Personal Growth on TorchTree

The Kelly Criterion: Finding the Optimal Bet Size in an Uncertain World

Wed, 20 May 2026 08:03:37 +0800

What problem does the Kelly Criterion solve?

Most decision dilemmas come down to two questions: whether to do something, and how big to do it. The former is about direction; the latter is about sizing. The Kelly Criterion focuses on the latter.

In 1956, John L. Kelly Jr. of Bell Labs derived a mathematical framework while studying noise in information transmission. He discovered that if you treat signal transmission as analogous to gambling, there is an optimal stake size that maximizes the geometric growth rate of long-term wealth.1

The discovery was later introduced to the casino and Wall Street by Edward Thorp. In Beat the Dealer, published in 1962, Thorp showed that managing bet sizes in blackjack with the Kelly Criterion provides a mathematically stable edge. Since then, the Kelly Criterion has become a foundational tool in quantitative investing and money management.2

What does the formula look like

The standard form of the Kelly Criterion:

f* = (bp - q) / b

where:

f* — the fraction of total capital to bet each time
p — probability of winning
q — probability of losing (q = 1 - p)
b — the odds (net profit on a win relative to the amount wagered)

Here’s a concrete example: suppose you’re in a game where a coin toss decides the outcome. You judge the probability of heads at 55% (p = 0.55), and winning pays 2x return (b = 1, i.e., net gain of 1x). Plug into the formula:

f* = (1 × 0.55 - 0.45) / 1 = 0.10

This means you should stake 10% of your total capital each time. Betting more won’t make you grow faster—in fact, the added volatility lowers your long-term growth rate. Betting less wastes your edge.

If the probability of heads is only 45% (below 50%), the formula returns a negative value. The meaning is clear: no edge, don’t bet.

Why “optimal” rather than “maximum”

Here’s a counterintuitive key point. Suppose you have a 60% win rate and 1:1 odds. Intuition says “since I’m favored to win, I should bet a lot.” But the Kelly Criterion says to stake 20% each time.

What if you bet 40%? In the short run, you might earn more. But over the long run, because wins and losses alternate, over-concentration causes wild swings in your capital curve. A single large loss can wipe out many rounds of accumulated gains.

Mathematically, one can prove that in repeated games, the Kelly Criterion maximizes the long-term growth rate (the geometric growth rate) of capital.3

Let’s illustrate with a simplified model. Assume total capital of 10,000 yuan, stake fraction f each round, 60% win rate, 1:1 odds. After 100 rounds, the expected capital at different stake sizes compares as follows:

Stake size	Expected capital after 100 rounds (60% win rate)
10% (Half-Kelly)	~18,000 yuan
20% (Full Kelly)	~22,000 yuan
30%	~19,000 yuan
40%	~12,000 yuan
50%	~4,000 yuan

Note: beyond the Kelly fraction, capital doesn’t decline gradually—it deteriorates sharply. That’s another layer of the Kelly Criterion: it isn’t just the “earn the most” plan; it’s also a “don’t go broke” safety boundary.

Half-Kelly and conservative strategies in real investing

Theoretically the Kelly fraction is optimal, but almost nobody in real investing runs the full Kelly. There are three reasons:

First, probability estimates themselves contain error. You think your win rate is 60%, but the true win rate might be 55% or 52%. Small misjudgments of parameters get amplified under full Kelly.

Second, odds in the real world aren’t fixed. Stock market odds (i.e., expected returns) keep changing with market sentiment, macroeconomics, and industry cycles.

Third, psychological tolerance is limited. Even if mathematically optimal, a 40% drawdown is enough to push most people into panicked capitulation.

That’s why in practice most people adopt “Half-Kelly” or even lower fractions. Half-Kelly sacrifices roughly 25% of the long-term growth rate but cuts capital volatility by about 50%. This tradeoff of “a little return for a lot less volatility” is far more sustainable for most investors.4

What the Kelly Criterion teaches us about life

The core logic of the Kelly Criterion applies not only to casinos and stock markets—it offers a general decision framework.

Identify your edge. The Kelly Criterion assumes you already know p and b. In reality, the first step is to honestly assess: do I have an edge in this domain? How big is it? If the answer is “no” or “uncertain,” the formula recommends zero commitment. That in itself is a valuable life principle: concentrate resources where you have a cognitive edge.

Sizing depends on the size of the edge. The bigger the edge, the more you invest; the smaller the edge, the less. This logic applies to career choices (what are your win rate and potential returns in this track), startup decisions (how strong is your product’s differentiation against competitors), even everyday time allocation (is the marginal return on time spent on this skill declining).

Avoid “all-in” thinking. The Kelly Criterion never recommends going all-in, even with a 90% win rate. It’s a mathematical reminder about humility: the world is always uncertain, and over-concentration is the shortest path to ruin. In career planning, this means keeping your skill set diversified and your income sources spread out; in investing, it means always leaving room to maneuver.

Accept volatility and focus on the long run. The Kelly Criterion optimizes the long-term geometric growth rate, not single-period returns. This requires accepting short-term volatility as the price of admission rather than trying to win every time. Many people lose money investing precisely because they try to eliminate all volatility, and end up accumulating fees and emotional attrition through over-trading.

Common misconceptions

Myth one: The Kelly Criterion guarantees you always win. It doesn’t. The Kelly Criterion only matters when you have a genuine edge. If your win rate is below the break-even point, the formula tells you not to bet.

Myth two: As long as you compute the probabilities correctly, you can use the Kelly Criterion well. The accuracy of your probability estimates is the whole bottleneck. In reality, human probability estimates are subject to systematic biases, and overconfidence is the most common trap.

Myth three: The Kelly Criterion only applies to gambling and investing. Its underlying logic is resource-allocation optimization. Any scenario involving “how much resource should I commit to maximize long-term returns” can draw on this framework.

Summary in one sentence

The core capability the Kelly Criterion teaches us is this: in an uncertain world, translate your judgment of an edge precisely into the intensity of action—neither wasting opportunity through caution, nor destroying yourself through recklessness.

References:

Kelly criterion - Wikipedia
Edward O. Thorp - Wikipedia
Kelly criterion, Wikipedia, “Practical use of Kelly criterion” section
The Kelly Criterion - Investopedia

Every Generation Thinks They Can't Get Ahead—300 Years of Data Says They're All Wrong

Thu, 14 May 2026 12:56:55 +0800

HuXiu recently republished an article from the WeChat account 「不懂经」 introducing the work How to Get Rich in American History by American historian Joseph Moore. The original article’s core argument is clear and powerful: every generation complains that “it’s hardest to get ahead right now,” yet 300 years of data repeatedly prove they were all wrong each time.

That conclusion is striking enough, but the original interpretation stops at a “believe and opportunity will come” level—like a bowl of lukewarm chicken soup. If we push Moore’s framework further, we uncover several key insights the original didn’t fully develop, and they have far more concrete relevance to today than the four words “stay optimistic.”

The real cost of “can’t get ahead” isn’t poverty—it’s hijacked attention

Moore introduces the concept of the “Big Woe” complex: a vast industry—built from academics, media, podcasts, and short videos—whose core business is convincing you that “you can’t get ahead.” The original article’s explanation was “they’re wrong, so you shouldn’t believe them.”

But that explanation ignores a sharper question: why is the human brain so easily captured by pessimistic narratives?

The neuroscience answer is that the human brain is naturally two to three times more sensitive to negative than positive information. This is called the “negativity bias,” a legacy of evolution. In the wild, ignoring good news only means missing a meal; ignoring bad news could cost you your life. To keep you alive, the brain weights negative information far above positive.

This means the “woe industry” Moore describes works not because its content is especially persuasive, but because it surgically exploits the underlying architecture of the human nervous system. Every analysis of “class hardening,” every tweet about “K-shaped divergence,” every podcast about “the middle class sliding back” is injecting extra anxiety signals into a brain already oversensitive to negative information. The real question isn’t “whether the pessimistic narrative is true”—it’s “once your attention is hijacked, how much cognitive capacity is left for action?”

Pew Research Center data shows 60% of young Americans believe “the American Dream is no longer possible.” Moore would counter that the data shows they’re wrong. But being “wrong” at the data level doesn’t change one fact: when a person spends a large slice of cognitive bandwidth on anxiety, the resources they’d use to spot opportunities, weigh risks, and execute plans shrink. The cost of pessimism isn’t that it makes you “disbelieve in opportunity”—it leaves you with no energy to do the things that create opportunity.

This is exactly the point Moore touches on when citing Consumer Financial Protection Bureau research: income and “financial well-being” are only weakly correlated—the strongest predictors are “optimism about the future” and “saving habits.” The causal chain goes like this: pessimistic narrative → hijacked attention → reduced capacity for action → missed opportunities → confirming the “can’t get ahead” prophecy.

“The past was easier” is a cognitive illusion, but today’s anxiety has real anchors

Moore uses 300 years of data to establish a stable cognitive pattern: every generation yearns for a time when “the past was easier.” Virginians in 1676 thought things were better 15 years earlier; textile workers in 1870 thought the colonial era was better; workers in 1929 thought the Gilded Age was better.

This pattern holds true in today’s China. Social media is full of nostalgia for “the era around 2010”: housing prices hadn’t taken off yet, the internet still had its dividend, and getting a civil-service job wasn’t so ruthlessly competitive. But China’s per-capita GDP in 1980 was only about $200, and most families couldn’t afford a single color TV. The so-called “easier past” simply doesn’t hold up in material terms. However, Moore’s framework has a blind spot: it conflates “material standard of living” and “opportunity structure.” Material standards have indeed been rising steadily. An ordinary white-collar worker in 2026 enjoys far better material conditions than a rural “ten-thousand-yuan household” of the 1980s. But changes in opportunity structure are nonlinear. In certain historical windows, the density and variety of opportunities contract sharply. For example:

In 19th-century America, if you could buy a plot of land and start a farm, you had a chance to accumulate capital. Land was cheap; the barrier was physical labor.
In the mid-20th century, if you could enter a big corporation and secure a lifetime employment contract, you could steadily build middle-class wealth. The barrier was a diploma and loyalty.
From the late 20th to early 21st century, if you could get into the internet industry and receive stock options, you might achieve upward mobility within a decade. The barrier was technical skill and timing.

By 2026, these three channels have all narrowed to varying degrees. Land is no longer an ordinary person’s ticket in; lifetime employment has become flexible labor; the internet’s dividend is over, and AI is reshuffling the deck. Moore would say “every generation thinks the channel is closed, but they’re wrong”—and that’s statistically true. But “statistically, someone can make it” and “most people can make it” are two different things. Here’s a key data point the original article missed: the research Moore cites shows that between 1820 and 1910, at most 28% of residents in US cities achieved upward mobility. The original called “28% a striking number” because it was far higher than Europe in the same period. But read the sentence the other way: 72% did not achieve upward mobility.

Most people stay where they are; a minority get ahead. What Moore proves is that “can’t get ahead” is wrong as a collective judgment, but as an individual experience it’s real for most people. Acknowledging that is more constructive than simply denying it.

Slow time and fast time: the real lesson isn’t “wait”—it’s “posture of readiness”

Moore’s “Slow Time” and “Fast Time” framework is the most practically valuable concept in the book. The original interpreted it as: accumulate during slow time, and when fast time arrives you’ll be ready to catch it.

That interpretation is too simplistic. Moore tells two stories to illustrate the framework:

Hollywood star Kim Basinger bought a small Georgia town in the 1990s, hoping to build a film studio. Her direction was right (the region later became one of the fastest-growing film production centers in the world), but she mistook slow time for fast time and went bankrupt waiting.

Norman McGhee bought more than 100 foreclosed homes with borrowed money during the Great Depression, and after the economy recovered became one of the most important Black businessmen of his generation. He kept slow-time patience during fast time.

The difference between these two stories isn’t patience—it’s cognitive precision. Basinger knew the direction was right but didn’t know the time scale. McGhee knew the direction was right and also understood that economic cycles have their own rhythm. The distinction: one was betting that fast time would come, the other was making structural preparations for fast time. To translate the framework into today’s language: “slow time” is the everyday you’re living through right now. Housing prices drifting down, discussions of AI replacing jobs, workplace rat-race, consumption downgrading. These changes are so slow they feel like they’ll never end.

“Fast time” is the moment some variable suddenly breaks through its threshold. It could be an AI capability breakthrough that makes certain jobs vanish overnight, a policy shift that opens a new wealth channel, or the start of a technology cycle that pays disproportionate returns to those who got positioned early. What Moore is really saying: you can’t predict when fast time will arrive, but you can place yourself in a structural position that can catch it when it does. That position is more than simply “saving up enough money.” It includes: whether the skills you hold will command a premium in the next wave of change; whether your network connects you to the sources of new opportunity; whether your cognitive framework is flexible enough to recognize the signal that “the rules have changed.”

In other words, preparation in slow time isn’t passive waiting—it’s a deliberate “posture adjustment.”

“The rules keep changing”: every generation navigates with the previous generation’s map

Moore uses 300 years of data to trace how the rules for getting rich have changed: farms and canal stocks in the 19th century, railroads and steel in the early 20th, stocks and real estate in the mid-20th, corporate equity and the internet in the late 20th, and today compute capacity, attention coordinates, and algorithmic pricing power. The original summarized this insight as “most people operate with the previous generation’s map,” but didn’t ask a deeper question: why does every generation walk around with an old map? The answer lies in the “anchoring effect” from cognitive science. The human brain tends to use the most recent success as the template for the future. If someone profited from real-estate speculation in 2015, their brain encodes “buying property = making money” as a reliable rule. When the environment shifts (prices no longer rise broadly, liquidity tightens), their first reaction isn’t “the rules changed”—it’s “this is only temporary.”

That’s why Moore’s data is so persuasive: it isn’t saying something abstract like “opportunity always exists.” It’s saying “the rules switch far more frequently than human intuition expects.” Over 300 years, the rule for getting rich has undergone a fundamental transformation roughly every 30 to 50 years. And the psychological age of humans (how fast their cognitive framework updates) can’t keep up with that pace.

Placing this pattern in the 2026 Chinese context: the mainstream wealth path of the past 20 years was “college → big tech / civil service → buy a house → financial security.” That path was largely effective before 2020. But now, layoffs at big tech have become routine, the civil service is cutting salaries while expanding headcount, and housing prices are flat or drifting down. The old map is failing. But Moore’s data also tells us: the moment the old map fails is precisely the moment a new map becomes most valuable. At every rule switch, a small group—precisely because their cognitive framework is flexible enough—identifies and executes the new rules first. What they share isn’t good luck; it’s that they started exploring the new rules while the old ones were still working.

Marriage as a financial strategy: a badly underrated variable in the Chinese context

The original article mentioned that Moore sees marriage as a “badly underrated financial strategy,” but only gave a superficial treatment. In today’s Chinese context, this point deserves its own unpacking.

All the examples Moore cites point to the same mechanism: a dual-income household lives on one income and pours the other into asset accumulation. Flagler, Bell, Stanford, and Sam Walton all launched with capital from their spouse’s family. This was common knowledge in 19th-century America, but in today’s Chinese discourse, “marriage” is mostly discussed as a matter of emotion and consumption (bride price, weddings, adding a name to the property deed), and almost never as a synergy.

The core mechanism is this: a single-worker financial model is linear (time for money); a dual-worker model is, ideally, exponential (one income covers living costs, the other goes entirely into compounding assets). Moore makes a sharp remark in the book: “Having one person support the family is the real invention—and it’s a terrible one.”

This isn’t advocating “marrying for money.” Moore’s point is that marriage, as a long-term shared financial unit, yields compounding effects far beyond what most people recognize. One person’s spending impulses are random; two people’s spending decisions, after negotiation, tend to be more rational. One person’s investment perspective is singular; two people’s combined knowledge and judgment cover more ground. This isn’t psychology chicken soup—it’s portfolio theory applied to the household.

The real takeaway: belief is the infrastructure for action

Moore’s own transformation story is the most persuasive part of the book. Born into a Southern working-class family and a committed Marxist, he taught students each semester that “the American Dream is a myth.” After the 2008 financial crisis, he began personally testing the money-making methods he’d excavated from historical sources. Eventually, he became a millionaire.

The original read this transformation as “belief changes destiny,” but that’s only the surface of the story. A more accurate formulation: belief is the infrastructure for action.

When Moore believed “you can’t get ahead,” his behaviors were: studying history, analyzing inequality, criticizing capitalism in the classroom. These behaviors were cognitively self-consistent but financially inert. When he changed that underlying belief, his behaviors changed with it: he started buying property, running things by market logic, making decisions with business reasoning. The key isn’t “optimism” itself—it’s that optimism changes the search space of behavior. A pessimistic person chooses among known, low-risk options. An optimistic person searches a larger space of options, including those with higher risk and higher reward. Over the long run, a larger search space means a higher expected value.

This aligns with another study Moore cites: optimistic views of the future significantly predict long-term saving behavior. Saving isn’t the “result of optimism”—saving is “a manifestation that optimism has changed behavior patterns.” What actually changes is that the person begins making decisions on longer time horizons, directing resources toward the future instead of burning them in the present.

Closing thoughts

Moore uses 300 years of data to prove three things: every generation thinks it’s living through the hardest era, but the data says they’re wrong; the rules for getting rich keep changing, and most people navigate with an old map; fast time will always come—you can’t predict when, but you can position yourself to catch it.

The original interpretation stops there. But push one level deeper, and Moore’s real contribution isn’t the data or the framework—it’s a more basic observation: human judgments about their own situation systematically deviate from reality.

And that deviation almost always runs toward pessimism. That bias itself is the source of opportunity.

Because while most people are consuming cognitive resources in pessimism, those who keep their capacity for action face less competition.

Reference: 不懂经 - It’s easier to get ahead today than ever before