Ergodicity in Action: A Story of Ski Racing and Investment

July 23, 2024

In their last episode, Taylor, Carlisle and Luca Dellanna discussed “Ergodicity in Action: A Story of Ski Racing and Investment.” Here is an excerpt from the episode:

Luca: Yes. So the trick is not to define it, but to give an example first. Because when people hear the story, they understand it immediately. And in the book I talk about the story of my cousin, who was a great skier since a very, very young age. He even made it to the World Championships in his age group. But then, unfortunately, due to one leg injury after another, he had to give up professional skiing before he even turned 18.

From him I learned that it is not the fastest skier who wins the race, but the fastest skier among those who make it to the finish line. (Tobias laughs) I don’t want to make the banal point here that survival is important for performance, but I want to show that it is more important than performance, especially in the long term. And that’s why I’m giving this numerical example, which is a very quick puzzle. It goes like this:

Imagine my cousin is a very good skier. He is taking part in a skiing championship with 10 races. My cousin is an excellent skier and has a 20% chance of winning each race, but he also takes a lot of risks, so the probability of breaking his leg in each race is 20%. The question is: how many races is he likely to win in a championship with 10 races? The naive answer is: two races. Because we assume that in 10 races the chance of winning each is 20%. 10 times 20% is 2. But if you do the math, you only get 0.71. This big difference between 2 and 0.71 is because if my cousin breaks his leg in one race, he will not only lose that race, but also all the following ones because he cannot take part in them.

So that’s the principle that irreversibility absorbs future gains. We see that in skiing, we see it in investing. If you have $1,000 and you lose 50%, you’ve not only lost $500, you’ve lost all the future gains that $500 could have produced. This principle that irreversibility absorbs future gains is the core of ergodicity. In particular, we say that if a context is ergodic, there is non-irreversibility; we call it ergodic, and you can use averages. But in most of the real world, if not the entire real world, losses are irreversible, which means you can’t use averages, you can’t rely on averages, and those are contexts that we call non-ergodic.

Jake: Beautiful. That’s surprising. I think it’s probably one of the most underappreciated concepts and yet the most important to understand in the world of finance.

Luca: Yes, exactly. I think it’s terrible that it has such a bad reputation. The name ergodicity is terrible for…

Jake: Yes, the marketing team.

Luca: -I understand. To the market.

Jake: That’s why he’ll definitely get fired.

Luca: (laughs) Exactly. And then the problem is that everyone tries to either define it or explain it in mathematical terms, which is a terrible idea to make the idea understandable. What I did in the book was I imposed two restrictions on myself. The first, I will not use mathematics at all. And the second, I will not define the concept until we get into the second – Oh, my God, what is happening here? until we get into the –

(Laugh)

Luca: Sorry. Until we get to the second half of the book. Yes.

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