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This would be easier to understand if the author used standard vocabulary. Forget about "time average" and "ensemble average", what's important here is distingu

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.
- kgwgk 6y ago
  > "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 6y ago
    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 6y ago
      “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.
- loup-vaillant 6y ago
  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 6y ago
nonameiguess 6y ago
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.
- coldtea 6y ago
  >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. It's funny how you say "the author made it a bit more complicated than it needs to be" and then proceed to explain it with even more jargon ("Ergodicity for a stochastic process just means the joint distribution of random variables that make up the sample space is time invariant").

__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.

kgwgk 6y ago
> "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 6y ago
  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 6y ago
    “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.
loup-vaillant 6y ago
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 6y ago

nonameiguess 6y ago

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.

coldtea 6y ago
>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. It's funny how you say "the author made it a bit more complicated than it needs to be" and then proceed to explain it with even more jargon ("Ergodicity for a stochastic process just means the joint distribution of random variables that make up the sample space is time invariant").