The Kelly Criterion: Cognitive Monetization in a Multiplicative World

The thinking tool for this session is the “Kelly Criterion.” Many people treat it as a capital preservation rule for investment, but its utility is far more aggressive and can guide life decisions—the Kelly Criterion is essentially a machine that turns cognition into monetization.

As we discussed earlier, people spend a lot of time on valueless information. The problem the Kelly Criterion solves is that when faced with valuable information, your actions do not match it.

For instance, some people in the stock market stare at the screen all day, trading in and out for a few cents of fluctuation, even using high leverage. Yet, when they encounter a new career path or entrepreneurial opportunity with massive potential and extreme certainty, they repeatedly calculate sunk costs, hesitate, and only dare to invest a tiny bit of their spare time. That’s simply because stock price fluctuations feel more “familiar.”

In the words Cao Cao used to evaluate Yuan Shao: “He is fierce of countenance but shrinking at heart; fond of schemes but lacking decision. He spares his person when great things are at stake, but risks his life for small gains. He is no hero.”

Betting too much and not daring to bet at all are both mistakes. This isn’t an information problem; it’s a position-sizing problem.

How do you grasp the right “degree” for taking action? This is the basic use of the Kelly Criterion.

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John L. Kelly Jr. (1923–1965) was a scientist at Bell Labs. His famous 1956 paper [1] originally researched communication problems.

You see, because of noise, information always encounters errors during long-distance transmission. Claude Shannon, the father of information theory, had already proven that you could reduce error rates by expanding the channel capacity, but you couldn’t eliminate errors absolutely. So Kelly thought: information with an error rate is still useful; can I calculate exactly how useful it is?

Kelly envisioned a scenario: Suppose a gambler outside a racetrack receives inside information through a noisy private telephone line. The person on the other end can 100% predict which horse will win.

However, due to noise on the line, the result the gambler hears is accurate with probability $p$ and incorrect with probability $q = 1 - p$. How should the gambler bet using this noisy inside information?

If he goes “all-in” every time, he can maximize single-bet winnings, but one wrong hearing and he goes bankrupt. But if he bets too little, he wastes this precious inside line.

Through mathematical derivation, Kelly proved an extremely elegant conclusion: To never go bankrupt and to maximize the long-term compound growth rate of his funds, the gambler has an optimal betting fraction $f^*$. Under this betting strategy, the maximum long-term logarithmic growth rate of the funds is mathematically identical to the “information transmission rate” of the telephone line!

This is the Kelly Criterion.

This paper linked information theory with capital compounding—one being “how much effective information you possess about the world,” and the other being “how your capital grows over the long term amidst random fluctuations.” Kelly’s conclusion, put simply, is:

Your growth ceiling is limited by your cognitive bandwidth.

People immediately realized this formula should be applied to investing—it’s the most fundamental principle.

The core problem it solves is: in an uncertain, multiplicative-settlement, repeatable world—where in each round you either win a bit or lose a bit, and wins/losses accumulate into compound interest—how exactly should you bet to maximize the long-term growth rate?

Kelly solves not “how to win,” but “how to keep winning.”

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Let’s take a look at the formula:

$$ f^* = \frac{pb - q}{b} $$

Suppose you bet a fraction of your current total assets each time. If you win, you earn the profit according to the odds; if you lose, you lose the amount bet. In the formula:

• $p$ is the probability of winning this bet;
• $q = 1-p$ is the probability of failure;
• $b$ is the odds (the net profit per $1 bet if you win).

Then $f^*$ is the optimal betting fraction calculated by Kelly. For example, if you judge the winning probability $p=0.6$ and you double your money on a win ($b=1$), the Kelly Criterion says you should bet 20%.

How much you should bet depends neither on how much you want to win nor on how adventurous you are, but on your winning probability and the odds.

In casino slang, the Kelly Criterion is often simplified into a mnemonic:

$$ Kelly = \frac{Edge}{Odds} $$

The denominator $odds = b$ is the odds provided by the market. Intuitively, you might think higher odds warrant a larger investment, but the market matches return with risk: the return is already reflected in the numerator [2], and the denominator here considers the risk. “Odds” reflect the market’s consensus on risk: high odds mean the market thinks it’s risky.

If a person bets only based on odds, they can only earn the market’s average return—and the average return in an efficient market is infinitely close to zero. In a casino, the average return is even negative.

What should truly move you is the numerator, the edge—I suggest you remember this English word because it’s cool—which equals $bp - q$, the expected net profit per $1 bet. The “edge” is your cognitive advantage!

A more intuitive understanding is: why should you win? Because you have “inside information” and you believe the winning probability should be $p$. As long as that $p$ value makes $edge > 0$, you believe this event is more worth betting on than the market estimates. The “edge” is actually the cognitive bias between your own estimate of the situation and the general market judgment [3].

What the Kelly Criterion helps you earn is the money that exceeds public cognition. However much smarter you are than the public, Kelly can help you earn that much—but no more.

And if you do not possess an edge greater than zero, Kelly requires that you do not bet. Simply put: if you aren’t smarter than others, don’t play.\

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None of this has to do with your personality or courage. Whether to act should only depend on how much better your opportunity is than the “market price” given by the world.

But let’s think back to Yuan Shao. In the fifth year of Jian’an, Cao Cao went to crusade against Liu Bei, leaving his home base, Xudu, empty. Yuan Shao faced a superb strategic window. His strategist Tian Feng immediately suggested sending troops to attack Xudu, saying the world could be settled in one battle. In the language of this session, Yuan Shao had a massive “edge” before him!

As it turned out, Yuan Shao said his youngest son was seriously ill, he was mentally distressed, and had no heart for war! … It perfectly matched the description: “careful of his own skin when great things are at stake.”

If heaven offers and you do not take, what more can Tian Feng and Kelly say? Yuan Shao subsequently had to face Cao Cao’s rapid return of troops, then the Battle of Guandu, where his entire army was wiped out.

If edge is good, taking action is an obligation.

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The Kelly Criterion isn’t just used in communication and investment; it’s useful in many fields. I saw a most interesting example from biology.

Biological reproduction is usually a very dangerous behavior. If the environment fluctuates violently and all your individuals gather to reproduce at once, and reproduction fails, it could lead to the extinction of the entire population. Therefore, some single-celled organisms have invented a coping strategy called “Phenotypic Bet-Hedging.”

Simply put, it involves letting a portion of individuals remain in a dormant state while letting the rest “bet” on reproduction. Salmonella and Bacillus subtilis both have this ability. Even if the population has identical genes (clones), they automatically diversify their bets. This way, even if the environment turns bad several times or encounters antibiotic attacks, the population can continue to survive.

Amazingly, researchers found [4] that the proportion of cells entering the dormant state is mathematically equivalent to the “unbet” proportion in the Kelly Criterion.

This is a miracle of evolution and the power of mathematics. Perhaps biodiversity is the “Kelly position” the system pays to cope with uncertainty.

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If you imagine life as a series of investment adventures, the Kelly Criterion gives you three most fundamental pointers.

First, do not let yourself be zeroed out.

As long as $p$ is not strictly equal to 1, the Kelly Criterion requires that you do not go all-in. And as we said before, Bayesians never assume a probability is 0 or 1. Furthermore, don’t forget that your estimate might have errors, so it’s better to leave a little more margin. No matter how great the temptation, you must preserve your capital and ensure you always stay at the table.

Second, cognition first: ask how much of an “edge" you have before thinking about how big a bet to place.

Many people do exactly the opposite: they first have a strong urge to place a big bet, and then look everywhere for reasons to prove they have an advantage. We monetize cognition; we don’t use cognition to explain monetization. If this round isn’t suitable, wait for another opportunity. Warren Buffett never buys stocks he doesn’t understand; the strength of your action must match the depth of your cognition.

Third, pursue compound interest, not a single win.

The entire derivation and setup of the Kelly Criterion is that you are going to play many rounds in this continuous multiplicative game: sometimes you win, sometimes you lose; it seeks to maximize your accumulation in the long run. This is a system, not a single bet. As long as you believe in the math, single successes or failures aren’t important; your outcome will eventually be good.\

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You don’t need to perform precise calculations with this formula every time; sometimes a rough estimate or a fuzzy sense of intensity is good enough. Let’s look at a few application scenarios in life.

First is career planning. We have already discussed "track selection" and the trade-off between "exploration and exploitation," and the Kelly Criterion can make your decisions a bit more refined. For example, if you are not very satisfied with your current job and an opportunity arises where you can change jobs or even change careers—do you switch?

Many people treat a career track as a one-time choice—doing one line of work for a lifetime and not daring to switch even if they chose wrong. Others do the opposite, treating track-switching as an emotional release, resigning in a fit of rage and going all-in.

But a career is a typical multiplicative asset: your skills and reputation are accumulable and suitable for multi-round betting. Kelly’s suggestion would be:

• First, ask about the "edge": Are you really better than others in that direction? Can you learn faster, deliver better, and endure more?
• Then, look at the "odds": What is the reward structure of that direction? Is the ceiling high?
• Finally, talk about the "position": If an "edge" exists but you can’t get an exact $p$, don’t bet too much—you can try what is called "Half-Kelly" or "Fractional Kelly" to test the waters. That is, start as a side hustle, run a few small projects for a round, and see the feedback.

From the perspective of the Kelly Criterion, it’s not about being "stable" versus "aggressive"; it’s just a choice of degree.

Similarly, you can treat your effective energy and health as an asset, even if it doesn’t easily appreciate. Doing different things every day, like learning, working, exercising, socializing, and entertaining, is equivalent to betting. These bets might make you "win"—making you healthier or bringing other positive returns—but they might also make you "lose"—making you feel worse or harming your health.

Maybe scrolling through short videos has no value, while learning and working have value—but those values are just "odds," not "edge." If you are already very tired but have an important job at hand, is it necessary to stay up late to finish it? Maybe your family says don’t do it, your boss says you should, but Kelly would say: calculate the "edge."

The most important Kelly strategy for time management is "betting less": you should concentrate your position on the few variables that can produce compound interest.

Another multiplicative asset is trust. Whether it’s your credit in society or the trust you give to others, it accumulates compound interest slowly through repeated interactions and carries the risk of rapid zeroing. Even if someone does 15 good deeds, you might not necessarily dare to tell them your bank password; but if they do just one bad deed, you might never trust them again.

So the Kelly strategy for interpersonal relationships is:

1. When you first meet, start with small bets—give them some small tasks; if they do well, increase the $p$ value.
2. If they perform well, you can gradually increase your position, but always ensure the odds and returns are symmetrical. That is, the potential gain of a project must be worthy of your potential loss—why shouldn’t you tell someone your bank password? Because the potential gain is very small, and the potential loss is very large.
3. Always maintain a stop-loss mechanism to prevent zeroing: no matter how good the relationship is, do not hand over permissions that could bankrupt you instantly.\

This sounds cynical, and I don’t advocate using a formula for all social interactions… but if you often travel the world, you need to manage your trust assets well.

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In summary, the Kelly Criterion is not just about calculating positions; it actually gives a philosophy of action for a multiplicative world: it translates your cognitive advantage into betting size.

How much of an "edge" you have determines how much you monetize.

The Kelly Criterion requires us to view life as a multiplicative marathon: do not worry about the gains or losses of a single city or battle, but believe in the gradual, though fluctuatable, accumulation—as long as you stick to the correct strategy. And then, no matter what, don’t reach zero.

The Kelly Criterion makes us think about what courage is. In recent years, people often say courage is the scarcest virtue, but what is courage? If the Kelly Criterion says you should bet 20%, but you bet 25%, is that courage? That might be out of your virtue—perhaps you want to self-sacrifice for a charitable cause—but it could also just be an impulse from alcohol, or more likely, from ignorance.

Considering that most people are result-oriented—timid after a loss and arrogant after a win—perhaps not being swayed by temporary wins or losses and being able to act soberly according to the Kelly Criterion is true courage. Courage is the overcoming of instinct.

Courage brings you freedom. And in the view of the Kelly Criterion, the fundamental freedom of life is that you always have the ability to make the next bet.

The world neither rewards your efforts nor punishes your motives. It only settles your probability quality.

Notes

[1] Kelly, J. L. Jr. “A New Interpretation of Information Rate.” Bell System Technical Journal 35, no. 4 (1956): 917–926. [2] Here we adopt a popular interpretation, but if I rewrite the formula as $f^* = p - q/b$, the role of the odds becomes very intuitive. [3] This has a precise mathematical meaning: $edge = (p - P_m)/P_m$, where $P_m = 1/(b+1)$ is the market’s judgment of the winning probability. [4] Kussell, Edo, and Stanislas Leibler. "Phenotypic Diversity, Population Growth, and Information in Fluctuating Environments." Science 309, no. 5743 (2005): 2075-2078.