by From Lars Syll A couple of years ago yours truly had an interesting discussion — on the Real-World Economics Review Blog — with Paul Davidson on ergodicity and the differences between Knight and Keynes re uncertainty. It all started with me commenting on Davidson’s article Is economics a science? Should economics be rigorous? : LPS: Davidson’s article is a nice piece – but ergodicity is a difficult concept that many students of economics have problems with understanding. 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. Time is what
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from Lars Syll
A couple of years ago yours truly had an interesting discussion — on the Real-World Economics Review Blog — with Paul Davidson on ergodicity and the differences between Knight and Keynes re uncertainty. It all started with me commenting on Davidson’s article Is economics a science? Should economics be rigorous? :
LPS:
Davidson’s article is a nice piece – but ergodicity is a difficult concept that many students of economics have problems with understanding. 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.
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.
When you assume the economic processes to be ergodic, ensemble and time averages are identical. Let me give an example: 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 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.
Just in case you think this is just an academic quibble without repercussion to our real lives, let me quote from an article of physicist and mathematician Ole 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?”
PD:
Lars, if the stochastic process is ergodic, then for an infinite realization, the time and space (ensemble) averages will coincide. An ensemble a is samples drawn at a fixed point of time drawn from a universe of realizations For finite realizations, the time and space statistical averages tend to converge (with a probability of one) the more data one has.
Even in physics, there are some processes that physicists recognize are governed by nonergodic stochastic processes. [see A. M. Yaglom, An Introduction to Stationary Random Functions [1962, Prentice Hall]]
I do object to Ole Peters exposition quote where he talks about “when risks increase”. Nonergodic systems are not about increasing or decreasing risk in the sense of the probability distribution variances differing. It is about indicating that any probability distribution based on past data cannot be reliably used to indicate the probability distribution governing any future outcome. In other words even if (we could know) that the future probability distribution will have a smaller variance (“lower risks”) than the past calculated probability distribution, then the past distribution is not a reliable guide to future statistical means and other moments around the means.
LPS:
Paul, re nonergodic processes in physics I would even say that most processes definitely are nonergodic. Re Ole Peters I totally agree that what is important with the fact that real social and economic processes are nonergodic is the fact that uncertainty – not risk – rules the roost. That was something both Keynes and Knight basically said in their 1921 books. But I still think that Peters’ discussion is a good example of how thinking about uncertainty in terms of “rational expectations” and “ensemble averages” has had seriously bad repercussions on the financial system.
PD:
Lars, there is a difference between the uncertainty concept developed by Keynes and the one developed by Knight.
As I have pointed out, Keynes’s concept of uncertainty involves a nonergodic stochastic process. On the other hand, Knight’s uncertainty — like Taleb’s black swan — assumes an ergodic process. The difference is the for Knight (and Taleb) the uncertain outcome lies so far out in the tail of the unchanging (over time) probability distribution that it appears empirically to be [in Knight’s terminology] “unique”. In other words, like Taleb’s black swan, the uncertain outcome already exists in the probability distribution but is so rarely observed that it may take several lifetimes for one observation — making that observation “unique”.
In the latest edition of Taleb’s book, he was forced to concede that philosophically there is a difference between a nonergodic system and a black swan ergodic system –but then waves away the problem with the claim that the difference is irrelevant.