2025-04-13
Erogodicity (or lack thereof) in random systems can lead to counter-intuitive results. I became interested in ergodicticity after reading the book “Skin in the Game” by Nassim Nicholas Taleb. The book discusses the concept of ergodicity in the context of risk and decision-making. When dealing with systems with some randomness baked in, it is quite crucial to understand if we can say something about all such systems by looking at the lifetime of only one instance. If the system is ergodic then we can. However if the principle of ergodicity does not hold for a system then a particular instance can behave quite differently than what is suggested by the average of all such systems.
The wealth simulator shows one such system based on an equation for “Random Multiplicative Processes” also called “the equation of life”. This equation is a contrived example of a non-ergodic system. Let us imagine a quantity which increases or decreases by one of two predetermined factors with a random chance of 50%. And then we repeat this process over and over again.
In the specific case imagine that you start with 100/- credits. Now we toss a coin and with heads we multiply the credits with 1.5 i.e. a 50% increase, and with tails we multiply it with 0.6 i.e. a 40% decrease. Now we repeat this process several times, each time using the credits from the previous run.
\(x_{n} (t+1) = \begin{cases} 1.5 x_{n}(t) & p=1/2\\0.6 x_{n}(t) & p=1/2\end{cases}\)
where,
\(x_{n} (t+1)\) is the new value,
\(x_{n} (t)\) is the previous value
I first came across this particular example in the videos on the YouTube channel “Erogodicity TV”. I’ve embedded the video by Alex Adamou starting at 13:11 where the equation is introduced.
What is interesting about this example is that almost every time we start with 100/- and do this we end up losing our credits and the value decays to zero. However if we calculate the expected value of all possible runs of this process, or even a large number of such processes then we see that the credits keep increasing by a steady rate of 5% per time interval.
I like the name “equation of life” that Alex Adamou uses in the video. It sounds more poetic than “random multiplicative process”.
I later discovered more detailed descriptions of this equation by “Ole Peters” - see his blog post here https://ergodicityeconomics.com/2023/07/28/the-infamous-coin-toss/.
The equation is a nice way to demonstrate how in some cases the expected value of all possible systems is quite different the value of one system. However the equation uses some set values. I wanted to try out different values for the parameters to see what would be the effect.
It is interesting to think of the value increasing and decreasing in this equation as wealth, and then imagine a persons wealth increasing and decreasing let’s say every year by one of two random multipliers.
\(x_{n} (t+1) = \begin{cases} m_{1} x_{n}(t) & p=p_{1}\\m_{2} x_{n}(t) & p=1-p_{1}\end{cases}\)
where,
\(x_{n} (t+1)\) is the new value,
\(x_{n} (t)\) is the previous value,
\(m_{1}\) is the first (gain) multiplier,
\(m_{2}\) is the second (loss) multiplier,
\(p_{1}\) is the probability of gain.
The simulator on the main page has the following parameters that you can play around with, and press the “Run Simulation” button to see the resultant plot.
For example, here’s the simulation for the values in the equation of life section.