[HN Gopher] Kelly Can't Fail
___________________________________________________________________
Kelly Can't Fail
Author : jmount
Score : 341 points
Date : 2024-12-19 23:07 UTC (23 hours ago)
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| dado3212 wrote:
| Very cool writeup, would've benefited from some LaTeX formatting.
| jmount wrote:
| Thank you, and sorry. The Wordpress/Markdown path seems to be
| getting worse over time.
| smcin wrote:
| How is the Wordpress/Markdown path getting worse over time?
| jmount wrote:
| Mathjax used to be available more places. Jupyter used to
| respect spacing around h2 labels. Things like that.
| Vecr wrote:
| Checks out with multiple RNG seeds.
|
| It shouldn't be a problem because the RNG is advanced each run.
| Might save someone a check though.
| jmount wrote:
| I love that. Gaming the seed is always a possibility in demos.
| hawkjo wrote:
| Very cool to see no variance in the outcome. But that also makes
| it feel like there should be a strategy with better expected
| return due to the unique problem structure. Do we know if the
| Kelly strategy is optimal here?
| jmount wrote:
| The book claims it is optimal for a set of strategies they
| called "sensible." I didn't think the argument flowed as well
| as the zero variance part of the proof, so I didn't work it in.
| I think the source also hinted at a game-theory proof as they
| called the sub-strategies in the portfolio "pure strategies."
| rahimnathwani wrote:
| Do we know if the Kelly strategy is optimal here?
|
| What do you mean by optimal? Do you mean you're willing to risk
| going bankrupt, if it means a higher expected value?
| scotty79 wrote:
| Surely there's some space between risking to go bankrupt and
| risking of getting less than 9.08 return guaranteed by Kelly
| strategy.
|
| If you are willing to take some risk in exchange for
| possibility of higher payout just bet a bit more then Kelly
| recommends. That's your "optimal" strategy for the amount of
| risk you are willing to take. I imagine it's expected return
| is the same as Kelly and calculating it's variance is left as
| the exercise for the reader.
| rahimnathwani wrote:
| I imagine it's expected return is the same as Kelly
|
| Given two options with the same expected return, most
| people would prefer the lower variance.
|
| Accepting higher variance with no increase in expected
| return has a name: gambling.
| barbegal wrote:
| It is optimal for expected returns yes.
| travisjungroth wrote:
| I have a feeling it's the highest EV. I tried a strategy of
| flipping all the cards until there's only one color left and
| then betting it all every time. Ran a million trials and got
| 9.08.
|
| I was thinking these are very different strategies, but they're
| not exactly. The Kelly strategy does the same thing when
| there's only one color left. The difference is this strategy
| does nothing before that point.
|
| Still, they feel like limit cases. Betting it all with only one
| color left is the only right move, so it's what you do before
| that. Nothing and Kelly seem like the only good strategies.
| foota wrote:
| Ah, but these aren't the same. The Kelly strategy has zero
| variance, whereas this strategy likely has very high
| variance.
|
| It would be interesting to do the math and show why they're
| equal. It seems like you should be able to make the same sort
| of portfolio probability argument.
| foota wrote:
| To start, your minimum return is 2x, and depending on how
| many cards of a single color are left at the end, you get a
| return of 2^N. You could take the summation of those N card
| returns, times the probability of each, and that must come
| out to 9.08 on average.
|
| I guess the number of possible arrangements of cards with N
| of one color remaining is... The number of permutations of
| N times 2 times the number of permutations of 52 minus N
| times 26 choose N?
|
| Ah, yes this works, you can see it here: https://www.wolfra
| malpha.com/input?i=%28summation+of+N%21+*+....
|
| That is: (summation of N! * (52 - N)!* (26 choose N) *
| 2^N/52! from N=0 to 26 (for some reason the * 2 for
| different suits was over counting, so I removed it. Not
| sure why? Also it seems like it should be from 1 to 26, but
| that also doesn't give the right answer, so something is
| whack)
| travisjungroth wrote:
| Of course they're not the same. They have the same EV and
| the strategies do the same thing in a condition that always
| happens: there's only one color left. The variance is
| wildly different.
| lupire wrote:
| The Kelly criterion _is_ the strategy with better return due to
| the uniquely problem structure.
| OscarCunningham wrote:
| In this game, all strategies have the same expected value, so
| long as they follow the rule 'if the remaining deck is all the
| same colour, then you should bet everything you have on that
| colour'.
| raydiak wrote:
| As a guy named Kelly, I appreciate the vote of confidence!
| pvg wrote:
| I think you're underselling it a bit, it's a decree of
| confidence rather than a mere vote.
| barbegal wrote:
| It would have been a better demo if reduced to more manageable
| numbers e.g. a deck of 2 black and 2 red cards.
|
| Turn 1 r = b so no bet
|
| Turn 2 bet 1/3 on whichever card wasn't revealed in turn 1.
|
| Turn 3 either you were wrong on turn 2 and you now have 2/3 of
| your stake but you know the colour of the next two cards so you
| can double your stake each time to end up with 4/3 after turn 3
| or you were right and you have 4/3 of your stake but have one of
| each red or black left so you don't bet this turn.
|
| Turn 4 you know the colour of the final card so you double your
| money to 8/3 of your original stake.
|
| And then the exercise to the reader is to prove optimality (which
| is fairly straightforward but I don't believe there is a short
| proof)
| stevage wrote:
| Agreed, I could follow the general argument but not enough to
| be convinced about why the result is exactly the same
| regardless of the order of cards.
| libraryofbabel wrote:
| Yes. Although four cards has only one nontrivial branch, on
| turn 3. So, start out with the four cards example, and then
| show tree diagrams for the 5 and 6 cards cases (still
| manageable numbers) to build intuition for induction to the
| general case.
| siavosh wrote:
| Can anyone comment on the universal portfolio article linked in
| the conclusion? Asking for a friend.
| robbomacrae wrote:
| It's a theory on how to optimally rebalance your investment
| portfolio every day.
|
| Original paper by Thomas Cover:
| https://isl.stanford.edu/~cover/papers/paper93.pdf
|
| A good breakdown with links to code examples by Andy Jones:
| https://andrewcharlesjones.github.io/journal/universal-portf...
| malisper wrote:
| I need to do some Math, but I wonder if there's a better strategy
| than Kelly betting. An assumption made for Kelly betting is the
| bets are independent of each other. That's not the case in the
| problem given.
|
| After making a bet, you gain information about the contents of
| the rest of the deck of cards. I could see it being possible to
| do better by pricing in that information into your bet.
| amluto wrote:
| But the information gained in this game is independent of your
| bet. The multi-armed bandit problem is a famous example of the
| opposite situation.
| necovek wrote:
| That seems to be exactly what this strategy is doing: at every
| step, you account for the probability of the red or black card
| coming up, and bet accordingly (both the sum and the colour).
| tooblies wrote:
| It's really disappointing that the code examples aren't given in
| PyGyat.
| pcthrowaway wrote:
| Note that you need to be able to infinitely divide your stake for
| this to work out for you all the time.
|
| For example, if the deck has 26 red cards on top, you'd end up
| dwindling your initial $1.00 stake to 0.000000134 before riding
| it back up to 9.08
| jmount wrote:
| Very good point. I did some experiments and the system is very
| sensitive to any sort of quantization or rounding of bets. You
| get the expected value about the right place, but the variance
| goes up quickly. So in addition to your important case, things
| are a bit dicey in general.
| boothby wrote:
| If you start out with a $1e12 stake, you're able to avoid
| catastrophic rounding errors even in the worst case. There's
| probably a life lesson here.
| fragmede wrote:
| Is the lesson: choose to be born to wealthy parents?
| darkerside wrote:
| Or is it to choose appropriate betting amounts based on
| your capacity for risk
| Onavo wrote:
| IEEE-754 isn't precise enough for my capacity :( I too
| need rich parents.
| laidoffamazon wrote:
| I guess it then does follow that having rich parents does
| expand your capacity for risk!
| darkerside wrote:
| This is a truism
| User23 wrote:
| The lesson I'm taking away is "learn math and how to use
| it."
| barrenko wrote:
| Learn math and discover poignantly all the situations
| where it is effectively useless.
| paulluuk wrote:
| Applying math to a more practical betting situation, like
| poker, is a lot harder. You'd have to be able to
| calculate your exact odds of winning given only a small
| amount of information, without a calculator and without
| it taking so long that the other players notice, and then
| also factor in the odds that the other players are
| bluffing and the advantages that you might have from
| (not) bluffing.
| Etheryte wrote:
| Or is it to choose appropriate betting amounts based on
| your parents?
| croes wrote:
| It's easier to make money if you already habe money
| mannykannot wrote:
| It would really help if your parents know someone who can
| and will take the other side in this game.
| renewiltord wrote:
| A popular view is that having wealthy parents gives one a
| great advantage. Another popular view is that working
| extraordinarily hard for money is a waste of one's life
| even if one gets the money. But the two are only consistent
| if one believes that one's own life is the optimization
| target. If I live a life of misery so that my children live
| a life of prosperity that would strike me as a phenomenal
| result.
|
| So another reading is "choose to give your children wealthy
| parents".
| cbsks wrote:
| My simulation shows that with a 52 card deck, if you round
| the bet to the nearest $.01 you will need to start with
| $35,522.08 to win a total of $293,601.28.
|
| If you start with $35,522.07 or less, you will lose it all
| after 26 incorrect cards.
| boothby wrote:
| Nearest rounding does seem like a mistake here. Rounding
| down is quite safe: rather than lose it all, you end up
| with at least 2^26 pennies.
| kamaal wrote:
| >>Note that you need to be able to infinitely divide your stake
| for this to work out for you all the time.
|
| This is what most people discover, you need to play like every
| toss of the coin(i.e tosses over a very long periods of time).
| In series, like the whole strategy for it to work as is. You
| can't miss a toss. If you do you basically are missing out on
| either series of profitable tosses, or that one toss where you
| make a good return. If you draw the price vs time chart, like a
| renko chart you pretty much see a how any chart for any
| instrument would look.
|
| Here is the catch. In the real world stock/crypto/forex trading
| scenario that means you basically have to take nearly trade.
| Other wise the strategy doesn't work as good.
|
| The deal about tossing coins to conduct this experiment is _you
| don 't change the coin during the experiment_. You don't skip
| tosses, you don't change anything at all. While you are trading
| all this means- You can't change the stock that you are
| trading(Else you would be missing those phases where the
| instruments perform well, and will likely keep landing into
| situations with other instruments where its performing bad),
| you can't miss trades, and of course you have to keep at these
| for very long periods of time to work.
|
| Needless to say this is not for insanely consistent. Doing this
| day after day can also be draining on your mental and physical
| health, where there is money there is stress. You can't do this
| for long basically.
| teo_zero wrote:
| While I don't agree on nearly anything you stated, I enjoyed
| your prose: I suppose you left out words here and there as a
| metaphorical proof of your claim that you can't miss a single
| toss, didn't you?
| kamaal wrote:
| >>I suppose you left out words here and there as a
| metaphorical proof of your claim that you can't miss a
| single toss, didn't you?
|
| You must always practice in real world conditions. Notice
| in the experiments conducted in programs, you are taking
| series of tosses as they come, even if they are in
| thousands in numbers, one after the other, without missing
| a single one. Unless you can repeat this in a live
| scenario. This is not a very useful strategy.
|
| Kelly criterion is for people who are planning to take
| large number of trades over a long period of time, hence
| the idea is to ensure failures are not fatal(this is what
| ensures you can play for long). As it turns out if you play
| for really long, even with a small edge, small wins/profits
| tend to add to something big.
|
| If you remove all the math behind it, its just this. If you
| have a small edge to win in a game of bets, find how much
| you can bet such that you don't lose your capital. If you
| play this game for long, like really really long, you are
| likely to make big wins.
| teo_zero wrote:
| You are conflating 2 concepts: a) that the reality
| converges to what the theory predicts only after a great
| number of samples; b) that if you skip some events the
| results will vary.
|
| Now, b) is false. You can change the code to extract 3
| random numbers each time, discard the first 2 and only
| consider the third one, the results won't change.
|
| Instead a) is generally true. In this case, the Kelly
| strategy is the best strategy to play a great number of
| repeated games. You _could_ play some games with another
| strategy and win more money, but you 'll find that you
| can't beat Kelly in the long term, ideally when the
| repetitions approach infinity.
| kamaal wrote:
| >>Now, b) is false. You can change the code to extract 3
| random numbers each time, discard the first 2 and only
| consider the third one, the results won't change.
|
| Might be in theory. In practice, this is rarely true.
|
| Take for example in trading. What happens(is about to
| happen), depends on what just happened. A stock could
| over bought/over sold, range bound, moving in a specific
| direction etc. This decides whats about to happen next.
| Reality is rarely ever random.
|
| Im sure if you study a coin toss for example, you can
| find similar patterns, for eg- if you have tired thumb,
| Im pretty sure it effects the height of the toss,
| effecting results.
|
| >>Instead a) is generally true. In this case, the Kelly
| strategy is the best strategy to play a great number of
| repeated games.
|
| Indeed. But do make it a point to repeat exact sequences
| of events you practiced.
| auc wrote:
| If you assume coin tosses are independent, it shouldn't
| matter if you miss coin tosses.
| kamaal wrote:
| Coin tosses are not independent. Unless the premise is
| coins toss themselves.
|
| A person tosses a coin, so tosses are are connected to each
| other.
|
| Ask yourself this question- Would your thumb hurt if you
| toss a coin 5000 times? If so, would that change the
| results?
| PaulHoule wrote:
| Naturally tossed coins tend to land on the same side they
| started with 0.51 of the time, see
|
| https://www.stat.berkeley.edu/~aldous/157/Papers/diaconis
| _co...
| aidenn0 wrote:
| Linked paper does not state that; it states that tossed
| coins tend to be _caught_ on the same side they stared
| with slightly more than half the time. The results
| explicitly exclude any bouncing (which will happen if a
| coin lands on a hard surface).
|
| The paper does discuss coins allowed to land on a hard
| surface; it is clear that this will affect the
| randomness, but not clear if it increases or decreases
| randomness, and suggests further research is needed.
| tgma wrote:
| Yup, the dual would be saying Martingale can't fail with
| infinite money.
| aidenn0 wrote:
| It's not because there is a finite amount of money at which
| this can't fail, which is never the case for martingale.
| Martingale is actually likely to bankrupt you against a
| casino that is much more well staked than you _even if_ you
| have a small advantage.
| ab_goat wrote:
| Finally a real world use case for bitcoin!
| nyeah wrote:
| It's a good point. I think it affects the realism of the model.
| When the stake is very low, finding a penny on the street gives
| an astronomical improvement in the end results. At the high
| end, it's possible the counterparty might run out of money.
| ed-209 wrote:
| why use a static seed on the random generator and could that be
| making this appear more interesting than it might otherwise?
| jfengel wrote:
| The idea is sound. The static seed is presumably so the results
| are repeatable, but it works for true randomness. (Assuming you
| were permitted to do it this way, which you wouldn't be.)
| fancy_pantser wrote:
| A very similar card game played by deciding when to stop flipping
| cards from a deck where red is $1 and black is -$1 as described
| in Timothy Falcon's quantitative-finance interview book (problem
| #14). Gwern describes it and also writes code to prove out an
| optimal stopping strategy: https://gwern.net/problem-14
| jmount wrote:
| That is a nice game and writeup.
| snthpy wrote:
| Nice!
|
| Only quibble i have is that black should be +$1 and red -$1 to
| follow standard finance conventions, i.e. be in the "black" or
| "red".
| JohnMakin wrote:
| Kelly criterion is one of my favorite game theory concepts that
| is used heavily in bankroll management of professional gamblers,
| particularly poker players. It is a good way to help someone
| understand how you can manage your finances and stakes in a way
| that allows you to climb steadily forward without risking too
| much or any ruin, but is frequently misapplied in that space. The
| problem is kelly deals with binary results, and often situations
| in which this is applied where the results are not binary (a
| criteria for applying this) you can see skewed results that look
| almost right but not quite so, depending on how you view the math
| amluto wrote:
| > particularly poker players
|
| The Kelly criterion seems excellent for many forms of gambling,
| but poker seems like it could be an exception: in poker, you're
| playing against other players, so the utility of a given
| distribution of chips seems like it ought to be more
| complicated than just the number of chips you have.
|
| (I'm not a poker player.)
| tempestn wrote:
| It's used for bankroll management (basically deciding what
| stakes to play) rather than for sizing bets within a
| particular game.
| fernandopj wrote:
| Chris "Jesus" Ferguson "proved" an application of this theory
| back in ~2009 [1]. He was a the time promoting Full Tilt and
| commited to turn $1 dollar bankroll to $10000 by applying a
| basic strategy of never using more than a low % of his
| bankroll into one tournament or cash game session.
|
| So, if one's skill would turn your session probability to
| +EV, by limiting your losses and using the fact that in poker
| the strongest hands or better tourney positions would give
| you a huge ROI, it would be just a matter of time and
| discipline to get to a good bankroll.
|
| Just remember that for the better part of this challenge he
| was averaging US$ 0.14/hour, and it took more than 9 months.
|
| [1] https://www.thehendonmob.com/poker_tips/starting_from_zer
| o_b...
| kelnos wrote:
| > _Just remember that for the better part of this challenge
| he was averaging US$ 0.14 /hour, and it took more than 9
| months._
|
| But consider the rate of return! He turned $1 into $10,000
| in 9 months. Could he then turn that $10k into $100M in
| another 9 months?
|
| Or if he'd started with $100 instead of $1, could he have
| turned that into $1M in 9 months? That would still be
| incredibly impressive.
|
| Certainly the game changes as the bets and buy-ins get
| higher, but even if he couldn't swing the same rate of
| return with a higher starting point and larger bets (though
| still only betting that same certain low percent of his
| bankroll), presumably he could do things like turning $5k
| into $1M. Even $100k into $1M would be fantastic.
| lupire wrote:
| I think the challenge is that the larger you bet, the
| harder it is to find people who are bad at basic strategy
| poker but willing to bet against you for a long series of
| bets.
| bloodyplonker22 wrote:
| You are right that Kelly criterion deals with binary results.
| This won't work for poker. In poker, we use expected value
| because wins and losses are not binary because of the amount
| you win or lose. Once you figure out your approximate EV, you
| use a variance calculator in addition to that (example:
| https://www.primedope.com/poker-variance-calculator/) to see
| how likely and how much it is you will be winning over a
| certain number of hands in the long run.
| peter_retief wrote:
| Could this work with roulette betting on color? Seems like you
| could spend a lot of time not winning or losing
| plorkyeran wrote:
| Roulette results are uncorrelated and you have the exact same
| chance of winning each time, so the Kelly criterion isn't
| applicable. Betting on a color has a negative edge and you
| don't have the option of taking the house's side, so it just
| tells you the obvious thing that you should bet zero.
| dmurray wrote:
| > exact same chance of winning each time, so the Kelly
| criterion isn't applicable.
|
| Actually, the main assumption that leads to the Kelly
| criterion is that you will have future opportunities to bet
| with the same edge, not constrained by the amount.
|
| For example, if you knew this was your last profitable
| betting opportunity, to maximise your expected value you
| should bet your entire stake.
|
| I'm slightly surprised it leads to such a nice result for
| this game - I don't see a claim that this is the optimal
| strategy for maximizing EV zero variance is great, but
| having more money is also great.
|
| Of course you are right about roulette and, if you are
| playing standard casino roulette against the house, the
| optimal strategy is not to play. But that's not because
| bets are uncorrelated, it's because they are all negative
| value.
| Tepix wrote:
| What makes 0 better than the other numbers?
| Vecr wrote:
| Can't bet negative in that kind of game. If a game is
| expected to lose you money, don't play.
| lupire wrote:
| $0, not 0 on the wheel.
| amluto wrote:
| > The problem and solution appear to come from Thomas Cover.
|
| I don't recall this specific example, but I learned about the
| Kelly criterion in a class that Thomas Cover taught. He was one
| of my favorite teachers, and any discussion with him was
| guaranteed to be interesting and worthwhile. RIP.
| moonlion_eth wrote:
| I was like "oooh fun a card game" then was like "oh shit I'm too
| dumb for this math"
| IAmGraydon wrote:
| You aren't dumb. You just don't have enough exposure to the
| prerequisites.
| necovek wrote:
| It could also be both: though it's not necessarily that they
| are "dumb", but that the language of mathematics is something
| they can't get their head around, even if they can understand
| the concepts when described in spoken language.
|
| Eg. it's probably pretty easy to convince them that with 15
| cards in a deck, out of which 5 are red and 10 are black,
| chances are bigger (and in particular 10/15 or ~67%) that
| they'll pull out a black card, and that you should bet more
| on this happening. If you happen to miss, you should only bet
| even more on black since the chances grow further -- to be
| able to maintain this strategy, you only need to never bet
| too much so you have enough "funds" to bet all the way
| through (eg. in the worst case where the least likely thing
| happens: in my example, that would be 5 red cards coming up
| first).
|
| Putting all this reasoning into formulae is what math is, and
| I do believe some struggle with abstracting these more than
| others (which is why the divide does exist and why many
| people believe those good at math are "smart", which is very
| much not so -- seen plenty of "stupid" mathematicians, even
| professors). Does not make them "dumb", but might make them
| "modern math dumb". A signal that someone can be good at math
| today is that they are unfazed with more-than-3-dimensional
| spaces (you need to stop tying things to physical world).
| lupire wrote:
| Corollaries, by considering different deck shufflings, such as
| perfectly interleaved as perfectly separated:
| 9.08 ~ 52/52 x 52/51 x 50/50 / 50/49 x ... 2/2 x 2/1
| = 52/51 x 50/49 x ... x 2/1 = 2^52 x 26!2 / 52!
| = (52/52 x 50/51 x ... x 2/27) x (52/26 x 50/25 x ... x 2/1)
|
| and these equalities can also be directly verified algebraically
|
| This also points to a non-"many worlds"/portfolio version of the
| prod of zero-variance.
|
| Every bet is e/d, where e is current edge and d is current deck
| size. So every outcome multiplies the stack by (d + e x
| (-1)^i)/d, where is +-1, depending on win or lose.
|
| Note that the product of all the values of d is constant, so we
| can ignore the denominator.
|
| Since we know (from the OP proof) that the product of these
| numbers is constant for all shuffles of the deck, we can split a
| shuffled deck anywhere such that both parts are balanced
| red=blue, and the total (multiplicative) return over each part of
| the deck is constant across all shuffling of that part of the
| deck. (There are at least two ways to prove this part!)
|
| This is gives a further hint toward another fascinating fact:
| over any span of the deck between points where the deck is
| balanced, the numerators of the bet results double-cover all the
| even numbers between the starting and ending deck size.
|
| To see why:
|
| * A loss after a loss has a numerator (deck minus edge) of 2 less
| than the previous bet, as the deck size decreased by 1 and the
| edge has inccreased by 1.
|
| * A win after a win also has a numerator (deck plus edge) of 2
| less than the previous bet, as the deck size decreased by 1 and
| the edge has decreased by 1.
|
| * A win after a loss, causes a big swing in the numerator,
| exactly back to the largest not yet double-covered numerator that
| started the streak that just ended. Then the new win streak
| continues making the second cover of even numerators, until... a
| loss after a win jumps the numerator back to continuing the
| sequence of decreasing even numberators, which will get their
| second cover later when the later wins come.
|
| Since the deck is balanced, the number of wins always equals the
| number of losses, as long as we consider the 0 wager on a
| balanced subdeck to be a loss, since it increases the edge like
| non-degenerate losses do.
|
| (When the deck is balanced, edge is 0, so the return of no-bet is
| same as a win is same as a loss)
|
| You can visualize the numerator changes like so: a crane is
| driving from 52 to 0. Its arm is pointing either forward or
| backward, and there is a counterweight of the same length
| pointing in the opposite direction. At each step, the crane arm
| is either pointing toward 0 and stretches another step toward 0,
| or points backward to 52 and shrinks (toward 0 milestone and
| toward 0 arm length), or it swings to the other direction.
| Whenever the crane stretches toward 0, the counterweight
| stretches backward, its end not moving relative to the ground.
|
| Because the deck is balanced at start and empty deck is balanced,
| the crane starts and ends with a 0-stretch arm. The front side is
| either the frame arm stepping 2 steps forward at a time relative
| to the ground, or holding still while the backside crane arm
| shrinks closer, and the crane arm occasionally flips back and
| forth pointing forward or ackward. And vice versa for the
| counterweight.
|
| Over the course of the drive, the crane arm end reaches every
| even milestone once pointing forward and once again pointing
| backward.
| lupire wrote:
| Intuition for the bet size:
|
| When the deck has d cards left, it is sensible to make d bets of
| 1/d your stack, where each bet is that one specific card is next.
| If there are r reds and b=r+e blues, r of these bets simply
| cancel out r other bets, leaving e (times 1/d) remaining to be a
| nontrivial bet.
| andrewprock wrote:
| In practice, there are a number of factors which make using Kelly
| more difficult than in toy examples.
|
| What is your bankroll? Cash on hand? Total net worth? Liquid net
| work? Future earned income?
|
| Depending on the size of your bankroll, a number of factors come
| in to play. For example, if your bankroll is $100 and you lose it
| all it's typically not a big deal. If you have a $1 million
| bankroll, then you are likely more adverse to risking it.
|
| What is the expected value? Is it known? Is it stationary? Is the
| game honest?
|
| Depending on the statistical profile of your expected value, you
| are going to have to make significant adjustments to how you
| approach bet sizing. In domains where you can only estimate your
| EV, and which are rife with cheats (e.g. poker), you need to size
| your wagers under significant uncertainty.
|
| What bet sizes are available?
|
| In practice, you won't have a continuous range of bet sizes you
| can make. You will typically have discrete bet sizes within a
| fixed range, say $5-$500 in increments of $5 or $25. If your
| bankroll falls to low you will be shut out of the game. If your
| bankroll gets too high, you will no longer be able to maximize
| your returns.
|
| At the end of the day, professional gamblers are often wagering
| at half-kelly, or even at quarter-kelly, due in large part to all
| these complexities and others.
| zahlman wrote:
| > In practice, you won't have a continuous range of bet sizes
| you can make.
|
| You may also be required to pay for the privilege of placing a
| bet (spread and commissions in trading; the rake at a casino
| table).
| ilya_m wrote:
| Beautiful, thanks for sharing it!
|
| I think the portfolio argument is an unnecessary detour though.
| There's a two-line proof by induction.
|
| 1. The payoff in the base case of (0,1) or (1,0) is 2.
|
| 2. If we are at (r,b), r >=b , have $X, and stake (r-b)/(r+b) on
| red, the payoff if we draw red and win is X * (1+(r-b)/(r+b)) *
| 2^(r+b-1) / (r+b-1 choose r-1) = X * 2^(r+b) * r / ((r+b) *
| (r+b-1 choose r-1)) = X * 2^(r+b) / (r+b choose r).
|
| Similarly, if we draw black and lose, the payoff is X *
| (1-(r-b)/(r+b)) * 2^(r+b-1) / (r+b-1 choose r) = X * 2^(r+b) * b
| / ((r+b) * (r+b-1 choose r)) = X * 2^(r+b) / (r+b choose r). QED
| lupire wrote:
| Why isn't your inductive proof an unnecessary detour?
| lordnacho wrote:
| Interesting side note on Kelly:
|
| In probability theory, Proebsting's paradox is an argument that
| appears to show that the Kelly criterion can lead to ruin.
| Although it can be resolved mathematically, it raises some
| interesting issues about the practical application of Kelly,
| especially in investing. It was named and first discussed by
| Edward O. Thorp in 2008.[1] The paradox was named for Todd
| Proebsting, its creator.
|
| https://en.wikipedia.org/wiki/Proebsting%27s_paradox
| dominicrose wrote:
| Quoting the same page: One easy way to dismiss the paradox is
| to note that Kelly assumes that probabilities do not change.
|
| That's good to know. Kelly is good if you know the
| probabilities AND they don't change.
|
| If you don't know or if they can change, I expect the right
| approach has to be more complex than the Kelly one.
| cubefox wrote:
| In particular, then the right approach has to be more risk
| averse than Kelly would recommend. In reality, most
| probabilities can only be estimated, while the objective
| probabilities (e.g. the actual long run success rate) may
| well be different and lead to ruin. That's also what makes
| the title "Kelly can't fail" more wrong than right in my
| opinion.
| LegionMammal978 wrote:
| For the issue in Proebsting's paradox, one simple approach
| I've found successful is to gradually accumulate your full
| bet as the betting lines progress to their ultimate
| positions. This works even in illiquid markets where your
| bets affect the lines, since it gives the other
| participants less room to suddenly react to what you're
| doing. (Though you always have the slight worry of a huge
| last-second bet that _you_ can 't react to, eBay-auction
| style.)
|
| As for the actual probability being different from the
| expected probability, that's not too difficult to account
| for. Just set up a distribution (more or less generous
| depending on your risk tolerance) for where you believe the
| actual probability may lie, and work out the integrals as
| necessary, recalling that you want to maximize expected
| log-value. It's not the trivial Kelly formula, but it's
| exactly the same principle in the end.
| cubefox wrote:
| I think the problem with estimating a distribution is the
| same, it might simply not match reality (actual unknown
| success rates, actual unknown variance of your estimation
| of success rates being correct). In particular, if you
| are too optimistic relative to reality, Kelly betting
| will lead to ruin with high objective probability.
| lupire wrote:
| The title is gentle clickbait applies to one specific game
| with 0 variance, not to all uses of Kelly.
| csours wrote:
| Unfortunately, in the real world, playing the game changes
| the game.
|
| For instance, casinos have different payout schedules for
| Blackjack based on minimum bet size and number of decks in
| the shoe. Payouts for single deck Blackjack are very small in
| comparison to multi-deck games, as well as requiring larger
| minimums (and they shuffle the deck after only a few hands).
| PaulHoule wrote:
| When I was a teen I discovered that I could always guess more
| than half the cards right using card counting to determine what
| color is more common in the deck. I programmed my
|
| https://en.wikipedia.org/wiki/TRS-80_Model_100
|
| to simulate it and it never failed. Recently I thought about it
| again and wrote a Python script that tried it 30 million times
| and... it never failed.
|
| I've been thinking about what to do with it and came up with the
| options of (i) a prop bet and (ii) a magic trick, neither of
| which seemed that promising.
|
| As a prop bet I can offer $1000 to somebody's $10 which is not
| the route to great prop bet profits, also I worry that if I make
| a mistake or get cheated somehow I could be out a lot of money.
| (Now that I think of it maybe it is better if I re-organize it as
| a parlay bet)
|
| As a magic trick it is just too slow paced. I developed a patter
| to the effect that "Parapsychologists were never able to reliably
| demonstrate precognition with their fancy Zener cards, but I just
| developed a protocol where you can prove it every time!" but came
| to the conclusion that it was not entertaining enough. It takes a
| while to go through a deck which doesn't seem like a miracle, you
| will have to do it 7 times in a row to exclude the null
| hypothesis at p=0.01. Maybe somebody with more showmanship could
| do it but I gave up.
| jdhwosnhw wrote:
| That reminds me of my favorite algorithm, which can find the
| majority element in a list with any number of distinct entries
| _while using O(N) time and O(1) space_ (provided a majority
| element exists). I sometimes pose deriving this algorithm as a
| puzzle for people, no one has ever solved it (nor could I).
|
| https://en.m.wikipedia.org/wiki/Boyer%E2%80%93Moore_majority...
| barapa wrote:
| That is really cool
| lupire wrote:
| What's great about that is that the assumption (or O(n)
| check) that the majority exists is incredibly powerful,
| enabling the algorithm, which is nearly the dumbest possible
| algorithm, to work.
|
| The one flaw in the magic is that "2nd pass to verify" is a
| significant cost, transforming the algorithm from online
| streaming O(1) space to O(n) collection-storage space.
| im3w1l wrote:
| This article uses _stake_ to mean _bankroll_ , but usually it
| denotes _bet size_.
| bell-cot wrote:
| Interesting as a mathematical puzzle - but note that it's
| difficult to find cooperative, solvent counter-parties for "I
| can't lose" betting games.
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