Uncertainty and ergodicity – the important difference between Keynes and Knight

30 Mar, 2012 at 14:34 | Posted in Economics, Statistics & Econometrics, Theory of Science & Methodology | 8 Comments

In the week I’ve had an interesting discussion with Paul Davidson – founder and editor of the Journal of Post Keynesian Economics – on uncertainty and ergodicity, on the Real-World Economics Review Blog. 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 for an infinite realizations, 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 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 irrelevent.

LPS:

Paul, on the whole, I think you’re absolutely right on this. Knight’s uncertainty concept has an epistemological founding and Keynes’s definitely an ontological founding. Of course this also has repercussions on the issue of ergodicity in a strict methodological and mathematical-statistical sense. I think Keynes’s view is the most warranted of the two.

BUT – from a “practical” point of view I have to agree with Taleb. Because if there is no reliable information on the future, whether you talk of epistemological or ontological uncertainty, you can’t calculate probabilities.

The most interesting and far-reaching difference between the epistemological and the ontological view is that if you subscribe to the former, knightian view – as Taleb and “black swan” theorists basically do – you open up for the mistaken belief that with better information and greater computer-power we somehow should always be able to calculate probabilities and describe the world as an ergodic universe. As both you and Keynes convincingly have argued, that is ontologically just not possible.

PD:

Lars, your last sentence says it all. If you believe it is an ergodic system and epistemology is the only problem, then you should urge more transparency , better data collection, hiring more “quants” on Wall Street to generate “better” risk management computer problems, etc — and above all keep the government out of regulating financial markets — since all the government can do is foul up the outcome that the ergodic process is ready to deliver.

Long live Stiglitz and the call for transparency to end asymmetric information — and permit all to know the epistemological solution for the ergodic process controlling the economy.

Or as Milton Friedman would say, those who make decisions “as if” they knew the ergodic stochastic process create an optimum market solution — while those who make mistakes in trying to figure out the ergodic process are like the dinosaurs, doomed to fail and die off — leaving only the survival of the fittest for a free market economy to prosper on. The proof is why all those 1% far cats CEO managers in the banking business receive such large salaries for their “correct” decisions involving financial assets.

Alternatively, if the financial and economic system is non ergodic then there is a positive role for government to regulate what decision makers can do so as to prevent them from mass destruction of themselves and other innocent bystanders — and also for government to take positive action when the herd behavior of decision makers are causing the economy to run off the cliff.

So this distinction between ergodic and nonergodic is essential if we are to build institutional structures that make running off the cliff almost impossible. — and for the government to be ready to take action when some innovative fool(s) discovers a way to get around institutional barriers and starts to run the economy off the cliff.

To Keynes the source of uncertainty was in the nature of the real – nonergodic – world. It had to do, not only – or primarily – with the epistemological fact of us not knowing the things that today are unknown, but rather with the much deeper and far-reaching ontological fact that there often is no firm basis on which we can form quantifiable probabilites and expectations.