The Future is: NonErgodic. Uncharted. Unpredictable

The Future is Non-Ergodic, uncharted: (of an area of land or sea) not mapped or surveyed. "these are unchartered waters"

...the average outcome for a group is not necessarily the same as the average outcome for an individual over time

What Does Non-Ergodic Mean?

In simple terms, ergodicity is the idea that given enough time, a system will experience all possible states or outcomes available to it. In an ergodic system, the long-term average behavior of a single trial is the same as the average across many trials run in parallel. In contrast, a non-ergodic system does not visit all possible states; its history matters and it can get “locked” into specific outcomes (Edge.org). Another way to say this: in a non-ergodic process, what happens over time to one individual path won’t necessarily match the probabilistic expectations derived from many possible paths (Ergodic descriptors of non-ergodic stochastic processes - PMC ). This challenges the traditional probability models that implicitly assume ergodicity – the idea that time averages equal ensemble (group) averages (The ergodicity problem in economics | Exploring Economics). When a model blindly uses the expected value (the average over all hypothetical parallel worlds) as a predictor for a single real-world timeline, it may be making an ergodicity assumption that doesn’t hold.Example: Imagine a jar filled with two types of balls, red and blue, and you repeatedly draw one ball, note its color, and put it back. If the process is ergodic, then given enough draws, the fraction of red vs. blue you get will converge to the true fraction in the jar. In a non-ergodic scenario, however, the composition of the jar might change with each draw (for instance, if you don’t replace the ball or if drawing a red somehow increases the chance of more reds in future). The path you take (which colors you drew early on) can influence what you get later. Over time, one sequence of draws might end up almost all red, while another sequence ends up mostly blue – and they won’t “average out” to the same mix. Each history is unique.

Time Averages vs. Ensemble Averages

A key to understanding non-ergodicity is distinguishing time averages from ensemble averages. An ensemble average is the result we’d get by imagining many parallel worlds or many identical copies of a system and averaging their outcomes at a given time. A time average is what you get by following one system over a long time and averaging its outcomes. If a process is ergodic, these two averages are the same. If it’s non-ergodic, they diverge – sometimes drastically (Ergodic descriptors of non-ergodic stochastic processes - PMC ). In non-ergodic systems, the average outcome of a large group is not a reliable guide to what a single individual will experience over time (Ergodic descriptors of non-ergodic stochastic processes - PMC ).Why doesn’t a non-ergodic system “average out” over time? Because such systems often have path dependence or absorbing states. Path dependence means early random events send the system down a particular trajectory that later events cannot undo. An absorbing state is a condition you can get stuck in (like going broke or an extinction event) which, once reached, skews the future outcomes. In these cases, averaging over possibilities fails because some possibilities effectively get eliminated or overemphasized as time goes on. The system doesn’t “forget” its history. As one research summary puts it, in non-ergodic processes the assumption that one can substitute long-term time behavior with a simple probabilistic expectation is invalid (The ergodicity problem in economics | Exploring Economics). Time is no longer “eliminated” from the model – the sequence of events matters.

The Gambler’s Ruin Example: Why Averages Can Mislead

To make this concrete, consider a simple betting game. You have $100 and you repeatedly bet your money on a coin flip. If it’s heads, you win 50% of your current money; if it’s tails, you lose 40% of your money. This game has a positive expected payoff – on average you gain 5% each toss (since 50% of the time you gain half, and 50% you lose 40%, the net expectation is +5% of your money each round). If we only look at this ensemble average, it sounds like your wealth will grow and follow an exponential curve upward. In fact, if many players played once, the average outcome would indeed be a 5% gain, and if many players played 10 times, the average wealth would be about 1.05^10 ≈ 1.63 times the starting value, an exponential increase.However, what happens to an individual player over time in this non-ergodic game is very different. Let’s play the game twice and look at all possible outcomes:

• You might win both flips. Then your $100 grows to $150 after the first flip, and $225 after the second flip.
• You might win once and lose once (in any order). If you win then lose, your $100 goes to $150, then loses 40% to end at $90. If you lose then win, your $100 drops to $60, then increases by 50% to end at $90. Either way, one win/one loss leaves you with $90.
• You might lose both flips. Then $100 goes to $60 after the first loss, and $36 after the second.

Out of these four equally likely paths, three leave you with less than $100. In 3 out of 4 cases you lost money, even though the average outcome is about $110.25 (probability - The ergodicity problem in economics - Cross Validated). The expected value (+10.25% over two flips) is not indicative of what happens typically (probability - The ergodicity problem in economics - Cross Validated). In fact, most single players’ wealth will shrink over time; only a lucky few will hit the big wins. If you play this game 50 times in a row, there’s a good chance your wealth dwindles significantly, and if you play 1,000 times, you are very likely to end up effectively broke (hovering near zero) – even though the game was “favorable” on average (Ergodicity: the Most Over-Looked Assumption – Neurabites) (Ergodicity: the Most Over-Looked Assumption – Neurabites)!What’s going on? This betting game is non-ergodic. The reason is that losses hurt you more, in the long run, than gains of the same percentage help you. Mathematically, your wealth each round is being multiplied by either 1.5 (heads) or 0.6 (tails). The arithmetic mean of these factors is 1.05 (a 5% gain, suggesting exponential growth), but the geometric mean (the typical factor per round for one sequence) is √(1.5 × 0.6) ≈ 0.95 – indicating an overall decline. Over time, a single sequence multiplies together many such factors, and the geometric mean governs the long-run fate. Here it’s less than 1, so the typical single-player trajectory heads toward zero. The ensemble of many players can have a few big winners that drive the average up, giving the illusion of steady growth, but most individuals do not share in that growth (probability - The ergodicity problem in economics - Cross Validated). As one commentator summed up: “the expected value of gambling is very much not indicative of what happens typically for longer trajectories” (probability - The ergodicity problem in economics - Cross Validated). In non-ergodic situations, risk and variability change the game – you can’t just multiply the average outcome by time and expect to predict an individual’s fate.

Real-World Examples of Non-Ergodicity

Non-ergodicity isn’t just a quirk of gambling simulations; it appears in many real-world processes. Here are a few domains where recognizing non-ergodicity is crucial:

• Economic Inequality: Wealth dynamics in an economy are often non-ergodic. If everyone has the same average rate of return, you might expect everyone’s fortunes to just grow uniformly – but in reality, random advantages compound. Those who get ahead early (even by luck) often have more to invest and can end up even further ahead; those who suffer losses can fall behind and might never catch up. In technical terms, “wealth is a growing quantity, and hence it is non-ergodic” – meaning the average across people doesn’t reflect what happens through time for each person (). In fact, studies suggest that wealth dynamics tend to be non-ergodic even after adjusting for growth, so using the average wealth as an indicator can be misleading () (). This helps explain why economic inequality can widen: the distribution of outcomes spreads out over time instead of converging. Only a few individuals or families might accumulate a huge fortune (skewing the average up), while many others see much lower growth or even losses in wealth. The system doesn’t “share” the gains equally over time – it’s path-dependent, with rich-get-richer effects.
• Investment and Risk Management: When investing, the distinction between ensemble and time averages can mean the difference between a strategy that looks great on paper and one that actually secures your future. For example, an investment might have a high average return but also a small chance of catastrophic loss. Traditional models that assume ergodicity would focus on the high average return and suggest an investor will do very well “in the long run.” But if a single catastrophic loss can wipe you out (a non-ergodic scenario), then the long run may never arrive for you. In practice, markets and investments are non-ergodic because they involve compounding and risk. One cannot assume that because an asset has, say, a 7% average yearly return, every investor will actually realize 7% every year. Some years might be -50%, others +30%, and the sequence matters immensely. An investor who experiences the bad years early may never recover – an outcome quite different from the smooth exponential growth assumed by averages. This insight has practical implications: it leads to strategies like the Kelly Criterion, which focuses on maximizing the time-average growth rate of wealth rather than the one-time expected value. In essence, Kelly betting (used in gambling and portfolio theory) acknowledges non-ergodicity – it seeks a balance that avoids ruin and achieves the highest typical growth over time, rather than chasing the highest possible average return that only works in an ensemble sense (The ergodicity problem in economics | Exploring Economics). The takeaway is that understanding non-ergodicity helps investors focus on avoiding ruin, managing risk, and securing steady growth, because no one lives in the average of parallel universes – we each live out one timeline.
• Biological Evolution: Evolution and natural selection are profoundly non-ergodic processes. The path that life on Earth has taken is just one of countless possibilities – and many viable forms of life were never realized because of historical chance events. As biologist Stuart Kauffman notes, the evolution of life in our biosphere “is profoundly non-ergodic and historical. The universe will not create all possible life forms” (Edge.org). Only a subset of potential organisms and adaptations actually appear, and once a lineage goes extinct or a genetic trait is lost, that particular path is closed off. There is no “averaging” over all possible evolutionary trajectories; history matters. If you replayed the tape of life, you could get a completely different set of species. Likewise, within a single evolving population, certain random mutations or events can lock in a trajectory – for example, a chance mutation might give one group of organisms a slight edge, allowing them to dominate and making other possibilities unreachable. Evolution shows that even though we can talk about average mutation rates or average fitness, the realized history is one specific sequence of events, not an average of many. This non-ergodicity in evolution is why there is such a rich diversity shaped by contingency, and why simplistic models of steady exponential growth in biological complexity or population often fail – real populations might boom and bust, or get stuck in evolutionary “dead ends.” Non-ergodicity gives us history and individuality in the biological world (Edge.org).

There are many other examples too. Cultures and languages evolve non-ergodically (each with a unique history), and even certain physical systems (like glassy materials that get stuck in one configuration and never explore all others) exhibit non-ergodicity. The common theme is that you can’t assume a system will explore all possibilities or that outcomes will smooth out over time. Sometimes, where you end up depends on the one path you took, not on the average of all paths.

Non-Ergodic Dynamics vs. Exponential Growth Assumptions

A lot of our classic models and intuitions assume exponential growth – for instance, the idea that an economy can grow at a steady percentage every year, or an investment will compound at a fixed rate. Exponential growth is a hallmark of ergodic-like thinking, because it implies a consistent multiplicative process where fluctuations even out. However, non-ergodic dynamics often break the expectations of exponential growth. Why? Because exponential growth assumes that each little increment multiplies your current state by some factor independently of the past. In non-ergodic processes, the increments are not independent of the past – in fact, the past may change the rules of growth.Consider again the betting game example: If it were ergodic, each 5% expected gain would accumulate into a nice exponential curve (in fact, the ensemble average wealth does grow exponentially as 1.05^t in that toy model (probability - The ergodicity problem in economics - Cross Validated)). But the actual single-player outcome doesn’t follow that curve; it typically crashes to zero (Ergodicity: the Most Over-Looked Assumption – Neurabites). In general, in a non-ergodic setting, you often find that the ensemble (average) grows exponentially while the typical individual trajectory does not (). Many exponential models fail in non-ergodic settings because they don’t account for things like absorbing states (e.g. bankruptcy or extinction) or fat-tailed fluctuations. For instance, a simple exponential growth model of a population might predict limitless growth, but if there’s a non-zero chance of a catastrophe at each step, the median population might actually stagnate or die out, even if the average (weighted by rare huge successes) suggests growth. Real-world systems often have saturation points, crashes, or resets that make pure exponential trends short-lived.Exponential growth assumptions can be dangerously misleading if applied when non-ergodicity is present. People might assume house prices will always rise 5% a year on average, or that a nation’s GDP will keep growing steadily, or that a pandemic will spread at a constant exponential rate. But non-ergodicity reminds us that hidden in the “average” growth rate are radically diverging individual trajectories. For example:

• If 90% of companies in a market slowly go bankrupt while 10% become massively successful, the average might show healthy growth (thanks to that 10%), yet most companies experienced decline or failure. Plotting an exponential trend through the average ignores the non-ergodic reality that a typical firm’s experience is very different.
• In personal finance, earning an average return of +15% a year means little if one year of -100% wipes you out. The exponential growth model (which would naively go up by 15% every year) fails because once you hit zero, you can’t continue – a non-ergodic outcome (you can’t “average” your way out of bankruptcy).

In summary, exponential models assume a kind of stability and repeatability that non-ergodic systems don’t have. Exponential growth curves smooth over the risk, the volatility, and the path dependency that are often present in real processes. That’s why economists like Ole Peters have pointed out that many economic models wrongly assume ergodicity and therefore overestimate growth or underestimate risk (The ergodicity problem in economics | Exploring Economics). Recognizing non-ergodicity forces us to revise those models to account for the fact that time changes things – you can’t just treat the future as a scaled-up version of the present on an exponential trajectory, especially when random shocks can knock the system onto a different path.

Why Understanding Non-Ergodicity Matters (Practical Implications)

Grasping non-ergodicity isn’t just an academic exercise – it has real implications for decision-making in finance, economics, and science:

• Better Financial Decisions: Individuals and firms who understand non-ergodicity tend to focus on risk management and survival. Instead of only looking at expected returns, they pay attention to worst-case scenarios and the distribution of outcomes. This leads to strategies like diversification and not over-leveraging, because one unlucky streak can be devastating if you bet too much. It also underpins the use of Kelly-optimal betting/investment, which maximizes long-term growth by balancing risk and reward, effectively acknowledging that you live with the time-average outcome, not the ensemble-average (The ergodicity problem in economics | Exploring Economics). In short, knowing about non-ergodicity teaches you not to put all your eggs in one basket and not to be seduced by a rosy average. As the saying goes, “Don’t cross a river that is on average 4 feet deep” – in a non-ergodic world, that average hides the fact that parts of the river might be 8 feet deep and you could drown. Understanding this prompts more prudent, robust financial planning.
• Economic Policy and Inequality: Policymakers aware of non-ergodicity realize that focusing on aggregate metrics (like average income or GDP per capita) can miss what’s happening to individuals over time (). An economy might be growing in the aggregate (exponential-looking GDP), yet many people are seeing stagnant or declining fortunes. Policies that only optimize for the average risk leaving large segments of the population behind. Recognizing non-ergodicity supports policies that address path dependency – for example, helping people recover from bad shocks (job loss, health crisis) so that one unlucky event doesn’t permanently derail their trajectory. It also sheds light on why wealth tends to concentrate: if returns compound non-ergodically, simply redistributing or “resetting” might be needed to avoid ever-widening inequality. In practical terms, non-ergodicity in economics suggests we should care about median outcomes and the breadth of the distribution, not just averages. It provides an argument for safety nets and insurance: these tools effectively remove or mitigate the absorbing barriers (like absolute poverty or bankruptcy) that make economic dynamics non-ergodic for individuals.
• Scientific and Statistical Modeling: In fields like ecology, epidemiology, or climate science, understanding non-ergodicity is key to making accurate predictions. It encourages scientists to model many scenarios rather than just rely on an expected scenario. For instance, when projecting climate change impacts or virus spread, the ensemble average might smooth out critical extreme events (like tipping points or super-spreader events) that a single real trajectory will feel. By accounting for path dependence and potentially irreversible changes (like ice sheet collapse or species extinction), scientists can provide better guidance – often emphasizing the need for precaution (because you can’t just assume things will revert to a nice average trend if something goes awry). Statisticians and psychologists also note that many experiments assume ergodicity (that an average over participants equals what holds over time for one person), which may not be true (Ergodic descriptors of non-ergodic stochastic processes - PMC ). This has led to re-evaluating how we interpret data in social sciences and medicine: e.g. a treatment might have a good average effect, but perhaps individuals’ responses over time vary widely and are dependent on their unique path (their health history).

In everyday life, an awareness of non-ergodicity instills a healthy respect for uncertainty and variability. It reminds us why “on average” can be a dangerously misleading phrase. Often we care about the chance of ruin or the variability of outcomes. Two investments might both yield 7% on average, but if one of them has a tiny risk of total loss, a non-ergodicity-aware investor will treat it very differently than the other. Likewise, if someone says the “average person” will do okay under some policy, we should ask: what about the distribution of outcomes? Are there scenarios where some people fare much worse or better? Non-ergodicity teaches us to think in terms of distributions and individual trajectories, not just expectations.

Conclusion

Non-ergodicity is a fundamental concept that challenges the way we traditionally think about probability and averages. In an ergodic world, the law of large numbers and long-term averages would reliably guide us – eventually, every sequence would resemble the ensemble. But the world we live in – with its unequal wealth distribution, risky investments, evolutionary twists and turns, and unpredictable events – is often non-ergodic. This means the story of outcomes is richer (and sometimes harsher) than a single average number can tell. By recognizing non-ergodicity, we gain insight into why “expected” results often don’t materialize for most people, why history and luck can have lasting effects, and why exponential growth can falter despite rosy averages. This understanding ultimately leads to better decision-making: it urges caution in extrapolating trends, highlights the importance of managing downside risks, and fosters strategies that are resilient over time rather than optimized to an imaginary average. In a non-ergodic world, how you play the game each step of the way matters enormously – and acknowledging that can make all the difference in economics, finance, and science.

Sources:

1. Peters, Ole (2019). The ergodicity problem in economics. Nature Physics – (Summary on Exploring Economics) Discussion of how economics has wrongly assumed ergodicity and how focusing on time-average growth resolves many puzzles (The ergodicity problem in economics | Exploring Economics) (The ergodicity problem in economics | Exploring Economics).
2. Mangalam, M. & Kelty-Stephen, D. (2022). Ergodic descriptors of non-ergodic processes. Royal Society Open Science – Explanation that non-ergodicity means group averages don’t predict individual time outcomes (Ergodic descriptors of non-ergodic stochastic processes - PMC ).
3. “Ergodicity: the Most Over-Looked Assumption.” Neurabites blog – Illustrative coin-flip betting example where individuals go broke despite a positive average growth rate, demonstrating non-ergodicity (Ergodicity: the Most Over-Looked Assumption – Neurabites).
4. Kauffman, Stuart (2017). “Non Ergodic.” Edge.org response – Describes ergodic vs non-ergodic and notes that biological evolution is non-ergodic (not all potential forms are realized) (Edge.org) (Edge.org).
5. Cross Validated (StackExchange) discussion (2019) on Ole Peters’s paper – User’s coin-flip game analysis showing expected value is not representative of typical outcomes; most runs lose money despite positive expected growth (probability - The ergodicity problem in economics - Cross Validated) (probability - The ergodicity problem in economics - Cross Validated).
6. Stojkoski, V. et al. (2022). Ergodicity breaking in wealth dynamics – Finds that wealth in an economy is non-ergodic: ensemble (average) wealth can grow exponentially while typical wealth does not; warns about interpreting average wealth in inequality studies.