In 2022, Liu Run published The Underlying Logic, using philosophical thinking to dissect the operating rules of the business world and teaching people "how to think." The book caused a sensation in the Chinese-language business book market, becoming that year's bestseller.
A year later, in 2023, he released the sequel The Underlying Logic 2. This time, he swapped his weapon from "philosophy" to "mathematics." No longer qualitative analysis, but quantitative analysis. No longer just telling you "what the direction is," but teaching you "how to calculate the answer with numbers."
The first book gives you a telescope to see the direction; the second gives you a calculator to compute the path.
Liu Run's ambition is to let every ordinary person see business through the eyes of a mathematician. He guarantees that only middle/high school-level math is needed to understand every concept in this book.
You might think addition, subtraction, multiplication, and division are too simple -- even a child can do them. But Liu Run says that behind arithmetic lie the four most fundamental relationship patterns in the business world.
Two people working together in the same dimension produce a "1+1" effect. For example, two people carrying bricks together double the output. This is the most basic form of cooperation, but also the least efficient -- because it only produces linear growth.
Two people competing for the same resource in the same dimension -- what you gain is what I lose. This is a zero-sum game. Price wars and market share battles are all subtraction thinking.
When two capabilities from different dimensions combine, the effect isn't additive but multiplicative. A tech expert and a sales expert working together don't produce 1+1=2, but potentially 3x5=15. This is why cross-disciplinary teams are more powerful than homogeneous ones.
When you use higher-dimensional weapons against a lower-dimensional opponent, this isn't subtraction (zero-sum) but division (dimensional strike). E-commerce vs. brick-and-mortar stores, smartphones vs. feature phones, streaming vs. DVDs -- these aren't about "stealing customers" but about eliminating the opponent's dimension entirely.
| Operation | Business Meaning | Effect | Typical Example |
|---|---|---|---|
| Addition | Same-dimension cooperation | Linear growth (1+1=2) | Hiring more salespeople |
| Subtraction | Same-dimension competition | Zero-sum game (your gain, my loss) | Price wars |
| Multiplication | Cross-dimension cooperation | Exponential growth (3x5=15) | Tech x Marketing cross-functional team |
| Division | Cross-dimension competition | Dimensional strike | E-commerce vs. brick-and-mortar |
Liu Run further points out that financial statements are essentially arithmetic. Revenue - Cost = Gross Profit (subtraction); Gross Margin = Gross Profit / Revenue (division); Profit x Turnover = Return on Investment (multiplication). When you view financial statements through the lens of arithmetic, numbers are no longer cold spreadsheets but a living business story.
Use multiplication to find partners, use division to compete. Most people spend their entire lives doing business with addition and subtraction -- hire one more person, grab one more customer. But true masters seek multiplication partnerships (cross-dimensional combinations) and division competition (dimensional strikes). Addition keeps you alive; multiplication makes you explode.
Cartesian coordinates -- the X-Y axes you learned in middle school. Liu Run says this seemingly simple tool is the ultimate weapon for "dimensional thinking."
When you encounter a knot in one-dimensional space (on a line), you can't untie it no matter what. But if you "ascend to two dimensions" (a plane), the knot resolves itself -- because you can go around it from above.
Business problems work the same way. When a problem seems unsolvable, it's probably not that the problem is too hard, but that your thinking dimension is too low.
If you only use one dimension -- "Is this person capable?" -- to hire, you'll never do it well. Liu Run suggests drawing a two-dimensional matrix: X-axis for ability, Y-axis for attitude.
| Good Attitude | Bad Attitude | |
|---|---|---|
| High Ability | Star employee -- promote | Wild dog -- eliminate immediately |
| Low Ability | Naive rabbit -- train or reassign | Let go -- don't hesitate |
The most dangerous people aren't those with low ability and bad attitude (they're easy to spot), but the "wild dogs" with high ability and bad attitude -- they can produce short-term results but will poison team culture in the long run.
Place all your business lines on a two-dimensional coordinate system: X-axis for "revenue scale," Y-axis for "profit margin." Businesses in the upper right (high revenue and high profit) are star businesses; those in the lower left should be cut decisively. This is far more efficient than analyzing each business line one by one.
The Gini coefficient measures income inequality. Liu Run applies it to team management: if the team's bonus distribution Gini coefficient is too low (everyone gets the same), top performers will leave; if it's too high (extreme wealth gap), the team will fracture. Finding the optimal Gini coefficient sweet spot for motivation is a manager's core task.
When you can't untie a knot in one dimension, ascending to two dimensions solves it automatically. When facing difficulties, don't grind away in the same dimension -- try adding one. Add an "attitude" dimension to hiring, a "profit margin" dimension to business evaluation, a "fairness" dimension to compensation. When you add a dimension, the answer reveals itself.
"Compound interest is the eighth wonder of the world" -- this quote is attributed to Einstein (though he probably never said it) and cited in countless financial articles. But Liu Run says most people fundamentally misunderstand compound interest; they simply treat it as a synonym for "slowly getting rich."
Legend has it that the inventor of chess asked the king for a reward: 1 grain of wheat on the first square, 2 on the second, 4 on the third, doubling each time. The king thought this request was too modest and gladly agreed.
The result? The 64 squares of the chessboard require a total of 2^64 - 1 = 18,446,744,073,709,551,615 grains. This number is hundreds of times the world's entire wheat production at the time.
This is the power of exponential growth -- it looks negligible at first, but once it passes the inflection point, the growth rate devours everything.
But Liu Run raises a critical warning: Compound interest only works when each step has a sequential correlation with the previous one.
If you study English today, painting tomorrow, and programming the day after, with no connection between them, compound interest won't kick in. Only when each step builds on the foundation of the previous one will compounding activate.
Compound interest isn't about "doing many things" -- it's about "continuously stacking on the same thing."
This is one of the most important concepts in the entire book. Liu Run explains two fundamentally different worlds:
Why does 80% of wealth concentrate in 20% of hands? Why do 80% of sales come from 20% of products? Because wealth and sales exist in a power law distribution world. In power law distributions, inequality isn't an exception -- it's a mathematical certainty.
In a power law world, the head effect determines everything. Choosing the right track matters more than effort. If you work hard in a normally distributed field (like screwing bolts in a factory), you'll be at most two or three times better than the average worker. But if you become a top player in a power law field (like content creation, investing, or entrepreneurship), you can be ten thousand times better than the average. The question isn't "how hard you work" but "which distribution you're working in."
Expected value tells you "what happens on average," but it doesn't tell you "how much fluctuation there is." And in the real business world, volatility is what kills you.
Imagine two investment plans:
50% chance of earning $1M
50% chance of earning $0
Expected value = $500K
Standard deviation = $500K
100% chance of earning $500K
Expected value = $500K
Standard deviation = $0
Both plans have identical expected values, but the risk is worlds apart. Plan A might leave you with nothing; Plan B guarantees you $500K. Variance and standard deviation are the tools that quantify this "fluctuation."
Liu Run points out that people who truly understand investing don't chase "the highest returns" -- they pursue the highest "risk-adjusted return." In financial terms, this is the Sharpe Ratio:
Sharpe Ratio = (Return - Risk-Free Rate) / Standard Deviation
A high return doesn't mean you're skilled; a high return with low volatility is true mastery.
This tool isn't just for investing. You can use standard deviation to evaluate:
Masters don't chase the highest returns -- they chase the highest "risk-adjusted returns." Ordinary people only look at gains; masters look at both gains and volatility simultaneously. An investment with 30% annualized returns but wild swings may be worse than one with 15% annualized returns that's rock-solid. Because high volatility means high blow-up risk, and blowing up once ends the game. Surviving longer matters more than running faster.
This is the core chapter of the entire book and the section Liu Run devotes the most space to. He says: "Probabilistic thinking is the dividing line between experts and ordinary people."
Before making any decision, calculate the expected value. For example, a business has a 30% chance of earning $1M and a 70% chance of losing $200K. Expected value = 0.3 x 1,000,000 + 0.7 x (-200,000) = +$160K. The expected value is positive, so it's worth doing in the long run.
But the magic of expected value only manifests when you "do it enough times." Flip a coin once -- heads or tails is 50/50, but with a single flip, the result is either 100% heads or 100% tails. Flip it ten thousand times and the proportion steadily approaches 50%.
The implication for entrepreneurs: Don't bet all your chips on a single opportunity. You need to create enough "coin flip" opportunities to let the law of large numbers work.
The probability of someone having a rare disease is one in a hundred thousand. But if their genetic test comes back positive, that probability changes dramatically. This is conditional probability -- new information changes the odds.
This is Liu Run's most brilliant metaphor in the entire book. The core of Bayesian theorem is:
P(Hypothesis|Evidence) = P(Evidence|Hypothesis) x P(Hypothesis) / P(Evidence)
In plain language: Your belief about something should be continuously updated with new evidence. Prior probability (your original judgment) + new evidence = posterior probability (your updated judgment).
Liu Run says that masters are people who live their lives as Bayesian theorems:
Liu Run redefines entrepreneurship as a probability management problem:
The entrepreneurship formula = Increase per-attempt success rate x Multiply number of attempts x Control per-attempt loss. Optimize all three variables simultaneously, and you transform entrepreneurship from "gambling" into "investing."
Why do venture capitalists dare to invest in projects with a 90% failure rate? Because they're not betting on a single project -- they're playing a statistics game. Invest in 100 companies; as long as one becomes a unicorn (100x return), it covers the losses of the other 99 and then some. This isn't courage -- it's math.
Probabilistic thinking is the dividing line between experts and ordinary people. Ordinary people think in "right or wrong": Is this right or wrong? Experts think in "probability": What's the probability of success? What's the cost of failure? Is the expected value positive or negative? When you convert every decision into a probability problem, you won't be swayed by emotions anymore, and you'll stop saying "I think" -- you'll say "the data shows."
Game theory doesn't teach you how to "win" -- it teaches you to understand: When everyone pursues their own maximum benefit, what happens?
The basic tool of game theory is the payoff matrix -- listing every choice and corresponding outcome for both parties in a single table. When you place business decisions into a payoff matrix, many "seemingly complex" problems suddenly become clear.
A dominant strategy is the strategy you should choose regardless of what the other party does. If you have a dominant strategy, the decision becomes extremely simple -- just pick it, no need to guess what the other side will do.
A Nash equilibrium is a "stable state": in this state, any party that unilaterally changes their strategy will make themselves worse off. So nobody wants to change.
There are countless Nash equilibria in the business world: Why do gas stations always cluster together? Why is fast food priced similarly? Not because they conspired, but because each party pursuing maximum benefit naturally arrived at this equilibrium point.
The classic prisoner's dilemma: if both suspects stay silent, each gets 1 year; if one confesses while the other stays silent, the confessor goes free while the silent one gets 10 years; if both confess, each gets 5 years.
| B Stays Silent | B Confesses | |
|---|---|---|
| A Stays Silent | Each gets 1 year (best cooperative outcome) | A gets 10 years, B goes free |
| A Confesses | A goes free, B gets 10 years | Each gets 5 years (Nash equilibrium) |
Under rational analysis, both parties will choose to confess (because regardless of what the other does, confessing is better for yourself). But this leads to a worse outcome than if both had cooperated.
The prisoner's dilemma is everywhere in business: price wars are the classic example -- if neither company cuts prices, both make money; but if one cuts prices to steal customers, the other loses market share. Result: both cut prices, both see profits decline.
The best competitive strategy isn't beating your opponent -- it's changing the rules of the game. Desperately "optimizing strategies" within a prisoner's dilemma is futile because the structure determines the outcome. True masters step outside the current game and redesign the rules -- for example, building brand differentiation (making price wars ineffective), building ecosystems (turning opponents into partners), or creating entirely new tracks (making former opponents irrelevant).
Zhuge Liang didn't start with a perfect plan. Before the Longzhong Plan, he had spent years observing the state of the world -- this was his "prior probability."
Then, new evidence kept flowing in:
From allying with Wu against Cao to taking Jingzhou, entering Sichuan, and the Northern Expeditions, every step was "adjusting prior probability based on new evidence." The Longzhong Plan wasn't a static perfect plan but a dynamic Bayesian reasoning process. Zhuge Liang was called a "genius for the ages" not because he got everything right from the start, but because he was always updating his judgments with new information.
Sun Tzu's Art of War says: "Victorious warriors win first and then go to war; defeated warriors go to war first and then seek to win." In plain language: great generals ensure they've already won before fighting; failed generals charge in first and figure out how to win later.
This is the ancient version of probabilistic thinking:
Liu Run's "entrepreneurship is managing probability" and Sun Tzu's teaching are exactly the same thing. Military strategy from 2,500 years ago has probability theory as its underlying logic.
Most people look for partners who are "similar" to them -- same industry, same skills, same background. But that's additive cooperation, at best 1+1=2.
True masters look for people in a "different dimension." A tech genius and a sales prodigy working together don't produce 5+5=10, but 5x5=25. A content creator and a supply chain expert partnering up don't add their capabilities -- they open up an entirely new business space.
Action guide: Next time you're looking for a partner, ask yourself: "Are our capabilities additive or multiplicative?" If the answer is additive, keep looking.
Working hard in a normally distributed track, you'll be at most two or three times better than average. A factory worker's per-hour bolt-screwing output can never be a hundred times the average, no matter how skilled.
But in a power law distributed track, top players can be ten thousand times better than average. A top YouTuber's income is a hundred thousand times that of an ordinary YouTuber. A successful app's downloads are a million times those of an ordinary app.
Action guide: Ask yourself: "Is my current track normally distributed or power law distributed?" If it's normally distributed, consider switching tracks. If you're already on a power law track, go all-in toward the top, because the gap between second place and last place may be smaller than the gap between first and second.
The most dangerous businesspeople aren't the unintelligent ones, but the intelligent yet stubborn ones. They have strong prior beliefs ("I think the market should work this way") but refuse to update with new evidence.
People who live as Bayesian theorems have three characteristics:
Action guide: Every quarter, conduct a "belief audit" -- list your three most important business judgments, then ask yourself: "What new evidence supports or refutes these judgments?" If there's strong refuting evidence, update immediately.
Liu Run provides the mathematical formula for entrepreneurship:
Entrepreneurial Success = Increase per-attempt success rate x Multiply number of attempts x Control per-attempt loss
How to operate each of the three variables:
Action guide: If you're currently running or preparing to start a business, write these three variables on your wall. Before every decision, ask yourself: "Which variable does this decision optimize? Does it simultaneously worsen another variable?"
The Underlying Logic 2 is not a mathematics book -- it's a business philosophy book written in the language of mathematics.
What Liu Run wants to tell us is: the business world may seem chaotic, but its underlying operating rules are mathematical. Arithmetic reveals the nature of cooperation and competition, Cartesian coordinates teach you to think in higher dimensions, exponential and power law expose the secrets of wealth distribution, variance teaches you to measure risk, probability theory turns every decision into a calculable math problem, and game theory explains why rational people make seemingly irrational choices.
The first "Underlying Logic" teaches you to see the world's hidden cards; the second teaches you to calculate the numbers behind those cards. See the hidden cards, and you won't be deceived; calculate the numbers, and you won't make mistakes.
Perhaps this book's most profound insight is: You don't need to become a mathematician, but you need to think like one. Not every formula needs to be computed, but every decision has a computable formula behind it. When you develop the habit of "calculate first, act second," you've already beaten most people who act on instinct alone.
Because in this world, instinct can deceive you, but numbers won't.