Ergodicity: Players Fear Variance, the House Loves It

We have discussed many principles and methods regarding compound interest. In this session, I want to offer some "anti-chicken soup": accumulating compound interest isn’t something you can achieve just by understanding the logic.

I once owned 16 Bitcoins. I bought them in 2013 for about $700 each. If I had held them until 2025, when the price exceeded $100,000, that would be $1.6 million…

But as you might guess, I didn’t hold. I held them for perhaps less than a month; when the price dipped slightly instead of rising, I sold them immediately. I chose a more "exciting" game and used that money to buy stock options, which failed and eventually wiped me out completely.

I’ve shared this story with several people, and no one mocked me. We all share the same sentiment: almost no one can hold.

At the end of 2015, Bitcoin was only about $300. If you had bought it then and held until today, it would indeed be a fortune. But what would you have experienced?

• By late 2017, Bitcoin hit $19,000. Could you resist selling after a 60x gain?
• By late 2018, it plummeted to $3,200. If you hadn’t sold the year before, could you avoid regret?
• By late 2021, the price reached $69,000. If you had regretted not selling before, how would you feel now?
• By late 2022, the price dropped back to $15,000… and then the roller coaster continued: $100,000 in 2025, back to $60,000 in 2026…

Only two types of people can hold for that long: those in prison who have no way to trade, and those who have so much money that they don’t need or care about that specific sum.

The mental model for this lecture is called "non-ergodicity." It explains why the latter group is best suited for investing—and for handling other multiplicative risks—and what you should do if you aren’t in that group.

Many stock market experts have annualized returns exceeding 15%, yet no fund in the world dares to promise you a guaranteed minimum annual return of 15%. While the long-term trend might rise that much, the process involves ups and downs. In the language of probability theory, the returns have high "variance."

Only two factors hinder your accumulation of compound interest: your initial capital and variance.

To understand this, let’s play a thought experiment. This is a simple coin-toss gamble with a 50/50 chance of heads or tails. The game requires 100 rounds: in each round, if it’s heads, your total wealth increases by 50%; if it’s tails, your total wealth decreases by 40%. Would you play?

Simple probability tells you the mathematical expectation for each round is: $0.5 \times (+50%) + 0.5 \times (-40%) = +5%$. This is a positive return—who wouldn’t play?

Now, suppose you enter the game with $1 million. Let’s assume your luck is average: exactly half the time you get heads, and half the time tails.

1. First round, heads: your money becomes $100 \times 1.5 = 1.5$ million.
2. Second round, tails: your money becomes $150 \times 0.6 = 0.9$ million.
3. Third round, heads: $90 \times 1.5 = 1.35$ million.
4. Fourth round, tails: $135 \times 0.6 = 0.81$ million.

…Wait, something is wrong! After every cycle of one win and one loss, your wealth becomes $1.5 \times 0.6 = 0.9$ times its previous value—a 10% shrinkage! After 100 rounds, your $1 million principal would dwindle to less than $10,000. You are bankrupt.

But think about it—this is still confusing! This is clearly a game with a +5% expectation every time. Theoretically, the "house" should be losing money. So who is winning?

The answer: people with extraordinary luck.

Imagine 100,000 people playing this game simultaneously. After the first round, 50,000 people have $1.5 million and 50,000 have $0.6 million. The total wealth of these 100,000 people has indeed increased by 5%. This +5% expectation is the "ensemble average." Over many rounds, a few extremely lucky individuals among the 100,000 will win many times and lose very few. These people will possess astronomical wealth, which is why the ensemble average is always growing.

Unfortunately, as an average person, your wealth decreases. This is because an individual experiences the "time average." Mathematically, your growth coefficient is the geometric mean of the gains and losses. In this case, $\sqrt{1.5 \times 0.6}$—and unfortunately, this value is less than 1.

As an individual moving step-by-step through time in this multiplicative game, a few consecutive losses will deal a devastating blow to your principal. Your account might hit zero, forcing you out of the game. This is called hitting an "absorbing barrier."

Your role is to add to the denominator of the "ensemble average," while the contribution to the numerator is provided by those few lucky individuals.

Simply put, if a system’s ensemble average equals an individual’s time average, the system is "ergodic." If the overall expectation does not represent the long-term fate of the individual, the system is "non-ergodic."

For example, your daily commute might involve occasional heavy traffic or all green lights, but over many trips, your experience will be roughly the same as everyone else’s. This is an "ergodic" system. The city’s average commute time is a meaningful number for you.

However, for non-ergodic systems like investing, the average value is largely meaningless—there’s no point in averaging your wealth with Elon Musk’s.

The concept of ergodicity is not new in statistics, but its application in economics is far from common knowledge. As recently as 2016, physicist Ole Peters of the London Mathematical Laboratory pointed out [1] that traditional economics often assumes human economic activity is ergodic without proof. This leads many to be overly optimistic and jump into games blindly, unaware that in a game with positive expectation, many single players are destined for bankruptcy.

Nassim Taleb also warned against non-ergodic risks in his book Skin in the Game [2].

In short: mathematical expectation is the average from a "God’s eye" view; time average is the fate from a mortal’s view.

These warnings might sound abstract, so let’s look at reality. In any country’s stock market, the long-term returns of the vast majority of retail investors fail to beat the market index. Not only do individuals underperform, but even many professional Wall Street investment funds struggle to beat the S&P 500 [3].

Some commentators claim this is because retail investors don’t understand technical analysis, value investing, or have the wrong mindset. But the true mechanism is mathematics: the stock market is a typical non-ergodic system. A market index like the S&P 500 is an ensemble; it enjoys the ensemble average of all major stocks and has a mechanism for regularly removing "trash" stocks and including high-quality ones. Retail investors, however, hold only a few stocks—they experience the time average [4].

It is a counter-intuitive phenomenon, but it follows the same logic as the coin-toss experiment.

So, as individuals, how do we deal with non-ergodic risk? Here are four strategies.

The first strategy is to reduce transaction frequency.

This is the strategy of the Bitcoin holder in prison. If you truly have confidence and want to lock in long-term gains, you must hold and not move.

Finance professors Brad Barber and Terrance Odean at the University of California analyzed over 60,000 retail accounts and found that the more frequently investors traded, the lower their returns [5]. Those who bought and never looked at their accounts again (perhaps because they forgot their passwords) had the best average returns.

There are psychological reasons: frequent traders are prone to "chasing highs and selling lows" due to emotional fluctuations. There are also reasons inherent to market fluctuations: most of the long-term returns of stocks are contributed by a very small number of explosive growth days. To capture the gains of those few days, you have to be holding all along…

But at the fundamental level, it’s about the non-ergodicity of the market. Every time you trade, you are tossing the coin again. The more you trade, the closer your outcome gets to the time average, increasing your probability of hitting the absorbing barrier. It’s better to trade less; you might actually get lucky.

The "world is dangerous," and those with small capital have no right to mess around. Betting once you are certain and using time to smooth out extreme variance is the survival wisdom of the poor. This is essentially the mathematical principle of "long-termism" and the scope where "loyalty" serves as a virtue.

The second strategy is Taleb’s famous "barbell strategy" [6]. This involves putting 90% of assets in extremely low-risk areas—such as cash, treasury bonds, and unleveraged real estate—and 10% in aggressive, high-risk areas to capture "heavy-tail" dividends.

The barbell strategy advocates ignoring medium-risk projects and sticking to the two ends: either low risk or high risk. Its advantage is that it retains the possibility of striking it rich while ensuring you never touch the absorbing barrier.

You might ask, what is the scientific basis for this? The barbell strategy is actually a variant of the "Kelly Criterion" we discussed in the previous session.

The biggest problem with the coin-toss thought experiment was that you went "all-in" every time. The barbell strategy requires you to bet only a tiny proportion each time, and the Kelly Criterion specifically opposes "all-in" bets. Let’s calculate: in that game, the probability $p = 0.5$ and the odds $b = 50 / 40 = 1.25$. Plugging this into the Kelly formula, the optimal betting ratio $f^* = 10%$—perfectly aligned with the barbell strategy.

Guess what? In 2011, Ole Peters theoretically proved that if you use the Kelly Criterion for every bet, you can defeat non-ergodicity [7]!

It sounds simple, but it was a major event in academia. Intuitively, the mathematical role of both the barbell strategy and the Kelly Criterion is to shrink the variance of transactions. As long as your betting ratio is small and you don’t go "all-in," you get a similar effect. For example, you could take a fixed, small portion of your salary every month to invest in the stock market; neither Taleb nor Kelly would object.\

The third strategy is to become the "House."

Since most retail investors can’t beat the index, why not be the index? Why not enjoy the ensemble average?

Of course, the prerequisite for building your own "ensemble" is having a lot of money. Consider the relationship between Venture Capitalists (VCs) and entrepreneurs [8].

Startups are a highly non-ergodic field. If you are an individual entrepreneur, you can only bet on one company, perhaps wagering your reputation, your home, and all your cash—and there is a massive probability of failure. But the VC model involves diversifying investments across dozens or hundreds of companies: even if most go to zero, if one or two succeed or become "unicorns," it is enough to cover all other losses and leave a handsome surplus.

VCs are not only unafraid of individual project failures; they actually crave extreme variance! They want you to take risks. Mediocre companies are useless to a VC.

It’s like the parent of a child playing soccer: you probably don’t want the child’s growth path to have high variance; you just want them to be safe. Even if they don’t become famous, you don’t want them to get injured and disabled. But if you are a youth soccer academy, you want the variance in your team to be as high as possible: even if 99 out of 100 players are injured or retire, as long as one becomes a superstar, you’ve struck gold.

Individual players fear variance, but the House loves variance.

This is a massive asymmetry. However, as an ordinary person, you can choose to stand with the "House" by investing in index funds.

The fourth strategy is "Risk Pooling" [9]. Since individuals lose out from non-ergodicity while the House benefits, why don’t individuals unite to form a collective "House"?

This is essentially what insurance is. For an insurance company, the mathematical expectation of your house burning down is certainly less than the premium it collects, so the insurance business is profitable. Why do you still buy insurance? The fundamental reason is that for you as an individual, a house fire is an "all-in" risk.

Major disasters are non-ergodic; this is the rationale for the existence of insurance companies.

In fact, venture capital can also be understood as a form of insurance. Entrepreneurs go out and do business boldly; if they lose money, it’s the VC’s money. The "House" is effectively providing a safety net for you.

This is exactly the greatness of the "Limited Liability Company" system! Businesspeople can venture out boldly; if they win, everyone shares the money; if they lose, they only bear limited liability. It won’t drag in their families, they won’t lose everything, and they certainly won’t be sold into slavery…

More deeply, families, clans, and various mutual aid networks in human society can be understood as using cooperation to "force ergodicity" onto the world.

In summary, the multiplicative world is full of non-ergodic risks. It is unfavorable to individuals but favorable to the House. Therefore, individuals must invest less, operate less, control variance as much as possible, and ideally unite to use structure against fate and scale for stability.\

Finally, allow me a small critique. Because non-ergodic risk is highly asymmetrical, mature societies try to protect individuals and let institutions bear the risk.

For example, in some countries, when you buy a house, you wait for it to be built, and only after you move in do you start repaying the mortgage. If house prices crash during the repayment period and you can’t afford it due to unemployment, you can apply for personal bankruptcy, walk away, and get a chance to start over.

In other places, however, the system protects banks and developers and shifts the risk down to the common people: you have to start repaying the mortgage before the house is even built; if the project is left unfinished, you still have to keep paying; and no matter what happens in the future, this debt follows you forever… I know of no modern mental model that advocates for such a practice.

Notes

[1] Peters, Ole, and Murray Gell-Mann. 2016. “Evaluating Gambles Using Dynamics.” Chaos 26(2): 023103. [2] Elite Lesson Season 2, "Skin in the Game" 8. "Ergodicity" and "Tail Risk" [3] John Paulos, A Mathematician Plays The Stock Market, 2004. See also Elite Lesson Season 2, Math Problem: Why the Vast Majority of Investors Lose to the Market. [4] Bessembinder, Hendrik. 2018. “Do Stocks Outperform Treasury Bills?” Journal of Financial Economics 129(3): 440–457. [5] Barber, Brad M., and Terrance Odean. 2000. “Trading Is Hazardous to Your Wealth: The Common Stock Investment Performance of Individual Investors.” Journal of Finance 55(2): 773–806. [6] Taleb, Nassim Nicholas. 2012. Antifragile: Things That Gain from Disorder. New York: Random House. [7] Peters, Ole. "Optimal leverage from non-ergodicity." Quantitative Finance 11, no. 11 (2011): 1593-1602. [8] Hall, Robert E., and Susan E. Woodward. 2010. “The Burden of the Nondiversifiable Risk of Entrepreneurship.” American Economic Review (Papers & Proceedings). [9] Cronk, Lee, and Athena Aktipis. 2021. “Design Principles for Risk-Pooling Systems.” Nature Human Behaviour 5: 825–833.