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What is ergodicity?

from Lars Syll

Time to explain ergodicity …

The difference between 100 people going to a casino and one person going to a casino 100 times, i.e. between (path dependent) and conventionally understood probability. The mistake has persisted in economics and psychology since age immemorial.

Consider the following thought experiment.

skin_in_the_gameFirst case, one hundred persons go to a Casino, to gamble a certain set amount each and have complimentary gin and tonic … Some may lose, some may win, and we can infer at the end of the day what the “edge” is, that is, calculate the returns simply by counting the money left with the people who return. We can thus figure out if the casino is properly pricing the odds. Now assume that gambler number 28 goes bust. Will gambler number 29 be affected? No.

You can safely calculate, from your sample, that about 1% of the gamblers will go bust. And if you keep playing and playing, you will be expected have about the same ratio, 1% of gamblers over that time window.

Now compare to the second case in the thought experiment. One person, your cousin Theodorus Ibn Warqa, goes to the Casino a hundred days in a row, starting with a set amount. On day 28 cousin Theodorus Ibn Warqa is bust. Will there be day 29? No. He has hit an uncle point; there is no game no more.

No matter how good he is or how alert your cousin Theodorus Ibn Warqa can be, you can safely calculate that he has a 100% probability of eventually going bust.

The probabilities of success from the collection of people does not apply to cousin Theodorus Ibn Warqa. Let us call the first set ensemble probability, and the second one time probability (since one is concerned with a collection of people and the other with a single person through time). Now, when you read material by finance professors, finance gurus or your local bank making investment recommendations based on the long term returns of the market, beware. Even if their forecast were true (it isn’t), no person can get the returns of the market unless he has infinite pockets and no uncle points. The are conflating ensemble probability and time probability. If the investor has to eventually reduce his exposure because of losses, or because of retirement, or because he remarried his neighbor’s wife, or because he changed his mind about life, his returns will be divorced from those of the market, period.

Nassim Taleb

Taleb’s excellent example shows why the difference between ensemble and time averages is of such importance in economics.

multiverseAssume we have a market with an asset priced at €100.​ Then imagine the price first goes up by 50% and then later falls by 50%. The ensemble average for this asset would be €100 – because we here envision two parallel universes (markets) where the asset price​ falls in one universe (market) with 50% to €50, and in another universe (market) it goes up with 50% to €150, giving an average of 100 € ((150+50)/2). The time average for this asset would be 75 € – because we here envision one universe (market) where the asset price first rises by 50% to €150 and then falls by 50% to €75 (0.5*150).

From the ensemble perspective nothing really, on average, happens. From the time perspective lots of things really, on average, happen. Assuming ergodicity there would have been no difference at all.

  1. A.J. Sutter
    July 16, 2019 at 4:24 am

    Sorry: Taleb’s example I understand, but I don’t follow your description of ensemble value in your own example. That example has time dependence built into its problem statement: “first … and then later ….” What’s the justification for envisioning this as two separate markets, with a base asset price of €100 in each case?

  2. Dave Raithel
    July 16, 2019 at 4:31 am

    Hey, I am going to post a link to a news article that I believe might be amusing to some people, and since Lars Syll is someone I usually read and etc., I picked this place to put the article. (I do not know how else to submit anything.) It is about Von Mises, the University of Missouri, and an ex governor ex-attorney general ex-state senator… oh, and money and ideology. Sure, I show up in the comments, but this ain’t about me. It is about me embarrassing the place I live.


    • Rob
      July 16, 2019 at 6:33 am

      This is spam

  3. Dave Raithel
    July 16, 2019 at 4:33 am

    https://www.columbiamissourian.com/news/higher_education/meet-the-conservative-college-trying-to-take-millions-from-mizzou/article_a7131c64-a4c1-11e9-88a9-9f8b0d7ec1de.html So, just as a teaser: How many professors does it take to teach what Von Mises thought? Mizzou has 4, and that is not enough….

    • Jeff
      July 16, 2019 at 6:08 am

      “How many professors does it take to teach what Von Mises thought?”

      It really depends on how Miserly they are, doesn’t it?

  4. Ikonoclast
    July 16, 2019 at 7:35 am

    In my lexicon “skin in the game” means you risk life and limb. A worker who risks life and limb in a dangerous job has skin in the game. A capitalist who risks money only has money in the game: a risk much less worthy of reward.

  5. deshoebox
    July 16, 2019 at 4:33 pm

    There appears to be an error in the original example. The outcome of every casino game depends at least partly on how you play and on the nature of the game. Inferring from the fact that Player #28 (out of 100) went bust that there is a one percent probability any particular gambler will go bust ignores this. Player #28 may have used a bad strategy. He/she may have been subject to one or more gambler’s fallacies. She/he may have played well but had very bad luck. Casinos operate on the principle that they can predict fairly accurately what percentage of the money that comes in the door will go back out, but this is based on thousands and thousands of gamblers playing hundreds of different games using many individual strategies, mostly losing ones. I don’t see that this example really illuminates anything.

  6. Ken Zimmerman
    July 22, 2019 at 3:14 pm

    Psychologists (primarily) have been examining ergodicity as it relates to the standard social science and medical research method of using case study results to apply conclusions to populations. They’ve raised concerns this is possible in only narrowly limited circumstances.

    • August 4, 2019 at 10:03 am

      Ken, Please could you supply a reference, preferably accessible to a mathematician?

      (I’m wondering how their concerns compare with ours.)

      • Ken Zimmerman
        August 4, 2019 at 3:26 pm

        Dave, check these out.

        Hurtado-Parrado and López-López. Single-Case Methods (SCMs) could be a valid alternative to Null Hypothesis Significance Testing (NHST). Integrative Psychological and Behavioral Science, 2015.

        The Idiographic Approach in Psychological Research. Salvatore and Valsiner Theory & Psychology, 20(6), 817-833, 2010; Valsiner Cultural & Psychology, 20(2), 147-159, 2014; Salvatore Culture & Psychology, 20(4), 477-500, 2014.

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