Ergodicity: Path Dependence and Why The Average Is Lying to Us.
It would be easy to get overly technical with this topic, so I’ll keep the mathematics to a minimum and focus instead on where the concept of ergodicity actually matters. If you want to go deeper, I highly recommend the work of Luca Dellanna, who has written extensively on the subject. His appearances on the Value After Hours podcast are also worth a listen. That said, let’s dive in.
Imagine a simple game. You roll a die ten times and take the average of the results. That average is your score. Now let’s add a financial incentive: it costs $2.50 to play a game (ten rolls), and you receive your score as a payout. Would you play?
Play enough times, and the average of your scores will converge toward 3.5: the expected value of a fair die. That’s an expected profit of $1.00 per game. Not bad.
Now imagine that instead of playing alone, you and nine friends all roll ten dice simultaneously. You average everyone’s results to get a collective score. Each player still pays $2.50 to play, and each player wins the collective total score. Play enough times, and once again the average converges to 3.5: still an expected profit of $1.00 per person per game.
So far, so good.
Now let’s introduce a small rule change. If anyone rolls a 1, that roll ends their game, and all remaining rolls needed to reach ten are counted as 1s. Would you still play?
At first, nothing seems different. You and your nine friends play, average the results, and once again arrive at… 3.5. Same expected profit. Unimpressed but satisfied, five of your friends leave the table.
You and the four remaining players keep playing, but something unexpected happens. While earlier games reliably averaged a $1.00 gain per person, your realized profit now hovers closer to $0.65 per game. Shrugging it off as bad luck, your remaining friends eventually leave too. You continue alone.
First game: 0.70
Second: 1.85
Third: 1.70
After one hundred games, your average never recovers above two. You’re now $50.00 underwater.
Were your friends right about luck running out?
Not quite. What you’ve encountered is non-ergodicity.
Ergodicity is a property of systems where time averages and ensemble averages converge, where outcomes are independent of the path taken to get there. This is how we’re usually taught to think about probability. It’s why saying “the next coin flip must be tails” after three heads is considered a fallacy.
But most real-world systems are not ergodic.
By introducing a rule where early bad results permanently degrade future outcomes, our dice game became path-dependent. Playing as a group masked that fragility by pooling outcomes. As the group shrank, individual exposure increased, and the time average diverged sharply from the expected value. Eventually, the positive expectation vanished altogether. (If the idea of collaboration as a way to stay closer to ergodic outcomes interests you, The Ergodic Investor and Entrepreneur by Graham Boyd and Jack Reardon is a worthwhile read.)
This is where our intuition often fails us. We instinctively assume life is ergodic, when in fact most of it isn’t.
Consider investing. A series of early losses leaves you with a smaller capital base that must now consistently generate above-average returns just to recover. Among traders, “blowing up” refers to losses so severe that the game ends entirely. This is common in leveraged strategies, where a 10% adverse move can wipe out 100% of your capital if you’re leveraged nine to one.
At that point, expected future returns are zero, regardless of how attractive the averages once looked, or how future market gains will be.
The heuristic is therefore simple:
When making decisions, don’t choose the option that looks optimal on average. Prefer the one that leads to acceptable outcomes across the widest range of scenarios, especially when failure is irreversible.
In other words, when ruin is possible, survival beats optimal.
