Knowing whether a process is ergodic or non-ergodic is critical in knowing how much risk to take. Investing and wealth are non-ergodic processes, which imply that our first thoughts on expected values are very wrong.
I've heard about ergodicity before, but wasn't quite able to understand it until I watched this video by Ergodicity TV. It's an important concept but seems more widely known in physics than in finance. I want to try explaining it in my own words below.
We're familiar with different types of "averages" - mean, median, mode 1. Let's focus on mean for today, and take it as the expected value of some random event 2. How we define expected value can give us dramatically different results, changing our mindset on the attractiveness of bets and how much to bet.
Ergodicity means that the ensemble average is the same as the time average. Something being non-ergodic means the opposite, that the ensemble average is not the time average.
Yeah, I'm not sure what ensemble and time here mean 3 either, so let's look at an example of tossing a coin.
Suppose some random dude tosses a coin 5 times, getting some heads and some tails. We can calculate the time average for this simulation by getting the average number of heads for one person across a period of time. There are 3 heads out of 5 tosses, so that's 0.6 heads (3 divided by 5).
Suppose we get a few more people to toss coins. We get something like below, where I'm representing heads as 1 and tails as 0 for convenience:
There are two types of averages we can use here. The first is the time average from before, where we get the average over some period of time for one person.
The second is the ensemble average, where we get the average over one period of time for multiple people.