Ergodicity Versus History: A Critical Commentary on the Work of Ole Peters

by Philip Pilkington

Article of the Week from Fixing the Economists

Lars Syll linked to a fantastic interview with the mathematician Ole Peters  that dealt with the topic of ergodicity and how it relates to economic and financial markets. First, a comment on the source.

The interview was conducted by Michael Mauboussin who is currently the Managing Director and Head of Global Financial Strategies at Credit Suisse but who was working with a hedge fund called Legg Mason Capital Management at the time of the interview. The latter firm was the one who published the interview.

The reason I call attention to this is because I think people working in the financial industry ‘get’ the fact that economic processes are non-ergodic far, far better than most economists. The reason for this is because they follow trends on a daily basis and anyone with any honest experience of this will appreciate that there is simply no way to believe that financial markets are ergodic if you experience them in real-time.

Take a simple example: what was the biggest news for the financial markets in the past 12 months? Undoubtedly, the Fed’s tapering program. Now, I’ve argued before that while the tapering program will likely have little impact on the real economy it has a massive impact on financial markets. But here’s the thing: you cannot quantify the taper. When Bernanke announced the taper the financial market reacted instantaneously.  And yet there is no immediate data-point for a Fed announcement so we cannot include this in a model except in the most arbitrary and farcical manner.

Further than this, and more fundamentally, the taper is a properly unique historical event. It is like, say, a defeat in an historical battle or the results of an election. It is not part of some deterministic, mechanical process — i.e. it is not subject to some law-like dynamic. Rather it is something that happens once, at one moment in time, has highly contingent effects and cannot be reversed.  It is not a shock to a system otherwise in equilibrium — like if we disturbed a pendulum which would behave chaotically for some time and then return to equilibrium — rather it is part of an unfolding, diverse and novel process.  In short, such events are more like moments in a narrative — in a film or a novel — than they are like logical moments in a mechanical system.

At the heart of this is, of course, the question of ergodicity. In this regard Peters gave what is perhaps the clearest exposition of ergodicity I have seen. I shall, for this reason, repeat it in full here.

In an ergodic system time is irrelevant and has no direction. Nothing changes in any significant way; at most you will see some short-lived fluctuations. An ergodic system is indifferent to its initial conditions: if you re-start it, after a little while it always falls into the same equilibrium behavior. For example, say I gave 1,000 people one die each, had them roll their die once, added all the points rolled, and divided by 1,000. That would be a finite-sample average, approaching the ensemble average as I include more and more people. Now say I rolled a die 1,000 times in a row, added all the points rolled and divided by 1,000. That would be a finite-time average, approaching the time average as I keep rolling that die. One implication of ergodicity is that ensemble averages will be the same as time averages. In the first case, it is the size of the sample that eventually removes the randomness from the system. In the second case, it is the time that I’m devoting to rolling that removes randomness. But both methods give the same answer, within errors. In this sense, rolling dice is an ergodic system. I say “in this sense” because if we bet on the results of rolling a die, wealth does not follow an ergodic process under typical betting rules. If I go bankrupt, I’ll stay bankrupt. So the time average of my wealth will approach zero as time passes, even though the ensemble average of my wealth may increase. A precondition for ergodicity is stationarity, so there can be no growth in an ergodic system. Ergodic systems are zero-sum games: things slosh around from here to there and back, but nothing is ever added, invented, created or lost. No branching occurs in an ergodic system, no decision has any consequences because sooner or later we’ll end up in the same situation again and can reconsider. The key is that most systems of interest to us, including finance, are nonergodic. (p2)

Although every word in the above passage is valuable if I were forced to highlight one sentence it would be the following: “Ergodic systems are zero-sum games: things slosh around from here to there and back, but nothing is ever added, invented, created or lost.” Now, let’s compare that with my characterisation of the structure of economic processes and financial markets as being similar to narratives.

In a narrative something is always added, invented, created and lost. Processes that have a narrative structure — that is, any and all historical processes — are by their very nature novel. They are the unfolding of novel events. If they were not and they were simply instances of past happenings projected into the future — as are ergodic processes — we could generate narrative structures by applying modelling techniques. So, we could use sophisticated modelling programs to generate historical and even fictional narratives. Computer programs could be created that would create both films and produce new historical research.

I have no doubt that some people think that this can be done — or will eventually be able to be done. But they are crackpots and cranks. And, as I have pointed out before, those who tackled this problem most directly — that is, those working in Artificial Intelligence (AI) research — have basically conceded today that their ability to produce systems that mimic human intellectual processes is extremely limited. With that, a comment or two on the rest of the Peters interview (I fear I will lose most of my audience from here, but however…).

Peters, quite sensibly, says that when building a portfolio in finance we should not use techniques that utilise “parallel universes” but rather techniques that try to imagine behavior through time in relation to given risks. The problem with the parallel universes picture, as I’ve pointed out before, is that it assumes normal or Gaussian distributions while financial markets do not look like this. Peters thinks that the solution lies in time-average maximisation — his way of trying to imagine behavior through time in relation to given risks,

In contrast [to the parallel universes idea], time-average maximization (geometric mean for multiplicative dynamics) doesn’t assume anything about the distributions. You stick in whatever distribution you like, crank the handle, and out comes your optimal investment strategy. (p8)

I think that there’s a problem with this. As I noted in the above linked piece the distribution of any given financial market is going to be different every time you take a reading of it. So, it cannot be projected forward. This gets back to the fact that financial markets and economic processes are characterised by novel events. The past cannot be projected forward to give us a picture, as if on a cinema screen, of the future.

This is the same problem encountered by enthusiasts of Bayesian methods in economic modelling: which priors do they use to confront the future if the future does not mirror the past? And if they posit arbitrary priors and keep updating them will they simply find themselves engaged in a perpetual and interminable regress, constantly searching for a formula that does not yet exist?

In actual fact, Peters recognises this to some extent. He says that the whole discussion of Gaussian distribution is an enormous distraction (I said the same thing in the above linked piece…). But then he gets bogged down in a rather technical issue — namely, the question of optimal leverage — and loses sight of the bigger picture. I don’t blame Peters in this regard. He is, after all, simply trying to navigate the market and build portfolios and I suspect that his advice on optimal leverage is sound. But he does not really deal in the interview with the Pandora’s Box that he has opened.

At the end of the interview Peters alludes that his thoughts on leverage — which are purely technical — have some bearing on the question of the housing market. We should read this, I assume, to imply that he is saying something meaningful about the 2008 financial crisis when he discusses optimal leveraging.  Let me be categorical here: he is not.  Peters is really not saying much of interest with regards to the financial crisis.

I don’t say this to denigrate Peters’ work; indeed, I am really glad to see a clearly articulate and intelligent mathematician talking about issues of ergodicity. But the fact is that the 2008 financial crisis was one generated at a properly macroeconomic level. The excess of leverage that we saw in the banking and financial system was truly an effect, a symptom, not a cause.

The real causes were to do with the new trade regimes that were established in the past three decades and the income inequality that accompanied them together with extremely confused government policies geared toward running public sector surpluses — all set against the backdrop of a deregulated credit system that expanded enormously to fill a macroeconomic black hole. These processes cannot be described by modelling. They are inherently historical and political.

Again, I am not saying that Peters thinks that fancy mathematical techniques can answer these questions. But I do think that less intelligent people will try to draw this conclusion from his work. Economists loath the idea that their discipline is not a clean science but rather a historical discipline intimately intertwined with politics. Most of them would love to hear that the whole thing can be explained by some silly geese confusing a time-average for an ensemble-average.

Equally well, this is music to the ears of certain financial technicians who think that they can provide a quick fix for the moribund financial industry. But these are merely illusions that will distract from the real issues facing us today. And I suspect that even finance, with its recent macro-political turn in the light of the chaos in the world economy since 2008, is beginning to wake up to this.