Ergodicity Fundamentals: Try Enough Times And You'll Fail

Ergodicity is the idea that a point of a moving system will eventually visits all parts of the space that the system moves in. So long as there are sufficient random samples, a process will represent the statistical average of the entire system. This can be the simple, every-day notion that smoke fills a room and it becomes smoke-filled, or that in an infinite range of coin flips heads and tails come up half the time. Ergodicity helps us understand how systems function over time, especially when random processes are involved.

While ergodicity is primarily used in thermodynamics, it has applications in probability and finance. This is because of a distinction in time averages and ensemble averages. A time average is the long-run average experienced by a single entity over time. The ensemble average, on the other hand, is the computed average by looking at many identical systems. In ergodic systems, these averages are equal. Many real-world processes, however, are non-ergodic. This means that a strategy that looks good “on average” might result in ruin over time. For instance, a risky investment might have a positive average return, but for a singular investor, the possibility of a catastrophic loss outweighs that average benefit.

In non-ergodic environments, the chance of experiencing a severe negative outcome gets hidden when you only consider ensemble averages. Take the instance of a gambler. They win occasionally. But over time, repeated risky bets deplete their bankroll. Even if the average outcome appears profitable, their little venture dries their pockets. As anyone trying to manage risk, you need to understand how repeated exposure to risk result in significantly different outcomes than the statistical mean. When you risk your life in a venture that has a 60 percent chance of killing you, you had a 60 percent chance of death. Now say you did that twice. The chance that, over that period, you could have died, was no longer 60 percent, but the aggregate of two bets against the reaper. Hence historical averages don’t accurately predict individual outcomes over time.

Real-world examples of ergodicity:

• Gambling: in a casino, the average payout of a game might be close to break-even when considering countless plays (ensemble average). However, for an individual gambler, the risk of a long losing streak is much higher, since they don’t have infinite funds to keep betting. Insight: the individual’s time average can be far worse than the ensemble average.

• Investing: a stock might have a positive average return based on historical data. But if an investor repeatedly takes large risks (e.g. heavy leverage), they might face periods of extreme losses that wipe out their portfolio. Insight: relying on ensemble averages without accounting for compounding risk over time can result in financial ruin.

• Business Decisions: a company’s project may have an average success rate that is appealing in aggregate. However, if one project failure has catastrophic consequences, the long-term average for the business is driven much lower. Insight: strategic decisions have to account for non-ergodic risks, ensuring that “Black swan” disasters are mitigated and planned for.

• Personal Life: consider lifestyle choices like health habits. While the average benefits of a healthy lifestyle are well-documented, for an individual, one extreme event (e.g. a heart attack due to poor habits) can dramatically diverge their life from the mean. Insight: personal decisions shouldn’t be evaluated just on aggregate data but also on the potential for severe deviations from the norm.

How you might use ergodicity as a mental model: (1) risk times time — when assessing an opportunity or choice, consider how the benefits in the aggregate data differ from the likelihood you will actually experience them over time (”If I repeated this many times, would I consistently get this outcome, or would I risk running into disaster?”); (2) consider the impact of repetition — focus on strategies that work over many iterations, not those that only look good on paper in a single instance; (3) flip off risk — identify decisions where outcomes vary widely over time, and take steps to limit potential negative impacts (build in safety margins, employ hedging strategies, diversify your portfolio, etc.); (4) consistency over intensity — take decisions that have more predictable, stable outcomes as time passes, rather than those that depend on favorable averages with hidden risks behind the scenes.