non ergodicity and path dependence why averages lie
An ergodic system is one where averaging across many people at one point in time gives the same result as averaging across many time periods for one person. Coin flips are ergodic. Markets are not.
This seemingly technical distinction has profound implications: most financial advice, most risk models, and most trading strategies implicitly assume ergodicity. They give you the ensemble average and claim it applies to your individual path. It doesn’t.
see also: gn13-non-ergodicity · gn15-risk-of-ruin · gn20-expected-value-edge-variance · gn17-position-sizing
the ensemble trap
Imagine a game: 100 people each bet 200), half lose (get 100 — breakeven. But the time average for any individual player is different: after one flip you either have 0. The ensemble average doesn’t describe your experience.
Now imagine 10 rounds. If you must bet your entire capital each round, most people will go to zero at some point, even though the ensemble average remains positive. The system is non-ergodic — the average across people doesn’t match any individual’s experience.
the ruin problem
Non-ergodicity creates the ruin problem: even if a strategy has positive expected value in the ensemble sense, any individual following it can still go to zero if they hit the wrong sequence.
This is why Kelly sizing exists. Kelly maximizes long-term growth by accounting for non-ergodicity — it sizes bets so that no single loss eliminates you from the game. The key insight: if you go to zero, you stop being part of the ensemble. The survivors create their own statistics.
path dependence
In non-ergodic systems, the order of outcomes matters. Two traders with the same strategy and the same final P&L can have completely different psychological experiences depending on whether the winning streak came first or last.
Path dependence also means:
- A 50% drawdown requires 100% gain to recover — the path matters
- Trading full Kelly (aggressive) means you’ll likely hit a -50% drawdown at some point, even with positive edge
- Backtests that randomize trade order (Monte Carlo) are essential because the actual sequence will differ
implications for position sizing
If trading were ergodic, you could maximize expected value without worrying about sequence. Since it’s not:
- Fixed fractional sizing (risk % per trade) is ergodic-friendly because it accounts for changing account size
- Martingale (doubling after losses) is anti-ergodic — it guarantees eventual ruin
- Full Kelly maximizes growth but accepts 30-50% drawdowns — you must survive those
- Half-Kelly is the standard compromise: 75% of maximum growth with 50% less drawdown risk
my take
Non-ergodicity is the single most important statistical concept traders don’t know. Every time I see someone say “this strategy averages 20% annually” I think: that ensemble average doesn’t describe your individual timeline. The question isn’t the average return — it’s the probability that your specific path survives long enough to realize that average.
I size for the worst path, not the average path. The average path is a fiction. The worst path is preparation.
linkage
- [[gn13-non-ergodicity]]
- [[gn15-risk-of-ruin]]
- [[gn20-expected-value-edge-variance]]
- [[gn17-position-sizing]]
- [[gn12-extremistan-vs-mediocristan]]
ending questions
is your position sizing designed for the average path or the worst path? if the answer isn’t “worst path,” you haven’t accounted for non-ergodicity.