Using ‘small-world’ models in a large world

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Lars Pålsson Syll
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from Lars Syll

Radical uncertainty arises when we know something, but not enough to enable us to act with confidence. And that is a situation we all too frequently encounter …

The language and mathematics of probability is a compelling way of analysing games of chance. And similar models have proved useful in some branches of physics. Probabilities can also be used to describe overall mortality risk just as they also form the basis of short-term weather forecasting and expectations about the likely incidence of motor accidents. But these uses of probability are possible because they are in the domain of stationary processes. The determinants of the motion of particles in liquids, or overall (as distinct from pandemic-driven) human mortality, do not change over time, or do so only slowly.

But most of the problems we face in politics, business (including finance) and society are not like that. We do not have, and never will have, the kind of understanding of human behaviour which emulates the understanding of physical behaviour which yields equations of planetary motion. Worse, human behaviour changes over time in a way that the equations of planetary motion do not …

Discourse about uncertainty has fallen victim to a pseudo-science. When no meaningful quantification is possible, algebra can provide only spurious precision, while at the same time the language becomes casual and sloppy. The terms risk, uncertainty and volatility are treated as equivalent; the words likelihood, confidence and probability are also used as if they had the same meaning. But risk is not the same as uncertainty, although it arises from it, and the confidence with which a statement is made is at best weakly related to the probability that it is true.

The mistake that Viniar of Goldman Sachs exemplified as the credit crunch bit was to believe that a number derived from a “small world” model—a simplification based on a historic data set—is directly applicable to the “large world,” complex and constantly evolving, in which we live. We are both strongly committed to the construction and use of models—we have spent much of our careers in academia and in the financial and business world doing exactly those things. But that has left us aware of the limitations of models as well as their uses.

John Kay & Mervyn King

Since yours truly thinks this is a great article — as is the authors’ book Radical Uncertainty (The Bridge Street Press, 2020) — it merits a couple of comments.

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. 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. Thinking about uncertainty in terms of “rational expectations” and “ensemble averages” has had seriously bad repercussions on the financial system.

Knight’s uncertainty concept has an epistemological founding and Keynes’ 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’ view is the most warranted of the two.

The most interesting and far-reaching difference between the epistemological and the ontological view is that if one subscribes to the former, Knightian view –- as Kay and King 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 Keynes convincingly argued, that is ontologically just not possible.

If probability distributions do not exist for certain phenomena, those distributions are not only not knowable, but the whole question regarding whether they can or cannot be known is beside the point. Keynes essentially says this when he asserts that sometimes they are simply unknowable.

John Davis

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 probabilities and expectations at all.

Sometimes we do not know because we cannot know.