18 ms·

Ergodicity, What's It Mean

125 points by simonebrunozzi 6y ago

amelius 6y ago
> 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 6y ago
  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 6y ago
    People might be happier with an equalitity of poverty (but has to be on a world scale).
    - Ericson2314 6y ago
      I get the Maslow argument, but after reading https://www.jasonhickel.org/blog/2019/2/3/pinker-and-global-poverty https://www.jasonhickel.org/blog/2019/2/3/pinker-and-global-... it sounds like one can explain one's intuitive sense of the world's happiness in both absolute and relative terms.
  - igorkraw 6y ago
    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 6y ago
    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.
    - nlitened 6y ago
      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.
    - jandrewrogers 6y ago
      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 6y ago
        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?
  - mensetmanusman 6y ago
    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).
    - 6gvONxR4sf7o 6y ago
      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.
    - ckw 6y ago
      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.
  - amelius 6y ago
    > 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?
- Ericson2314 6y ago
  That would mean banning inheritance and standardizing child-rearing or at expenses. Yes, the latter is a leftist boogieman. But the non-ergodicity of wealth is really scary! We got along fine pre agriculture (little wealth differences), and then with wars and famines to reset, but surely we banning private school is better than more of those?! Keep in mind non-ergodicity is also good. Socialized infrastructure like public transit for example: - The risk / lack of extractive fees to make profitability makes it impossible to do well privately - The benefits of pooled risk apply to even the richest people, who might with conventional econ think they are stuck at best with a noblesse oblige. Agricultural, colonialism, and industrialization are massive qualitative shifts that non-ergodic models also do better justice.
oh_sigh 6y ago
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 6y ago
  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 6y ago
    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 6y ago
      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.
nelsondev 6y ago
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 6y ago
  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 6y ago
    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.
thewayfarer 6y ago
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).
- fractionalhare 6y ago
  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 :)
- kpwagner 6y ago
  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.
6gvONxR4sf7o 6y ago
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:ANd9GcQa3gvhOr96M-82lDY8vaLj5XDyFj7cxLYTqMFA4iFhMQ6VWb0bk9bnqQdq221XUwvGfcA&usqp=CAU https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQa3gvh...
- nvader 6y ago
  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"!
- ssivark 6y ago
  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.
fractionalhare 6y ago
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 6y ago
  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.
contravariant 6y ago
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 6y ago
  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 6y ago
    I'm confused, what's that got to do with ergodicity?
- joe_the_user 6y ago
  Why isn't a dynamic system? A given point has current wealth as it's level and moves randomly up and down in that level according to a process that's random. https://en.wikipedia.org/wiki/Dynamical_system https://en.wikipedia.org/wiki/Dynamical_system https://en.wikipedia.org/wiki/Random_dynamical_system https://en.wikipedia.org/wiki/Random_dynamical_system
  - contravariant 6y ago
    Well dynamical systems aren't usually random for a start. I suppose you can make it fit the definition for the random dynamical system, though in that case the underlying dynamical system is ergodic.
nonameiguess 6y ago
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.
- alkonaut 6y ago
  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 6y ago
    > 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.
    - alkonaut 6y ago
      For a 50% chance you need odds of over 2.0 (regardless of bet size) to have positive EV. This is a bet of 0.4. With 2.0 odds the payout would be 0.8, and here it is 0.9? That looks like odds over 2.0 for 50/50 chance. So the individual bet is positive EV but the EV of an infinite sequence of such bets is still zero and that’s without the zero limit? There is just nothing in the article that begins to explain how that works. Maybe the video is better
pm90 6y ago
Honestly I would recommend the video the author linked to which to was much clearer in its explanation https://youtu.be/CCLtQHL-VUs https://youtu.be/CCLtQHL-VUs.
vladTheInhaler 6y ago
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.
jmount 6y ago
I have some notes I share on ergodicity here: https://win-vector.com/2012/02/04/ergodic-theory-for-interested-computer-scientists/ https://win-vector.com/2012/02/04/ergodic-theory-for-interes...
- jsweojtj 6y ago
  Note: the link in the post to the pdf doesn't work.
Ceezy 6y ago
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.
loup-vaillant 6y ago
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 6y ago
  If there's confidence in this distribution, you make a pool of people who take these bets & divide the result
  - loup-vaillant 6y ago
    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.)
- fractionalhare 6y ago
  "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.
pietrod 6y ago
this is indirectly related https://www.scientificamerican.com/article/is-inequality-inevitable/ https://www.scientificamerican.com/article/is-inequality-ine... is an huge takeout from the simple fact that (1+delta)*(1-delta) < 1 with 0<delta<1 here I attach a mini program where I show that of course the gini coefficient in the group growth (in this case the yield is always negative but total wealth of group constant): https://pastebin.com/L5sSM09V https://pastebin.com/L5sSM09V
throwaway99900 6y ago
Ergodicity can be explained very simply. If you've heard of Murphy's Law, you probably already understand the principle behind ergodicity. "If there's more than one way to do a job and one of those ways will end in disaster, then somebody will do it that way." When you increase the number of people doing something, the number of opportunities for that somebody to do it the wrong way goes up. These are the outliers versus the normal state of affairs. These don't have to be positive or even negative, simply improbable. Given enough players for a lottery, there will be a winner, but there will also be people that randomly get linear sequences of numbers, all primes, etc. Why this is coming from a throwaway is that this occurs for everything in life, including, for example, schools and police shootings. These are extremely outlying events when given the absolute number of interactions between students and police. The ergodic space allows for these to occur, and so they do, eventually. The problem is when people start taking infrequent outlying events as the norm rather than outlying properties of the system itself. If you give 800,000 cops guns and send them out to interact with the public tens of times per day, they will eventually shoot someone they shouldn't have. This is a normal part of the space but isn't probabilistically normal. The normal state of affairs is that they don't shoot anyone. In the rare case that they do shoot someone, it's generally because they are being shot at. Equally, if 100 million citizens have guns, eventually one will end up in the hand of one of approximately 7.5 million (probabalistically) male high school students that is willing to use it against other students. This too is part of the ergodic space, but isn't probablistically normal. The normal state of affairs is that school shootings don't happen. The normal state of affairs is that bullied and/or mentally ill students don't seek violent retribution against their classmates. These are simultaneously improbable and probable events. We know that they occur, but we know that they occur infrequently. Inevitably, given enough rolls of the dice, circumstances conspire to enable extremely improbable events to occur, but these should always be seen as the opposite of the norm.
einpoklum 6y ago
The author explains, using fancy math concepts, that if investments in Capitalism were like independent rounds of betting on what you have, capital would still tend to accumulate with few people rather than be distributed evenly. However, if you detach yourself from a distraction you would note that capitalist economies are not like independent bets at all, and much of the accumulation is due to confiscation, occupation, sabotage, undercutting to drive competitors out, propaganda (advertising), etc. - directly or by manipulating the government.
salamanderman 6y ago
My take away from this is don't go "all in" on every bet. There is a floor of $0, and the probability of N bets going the wrong way is always nonzero, even if it is small depending on the game. And with repeated "all in" betting the probability of N bad bets in a row goes to 100%.

> 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 6y ago
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 6y ago
  People might be happier with an equalitity of poverty (but has to be on a world scale).
  - Ericson2314 6y ago
    I get the Maslow argument, but after reading https://www.jasonhickel.org/blog/2019/2/3/pinker-and-global-poverty https://www.jasonhickel.org/blog/2019/2/3/pinker-and-global-... it sounds like one can explain one's intuitive sense of the world's happiness in both absolute and relative terms.
- igorkraw 6y ago
  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 6y ago
  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.
  - nlitened 6y ago
    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.
  - jandrewrogers 6y ago
    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 6y ago
      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?
- mensetmanusman 6y ago
  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).
  - 6gvONxR4sf7o 6y ago
    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.
  - ckw 6y ago
    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.
- amelius 6y ago
  > 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?
Ericson2314 6y ago
That would mean banning inheritance and standardizing child-rearing or at expenses. Yes, the latter is a leftist boogieman. But the non-ergodicity of wealth is really scary! We got along fine pre agriculture (little wealth differences), and then with wars and famines to reset, but surely we banning private school is better than more of those?! Keep in mind non-ergodicity is also good. Socialized infrastructure like public transit for example: - The risk / lack of extractive fees to make profitability makes it impossible to do well privately - The benefits of pooled risk apply to even the richest people, who might with conventional econ think they are stuck at best with a noblesse oblige. Agricultural, colonialism, and industrialization are massive qualitative shifts that non-ergodic models also do better justice.

oh_sigh 6y ago

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 6y ago
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 6y ago
  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 6y ago
    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.

nelsondev 6y ago

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 6y ago
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 6y ago
  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.

thewayfarer 6y ago

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).

fractionalhare 6y ago
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 :)
kpwagner 6y ago
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.

6gvONxR4sf7o 6y ago

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:ANd9GcQa3gvhOr96M-82lDY8vaLj5XDyFj7cxLYTqMFA4iFhMQ6VWb0bk9bnqQdq221XUwvGfcA&usqp=CAU https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQa3gvh...

nvader 6y ago
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"!
ssivark 6y ago
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.

fractionalhare 6y ago

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 6y ago
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.

contravariant 6y ago

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 6y ago
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 6y ago
  I'm confused, what's that got to do with ergodicity?
joe_the_user 6y ago
Why isn't a dynamic system? A given point has current wealth as it's level and moves randomly up and down in that level according to a process that's random. https://en.wikipedia.org/wiki/Dynamical_system https://en.wikipedia.org/wiki/Dynamical_system https://en.wikipedia.org/wiki/Random_dynamical_system https://en.wikipedia.org/wiki/Random_dynamical_system
- contravariant 6y ago
  Well dynamical systems aren't usually random for a start. I suppose you can make it fit the definition for the random dynamical system, though in that case the underlying dynamical system is ergodic.

nonameiguess 6y ago

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.

alkonaut 6y ago
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 6y ago
  > 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.
  - alkonaut 6y ago
    For a 50% chance you need odds of over 2.0 (regardless of bet size) to have positive EV. This is a bet of 0.4. With 2.0 odds the payout would be 0.8, and here it is 0.9? That looks like odds over 2.0 for 50/50 chance. So the individual bet is positive EV but the EV of an infinite sequence of such bets is still zero and that’s without the zero limit? There is just nothing in the article that begins to explain how that works. Maybe the video is better

pm90 6y ago

Honestly I would recommend the video the author linked to which to was much clearer in its explanation https://youtu.be/CCLtQHL-VUs https://youtu.be/CCLtQHL-VUs.

vladTheInhaler 6y ago

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.

jmount 6y ago

I have some notes I share on ergodicity here: https://win-vector.com/2012/02/04/ergodic-theory-for-interested-computer-scientists/ https://win-vector.com/2012/02/04/ergodic-theory-for-interes...

jsweojtj 6y ago
Note: the link in the post to the pdf doesn't work.

Ceezy 6y ago

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.

loup-vaillant 6y ago

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 6y ago
If there's confidence in this distribution, you make a pool of people who take these bets & divide the result
- loup-vaillant 6y ago
  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.)
fractionalhare 6y ago
"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.

pietrod 6y ago

this is indirectly related https://www.scientificamerican.com/article/is-inequality-inevitable/ https://www.scientificamerican.com/article/is-inequality-ine... is an huge takeout from the simple fact that (1+delta)*(1-delta) < 1 with 0<delta<1 here I attach a mini program where I show that of course the gini coefficient in the group growth (in this case the yield is always negative but total wealth of group constant): https://pastebin.com/L5sSM09V https://pastebin.com/L5sSM09V

throwaway99900 6y ago

Ergodicity can be explained very simply. If you've heard of Murphy's Law, you probably already understand the principle behind ergodicity. "If there's more than one way to do a job and one of those ways will end in disaster, then somebody will do it that way." When you increase the number of people doing something, the number of opportunities for that somebody to do it the wrong way goes up. These are the outliers versus the normal state of affairs. These don't have to be positive or even negative, simply improbable. Given enough players for a lottery, there will be a winner, but there will also be people that randomly get linear sequences of numbers, all primes, etc. Why this is coming from a throwaway is that this occurs for everything in life, including, for example, schools and police shootings. These are extremely outlying events when given the absolute number of interactions between students and police. The ergodic space allows for these to occur, and so they do, eventually. The problem is when people start taking infrequent outlying events as the norm rather than outlying properties of the system itself. If you give 800,000 cops guns and send them out to interact with the public tens of times per day, they will eventually shoot someone they shouldn't have. This is a normal part of the space but isn't probabilistically normal. The normal state of affairs is that they don't shoot anyone. In the rare case that they do shoot someone, it's generally because they are being shot at. Equally, if 100 million citizens have guns, eventually one will end up in the hand of one of approximately 7.5 million (probabalistically) male high school students that is willing to use it against other students. This too is part of the ergodic space, but isn't probablistically normal. The normal state of affairs is that school shootings don't happen. The normal state of affairs is that bullied and/or mentally ill students don't seek violent retribution against their classmates. These are simultaneously improbable and probable events. We know that they occur, but we know that they occur infrequently. Inevitably, given enough rolls of the dice, circumstances conspire to enable extremely improbable events to occur, but these should always be seen as the opposite of the norm.

einpoklum 6y ago

The author explains, using fancy math concepts, that if investments in Capitalism were like independent rounds of betting on what you have, capital would still tend to accumulate with few people rather than be distributed evenly. However, if you detach yourself from a distraction you would note that capitalist economies are not like independent bets at all, and much of the accumulation is due to confiscation, occupation, sabotage, undercutting to drive competitors out, propaganda (advertising), etc. - directly or by manipulating the government.

salamanderman 6y ago

My take away from this is don't go "all in" on every bet. There is a floor of $0, and the probability of N bets going the wrong way is always nonzero, even if it is small depending on the game. And with repeated "all in" betting the probability of N bad bets in a row goes to 100%.