Home > The Economics Profession > Ergodicity and randomness in economics (wonkish)

## Ergodicity and randomness in economics (wonkish)

from Lars Syll

To understand real world ”non-routine” decisions and unforeseeable changes in behaviour, ergodic probability distributions are of no avail. In a world full of genuine uncertainty – where real historical time rules the roost – the probabilities that ruled the past are not those that will rule the future.

When we cannot accept that the observations, along the time-series available to us, are independent … we have, in strict logic, no more than one observation, all of the separate items having to be taken together. For the analysis of that the probability calculus is useless; it does not apply … I am bold enough to conclude, from these considerations that the usefulness of ‘statistical’ or ‘stochastic’ methods in economics is a good deal less than is now conventionally supposed … We should always ask ourselves, before we apply them, whether they are appropriate to the problem in hand. Very often they are not … The probability calculus is no excuse for forgetfulness.

John Hicks, Causality in Economics, 1979:121

Time is what prevents everything from happening at once. To simply assume that economic processes are ergodic and concentrate on ensemble averages – and a fortiori in any relevant sense timeless – is not a sensible way for dealing with the kind of genuine uncertainty that permeates open systems such as economies.

Ergodicity is a difficult concept that many students of economics have problems with understanding. In the very instructive video below, Ole Peters – from the Department of Mathematics at the Imperial College of London – has made an admirably simplified and pedagogical exposition of what it means for probability structures of stationary processses and ensembles to be ergodic. Using a progression of simulated coin flips, his example shows the all-important difference between time averages and ensemble averages for this kind of processes:

[Why is the difference between ensemble and time averages of such importance? Well, basically, because when you assume the processes to be ergodic,ensemble and time averages are identical. Let me giva an example even simpler than the one Peters gives:

Assume 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 assetprice 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 assetprice 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.

Just in case you think this is just an academic quibble without repercussion to our real lives, let me quote from an article of Peters in the Santa Fe Institute Bulletin from 2009 – On Time and Risk - that makes it perfectly clear that the flaw in thinking about uncertainty in terms of “rational expectations” and ensemble averages has had real repercussions on the functioning of the financial system:

In an investment context, the difference between ensemble averages and time averages is often small. It becomes important, however, when risks increase, when correlation hinders diversification, when leverage pumps up fluctuations, when money is made cheap, when capital requirements are relaxed. If reward structures—such as bonuses that reward gains but don’t punish losses, and also certain commission schemes—provide incentives for excessive risk, problems arise. This is especially true if the only limits to risk-taking derive from utility functions that express risk preference, instead of the objective argument of time irreversibility. In other words, using the ensemble average without sufficiently restrictive utility functions will lead to excessive risk-taking and eventual collapse. Sound familiar?

Still having problems understanding the ergodicity concept? Let me cite one last example that hopefully will make the concept more accessible on an intuitive level:

Why are election polls often inaccurate? Why is racism wrong? Why are your assumptions often mistaken? The answers to all these questions and to many others have a lot to do with the non-ergodicity of human ensembles. Many scientists agree that ergodicity is one of the most important concepts in statistics. So, what is it?

Suppose you are concerned with determining what the most visited parks in a city are. One idea is to take a momentary snapshot: to see how many people are this moment in park A, how many are in park B and so on. Another idea is to look at one individual (or few of them) and to follow him for a certain period of time, e.g. a year. Then, you observe how often the individual is going to park A, how often he is going to park B and so on.

Thus, you obtain two different results: one statistical analysis over the entire ensemble of people at a certain moment in time, and one statistical analysis for one person over a certain period of time. The first one may not be representative for a longer period of time, while the second one may not be representative for all the people. The idea is that an ensemble is ergodic if the two types of statistics give the same result. Many ensembles, like the human populations, are not ergodic.

1. September 19, 2012 at 3:19 pm | #1

Determinism, in general, has its roots in wishful thinking.

2. September 19, 2012 at 3:22 pm | #2

Since I introduced the concept of ergodicity vs. nonergodic stochastic processes (the latter for uncertrainty) into the economics literature more than 3 decades ago in an article on the fallacious founbdations of rational expectstions, I would hope you would give me some credit oe at least mention in your essay.

After all three Nobel Prize winners have given me credit –
(1) John Hicks has written that I was able to convincve him that he should label his approach as nonergodic — and therefore the ISLM analysis is NOT a view of Keynes,
(2) Doug North has indicated that he applies my concept on nonergodicity to explain economic change and development, and (even)
(3) Bob Solow thinks my association of uncertainty with nonergodicity is innovative and good.

Since then I have demonstrasted that if the economy is governed by a nonergodic stochastic process, then (1) markets can NOT be efficient and (2) there is a role for government in improving the economic sysrem.
Only if the system is ergodic can a laissez faire philosophy be justified..

3. September 19, 2012 at 5:38 pm | #4

I don’t know what the avg reader is here, but starting with the park example would have been good for me.
And then giving some equally concrete examples of ergodic and non ergodic processes
as it is, the first 2/3rds of this piece are, for people like me, not that helpful.
in particular, i found the 100 euro market example really unhelpful, pedagogically, cause i got stuck on the idea of multiple universes.
i think a first rule of pedagogy is to not introduce new ideas in explaining a concept, untill all the previously menitoned new ideas are fully explained

4. September 20, 2012 at 2:01 am | #5

at a glance, i dont think you understand what you are talking about. sure, most likely ‘markets’ are nonergodic (eg marshall, brian arthur…).
but your example is bad. assume a random walk, starting at 0, w equal chances to go left or right in one dim. thats ergodic. and same as your very poorly worded and thought out example. but hey thats real world economics view. excess verbuage, a corrolary to theorem/proof in econometrica.