What is ergodicity?

12 Jul, 2019 at 17:22 | Posted in Economics | 22 Comments

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. Derivative index funds allow individuals to receive the ensemble average. When Taleb says “no person can get the returns of the market unless he has infinite pockets and no uncle points” I think he is forgetting that you can buy shares in an S&P index.
    If your asset goes up then down or down then up, you can bet on its volatility and come out with the ensemble average (for volatility at least).
    Thus using derivatives, individuals can experience ensemble averages. Such is the power of derivatives and economists should start talking about them in pieces such as this! (If you are serious about including finance as an integral part of models, you need to study derivatives.)

    • It is better to be a speculator than an investor… a speculator is one who runs risks of which he is aware and an investor is one who runs risks of which he is unaware.
      ̶ John Maynard Keynes. Cited in The Devil’s Derivatives.

      If you put lipstick (derivatives) on a pig (Ponzi scheme) it is still a pig. You are advocating putting lipstick on a pig and calling it Wagu beef (investment). This type of financial speculation is just the kind of finance that brought the world to the brink of financial collapse. Those who propose and espouse such forms of capitalism are nothing more than predators. They are not seeking to make our communities, societies, or nations, or the world a better place but to line their pockets at any expense everyone else, regardless of the long term consequences. They should be the first to be held accountable; time to jail the to the too big to jail and the break up the too big to fail. It is time to break up the banks and bring back some santity to the financial system. Elizebeth Warren is right, this must end, and it will, one way (peacefully through democracy and legal means) or the other (by violence and destruction). This kind of blind mathematical hubris that casts asisde the damage of such predatory exotic financial instruments and those who push them are and will be esposed to the average person for just what they are. The attempt of the shrewd and wicked to proft at the expense of the middle class and everyone else.

      On a chilly winter’s evening in 2003, I went out to an exclusive nightclub in London’s Knightsbridge district favored by bankers and hedge fund managers. My senses were assaulted by thumping dance music as I followed my friend who was weaving across a dance floor thronged with leggy Russian blondes and the men who love them. There were acquaintances under the strobe lights: I spotted the global head of interest rate trading at a big German bank shimmying up against a pair of microskirted brunettes who towered over him. We then went up some steps and came to the closed door of the VIP lounge—which had its own doorman. The door swung open and we continued our way to a low-ceilinged room, the VIP lounge within the VIP lounge. There, sprawled across low sofas and thick cushions were bankers celebrating their annual bacchanal, which is also known as “bonus season.”
      There were a few Brits and Americans there, but most of the revelers were continental Europeans wearing well-cut Italian suits and well-pressed dress shirts, with their Hermes ties long ago cast to the winds. They either sipped £30 whisky sours or topped off their glasses from £400 bottles of Belvedere vodka. This was London before the smoking ban, and the glowing tips of cigarettes could be seen tracing formulas in the air as bankers sketched out the key details of their wildly successful deals for one another. I knew about some of them: there was the head of financial institutions derivatives marketing who forgot which of his Italian supercars had been towed off to the car pound. There was the head of credit structuring notorious for preying on female staff and having his corporate credit cards stolen by prostitutes. These young men—and almost all of them were young, some shockingly so—were the avant-garde of the credit derivatives boom, enjoying their first, fifth, or tenth million; outside the door of the VIP lounge, the Eastern European blondes were waiting to pounce on them.
      There are many sobriquets for these young lions, but I like to think of them as the men who love to win. (The Devil’s Derivatives: The Untold Story of the Slick Traders and Hapless Regulators Who Almost Blew Up Wall Street. Read.)

      • “This is an attempt of the shrewd and wicked to proft at the expense of the middle class and everyone else.”

      • “esposed to the average person” -> exposed to the …

      • When there is one bank at the top, there need be no runs because the top bank can supply all the money demanded. The Fed is that bank and the world central bank currency swap network is a proxy for a world central bank that could supply other currencies than the dollar, if need be. The only real uncertainty is, will the top bank choose to meet money demand in a panic?
        We should use derivatives to insure individuals so I can pursue a monk-like existence and influence traders by my example to live more mindful lives. If you try to coerce them you are playing their game and they are better at it and will win politically.

        • Hogwash. When we make white collar fraud just as costly to personal

          • freedom consistently, which we have done before and can do again. The argument that the financial predatory bankers are to smart to be regulated or prosecuted is pretty lame, in my view.

            • There are better ways to change behavior. It is best for bankers to change voluntarily, because they become more knowledgeable. We should craft public policy to empower spiritual teachers so we can survive independently of fickle and arbitrary banker credit allocation. We don’t have to regulate them, we can use their tricks (derivatives) to free ourselves from their influence.

              • What do you mean by “craft public policy to empower spiritual teachers?”

                • Pay me a basic income so I can have access to resources without having to entangle myself too much in the banker’s world.
                  The Fed could sell derivatives such as panic insurance and inflation swaps to fund it. Banks would line up to voluntarily spend more on such public derivatives than you could tax out of them …

                • Sounds like a national Ponzi scheme to me 🙂 I note you did not answer my question.

    • That’s not really convincing as there is a counterparty risk: the company you buy the derivative from may go bust. In other words, it is this company now that has to have infinite ressources. Just note that no sane bank throws it’s full weight behind securing any of its derivative products but rather places handy circuit breakers between the bank and the investor. Bottom line: derivatives may alleviate some of the deep-pocket problems but not all and those problems it solves you have to pay for quite substantially.

      • Clearinghouses provide counterparty default insurance. The Fed is insurer of last resort. The Fed should sell explicit panic insurance upfront, so if a panic causes a run on liquidity, banks can get liquidity from the Fed’s explicit insurance. The Fed could sell shares in a suitably countercyclical index as insurance, such as the External Finance Premium described by Bernanke. When panic strikes and liquidity is hoarded and banks can’t get funding in markets they can sell the EFP shares to replace the lost liquidity and make good on their obligations.
        The Fed put is implicit. We should make it explicit.

        • You see, there is no easy way out of the notion that you cannot buy the average performance. The more you think about it the more overhead you need to get even close and in the end the FED shall absorb all the risk. That’s quite convenient and somewhat weird given that according to your scheme loss of employment is way worse covered than the derivative adventures.

          • You can buy the average performance of the S&P 500. (That was the original point: individuals can experience the benefits of the ensemble average.) You can hedge by triple-shorting the same index for much less of a down payment. You can also spend a little to further insure with volatility. The last two require more active involvement, but produce guaranteed returns no matter what the S&P does.
            My scheme includes a basic income to insure individuals. It might be funded by the Fed explicitly selling panic insurance and inflation swaps.

    • In the lead up to the GFC it seemed that derivatives largely were a credit burnishing tool which lowered risk factors which hence allowed more credit at better terms E.g. VoM for financial transactions with a plethora of ticket clippers at various transactional stages or bottlenecks.

      The drama was that the insurance was not mature yet and when things went south counter party risk went parabolic [in the Scientific sense], meaning a threshold was broached and sub 10% went full melt down [roll overs]. Strangely in some Scientific endeavor no one will have a bar of 2 to 5 percent fail rate due to the compounding effect.

      Yet this does not even broach the human agency, which ended up pushing VaR past its original intent – because of incentives.

      So were back to Milgram I suppose ….

      • The problem was a spreading panic. The non-government private label mortgage backed securities might have been uninsured but the problem was the panic over that non-systemic segment spread to other segments such as the insured Government Sponsored Entity mortgages and then to a freezing of nearly all repo. Or, you can tell the story as: dealers raised the price of liquidity to prohibitive levels as a signal that they wanted borrowers offering any collateral to go away. They preferred to hoard.
        The panic had nothing to do with science. Prices went up because of emotion and down because of irrational emotion and back up because of arbitrary valuations administered by traders and central bankers.
        The Fed today is not setting rates scientifically. Powell raised rates in December of last year because he was in a pissing contest with the President. This isn’t a mechanical process, it’s a personality-dependent arbitrary and fickle process of assigning prices on whims and ideological assumptions.
        Whether a mechanical process is actually as mechanical as Science assumes is a separate matter. I maintain that physics envy is silly because physics is just social consensus itself, but that is really irrelevant to economics being a fickle social consensus.

        • Are you arguing against the Taylor rule …

          • http://www.perrymehrling.com/2016/09/beyond-the-taylor-rule/
            “Instead of nostalgia for a simpler past, we need to be engaged with the project of designing resilient monetary policy frameworks for the complex future.”
            My framework: rates set at zero forever, index fully to maintain real purchasing power rather than nominal price stability, and fund basic income on central bank’s balance sheet (replacing employment mandate in Section 2A of the Federal Reserve Act).

            • My inference to the Taylor rule was its totem of the IS-LM, one cannot be against the prior and not have dramas with the latter and what that portends.

              Still anti UBI due to the historical back drop, debates about its implementation and extenuating circumstances requiring anti democratic administration. You may or may not be familiar with this territory.

  2. @ Rob – Socrates proposed he be maintained in the Prytaneum in the Apology. Public policy should guarantee everyone, including those inclined to teach spirituality and inquire into virtue, access to vast oversupply. A Ponzi scheme involves fraud but my scheme is forthright in instructing the Fed, or the world central bank proxied by the currency swap lines among major central banks, to insure everyone on an individual level. With one bank, there need be no runs, and no Ponzi scheme because there is no misrepresentation.

    • Thanks, sorry, I missed this, you did answer my question. Appreciated. I don’t see you proposal as realistic or ideal, but I see your idea better now.

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