One Night to Wealth: The Illusion of Quick Riches and Risk Management
Becoming wealthy overnight is the dream of every investor; however, many people are often misled by the experiences of a few successful individuals online, losing their rational judgment. This article, sourced from Xue’e and authored by DataCafe, teaches you how to manage risk. Compiled, translated, and written by Shenchao.
(Background: Legendary investor Paul Tudor Jones: Buy Bitcoin, gold, and stocks to combat the dollar’s potential 10% drop in the coming year.)
(Background Supplement: Is gold still worth investing in? Is the most well-known safe-haven asset outdated?)
We often feel that one more round could turn the tables, precisely because we mistakenly equate the average of a group with individual fate. Imagine participating in a coin toss challenge with an initial capital of 1,000 yuan, where you can choose to play continuously:
- In each round, you toss a coin.
- If heads, your wealth increases by 80%.
- If tails, your wealth decreases by 50%.
This sounds like a guaranteed profit game! But the reality is…
If you let 100,000 players participate in this game and have them each play 100 rounds, you will find that their average wealth indeed grows exponentially, yet the vast majority end up with less than 72 yuan, or even go bankrupt!
Why is the average wealth increasing while most players end up poorer? This exemplifies a typical non-ergodic trap. The feeling that one more game could turn things around stems from our erroneous belief that the group’s average reflects individual fate.
The Trap of Non-Ergodicity: Long-Term Average ≠ Your True Fate
What is ergodicity? The concept of ergodicity first appeared in statistical physics and has since had a profound impact in areas such as probability theory, finance, behavioral science, and machine learning. The core question it attempts to answer is: Is the long-term average applicable to individuals? When making decisions, should we trust the ‘long-term average’ or the reality of ‘repeated personal experiences’?
In the 19th century, physicist Ludwig Boltzmann proposed the ergodic hypothesis while studying the motion of gas molecules: If you observe a gas molecule for long enough, it will traverse all possible states. Imagine a closed gas container filled with countless gas molecules, each experiencing different velocity trajectories during collisions. The long-term trajectory of a single molecule and the statistical distribution of the entire gas are the same, meaning we can use the states of all molecules at a certain moment to infer the long-term trajectory of a single molecule. This is the famous Boltzmann ergodic hypothesis.
Mathematically, ergodicity means:
The left side represents time average: describing the average result obtained from an individual experiencing the same process multiple times over a sufficiently long period; the right side represents group average: describing the statistical expectation obtained from observing countless individuals at a certain moment. This means that when a system meets the conditions of ergodicity, the performance of an individual will eventually converge to the group’s “long-term average.”
If the world were ergodic, everyone’s wealth would eventually approach the average wealth level of society. In an ergodic world, everyone could experience all possible economic states (wealth, poverty, success, failure), and individual fates would ultimately converge to the group’s “long-term average.” However, real life is often non-ergodic: individuals have limited resources and often exit due to a single failure before experiencing all possible paths.
We often hear guiding statements like:
- “The average annual income in a certain industry exceeds one million.”
- “Someone achieved financial freedom at 30, starting a business that took only two years.”
- “A certain index fund has high long-term annualized returns; just keep investing to get rich.”
These seemingly reasonable statistics seem to tell us a definitive truth: as long as we act, the long-term average returns will apply to individuals. But these cases are part of a path-dependent, non-replicable non-ergodic process. Imitators cannot experience the same historical background, relational networks, lucky breaks, nor even know the hidden number of failures.
Data tells you the long-term average of the group, but reality is filled with short-term “cliff-like failures.” This is exactly the most insidious trap of non-ergodicity—
The average of big data statistics ≠ the true fate of individuals.
A single collapse could be irreparable for an individual; one failure might lead to complete exit, making it impossible to return to an “average state.” Each individual’s life path can only be experienced once and cannot, like in a casino, benefit from the long-term average of the group, waiting for probabilities to average out among countless gamblers.
Why is the Long-Term Fate of Individuals Often Worse Than the “Average”?
In non-ergodic systems, individual long-term performance often falls below the group average. This is not incidental but rather a systematic structural feature. The glamorous average is often propped up by the very few stories of successful entrepreneurs, wealthy investors, and those who made a comeback; the failures of the majority never enter the statistics.
Real systems are typically multiplicative and exhibit path-dependent characteristics—such as the compounding in investments, the decline in health, and the deterioration of reputation. A typical feature of such systems is: limited upside, but unlimited downside.
- A single bankruptcy could ruin a lifetime;
- One wrong decision might completely change fate;
- A breach of trust may utterly destroy confidence;
Yet, the wealth that can be earned, the performance that can rise, and the advantages that can be established are always limited.
This is why mathematically, the long-term growth rate of a multiplicative process does not equal the “average return,” but is closer to:
In contrast, the group average is typically represented by the arithmetic mean,
And since the logarithmic function is strictly concave, based on Jensen’s inequality, we have:
Therefore, the long-term growth rate of a multiplicative system (i.e., geometric mean) is always less than the arithmetic mean. The greater the volatility, the more pronounced this gap becomes. The arithmetic mean tells you what would happen ‘if you were forever lucky,’ while the geometric mean shows you ‘how much you have left after weathering the storms of the real world.’
This means that an individual’s long-term performance is always far below the “group average return,” not due to bad luck but rather due to structural reasons.
How to Make Optimal Decisions? The Golden Ratio of the Kelly Criterion
So, in life decisions, what can we do to avoid the fate of going to zero in a long-term game? How can we avoid bankruptcy while achieving long-term compounding?
The answer is: never go all-in; learn to bet using the Kelly Criterion!
The Kelly Criterion is an optimal betting strategy used in repeated games, aiming to maximize long-term returns while avoiding short-term bankruptcies. It was originally proposed by John L. Kelly Jr. in 1956 at Bell Labs to solve the problem of “how to allocate signal power in a noisy channel” to maximize information transmission efficiency.
Later, this theory quickly crossed over into other fields. American mathematician and investment genius Edward Thorp discovered that the Kelly Criterion could optimize wealth growth paths. He introduced the Kelly method to casinos and systematically defeated the blackjack dealer in “Beat the Dealer” and later brought it to Wall Street in “Beat the Market” to continue “harvesting.”
This principle is essentially equivalent to maximizing logarithmic expected returns (log-utility), thus balancing growth and risk dynamically. It helps you find an optimal balance point between “living long” and “earning enough.”
The Kelly Criterion:
Where the probability of success is p, the probability of failure is q = 1-p; the profit multiplier when successful (excluding the principal) is b, and the loss ratio when failing is a (usually 1, if you lose the entire bet amount).
Returning to the coin toss game mentioned at the beginning, you can choose to bet a certain proportion of your capital and keep playing, but how much should you bet each time?
In other words, the Kelly Criterion suggests you should invest 37.5% of your total capital each time. Betting too much, even with an advantage, could lead to a complete loss due to a few consecutive losses; betting too little means missing out on growth that should have been yours.
The significance of the Kelly Criterion lies in finding that point which maximizes long-term earnings while ensuring survival.
Additionally, the Kelly Criterion is very sensitive to winning probabilities and odds; however, in reality, these parameters are often uncertain or dynamically changing. Therefore, many prudent practitioners choose to use half of the Kelly suggested value (known as the half-Kelly strategy) to achieve a smoother return path.
Simulation Experiment: In a Coin Toss Game with 100,000 Players, How Many Can “Survive”?
To better understand how different betting strategies affect individual fates, I simulated 100,000 players participating in the aforementioned coin toss game, playing a total of 200 rounds independently.
The game rules remain: starting capital of 1,000, heads profit 80%, tails lose 50%. Players can choose fixed betting proportions: for example, bet all (100%), bet 65%, 37.5%, etc.
The result… Almost all players who bet 100% went extinct!
The final wealth showed a “power law distribution,” with very few becoming wealthy, but the vast majority of players went bankrupt.
We compared the wealth distribution of players using these four different betting strategies, with wealth distribution moving right indicating higher assets.
- a. 100% Bet: Almost everyone went bankrupt
- b. 65% Bet: Still polarized, with many going bankrupt
- c. 37.5% Bet (Kelly Criterion): Wealth grows steadily
- d. 10% Bet: Almost no one went bankrupt, but returns are too low
Under the all-in strategy, the final asset distribution displays a massive left poverty peak + a very thin right tail structure of wealth: most players went bankrupt, while a very few walked away with all the money, which reflects the true presentation of gaming asymmetry + survivor bias.
Under the Kelly betting strategy, the asset distribution clearly shifts right, with most players’ assets increasing and concentrated distribution, representing the optimal wealth accumulation model.
Without the bankruptcy distribution peak seen in the all-in scenario, the overall wealth is concentrated in low asset areas. In contrast, the 37.5% strategy would pull out a clear long tail on the right, achieving asset multiplication.
The Kelly betting strategy is the only one that balances “not going bankrupt in most cases” and “substantial appreciation” and is mathematically the optimal long-term survival strategy. This is the essence of the Kelly Criterion: it is not about winning the most, but ensuring you can live long enough.
The Philosophy of Life in the Kelly Criterion
The Kelly Criterion teaches us that the secret to long-term success is learning to control the proportion of “bets.” Life is not about who can make a big hit once, but about who can keep playing.
In your career, it’s not about impulsively quitting your job or sticking to your comfort zone, but rather continuously positioning yourself, improving your skills, daring to change paths, and keeping options open;
In investments, it’s about not going all-in to get rich, but controlling your position based on odds and preserving your chips;
In relationships, it’s not about investing all your emotions and values in one person, but rather investing while maintaining your individuality;
In growth and discipline, it’s not about relying on one explosive change but rather through stable, compound optimization of your life structure.
Life is like a long game; your goal is not to win once but to ensure that you can always keep playing. As long as you don’t exit, good things will happen.
