## Uncertainty and ergodicity – the important difference between Keynes and Knight

30 March, 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.

1. Doesn’t this presume a government with superior information relative to the rest of the economy? Or does non-ergodicity per se create some obvious and observable externalities that the government can fix? Doesn’t the government suffer from the same challenges as the private sector? And why can it then make better decisions?

• There is no presumption, neither in my, nor Paul’s and Keynes’s, argument, that governments should be God-like. Government acts have to be scrutinized just as private acts. What we maintain is that if the economic system is non-ergodic then the government CAN play a positive role – just as money and liquidity can play necessary and buffering roles in a monetary economy.
Well-functioning institutional structures and “third-parts” actors are often crucial for solving coordination problems and market failures. But they, self-evidently, come with no guarantee. And I’ve actully never heard any serious economist – of whatever ilk – say anything else.
To panglosssian market fundamentalists the possibility doesn’t even exist.

2. First, it is well known that the government can do a lot of good in a world with externalities. This follows from the first welfare theorem, and nobody, as you say, disputes this.

But what does that have to do with ergodicity? Can’t externalities exist in an ergodic world? Of course they can. So why is non-ergodicity important for the government’s possibility of playing a positive role?

And in what way does non-ergodicity infer externalities? Certainly, in a world with, as you call it, “genuine uncertainty” only agents whom have “seen the light” will be able to provide an guidance and correct any errors. And if the government is not God, she suffers from the same cognitive limitations as the agents in the economy themselves, and should presumably be equally incompetent in her decision-making.

It seems to me that both you and Paul Davidson are saying: Individuals wish to do x. But x is something stupid to do. Ergo, the government should regulate x.

Clearly, in lieu of clearly specified externalities, such an argument hinges on unlimited and clear insights by the government in an incredibly blurry world.

• In an ergodic world there is in a strict sense no room for consistent market failures. In the ergodic world – a non-existant model-universe – rational expectations, market clearing and equilibrium rule the roost. In a non-ergodic world – the world in which we live in – genuine uncertainty, “imperfect” rationality and disequilibrium rule the roost. In the former there is no need of any “government” since the efficient market and its price mechanism solve it all. In the latter we have abundant evidence of inefficient markets, financial crises and unemployment. In that world – in our world – “government” has had an important role to play. Is it perfect? Abolutely not! Is it necessary? Both economic history and relevant economic theory give us evidence for thinking so! Are there alternatives? Sometimes, but not often enough, and not potent enough, to dismiss “government”! Does “government” always succeed in wht it tries to do? No! But I rather go for an imperfect “government” that at least tries to do things in an imperfect world, than sit down with my arms crossed, believing that since we have efficient markets and “modern” macroeconomic models based on assumptions of ergodic processes ruling our economies, we are not really experiencing what we think we are experiencing, there are no crises, financial markets work nicely, rising inequality is just a mirage, etc etc.

3. Lars, have you ever studied microeconomics? The classics? Or are you simply too obsessed with Keynes to even bother?

Why don’t you take a look at, for instance, a basic microeconomic textbook such as Mas-Colell, Whinston and Green, and look up chapter 10 and 11. Or why not Varian’s excellent treatment in chapters 23 and 24? You’ll see nothing that resembles “non-ergodicity”, but you will also see a multitude of interesting economic problems littered with market failures and possible remedies. These range from externalities (climate change, anyone?), monopolies, asymmetric information, hold-up problems, and so on. These are indeed real world problems with real world solutions. And these are the questions you would like economists to reply “we simply do not know”, when we in fact do know! Take for instance the incredibly successful auction of telecom licenses in the UK spearheaded by Paul Klemperer with friends.

If you would have known economics, you would also have known that the role of a government is indeed an all-present possibility. And saying anything else is attacking an “economics” which doesn’t exist!

Second, how do you ex-ante evaluate policy in a “non-ergodic” world characterized by “genuine uncertainty”. Surely, any decision with relies on prices or future outcomes can not be tagged by a number, or the equivalent. So how can you say that policy A is better than policy B? Policy A carries a multitude of consequence for individuals, both in the present and in the future. Consequences which, according to you, can never be known. So how can you ever say that something is better than something else? How can you say that their are more “pros” associated with policy A than “cons”, while at the same time you can’t say how many “pros” it takes to outweigh a “con”?

I personally think that we need to be very clear with the assumptions that we use, and to which extent out results are direct artifacts of such assumptions. But when that is done, we can say that under those stipulated assumptions certain policies are better than other. In your utopia of economics we would never make such statements at all, but would ad nauseum retort to the mindless “we simply do not know”.

Why don’t you give me one non-trivial insight in social science which satisfies your criteria of non-ergodicity and genuine uncertainty which has change the way we view the world. For each of those, I’ll give you three which does not satisfy any such standards and still gives useful guidance. (oh, and saying that “in a world of true genuine uncertainty Keynes unambiguously proved for now and forever after that governments are good and people ar bad” does not count as an insight)

4. Hi Lars, found your site because i search for content related to non-ergodicity.

It has been a stressful summer for my synapsis. The introduction to Knightian uncertainty and through Ole Peters the concept of non-ergodicity has basically forced me to paradigm shift into how I view the Financial Markets and my approach to investing and risk management in general.

The use of Copulas and Monte Carlo analysis need be discredited as tools in portfolio construction, and I have developed a preference for income generating assets vis-à-vis investment into assets with prospect of return through price appreciation.

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