Here’s a question for you to take your mind off COVID-19: Who is your favourite bear?
Is it Paddington? Or maybe it’s Rupert. For many it is Yogi, who is, after all, smarter than the average. Personally, I always had a soft spot for Boo-Boo. But if my life depended upon it, I think I would have to choose Baloo, if only for his wise counsel that we should all “Look for the bare necessities.”
Wise words indeed. But wiser still would have been the bear that entreaties we look for the ergodicities. Yes, the simple ergodicities of life will come to you. But only if you plan for them. Sadly, that bear of my dreams has yet to be born, and so the majority of us, unable to rest at ease without the ergodicities, continue to live out our decidedly non-ergodic lives with nothing but the precautionary principle to console us. Allow me to explain.
Under the somewhat misleading heading of ‘Informal Discussion’, Wikipedia explains the simple ergodicities as follows:
“For a discrete dynamical system (X,T), where the space X is endowed with the additional structure of a probability measure space which is invariant under the transformation T, ergodicity means that there is no way to measurably isolate a nontrivial part of X which is invariant under T.”
Need I say more?
Okay, so let me put it this way. If you took 100 dice and threw them simultaneously, do you think (allowing for statistical fluctuation) that the total score would be the same as if you took just a single die and totalled the score after 100 throws? If the answer is yes, then you are dealing with an ergodic system.
Why does this matter? It matters because so much of risk assessment implicitly assumes ergodicity, since it seeks to predict the average performance over time using a probability density function that assumes ‘invariance under T’. In effect, the throwing of a die 100 times is treated as statistically equivalent to 100 dice thrown at the same time, thereby enabling the probabilistic calculations required for economic evaluations such as cost benefits analysis. In oh so many cases this is a perfectly safe assumption, but in oh so many more, it is not. Very often a throw of the die has a huge bearing on what follows, in which case there is no equivalent to the scenario of 100 simultaneous throws. Take, for example, a game in which throwing snake eyes means ‘game over’. With such, so-called ‘ruin’, scenarios the old economic game of calculating ‘equivalent values’ and performing costs benefits analyses doesn’t make much sense. Instead, the game is non-ergodic and we need a game play that takes into account the fact that, in order to attain the 100 die readings required to determine a total, one will need an extraordinary run of good luck. What we need instead is a precautionary approach.
As an example of this principle in action, take the case of house insurance. Every day of our lives we cast the die. We win when our house and contents are safe at the end of the day, but we lose when they are not. Furthermore, we are out of the game, because most of us cannot afford to replace everything in the event of catastrophe. The probability of that happening on any given day is low, but the probabilities stack up. So we take out insurance. With insurance we lose every day (by paying out a pro-rata daily insurance fee) but the losses are predictable and never enough to remove us from the game. A non-ergodic setup has been turned into an ergodic one. Of course, from the perspective of the insurance company the situation has always been ergodic. They are throwing dice simultaneously and paying out for housing disasters every day, but the premiums have been set up so that even on a statistically bad day, they can always take it on the chin and stay in business.
And so, in our daily lives, the successful management of risk depends critically upon identifying the ergodic and non-ergodic situations and acting appropriately. In the non-ergodic situation, it’s all about minimizing the maximum loss in order to keep oneself in the game so that one can reap the compensatory benefits of any upside further down the road. Ideally, one would always wish to transform the non-ergodic into the essentially ergodic, usually by means of a judicious risk transfer such as taking out insurance. Unfortunately, however, this is not always possible, and so other strategies are often required, normally involving the precautionary principle.
And then Along Comes Global Warming
Global warming is often cited as the classic ruin scenario leading to a non-ergodic game. Worse still, the human race cannot transfer the risk in order to contrive ergodicity because there is no one out there listening (apologies to all the religious readers). It should come as no surprise therefore that the precautionary approach is advocated. No one wants to talk about cost benefits analysis anymore because no one is deceiving themselves that ergodicity applies. In fact, no one even wants to talk about probabilities anymore. All that is required to invoke non-ergodicity, and the lurking precautionary principle, is to conceive a plausible worst case that takes us all out of the game. Now any cost can be justified to manage the risk – and I do mean any cost.
I might be going along with all of this if it were not for one thing: Even when probability is deemed immaterial, uncertainty still has a role in the game. It isn’t risk aversion anymore, because one needs to assess probabilities in order to assess levels of risk. But it is still ambiguity aversion because we are basing our decision-making upon the notion of plausibility – a concept heavily invested in epistemic uncertainty. This is a crucial point since, as soon as a plausible worst case scenario has been agreed, it forms the basis for a deterministic approach in which one proceeds on the premise that the worst case will happen. This is all very well but there is now an awful lot at stake based upon the somewhat problematic notion of plausibility. Nor is it simply a case of replacing a non-ergodic game with an ergodic one. Can we really be confident that we will stay in the game, considering the economic shock entailed when following Extinction Rebellion style ‘climate emergency’ management?
Hold on a minute! I’ve just remembered that my favourite bear is Sooty. He’s a rebellious little sod, so forget everything I’ve just said. What’s that Sooty? You’ve got a magic wand?