Ergodicity
The property of a system where the time-average of an individual path equals the ensemble average across all possible paths. Most real-world repeated-risk processes are non-ergodic — meaning the average outcome across many people is very different from the likely outcome for any one individual over time.
The Jackpot Paradox
A game can have positive arithmetic expected value and yet cause almost everyone to go broke.
Example: Coin flip — +50% on heads, -40% on tails.
- Arithmetic mean (average wealth across all outcomes): positive
- Geometric mean (median outcome over many flips): negative
Why? One bad streak wipes out many previous wins. The arithmetic mean is dominated by rare jackpot outcomes. The median path bleeds to zero.
Simulation: 25,000 people each flip 1,000 times → virtually all end up at ~$0, despite the game having positive expected value.
To break even, you need 570 heads out of 1,000 flips. All expected value is concentrated in the top 0.0001% of outcomes.
Arithmetic Mean vs Geometric Mean
| What It Measures | |
|---|---|
| Arithmetic mean | Average wealth across all possible outcomes (ensemble) |
| Geometric mean | Expected wealth for a single individual over time (time-average) |
In a multiplicative process (like compounded returns), you should care about the geometric mean, not the arithmetic mean. "In the world of compounded returns, the dose makes the poison."
Also called: ergodicity problem (Ole Peters / LML Economics), volatility drag (finance), or the Jackpot Paradox.
Three Wealth Preferences
| Preference | Behavior | Long-run fate |
|---|---|---|
| Log | Risk appetite shrinks as bankroll grows | Survives — Kelly-optimal |
| Linear | Same risk appetite at any wealth level | Goes to zero eventually |
| Exponential | Risk appetite grows with bankroll | Goes to zero fast |
SBF (FTX) held linear preferences. Su Zhu / 3AC held exponential preferences. Both collapsed.
Practical Implications
- Never bet a fixed fraction of your bankroll on positive-EV games if the volatility is high — you'll still go broke
- Kelly criterion or log-wealth sizing is the mathematically correct solution for ergodic survival
- Diversification works partly because it reduces variance of the geometric mean, even at the cost of arithmetic mean
- The scarcity mindset that drives people to chase jackpots (feeling behind, wanting to make it fast) is the enemy of compounding
"Build more edge rather than risk more size. Log wealth is what matters. Maximize the 50th percentile outcome."
Connection to Other Concepts
- Deliberate Practice is ergodicity-aware: consistent, calibrated effort produces far better long-run outcomes than high-variance swings
- Prerequisite Mastery applies the same logic: skipping steps is a high-variance bet that often leads to total failure
- alternative-histories makes the path-dependence explicit: the realized path is only one possible draw
- skewness-and-asymmetry explains why average outcomes can hide ruinous individual paths
Taleb Connection
fooled-by-randomness is not an ergodicity textbook, but it gives the trader's lived version of the same problem. A strategy can look successful across a short observed path while carrying hidden exposure to a rare event that would erase the player. The lesson is the same as the Jackpot Paradox: do not optimize for the attractive average if the individual path cannot survive variance.
Sources
- fooled-by-randomness - Alternative histories, hidden tail risk, and survival under randomness.
- the-jackpot-age — detailed quantitative exposition, cultural analysis