Ergodicity
I don't fully comprehend ergodicity or its significance. But the concept is sufficiently counter-intuitive to make it interesting to me. In this article I'll try to explain ergodicity, as far as I understand it, based on a simple game:
- You have some amount of wealth.
- A fair coin is tossed.
- On heads, your wealth increases 50%.
- On tails, your wealth decreases 40%.
Before reading any further, ask yourself — would it be rational to participate in the game..?
Ensemble Average — 5% Gain
First, let's see what happens on average when multiple players (an ensemble) participate in the game. The average player should gain 5% wealth on each round (on each coin toss):
EA = (1.5 + 0.6) / 2 = 1.05
Just to verify this formula, let's simulate an average player. Here we use 10,000 players (the ensemble) and run the game for 10 rounds. The vertical axis shows average wealth on a log scale:
Time Average — 5% Loss
Now, let's take the perspective of an individual player. An individual player should, on average, loose 5% wealth on each round:
TA = (1.5 * 0.6) / 2 = 0.95
To verify this formula, let's simulatate an individual player. Here we use one single player and run the game 10,000 rounds. The vertical axis shows wealth on a log scale:
Defining Ergodicity
Ergodicity have slightly different meanings in different contexts, but here's a commonly used defintion for our case:
A process is ergodic if EA = TA
Based on this definition, we can conclude that our game is non-ergodic — the ensemble average is not equal to the time average. The ensemble average is 5% gain, and the time average is 5% loss.
What I find counter-intuitive is that it's sometimes rational for a group (an ensemble) to participate in a game, but, at the same time, irrational for an individual. How can this be? How can wealth increase for an average player and decrease for an individual player at the same time? I understand the math, but my intuition find it paradoxical.
The Significance of Ergodicity
Most economic theories are implicitly based on the assumption that processes are ergodic. Economic decisions are considered rational if the expected utility is positive (and irrational otherwise). The expected utility is, by definition, the same as the ensemble average, but as we've seen, the ensemble average might be very different from the time average. When a process is non-ergodic, we can't use expected utility as a definition of economic rationality. In these cases, economic theories are wrong.
So the question becomes: Are economic processes typically ergodic? If not, our economic theories might be fundamentally inaccurate.
I don't know the answer, but investor Tylor Pearson argues that "pretty much every human system is non-ergodic". And advertisement executive Rory Sutherland suggests that risk-aversion-bias might not even exist (at least not to the extent that we've traditionally assumed), because decisions are almost always made in non-ergodic circumstances, and therefore, risk-avoidance must be considered rational, rather than a psychological bias.
Whatever the case, I recommend watching this talk by Ole Peters, where he explains the concept of ergodicity and its potential consequences for economic science.
The Kelly Fraction
To end this article, let's switch gears and have a look at the "Kelly Fraction".
The Kelly Fraction is used in finance and investment. It's somewhat abstract and difficult to understand, but based on our understanding of ergodicity, I believe we can reach a simple description.
Let's slightly modify the game so that a player can choose the amount to bet in each round. In this new scenario, it's in fact possible for an individual player to "get access" to some of the ensemble gains. The optimal amount to bet is calculated by the Kelly Fraction:
KF = P/A - Q/B
P = Probabilty of win
A = Fraction lost on loss
Q = Probability of loss
B = Fraction gained on win
For our game, the Kelly Fraction is:
KF = 0.5/0.4 - 0.5/0.5 = 0.25
To verify this formula, let's run the time average simulation again, but this time using the Kelly Fraction. Here we use a single player making a 25% bet in 10,000 rounds. The vertical axis shows wealth on log scale:
It's clearly valuable to understand the Kelly Fraction when making investments in non-ergodic environments.