This
| tendency is known as the Gambler's fallacy
| (https://en.wikipedia.org/wiki/Gambler's_fallacy).
| Jtsummers wrote:
| That's the gambler's fallacy in action. So long as each event
| is independent, the prior ones have no impact on the
| likelihood of future events. If you've flipped the coin 60
| times and they've all been heads, there's no reason to expect
| the next 40 will be tails. They still have better odds of
| being heads.
| travisjungroth wrote:
| If you see 60 heads in a row in the real world you've got a
| trick coin. The odds of that are 1/10^17.
| Jtsummers wrote:
| It's certainly low odds, but it's not impossible nor does
| it require a trick coin. I've seen people roll a 20 on a
| d20 10 times in a row, and then not a single 20 the rest
| of the session on the same die. Shit happens, it's
| probability and it may be improbable but it isn't
| impossible.
| lupire wrote:
| I don't believe you.
| Jtsummers wrote:
| 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.
| travisjungroth wrote:
| 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.
| Jtsummers wrote:
| 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.
| travisjungroth wrote:
| 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.
| [deleted]
| [deleted]
| dragonwriter wrote:
| 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.
| dragonwriter wrote:
| 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.
| mytherin wrote:
| The probability of a coin flip being heads or tails is
| completely independent from the previous flips. If the coin
| lands 6 heads in a row, the next coin flip still has a 60%
| chance of being heads, hence it is always unwise to bet on
| tails in this experiment. This is an example of the Gambler's
| fallacy [1].
|
| [1] https://en.wikipedia.org/wiki/Gambler%27s_fallacy
| ska wrote:
| > If you've just had 6 heads in a row the next 4 "should" be
| tails
|
| That's not how this works. Each toss is independent, so you
| should never pay attention to previous results if you know
| the true odds.
| deeg wrote:
| This wiki page can explain why better than me:
| https://en.wikipedia.org/wiki/Gambler%27s_fallacy
| sorokod wrote:
| 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.
| jfk13 wrote:
| No, the next toss still has a 60% chance of being heads. The
| coin doesn't remember how it landed last time.
| blowski wrote:
| If I'm expecting 60% of my flips to be heads, and I've
| already had 60%, isn't it more likely that the next one
| will be tails?
|
| I'm sure you can probably tell I know next to nothing about
| either maths or probability, so feel free to explain why
| I'm wrong.
| dragonwriter wrote:
| > 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.
| Jarwain wrote:
| 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
| EvanAnderson wrote:
| 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.
| ska wrote:
| Yes, you are wrong, but your confusion is very common.
| It's so common it even has a name "The Gamblers Fallacy".
|
| Over the long run, you expect 40% tails, but if you run
| the experiment an infinite amount of time there will be
| sequences of all-heads or all-tails.
|
| Because the events are independent, the previous flips
| don't change anything about what happens next.
| FredPret wrote:
| https://dilbert.com/strip/2001-10-25
| blowski wrote:
| So the probablity of 60% assumes infinite flips. Whereas
| I'm only flipping 10 times, so I won't necesssarily get
| 60% heads. I'd also need to know the probability that I'm
| in one of the cases where I get 60% of heads. Is that
| right?
| FredPret wrote:
| With a 60%-heads coin you can still get 10 straight
| heads. It's just that over many many flips, the average
| will gradually tend towards 60%.
|
| You can still have streaks of hundreds, thousands,
| millions of either heads or tails in a row.
| ska wrote:
| This gets into interesting stuff!
|
| So the situation described in that paper is that you are
| given the true odds of the coin, 60% heads. In this case
| it's just as I described - knowing previous results
| doesn't tell you anything useful.
|
| > Whereas I'm only flipping 10 times, so I won't
| necesssarily get 60% heads.
|
| This is true. In fact there is only about 25% chance of
| getting exactly 6 of the 10 to be heads (but nearly 70%
| chance of >= 6 heads). You can work this out with
| something called the binomial distribution. Chance of
| getting 10 heads in a row is .6%
|
| A more interesting aspect is when you don't know the odds
| (or don't trust what you've been told). In this case it's
| definitely important what the history is. So given your
| 10 flips, we can ask questions like "how likely is it
| that this coin is fair (50/50) given the 10 flips I just
| saw".
|
| It turns out the best estimation of the true probability
| is, pretty intuitively, (h+t)/h; this will jump aroudn
| for small N . In practice you are more often looking at
| something like P(0.55 < p < 0.65 | samples) , i.e. the
| probability that the true value lies between 0.55 and
| 0.65 heads, given the 10 flips I've seen).
|
| Obviously in these cases, the more samples you have seen
| the tighter the estimate get. You can also ask questions
| like how many flips do I need to see to be confident at a
| certain the coin is really 0.6 heads.
| gen220 wrote:
| 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).
| quickthrowman wrote:
| Previous tosses do not change the outcome of subsequent
| tosses, so no, it's not more likely to be tails, it has a
| 60% chance of being heads.
| antattack wrote:
| 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.
| WJW wrote:
| 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.
| baobabKoodaa wrote:
| 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.
| dplavery92 wrote:
| 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.
| curtainsforus wrote:
| 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?
| hansvm wrote:
| 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.
| tialaramex wrote:
| 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.
| function_seven wrote:
| Spot on.
|
| 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.
|
| I feel sorry for social science experimenters.
| btilly wrote:
| _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.
| tailwind wrote:
| "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.
| whatshisface wrote:
| Maybe once you've started to perceive the meta-patterns
| between psych experiments, you've taken too many tests to be
| a good subject.
| seoaeu wrote:
| 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.
| dragonwriter wrote:
| 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.
| lupire wrote:
| Biased coins are *impossible" to make if the coin is flipped
| not spun.
|
| I doubt any story about a biased coins in the real world.
| function_seven wrote:
| 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.
| lupire wrote:
| 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.
| shezi wrote:
| 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.
| function_seven wrote:
| 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.
| unanswered wrote:
| 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".)
| tedunangst wrote:
| 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.
| kortilla wrote:
| Sometimes, but they have a habit of lying about the
| purpose.
| tedunangst wrote:
| 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.
| syassami wrote:
| If you enjoyed this, I highly recommend reading Fortune's
| Formula.
| kqr wrote:
| And when you're done with that, the Kelly Capital Growth
| Investment Criterion is one of the better books I've read. But
| it's a much more advanced read.
| _e wrote:
| 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).
| ttty wrote:
| Kelly criterion (or Kelly strategy or Kelly bet) is a formula for
| bet sizing that leads almost surely to higher wealth. It was
| described by J. L. Kelly Jr, a researcher at Bell Labs, in 1956.
| The Kelly Criterion is to bet a predetermined fraction of assets,
| and it can seem counterintuitive. In one study, each participant
| was given $25 and asked to place even-money bets on a coin that
| would land heads 60% of the time. 18 of the participants bet
| everything on one toss, while two-thirds gambled on tails. The
| average payout was just $91.36.
|
| Kelly bet: Bet one-nineteenth of the bankroll that red will not
| come up. If gamble has 60% chance of winning, gambler should bet
| 20% of bankroll at each opportunity. In American roulette, the
| bettor is offered an even money payoff (even money payoff, with a
| 95% probability of reaching the cap) If edge is negative, then
| the formula gives a negative result, indicating that the gambler
| shouldn't bet. The top of the first fraction is the expected net
| winnings from a $1 bet, since the two outcomes are that you
| either win with probability or lose the $1 wagered, i.e. win $ +
| $1, with probability $+.1. For even-money bets, the formula can
| be simplified to the first simplified formula: b=p-q-q/p, b=1,
| and b=0.20.
|
| Probability of success is: If you succeed, the value of your
| investment increases. If you fail (for which the probability is):
| In that case, the price of the investment decreases. Kelly
| criterion maximizes the expected value of the logarithm of
| wealth. Using too much margin is not a good investment strategy
| when the cost of capital is high, even when the opportunity
| appears promising. The general result clarifies why leveraging
| (taking out a loan that requires paying interest in order to
| raise investment capital) decreases the optimal fraction to be
| invested, as in that case the odds of winning are less than 0.1.
| The expected profit must exceed the expected loss for the
| investment to make any sense. The Kelly criterion can be used to
| determine the optimal amount of money to invest.
|
| Kelly has both a deterministic and a stochastic component. If one
| knows K and N and wishes to pick a constant fraction of wealth to
| bet each time (otherwise one could cheat and, for example, bet
| zero after the Kth win, that the rest of the bets will lose), one
| will end up with the most money if one bets. In the long run,
| final wealth is maximized by setting the derivative to zero,
| which means following the Kelly strategy. The function is
| maximised when this derivative is equal to zero. For a rigorous
| and general proof, see Kelly's original paper[1] or some of the
| other references listed below. Some corrections have been
| published[12] .
|
| In the long run, Kelly always wins. This is true whether N is
| small or large. In practice, this is a matter of playing the same
| game over and over, where the probability of winning and the
| payoff odds are always the same. Betting each time will likely
| maximize the wealth growth rate only in the case where the number
| of trials is very large, and the odds of winning are the same for
| each trial. The heuristic proof for the general case proceeds as
| follows. In a single trial, if you invest the fraction f of your
| capital, if your strategy succeeds, your capital at the end of
| the trial increases by the factor 1-f+f(1+b) Exponential
| mechanism (differential privacy)
| dang wrote:
| 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.
| floatingatoll wrote:
| (This appears to be a copy-paste of the Wikipedia article.)
| jedberg wrote:
| Sadly this only works in games with a positive expected outcome,
| so it's not actually useful in a casino unless you're a card
| counter.
| hansvm wrote:
| Nah, you just use the technique to conclude that the optimal
| percentage of your bankroll to bet to maximize expected
| logarithmic wealth is 0% ;)
| anonu wrote:
| This has been posted 5 other times on HN with no real discussion
| [1].
|
| I'll add my 2 cents: I used to use the principles of kelly
| betting back when I designed systematic HFT strategies. It gives
| you a good framework to think about how much to bet based on the
| batting average of a particular pattern you recognize in the
| market...
|
| [1]
| https://hn.algolia.com/?q=https%3A%2F%2Fen.wikipedia.org%2Fw...
| voldacar wrote:
| > 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?
| nwsm wrote:
| 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.
|
| [1] https://en.wikipedia.org/wiki/John_Larry_Kelly_Jr.
| smabie wrote:
| Hi I work at a small hft firm and would love to discuss this
| more in detail, please contact me if you have the time.
|
| Thank you
| jared_buckner wrote:
| There's a bit of a discussion here:
| https://news.ycombinator.com/item?id=18484631
| fighterpilot wrote:
| How did you apply Kelly to a HFT strategy? Usually those strats
| don't have a binary outcome so standard Kelly wouldn't fit.
| dcolkitt wrote:
| For continuous payoffs, Kelly sizing reduces to the square of
| Sharpe ratio.
| howlin wrote:
| Kind of. Most simple models for continuous payoffs will
| assign a nonzero probability to losing all your wealth or
| your wealth going negative. The Kelly bet size for any
| thing with a nonzero chance of "ruin" is zero.
| dcolkitt wrote:
| 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.
| kgwgk wrote:
| A proper Sharpe ratio is calculated with arithmetic
| returns.
| fighterpilot wrote:
| Where did you see this?
| potatoman22 wrote:
| Not sure if it's how they did it, but there's this: https://e
| n.wikipedia.org/wiki/Kelly_criterion#Multiple_outco...
| kqr wrote:
| 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.
| fighterpilot wrote:
| Suppose we have 100 sequential bets with distribution
| U(-1,1.1) on each. How would we apply Kelly here?
| [deleted]
| hansvm wrote:
| 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.
| sigstoat wrote:
| 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.
| fighterpilot wrote:
| 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.
| sigstoat wrote:
| 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
|
| hahaha.
| fighterpilot wrote:
| Right so just do a simulation, no closed form solution.
| sigstoat wrote:
| that's not simulation.
|
| for that trivial case, there's going to be a closed form
| solution. your nearest copy of mathematica can derive it
| for you.
|
| not that having a closed form solution is relevant to
| anything. the answer is still the answer.
| [deleted]
| treesrule wrote:
| brb updating the stardew valley wiki
| https://stardewvalleywiki.com/Stardew_Valley_Fair
| sl8r wrote:
| 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".
| xiphias2 wrote:
| 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.
| barbazoo wrote:
| Do you have examples of how that would be used in real life
| decisionmaking?
| CodesInChaos wrote:
| 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.
| xiphias2 wrote:
| 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.
| xiphias2 wrote:
| 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.
| smabie wrote:
| I have an old blog post about the subject:
| https://cryptm.org/posts/2019/10/04/vol.html
|
| Optimal over my time period was 2.99x, but the expense
| ratio was not accounted for.
| lutorm wrote:
| 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?
| xiphias2 wrote:
| 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.
| breck wrote:
| 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.
| xiphias2 wrote:
| 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.
| joosters wrote:
| Betting with 'full' Kelly-calculated stakes is highly volatile.
| If I'm remembering correctly, if you get your
| probabilities/edge exactly right, you will still have a 50/50
| chance of losing half of your bank at some point in the future
| (i.e. after some number of future bets) It's very common to bet
| just some fraction of the Kelly stakes in order to smooth out
| the roller coaster ride.
| xiphias2 wrote:
| 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.
| kqr wrote:
| 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.)
| smabie wrote:
| 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)
| keithalewis wrote:
| A Nobel winning economist was not impressed by the Kelly
| criterion. http://www-
| stat.wharton.upenn.edu/~steele/Courses/434F2005/C...
| TameAntelope wrote:
| Kelly himself ended up using 1/n for his own personal portfolio
| management.
|
| Gerd Gigerenzer has a _lot_ to say about how harmful this model
| has been to finance and the world, because it creates "false
| certainty".
|
| https://news.ycombinator.com/item?id=26325425 has further
| discussion.
| fighterpilot wrote:
| 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.
| kqr wrote:
| I'm not sure what you mean by using 1/n, but the Kelly
| criterion optimised on past returns for common portfolios of
| thickly traded assets does suggest something very close to 1/n
| very often.
|
| I've always attributed this to market efficiency (if it
| suggested anything else, that's what investors would do until
| the mispricing went away) but maybe there's a deeper reason it
| happens.
| TameAntelope wrote:
| 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 "naive 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 naive and too simple, by McClatchy and VandenHul
| (2005) for example.
|
| http://arno.uvt.nl/show.cgi?fid=129399
|
| Gerd Gigerenzer has a number of books, the one I recently
| read was, "Risk Savvy" and he goes into some detail about the
| topic. All I'd do here is write a terrible book review, so if
| you're curious, I definitely recommend taking a look at the
| book. I'm not sure I totally agree with his arguments (I had
| a hard time understanding how he would suggest accounting for
| human bias), but they're definitely interesting.
| haltingproblem wrote:
| 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.
| TameAntelope wrote:
| 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!
| a11r wrote:
| For anyone actively managing investment portfolios, a deep
| understanding of the Kelley criterion is very important. For
| example, it is common practice to use "Half Kelly" to size
| positions, but most sources only provide a hand-wavey intuitive
| explanation. Thorp's paper[2] quantifies the benefits for any
| fraction of the "full Kelly" bet and its implications. In
| addition to Poundstone's book [1] I strongly recommend Ed Thorp's
| highly readable paper[2].
|
| [1] Poundstone, William (2005), Fortune's Formula: The Untold
| Story of the Scientific Betting System That Beat the Casinos and
| Wall Street, [2]https://wayback.archive-
| it.org/all/20090320125959/http://www...
| User23 wrote:
| For any kind of trading activity, the most important skill to
| have is risk management. You can get everything else completely
| wrong, but if you have your risk management down you're still
| in the game and can learn from your mistakes. If you don't
| you're liable to be ruined and be out of the game until you can
| build back a bankroll some other way.
| hodder wrote:
| 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.
| kqr wrote:
| 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.
| pushrax wrote:
| There are kind of two main points by Kelly:
|
| 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.
| kqr wrote:
| I would be surprised and perhaps a little disappointed if
| any professional investors think of E log X optimisation
| as the latter.
| pushrax wrote:
| 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.
| fighterpilot wrote:
| Second this. Nobody actually uses it. It's a bit too
| theoretical.
| howlin wrote:
| 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.
| andrewprock wrote:
| 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.
| anotheranon631 wrote:
| https://blog.alphatheory.com/2013/01/kelly-criterion-in-
| prac...
| robocat wrote:
| 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.
|
| [2] https://blog.alphatheory.com/2013/01/kelly-criterion-
| in-prac...
|
| [0] https://alphatheory.zendesk.com/hc/en-
| us/articles/3600356960... has an explanation of the "Alpha
| Theory" which I couldn't quickly find on the alpha theory
| site.
| mrfredward wrote:
| Take a look at the "Many Assets" subheading in the
| original post.
| robocat wrote:
| 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.
| hogFeast wrote:
| 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.
| anotheranon631 wrote:
| " 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.
| hogFeast wrote:
| Correct. That is the gap in understanding here.
|
| 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).
| anotheranon631 wrote:
| 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.
| throwaway823882 wrote:
| > It is important to understand how a tool works at a
| practical level.
|
| There needs to be a term for "This page you're reading is
| bogus horseshit theory, do not try to apply it
| practically".
| WanderPanda wrote:
| "Academic"
| throwastrike wrote:
| 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.
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