[HN Gopher] Ergodicity, What's It Mean
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Ergodicity, What's It Mean
Author : simonebrunozzi
Score : 77 points
Date : 2021-04-03 16:45 UTC (6 hours ago)
(HTM) web link (avoidboringpeople.substack.com)
(TXT) w3m dump (avoidboringpeople.substack.com)
| contravariant wrote:
| Unless I'm very mistaken the example they give is not a dynamic
| system, so asking whether it's ergodic doesn't make sense. For it
| to be a dynamic system it would need to have an invariant
| measure, which I can't even begin to figure out from their
| description.
|
| The process of simply flipping coins, even a biased coin, _does_
| correspond to a dynamic system where the space is all infinite
| sequences of heads and tails, the invariant measure is the joint
| probability, and the operator is the left-shift (i.e. if you
| sample 1 series of coin flips and throw away the first result you
| still end up with a sample of the same distribution).
|
| But this doesn't translate _at all_ to their proposed scenario of
| starting everyone at 1 and letting the results diverge from
| there.
| base698 wrote:
| An you example I always liked: baggage fees at airlines. On
| average a $25 bag fee is the same. For 4 business travelers.
| For a family of four while the dollar amount is the same it
| effects the family much more.
| contravariant wrote:
| I'm confused, what's that got to do with ergodicity?
| jmount wrote:
| I have some notes I share on ergodicity here: https://win-
| vector.com/2012/02/04/ergodic-theory-for-interes...
| jsweojtj wrote:
| Note: the link in the post to the pdf doesn't work.
| thewayfarer wrote:
| If the loss amount is adjusted to 33%, then on average over time
| individuals will make a net profit. A 50% win will more than
| compensate for a 33% loss (0.67 * 1.5 = 1.005).
| kpwagner wrote:
| The greater the edge, the more you can bet on one occurrence.
| Easier to understand if you look at binary events: double your
| bet with a win, lose your bet with a loss.
| fractionalhare wrote:
| Yes. Given 1000 players and 1000 turns, if each player starts
| with $100 in capital under your chosen parameters:
| import random l = 0.33 w = 0.5 c =
| 100 m = 1000 p = {k: c for k in range(m)}
| n = 1000 for k in range(n): for j in
| range(m): if random.choice([0,1]): p[j]
| += (w * p[j]) else: p[j] -= (l * p[j])
| print(sum([p[k] for k in p]) / len(p)) print(sum(1
| for k in p if p[k] > c) / len(p))
|
| I wrote this up quickly so there might be an error, but under
| your stated parameters the average wealth increases over time
| and most people end up wealthier than they started.
| Specifically, the number of people who will be wealthier at the
| end seems to converge to somewhere between 57-60%.
|
| NB: This assumes you bet your entire capital each round instead
| of a constant bet size. In the presence of non-ergodicity you
| wouldn't want to do this, but that just means it's an even
| stronger result that most people come out ahead.
|
| In fact 33% happens to be the maximum loss percentage this
| system (win rate, win percentage, bet = total capital) can
| tolerate while still exhibiting higher wealth for most players
| over time :)
| nonameiguess wrote:
| He actually picked bad numbers. That is a losing bet even on
| average. You can see that pretty easily by just assuming you get
| exactly the opposite result each time. 1.5 * 0.6 = 0.9, so no
| matter how you start, you're behind with an equal number of wins
| and losses. Naively, you might think "50% is more than 40%," but
| that isn't how percentages work. You need win and loss
| proportions with a geometric average of 1 for a breakeven
| expectation (percentages are multiplicative, not additive). In
| this case, 3/2 and 2/3 is what you'd need.
|
| This happens to not show after only 100 trials just because some
| tiny number of people get really lucky and draw up the ensemble
| average, but if you keep going, somewhere between 200 and 500
| trials, the ensemble average pretty quickly drops below the
| starting average wealth and stays there, asymptotically
| approaching 0.
| tryptophan wrote:
| OOf, yeah, expected value is a very misleading metric. Hard for
| me to wrap my head around it.
|
| After 2 tosses, the probable outcomes are 25% 2.25, 50% .9, and
| 25% .36, giving an expected value of 1.1025 interestingly
| enough. Overall a 75% chance of losing money.
| maximilianroos wrote:
| This is wrong and misses the point of the post.
|
| The median outcome is indeed negative, for the reason you give.
| But the mean outcome is positive, because some players become
| exceedingly rich.
|
| You can try it at home, here's some Julia which runs it over 1M
| people, each with 1K flips: using Distributions
| n = 1000 d = Binomial(n, 0.5) to_wealth(heads) =
| 1.5^heads * 0.6^(n - heads) rand(d, 1_000_000) .|>
| to_wealth |> mean
|
| You can keep running that, it's above 1 almost all the time.
|
| To look at this another way -- would you take the other side of
| the bet? Someone on average has to be making money, and the
| other side is clearly losing money.
| nonameiguess wrote:
| I did run this many times by adapting the OP's own code (and
| vectorizing it, which he asked someone to do). Here it is:
| import numpy as np import pandas as pd
| n_subjects = 100 n_trials = 1000 start = 10.0
| win = 1.5 loss = 0.6 prob = 0.5
| results = np.ones((n_trials, n_subjects)) * start for
| trial in range(1, n_trials): wins =
| np.random.binomial(1, prob, n_subjects) == 1
| results[trial, wins] = results[trial - 1, wins] * win
| results[trial, ~wins] = results[trial - 1, ~wins] * loss
| ax = pd.DataFrame(results).plot(legend=False,
| figsize=(18,10), logy=True, linewidth=0.5)
| ax.plot(results.mean(axis=1), color='red', lw=2,
| linestyle='--')
|
| The mean always trends to 0 and every single player
| eventually loses. There are never any winners at all past
| around 500 trials or so. Not sure how you're getting a
| different result as I have never used Julia and can't tell
| what your code is doing (except apparently something
| different).
| javitury wrote:
| Try these parameters
|
| n_subjects = 1000
|
| n_trials = 100
| pcmonk wrote:
| I think your n_subjects is too low. You need that to be
| high enough or you'll miss those low-probability winners
| that bring up the average.
| nonameiguess wrote:
| I did it with more subjects and it doesn't make a
| difference. The only reason I reduced to 100 is because
| the plot is unreadable otherwise.
|
| Looking at the Julia code, I think what he is doing wrong
| is making all wins worth $.50 and all losses worth $.40,
| but the bet computes a win or loss based on your current
| wealth, not your starting wealth. His formula would work
| if you were always betting $1 no matter what your
| bankroll was, but that isn't what the actual post
| stipulates.
| alkonaut wrote:
| It's just a really weird way of expressing the bet. This bet
| either has a positive EV or a negative one and if it's positive
| it's going to tend to infinity when repeated.
|
| There is one assymmetry at the zero point (assuming people
| can't recover from bankruptcy by borrowing another dollar) but
| that's easily fixed by adding a simple bet strategy e.g "bet at
| most 1/10 your bankroll on each bet".
| jsweojtj wrote:
| > This bet either has a positive EV or a negative one and if
| it's positive it's going to tend to infinity when repeated.
|
| This is wrong. The bet as described has a positive EV and the
| time average for a single player tends to zero as the bet is
| repeated.
|
| > There is one assymmetry at the zero point (assuming people
| can't recover from bankruptcy by borrowing another dollar)
| ...
|
| The result is not due to zero being an absorbing value. In
| the setup you can go arbitrarily small and come back without
| issue. The result is the same.
| jsweojtj wrote:
| The ensemble average is the expected value, and the expected
| value is positive.
|
| For a bet of $X, the expected value is: (1.5 * X) * 0.5 + (0.6
| * X) * 0.5 => 1.05 * X. The ensemble average per round is
| positive (1.05) and over multiple rounds smoothly tends to
| infinity with the number of bets. (Definition here:
| https://en.wikipedia.org/wiki/Expected_value).
|
| The time average for any specific person betting in this game
| is 0.95 * X (for the reasons you mention) and tends to zero
| with the number of bets.
|
| So let's go through a few specifics of your comment:
|
| > He actually picked bad numbers. That is a losing bet even on
| average.
|
| The point of this article is that "on average" is trickier than
| people tend to assume. There are different ways of taking
| averages. If you do the expected value calculation and get a
| positive number, you might (as other comments have said
| explicitly) expect that a participant repeatedly engaging such
| a bet would have his wealth trend toward infinity. But, they
| are wrong (as shown in the article).
|
| > This happens to not show after only 100 trials just because
| some tiny number of people get really lucky and draw up the
| ensemble average, but if you keep going, somewhere between 200
| and 500 trials, the ensemble average pretty quickly drops below
| the starting average wealth and stays there, asymptotically
| approaching 0.
|
| The ensemble average is positive and monotonically increases w/
| the number of rounds of betting.
| loup-vaillant wrote:
| This would be easier to understand if the author used standard
| vocabulary. Forget about "time average" and "ensemble average",
| what's important here is distinguishing the _expectation_ of a
| bet, from the _distribution_ of outcomes. In the bet he
| describes, the expectation is indeed greater than one, but the
| distribution means you 'll most probably end up poorer, while a
| lucky few get super-rich.
|
| Who here would take a bet where there's a 95% chance of losing
| their home and their well paying job, for a 5% chance of becoming
| a billionaire? I sure wouldn't.
|
| My take on this: don't stop at averages, look at the whole
| distribution.
| __s wrote:
| If there's confidence in this distribution, you make a pool of
| people who take these bets & divide the result
| loup-vaillant wrote:
| Successful societies do exactly that. It's called taxes.
|
| As for what a realistic bet would look like (you're founding
| a startup or something), I believe the expectation is often
| not much greater than 1, so one does not simply found 100
| startups and distribute the income of the 5 successful ones
| to everyone else. (And even if it _is_ , the people capable
| of founding startups often have steadier, though less
| impressive, means of increasing their wealth. Startups are
| often founded for reasons other than wealth, after all.)
| nonameiguess wrote:
| Although these notions are standard language, I still agree
| with you that the author made it a bit more complicated than it
| needs to be. Ergodicity for a stochastic process just means the
| joint distribution of random variables that make up the sample
| space is time invariant.
|
| For a coin toss example like this, the distribution of heads
| and tails in each trial is ergodic. The distribution of
| earnings is not. This isn't because of any difference between
| time average versus ensemble average. It's because the
| probability of winning each toss is time invariant but the
| amount you stand to win or lose isn't because it's a function
| of both the probability of winning and your current bankroll,
| and current bankroll is not time invariant.
|
| Although, ironically, because of the numbers he picked, all
| bankrolls tend to zero eventually, so over a large enough
| number of trials, wealth eventually becomes an ergodic process
| as well. Graphing out his scenario over more trials gives a
| sort of heat death of the universe plot, where some players
| stay alive longer than others, but in the long run, the enemy
| always wins.
| fractionalhare wrote:
| "Time average" and "ensemble average" _are_ standard vocabulary
| in the statistical mechanics literature. Your comment is
| essentially a restatement of the article 's point.
|
| I think it's uncharitable to say the article would be easier to
| understand if it didn't use the language of ergodicity. Its
| explicit goal is to show how non-ergodicity leads to an example
| like yours.
|
| So of course your comment seems easier to understand. But
| that's because you're just saying different distributions can
| be parameterized by the same mean. Ergodicity is about a lot
| more than that, and the language of ergodicity was the entire
| exercise here.
| loup-vaillant wrote:
| Thing is, I don't believe we even _care_ about the time
| average. What we care about is the evolution of the
| distribution of outcomes over time.
|
| More specifically:
|
| - The distribution of outcomes at certain points of interest
| in time (like the valuation of my company when I intend to
| sell it).
|
| - The probability that we cross a catastrophic threshold at
| some point (like bankruptcy).
|
| Time average is a _terrible metric_ to estimate those things.
| Heck, I 'm not sure it can measure anything of interest,
| besides our own mistaken intuitions. It should probably be
| called something like "time average fallacy".
| fractionalhare wrote:
| I'm a little confused - ergodic theory very much cares
| about the time average. Or do you mean the toy example of
| betting shouldn't care about it?
|
| It seems like you think the problem here is too
| unsophisticated for ergodic theory or something. Which,
| fine sure. But this isn't an article intended to teach you
| about betting. It's an article intended to teach you about
| ergodicity, using betting as a _toy example._ The author
| isn 't trying to introduce the best way to analyze betting
| strategies, they're trying to show what non-ergodicity is.
| And I think they basically succeed.
|
| Just meet the article where it is, for its intended usage.
| kgwgk wrote:
| > "Time average" and "ensemble average" are standard
| vocabulary in the statistical mechanics literature
|
| But their application to non-standard-mechanical things is
| very confusing.
|
| Of course wealth is not ergodic. Ergodicity would mean that
| the distribution is always the same. Every point in time
| would be identical to every other point in time and growth
| would be impossible.
| fractionalhare wrote:
| I agree it's not perfectly explained. But I think someone
| new to ergodic theory would find the article clearer (or at
| least more helpful overall) than your second paragraph
| here.
| kgwgk wrote:
| "What we're seeing is that even though the expected value
| is positive, and the ensemble average is increasing, the
| time average for any single person is usually decreasing.
| The average of the entire "system" increases, but that
| doesn't mean that the average of a single unit is
| increasing."
|
| Someone new to ergodic theory may understand from that
| article that if wealth was ergodic the average for every
| trajectory would increase like the average for the entire
| system. But that doesn't make sense.
| amelius wrote:
| > Wealth in this scenario is non-ergodic, since the wealth in the
| future depends on the wealth of the past (path dependence).
|
| Perhaps we should find a tax rule which makes wealth ergodic.
| hirundo wrote:
| That's a straightforward proposal. Making wealth non-path
| dependent means seizing and redistributing it periodically or
| continuously. That would tend toward economic equality, but
| since it has the side effect of suppressing the incentive to
| create wealth, it's an equality of poverty.
| andi999 wrote:
| People might be happier with an equalitity of poverty (but
| has to be on a world scale).
| amelius wrote:
| > That's a straightforward proposal. Making wealth non-path
| dependent means seizing and redistributing it periodically or
| continuously. That would tend toward economic equality, but
| since it has the side effect of suppressing the incentive to
| create wealth, it's an equality of poverty.
|
| Did you consider the case where A works twice as hard as B
| and ends up with twice the wealth of B?
| igorkraw wrote:
| I mean, all you'd need is a damping factor via a wealth tax,
| above some threshold that's still motivating enough. I doubt
| anyone would call everyone having 1 million dollars "equality
| of poverty", and you can make it so that if you are _really_
| good at business, you can outrun the wealth tax up to say 100
| million $. Oh, and 100% inheritance tax on estates above 1
| million, with a buyback right for familiy businesses that
| allows you to buy back the business at it 's current market
| value over X years (so you actually benefit from inflation).
|
| IFF implemented globally, would you truly argue this
| disincentivizes creation of wealth? Add inflation and
| exchange rate adjustments and I honestly don't buy that
| argument anymore.
|
| NOW, will you have selfish actors trying to game the system
| and evade these taxes through all means possible? Yes, but
| that's why I think anyone who supports the protection of
| private property through state violence and democracy at the
| same time needs to do some heavy gymnastics to justify tax
| evasion _and_ dynasty enabling tax policies (i.e., anything
| that doesn 't at least do the 100% inheritance/gift tax bit).
| And of course you'd need to implement it either globally or
| at least in economic powerhouse blocks like EU+US+Canada.
| otde wrote:
| I wonder if that's not necessarily a negative thing. Much of
| the beauty of the open-source ecosystem comes from its
| (typical but not guaranteed) lack of explicit wealth
| incentives. It feels to me like the idea that wealth
| redistribution _necessitates_ widespread poverty is almost
| purely speculative, though I'm happy to be corrected.
| jandrewrogers wrote:
| I think open source is an example of how any set of
| incentives cuts both ways, it is just prioritizing
| different things. Incentives are offsetting; things like
| wealth incentives are often valuable to encourage people to
| do important things they otherwise would have no
| incentivize to do in open source.
|
| For example, a well-known issue with open source data
| infrastructure is that it often has much lower performance
| and efficiency than equivalent proprietary software. There
| are many dis-incentives in the open source ecosystem to
| producing software that is highly performant and efficient,
| not the least of which is development complexity and
| sophistication level required to contribute. Open source
| developers do not pay the operational cost of wasteful data
| infrastructure but they _do_ pay the cost of their time,
| and prioritize accordingly. Proprietary data infrastructure
| is explicitly motivated by wealth incentives to be highly
| efficient, which is why companies invest in it even though
| open source equivalents exists.
|
| The relative wastefulness of open source in terms of
| computing resources is increasingly perceived as bad for
| the environment, so it isn't just a money motivation.
| Incentives are a powerful thing and it is evident that open
| source lacks incentives to produce some important outcomes.
| inimino wrote:
| This sounds like a just-so story about incentives and
| open-source software, and seems to ignore some of the
| significant motivations actually at play, as well as some
| fairly glaringly obvious historical examples. What are
| the specific examples you have in mind here?
| nlitened wrote:
| We could also say that much of the beauty of the open-
| source ecosystem comes from top 1% earners (software
| developers) having cushy jobs and some free time to invest
| in common good.
| mensetmanusman wrote:
| Not necessarily, society could decide what a value of "I win"
| income is.
|
| Maybe it's $1 billion, after which you hit a ceiling function
| or wealth becomes ergodic, and you earn a badge that says
| "you won" (in the context of this society).
|
| If you earn such a badge, society would call on you for
| advise (if you are a non-inheritor).
| ckw wrote:
| The value should be low enough as to make capture of the
| political system at the national level by small groups of
| wealthy individuals impractical.
| 6gvONxR4sf7o wrote:
| I love this idea of giving people a badge or even better a
| sticker if they hit some particular level above which we
| take their excess. It's hilarious.
| 6gvONxR4sf7o wrote:
| This one took me way too long to get intuition for and I don't
| think OP's explanation would have helped me grok it. I wonder if
| it's like monads, in that the intuition is really simple, but
| it's a very unique concept; so unusual that the intuition is hard
| to convey to the uninitiated. But the intuition really is simple,
| so the initiated feel compelled to try to convey how simple it
| is.
|
| My intuition for it is like a particular kind of spread out
| mixing. Imagine a giant bowl with a bunch of crazy high powered
| pinball bumpers [0] at the bottom, randomly jostling the pinballs
| around, sometimes kicking them out where they started, but they
| always come back eventually.
|
| If you roll a ball into it, it doesn't matter where you start.
| It'll get lost in the mix eventually. (it "mixes" sufficiently).
|
| And no matter where you start, those high powered bumpers will
| eventually happen to kick a pinball all the way out there again,
| given long enough. (The "mixing" spreads things out sufficiently
| and occasionally sends things all the way to any point in the
| bowl).
|
| By contrast, a bowl that's just high friction, where everything
| ends up stopped at the bottom wouldn't work, even though it makes
| it not matter where you started, because it doesn't "spread." It
| just sinks things to the same spot. An inverted bowl/a dome
| wouldn't work because starting on opposite sides means you'll
| just roll away from each other and never come together (no
| "mixing" at all). A bowl without bumpers would have you coming
| back where you started, but not "mixing" around to all the other
| spots.
|
| You need both elements. It has to not matter where you started
| specifically by getting back to where anyone started.
|
| Rereading my comment now, it really does come off as "a monad is
| like a burrito," doesn't it. But screw it, I'll hit post and
| maybe it helps somebody.
|
| [0] This kind of bumper: https://encrypted-
| tbn0.gstatic.com/images?q=tbn:ANd9GcQa3gvh...
| ssivark wrote:
| Here's the essence of non-ergodicity: the "typical" result
| (technically the mode, but practically the median is
| acceptable) is very different from the "mean" result. The
| language of "time" -vs- "ensemble" averages is simply about
| whether to include only the "typical" possibilities or to also
| include corner cases which have extremely low probability but
| extremely high payoffs. The technical point is that unless you
| get to play the game ("sample the process") absurdly many times
| (comparable to the total number of possibilities), you will
| never access the corner cases, and so the averaging over the
| "typical" results corresponds better to reality. The "mixing"
| idea means that if you run the process "long enough" then there
| won't be any corner cases -- the extreme possibilities will
| thoroughly mix with the typical possibilities, so both kinds of
| averaging will lead to the same (correct) answer.
|
| If you grok this, everything else is technical detail and
| window dressing.
| nvader wrote:
| I found this helpful. Not because it has helped me to grok
| ergodicity yet, but because it's another foothold on my way to
| climb the mountain. At some point it will click, and your
| pinball-bowl mixer will help, I'm sure.
|
| I'm also happy for the link to "a monad is like a burrito"!
| pm90 wrote:
| Honestly I would recommend the video the author linked to which
| to was much clearer in its explanation https://youtu.be/CCLtQHL-
| VUs.
| fractionalhare wrote:
| The thrust of the point is that while the wealth of a group will
| rise on average when playing a game with positive expected value,
| individuals with significant upfront losses will lose over time
| if the reward percentage is too close to the loss percentage.
| Because your future wins depend on your present capital, which in
| turn depends on your past wins. This becomes an optimization
| problem!
|
| This does _not_ mean that you shouldn 't play a game with
| positive expected value. Expected value is still the salient
| framework with which you should judge risk. It just means that
| the size of your bet needs to be considered in conjunction with
| your total capital, not just whether any individual bet is more
| likely to win than lose.
|
| The author states this seems to not be well known in finance, but
| in point of fact this is very well known in both literature and
| practice. A trading strategy with positive expected value has
| additional considerations before you execute on it, including
| your total capital and liquidity.
| SatvikBeri wrote:
| Well, not just upfront losses, since it's commutative. You
| break even if you have 10 wins for every 9 losses, which will
| happen to very few people with enough flips - but the amount
| you win if you have more victories than that threshold is quite
| large, while you can only lose $1 at most.
| vladTheInhaler wrote:
| I wrote up a jupyter notebook myself looking at this problem a
| couple years ago and I was thinking about cleaning it up into a
| blog post. Oh well. Guess that just goes to show the dangers of
| procrastination.
|
| For what it's worth, my takeaway was that the "paradox" is that
| we aren't accounting for the nonlinear utility of money.
| Therefore the exponentially unlikely probabilities of winning
| quadrillions of dollars have exponentially large weights. But a
| quadrillion dollars isn't a million times more useful to me than
| a billion dollars. So if you account for that saturation effect
| and take the expected _utility_ instead, the "paradox" goes
| away.
| oh_sigh wrote:
| ergodic example: Rolling a die. If you roll a die 1e6 times in a
| row, you will get each number approximately 1e6/6 times. If you
| roll 1e6 dies once all at the same time, you will get each number
| approximately 1e6/6 times. Same thing basically.
|
| non-ergodic example: Russian roulette. If you play russian
| roulette 1e6 times in a row, you will always be dead at the end.
| If 1e6 people play Russian roulette at the same time, 1e6*(5/6)
| people will be alive at the end, and only 1e6/6 people will be
| dead.
| ryebit wrote:
| Thank you! That's... well, beautiful isn't the right word,
| given nature of example... But I was hoping to find a simple
| non-ergodic example.
|
| The situation the article explores was interesting, but made
| the jump to something mathematically complex before I sunk my
| teeth into the fundamental bit.
| aliceryhl wrote:
| As I understand it, the difference is basically whether it is
| possible to end up in a situation where you are out of the
| game going forward. E.g. with repeated bets, you eventually
| hit zero money, and then you are stuck at zero forever.
| inimino wrote:
| There could also be attractor states that reduce the risk.
| For example in this case (if the numbers are taken to be
| such that the EV is positive) then you can also end up with
| a lucky player getting rich. The chance of that player
| going bust goes down much lower than the chance for a new
| player starting with $1. So while individual players may
| tend to go bust reliably, the total pool of wealth can
| still grow beyond any set upper boundary. Over time each
| player tends to get fabulously rich or go bust, so the game
| is mostly one of whether an initial run of luck gets you
| out of the danger zone before running into zero.
|
| If you added some effects on what kind of gambles are
| available to players at different levels, you can create
| several different attractor states.
|
| Ergodicity is a nice property of models like molecules of
| gas bouncing around a room, which means that statistical
| mechanics is practical. If one percent of the molecules
| tended to end up with all the kinetic energy, while the
| other molecules gradually one by one reached a complete
| standstill, then statistical mechanics wouldn't work.
|
| Since the very simple process shown in the article doesn't
| have this property, it means some familiar statistical
| tools can't be used naively with these models, or to
| extrapolate a little bit, to any model of any human
| activity that tends to these kinds of capturing, fixed-
| point, attractor outcomes.
| Ceezy wrote:
| Proving that a measure is invariant is usually hard or impossible
| (in physics). It's useful as toy experiment but usually not for
| real life example.
| nelsondev wrote:
| This example assumes you only have $1 to bet, and if you lose it,
| you're out of the game. I wonder what happens to the simulated
| outcome if you can keep betting, even if you lose.
| vitus wrote:
| Not quite.
|
| > Everyone starts with $1, gets 50% profit if they win, and
| pays 40% of their bet if they lose.
|
| From the wealth-over-time graph, it looks like the bet is sized
| such that it's always 100% of what you have (per some
| trajectories going as small as 10^-7).
|
| My read is that while each individual bet looks good in
| isolation, the fact is that one win and one loss puts you in
| the red overall -- when dealing with this iterated experiment
| you want to look not at E[X] = 0.05, but at E[log(X)] = -0.05
| to get a sense for how your assets evolve each round.
|
| For some intuition: if you win once and lose once, your net
| result is 1.5 * 0.6 = 0.9, so you've lost 10% of your starting
| money.
| vitus wrote:
| To expand on this further, and to plug the Kelly Criterion
| which the author mentions having written a former blog post
| on:
|
| Suppose instead of betting all your money every round, you
| instead decide to bet 10 cents each time. Now, instead of
| being essentially guaranteed long-term ruin, you can and most
| likely will be able to continue making money indefinitely.
| (In fact, your chance of ever dipping below, say, $0.50 is
| finite even when extending your rounds played arbitrarily.)
|
| The Kelly Criterion for this scenario actually dictates that
| you should bet 25% of your money each round. Using this
| betting strategy, somewhere around 70% of people end up
| making money off this game when run for 100 rounds (1% end up
| ending up with a respectable $25 or more, while about as many
| end up with <$0.15). You even have an opportunity for
| redemption -- when we drag out the horizon to 5000 rounds
| played, somewhere around 90% of individuals become
| _billionaires_ , even as 30% of people were behind after 100
| rounds (so 2/3 of those redeemed themselves).
|
| On the other hand, with the all-or-nothing solution outlined
| in the article, about 13% of the population coming out ahead
| (around 1% of the population gets really rich, ending up with
| >$200, while more than half end up with less than a penny).
| Meanwhile, the odds get worse as the game goes on, as at just
| 500 rounds, >80% of players have been reduced to less than a
| penny.
|
| That's a long-winded way of saying that the amount you bet is
| really important.
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