Why do gamblers ultimately lose everything? The survival rules in non-iterative systems.

Author: Xue'e, DataCafe

Imagine you start with 1000 yuan participating in a coin flipping challenge game, and you can choose to keep playing:

Toss a coin once each round,

Flipping to the front, wealth increases by 80%.

Flipping it over, wealth decreases by 50%.

Sounds like a surefire 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 does indeed grow exponentially, but the vast majority of people end up with less than 72 yuan, or even go bankrupt!

Why is the average wealth increasing, yet most people are getting poorer?

This is a typical non-reversible trap. We always feel that if we play one more round, we can turn things around, precisely because we mistakenly treat the average of the group as the fate of the individual.

Non-iterative traps: Long-term average ≠ your true destiny

What is traversability?

The concept of ergodicity first appeared in statistical physics and has had a profound impact in fields such as probability theory, finance, behavioral science, and machine learning. The core question it attempts to answer is: Does the long-term average apply to individuals? When making decisions, should we believe in the 'long-term average' or the reality of '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 long enough, it will explore all possible states.

Imagine a closed gas container filled with countless gas molecules, each undergoing different velocity trajectories during collisions. The long-term trajectory of an individual molecule is the same as the statistical distribution of the entire gas, which means we can use the state of all molecules at a given moment to infer the long-term trajectory of a single molecule.

This is the famous Boltzmann's ergodic hypothesis.

In mathematics, traversal means:

The left side is the time average: it describes the average result obtained after an individual experiences the same process multiple times over a sufficiently long period of time.

The right side is the group average: it describes the statistical expectation obtained by observing countless individuals at a certain moment. In other words, when the system meets the ergodic condition, the performance of a single individual will ultimately converge to the group's "long-term average."

If the world is traversable, everyone's wealth will eventually converge to the average wealth level of society. In a traversable world, everyone can experience all possible economic states (rich, poor, successful, failed), and an individual's fate will always converge to the group's "long-term average."

But real life is often non-browsable: individuals have limited resources and often exit directly after a failure before experiencing all possible paths.

We often hear some guiding statements like this:

"The average annual income in a certain industry exceeds one million."

"Someone became financially free at 30, and it only took two years to start a business."

"Some index funds have high long-term annualized returns; as long as you keep investing, you will become rich."

……

These seemingly reasonable statistics seem to tell us a certain truth. It appears that as long as we take action, the long-term average returns will apply to individuals. However, these cases belong to path dependence + non-replicable non-ergodic processes. Imitators cannot experience the same historical context, relational networks, luck nodes, and may even be unaware of the number of hidden losers.

Data tells you the long-term average of the group, but reality is full of short-term "cliff-style failures."

This is the most covert trap of non-traversability - the average value of big data statistics ≠ the true fate of individuals.

A single collapse may be irreparable for an individual, and a single failure could result in complete elimination, making it impossible to return to an "average state." Each of our life paths can only be experienced once; we cannot, like in a casino, benefit from the long-term average of a group, waiting for probabilities to average out among countless gamblers.

Why is the long-term fate of individuals mostly worse than the "average value"?

In non-iterative systems, individual long-term performance often falls below the group average. This is not coincidental, but a systemic structural characteristic. The shiny average is often lifted by the stories of a very few who succeeded in entrepreneurship, became wealthy through investment, or turned their fortunes around, while the failures of many have never entered the statistics.

Real systems are mostly multiplicative and exhibit path dependence characteristics—in areas such as investment compounding, health decline, and reputation damage. The typical features of such systems are: limited upside and no downside.

A bankruptcy can ruin a lifetime;

A single wrong decision can completely change one's destiny;

A single breach of trust can completely destroy confidence.

However, the wealth that can be earned, the performance that can be increased, and the advantages that can be established are always limited.

This is precisely why, in mathematics, the long-term growth rate of multiplicative processes is not equal to the 'average return', but is closer to:

In contrast, the group average is usually represented by the arithmetic mean.

Since the logarithmic function is a strictly concave function, based on Jensen's inequality, it follows that:

Therefore, the long-term growth rate of the multiplicative system (i.e., the 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 lucky forever', while the geometric mean tells 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 "average returns of the group"; it's not bad luck, but rather a result of the structure.

How to make optimal decisions? The golden ratio of the Kelly formula.

So in life decisions, what can we do to avoid the fate of going to zero in the long-term game? How can we avoid bankruptcy while still achieving long-term compounding?

The answer is: Never go All in, learn to use the Kelly criterion!

The Kelly Criterion is an optimal betting strategy used in repeated games, aimed at maximizing long-term returns while avoiding short-term bankruptcies. It was originally proposed by John L. Kelly Jr. in 1956 at Bell Labs, with the original intent of solving 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 formula could optimize wealth growth paths. He brought the Kelly criterion into casinos, systematically defeating the blackjack dealer for the first time in "Beat the Dealer," and then took it to Wall Street, continuing to "harvest" in "Beat the Market."

This principle is essentially equivalent to maximizing the logarithmic expected utility (log-utility), thereby balancing the dynamic relationship between growth and risk. It helps you find an optimal balance between 'living long' and 'earning enough.'

Kelly Criterion:

Among them, the probability of success is p, and the probability of failure is q = 1-p; the profit multiplier (excluding the principal) when successful is b, and the loss ratio when failing is a (usually 1, if the entire bet amount is lost).

Returning to the coin toss game mentioned at the beginning, you can choose to bet a certain percentage of your principal and keep playing, but how much should you bet each time to be reasonable?

In other words, the Kelly formula suggests that you should invest 37.5% of your total capital each time. Betting too much, even with an advantage, can lead to a direct liquidation after a few consecutive losses; betting too little may cause you to miss out on the growth that should have been yours.

The significance of the Kelly criterion lies in finding the point that allows one to earn the most in the long term while still being able to survive.

To add a point, the Kelly formula is very sensitive to the probability and odds, but in reality, these parameters are often uncertain or dynamically changing. Therefore, many prudent practitioners choose half of the Kelly recommended value (known as the half-Kelly strategy) in exchange for a smoother return path.

Simulation Experiment: In a coin-tossing betting game with 100,000 people, how many can "survive"?

To better understand the impact of different betting strategies on individual fate, I simulated 100,000 players participating in the initial coin tossing game, with a total of 200 rounds, each playing independently.

The rules of the game remain the same: a principal of 1000, with an 80% profit when facing up and a 50% loss when facing down. Players can choose a fixed betting ratio: for example, bet all (100%), bet 65%, 37.5%,...

As a result... almost all players who bet 100% are eliminated!

Wealth ultimately follows a "power law distribution"; although a very small number of people become wealthy, the vast majority of players go bankrupt.

We compare the wealth distribution of players using these 4 different betting strategies; the further right the asset distribution, the higher the player's assets.

a. 100% bet: Almost everyone goes bankrupt

The final asset distribution under an all-in strategy has a massive left-side poverty peak and a very thin right-side wealth tail structure: the majority go bankrupt, while a very small number take all the money; this is the true representation of asymmetry in the game + survivor bias.

b. 65% betting: Still polarized, with many people still going bankrupt.

c. 37.5% Bet (Kelly Criterion): Stable Wealth Growth

Under the Kelly betting strategy, the asset distribution shifts significantly to the right, with most people's assets growing and becoming concentrated, which is the optimal wealth accumulation model.

d. 10% bet: Almost no one goes bankrupt but the returns are too low.

There is no longer a bankruptcy distribution spike similar to full margin cases, but overall wealth is concentrated in the low asset area. In contrast, the 37.5% strategy will pull a clearly long tail to the right, achieving asset multiplication.

Kelly betting is the only strategy that balances "not going broke in most cases" and "considerable appreciation"; it is mathematically the optimal long-term survival strategy. This is the essence of the Kelly formula: it is not about winning the most, but about ensuring that you can survive long enough.

The philosophy of life in the Kelly formula

The Kelly criterion tells us that the secret to long-term success is learning to control the ratio of "betting." Life is not about who can land a critical hit once, but about who can keep playing consistently.

In a professional context, it's not about quitting impulsively out of passion, nor is it about clinging to your comfort zone. It's about continuously strategizing, enhancing your skills, having the courage to change paths, and keeping an option for yourself.

In investment, it's not about going all in to get rich quickly, but rather controlling your position based on the odds and keeping your chips.

In a relationship, it is not about placing all your emotions and values on one person, but rather investing while maintaining your own self;

In growth and self-discipline, do not rely on a single outburst for change, but rather optimize life structure steadily and in a compounding manner.

Life is like a long game, your goal is not to win once, but to ensure that you can keep playing. As long as you don't get eliminated, good things will definitely happen.