Foundations of the Precautionary Principle and What is Wrong with
Definition 2: Thick-tailed distributions are characterised by the dominance of a few number of events over the total statistical properties. Conventional
2. Different Probabilities: Assessing Risks Across Time not Space
I am going to start with the concept of time probability (in real life) as an extremely different animal distinct from ensemble probability (used in deci-sion theory). We will see the difference intuitively.
First, consider path dependence. If I wash my shirt and then iron it, visibly, it would be a very different outcome than if I iron my shirt first and then wash it. This may seem trivial, but if we look at it from a risk stand-point, some truth can emerge by introducing a sequence. As Warren Buf-fet says: ‘In order to succeed, you must first survive!’, which leads to some inseparabilities – you cannot separate returns from risks of ruin. You cannot say an investment strategy is a great strategy if it does not first guarantee survival. Cost-benefit analysis breaks down under some environments – those entailing ruin.19 That point can be easily seen in the following thought experiment (Figure 6). We select 100 people, give them an endowment, each one of them, and send them to a nearby casino to figure out their expected return from the casino, which may be positive. The 100 people proceed to gamble eight hours a day at a certain pace. Then person number 28 goes bust.
Will it affect person number 29? Of course not. That is what we call vertical (state-space) probability. The total return that we are going to get that way will be an arithmetic average of the returns – standard expectation operator.
By the law of large numbers, we can figure out what the expectation of that return would be and infer how the casino fares.
The story would be markedly different if we send one person for 100 days to the casino, instead of 100 people each for one day. So, one single person speculates across time, and of course, if on day number 28 the person is ruined, there is no more day 29, unlike in the other experiment.
19 For more technical, at length discussions, and precise definitions, see: Taleb, N. N. (2020). On the statistical differences between binary forecasts and real-world payoffs. International Journal of Forecasting 36(4), p. 1228-1240.
As a consequence, we individuals have to evaluate our risk process across time, not across space; that is completely different. This is what has been recently rediscovered and formalised by Ole Peters along with the late Murray Gell-Mann20; they realised that there is something a little shaky in the foundation of the way we treat probability because these are completely different probabilities. We traders dismiss averages with the expression
‘never cross a river if it is on average four feet deep’ – and traders and business-persons have a deep intuitive understanding of the notion.
Now if I take the expectation of something horizontally (over time), I get a completely different result. For example, if a gambler is exposed to a small probability of ruin, they will be eventually ruined no matter how small the probability. Why? Because over time it is a geometric average since you are multiplying, and if you multiply something by zero, you get zero.
So, the two probabilities are not the same, but they have been treated the same in the literature. The point was not fully formalised until Peters and Gell-Mann did so in 2016. Another way to see it is that, in the presence of an absorbing barrier, these had all the same return, but the paths are very different. To compare or equate time probability and the other one, of course you need some kind of function.
Visibly, the logarithm operator works well because it gives you the same payoff on both sides. It can translate between arithmetic and geometric because when a state corresponds to ruin, the log equals negative infinity, no longer zero. People mistakenly believe that the log is used for risk aversion;
it is not. It so happens that the log operator maximises things over time, using simple parameters such as growth – and what maximises things over time also prevents ruin because this business of multiplying by zero causes absorption at zero.
Contrary to what is said in the literature, the law of large numbers does not work over time (what we call horizontally). If you looked at your future outcomes (in a multiplicative way) on a binomial tree, you will realise that you rapidly start diverging from the average, with no eventual pull-back or what is called ‘reversion to the mean’. There is no law of large numbers for a single individual that way, no matter the number of periods involved.
It is just you – and you are not an average. So, the only way you can start equating strategies is if someone engages in a stream of payoff that sort of
20 Peters, O., & Gell-Mann, M. (2016). Evaluating gambles using dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(2), 023103.
completely eliminates ruin. There are classes of strategies that dynamically eliminate ruin that are acceptable because they give you the same or some very similar returns across ensemble and across time.
___________________________
The difference between 100 people going to a casino and one person going to a casino 100 times, i.e. between (path dependent) and conventionally understood probability.
The conflation has persisted in economics and psychology since time immemorial, partic-ularly with the use of nonergodic cost-benefit analyses.
Let us consider surviving risk takers in the real world. Take for instance the trading firm (and investment bank) Goldman Sachs. They must have engaged in these restricted classes of strategies because these people take monstrous
FIGURE 6
amount of risk, and they have been doing it for 149 years, and 149 years is a very long time: a small local (daily) probability of blow-up would have ensured total ruin. That something they are doing right is not taking small-tail risk; it is taking no-tail risk (or something of the order of 10-8). They try to be exposed to maximum risk without tail exposure, which is completely different from what people engage in if they follow something like the Markowitz mean variance optimization or similar standard business school methods. The only strategies that make you avoid the gambler’s ruin are part of the classes that have some specific properties either by using logs or by using dynamic hedging and/or the Kelly criterion, things that come from information theory rather than from standard economics or decision theory.21
To rephrase: owing to gambler’s ruin, eventually, you are going to go bust no matter what your space expectation is. If you have a small probability of going bust, you are going to go bust. At the stopping time, you are going to go to zero regardless of your return, unless you have zero probability of ruin. Consequently, if we start using these concepts exposed above, then you must start evaluating risk not by incident, like what is your risk of falling off a mountain or something, but by integrating over your lifespan.
We do it intuitively – see the works of Gerd Gigerenzer and the ABC group for the validity of intuition under some probabilistic representation.22