💰 quantitative finance

The gradual transformation of inland areas -- human plowing, horse plowing and equity incentives

This article argues that societies can utilize mathematical models such as lognormal distributions to optimize incentive systems for equal opportunity by examining historical governance strategies, such as the transition from human-drawn to horse-drawn plows and the introduction of various selection mechanisms, thereby strengthening civilizational stability and reducing internal conflicts.

Original authors: Hongfa Zi, Zhen Liu

Published 2026-04-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hongfa Zi, Zhen Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine civilization as a massive, complex machine that sometimes gets stuck in "autopilot." The authors of this paper argue that many modern societies are like people who have bought the newest, most expensive smartphone but still only use it to play a game from 2,000 years ago. They have the technology, but their "operating system" (how they govern and distribute wealth) is outdated, leading to stagnation and conflict.

Here is the story of the paper, broken down into three simple parts using everyday analogies:

1. The "Horse vs. Human" Upgrade: Why Speed Matters

The paper views history like a video game where the "level" of civilization depends on how fast and efficiently one can move.

The Old Way (Plowing with Humans): Imagine a village where everyone walks slowly. If a few greedy landlords try to take all the food, the slow villagers cannot stop them, so the landlords simply take more and more. This leads to poverty and rebellion. The paper suggests that ancient China got stuck here because rulers feared strong, independent people (like cavalry), and therefore "broke the legs" of the "horses" (the people) to keep them weak and obedient.
The Upgrade (Horse Plowing): Now imagine swapping these slow walkers for fast horses. A single rider on a horse is much faster and stronger than a hundred people on foot. The paper uses mathematics (the Lanchester equation) to prove that a fast cavalry unit can easily defeat hundreds of slow infantry soldiers.
- The Lesson: If a society allows "fast horses" (strong, independent cavalry or military power), it forces the system toward more fairness. Landlords can no longer simply intimidate people because the "horses" can fight back. This creates a society where rules are determined by voting and strength, not by a single tyrant.

2. The "Wealth Curve": How to Share the Cake Fairly

The paper argues that money is not distributed evenly like a flat line, but follows a lognormal distribution.

The Analogy: Imagine a room full of people holding sacks of gold. Most people have medium-sized sacks. A few have tiny sacks, and a very small number of people have huge sacks. This is the "lognormal" shape.
The Problem: If you allow the rich to become too rich (as in a "power law" where one person owns 99% of the gold), the system collapses. The poor become too desperate, and the rich become too arrogant.
The Solution (Equity Incentives): The authors suggest treating a country or company like a giant team. Instead of just paying people a salary, we should give them shares (equity) in the team.
- The Math: They use a formula to show that having more shares makes you richer. By adjusting the "spread" (standard deviation) of these shares, one can ensure that no one is too poor and no one is too rich.
- The 1:1:1 Rule: To maintain stability, the paper proposes splitting an employee's reward into three equal parts:
  1. Salary: To eat today.
  2. Bonus: For good work this year.
  3. Shares/Dividends: To stay with the team long-term.
- The Goal: This turns employees into "owners." If you own a piece of the company, you care about its success. It ends the "us versus them" struggle between bosses and workers.

3. The "Traffic Light" of Conflict

The paper states that conflict (such as disputes, protests, or even small uprisings) is not always bad. Think of it like tempering steel.

The Analogy: If you heat steel too much, it melts. If you don't heat it enough, it is brittle and breaks. You need the right amount of heat to make it strong.
The Application: A society needs a little pressure (conflict) to update its rules and become stronger. However, if the pressure gets too high (too much inequality or too much oppression), the entire system shatters.
The Solution: The paper suggests we must be like a smart traffic light. We must let some "traffic" (conflict) flow to keep the system moving, but we must stop it before a crash occurs. This is achieved by constantly adjusting the rules (such as who becomes the leader and how money is distributed) based on mathematics and history.

The Big Picture: The "Algorithm" of Civilization

The authors conclude that history is not just a story; it is a database of data.

The Mistake: Many leaders try to govern with "art" or "gut feeling," which leads to corruption and collapse.
The Solution: We should treat governance like a computer algorithm. We must:
1. Look at data from the past (history).
2. Use mathematics to find the perfect balance between distributing money to people and maintaining order.
3. Mix different methods for selecting leaders (such as elections, exams, and lotteries) so that no single family or group can seize power.

In short: The paper says that to survive and thrive, we must stop being "old societies in modern clothing." We must harness the speed of the "horses" (strong, independent forces), distribute wealth fairly through "equity" (so everyone is an owner), and use mathematics to keep the pressure just right so that society becomes stronger instead of falling apart.

1. Problem Statement

The paper addresses the stagnation of modern civilizations that possess advanced technology but fail to evolve their governance models, thus effectively remaining "old societies in technological disguise." The authors identify three fundamental systemic failures in historical and modern governance:

Government Rigidity: Dependence on outdated power structures (e.g., corruption, hereditary succession, or pure elections) leads to the solidification of classes and oligarchic rule.
Economic Inequality: Uncontrolled wealth distribution creates extreme social stratification, leading to internal conflicts, tax evasion by the wealthy, and peasant uprisings.
Imbalance Between Production and Suppression: Historical examples (particularly ancient China) demonstrate that prioritizing rule over production and suppressing efficient military/production modes (such as horse plowing) leads to vulnerability against external enemies and internal collapse.

The central problem is how to achieve an "optimal solution" for a civilization that aligns conflict and stability, efficiency and justice, and technological progress with institutional evolution.

2. Methodology

The authors apply a multidisciplinary approach combining historical analysis, mathematical modeling, and interdisciplinary synthesis:

Historical Comparison:
- Contrasting the "Human Plow Civilization" (characterized by inefficient manual labor, exploitation by large landowners, and bureaucratic corruption) with the "Horse Plow Civilization" (characterized by highly efficient cavalry, free trade, and meritocratic selection).
- Analysis of case studies, including the collapse of the Roman Empire, the Glorious Revolution in England, the mixed electoral system of the Republic of Venice, and the decline of the Qing Dynasty.
Mathematical Modeling:
- Lanchester Equations: Used to quantify the difference in combat efficiency between cavalry and infantry/landowner soldiers.
- Lognormal Distribution: Applied to model national income and wealth distribution. The authors derive that income growth follows a multiplicative process (compound interest), leading to a lognormal distribution rather than a normal distribution or a pure power-law distribution.
- Calculus of Equity Incentives: Uses differential calculus ( $dW/d\alpha = V > 0$ ) to prove the positive correlation between participation rates ( $\alpha$ ) and total wealth ( $W$ ).
- Statistical Integration: Uses the cumulative distribution function (CDF) of the standard normal distribution ( $\Phi$ ) to calculate the number of individuals within specific participation intervals under a truncated lognormal distribution.

3. Main Contributions

A. The Paradigm Shift to "Horse Plowing"

The paper argues that the transition from human plowing to horse plowing is not merely an agricultural upgrade but a civilizational governance shift.

Military Efficiency: Cavalry offers a massive speed advantage ( $\Delta v \approx 4.5 m/s$ ) and higher killing efficiency compared to infantry.
Impact on Governance: The ability of cavalry to suppress the "tax-resistant" class of large landowners forces a more centralized and fair tax system. The authors argue that the suppression of cavalry in ancient China (to prevent internal rebellions) led to vulnerability against external nomadic invasions, whereas European knight systems fostered competition and innovation.

B. Mathematical Framework for Wealth Distribution

The authors propose that national income follows a lognormal distribution ( $lnX \sim N(\mu, \sigma^2)$ ), where:

$\mu$ (Mean): Represents the average wealth level.
$\sigma$ (Standard Deviation): Represents the wealth gap.
Contribution: They show that a society can achieve sustainable growth by adjusting $\sigma$ (via policy) while maintaining $\mu$ , avoiding the extreme inequality associated with power-law distributions (which eliminate too many people).

C. The "Equity Incentives" Governance Model

The paper proposes a specific institutional design for enterprises and states:

Tripartite Compensation: A balanced ratio of 1:1:1 between salary (livelihood), bonuses/dividends (short-term performance), and equity appreciation (long-term tenure).
Democratic Administration: Managers should be elected by employees (one person, one vote) to ensure mutual checks and balances. This prevents the "boss allocation" system where managers exploit shareholders.
Internal Oversight: Since a state has no "external shareholders," internal oversight must be achieved through profit-sharing (equity dividends) within the bureaucrat/employee class to align interests.

4. Results

Military Quantification: Using the Lanchester equation with parameters $A_0=1$ (cavalry) and $B_0=1000$ (infantry), the model shows that a single cavalry unit can theoretically neutralize hundreds of infantry soldiers due to a kill coefficient ratio ( $\alpha/\beta$ ) of 2000:1. This mathematically validates the historical concept "one hundred cavalrymen destroy a country."
Proof of Income Distribution: The derivation confirms that if income growth is proportional and independent ( $Y_t = Y_{t-1}(1+g_t)$ ), the Central Limit Theorem dictates that $\ln(Y)$ tends toward a normal distribution, making $Y$ lognormally distributed. This fits empirical data (e.g., SCF surveys) better than normal distributions.
Correlation Between Participation and Wealth: The derivation $dW/d\alpha = V$ proves that as participation increases, wealth increases, provided the enterprise value ( $V$ ) is positive.
Optimal Conflict Threshold: The analysis suggests that civilizations thrive when conflict is a "controllable deterrent" (moderate pressure driving reform, such as the Glorious Revolution) rather than a systemic catastrophe caused by extreme inequality or total suppression.

5. Significance and Implications

Governance as an Algorithm: The paper redefines governance not as an art but as an optimizable algorithm. It suggests that history is a database of parameters that can be parsed to create "evolutionary algorithms" for future governance.
Sustainable Development: By adopting a lognormal distribution for equity incentives, societies can prevent the "elimination" of the lower class (a risk with power-law distributions) while maintaining motivation for innovation and wealth creation.
Institutional Flexibility: The proposed model advocates for a mixed electoral system (elections + examinations + lotteries) to prevent oligarchies, combined with equity incentives to align the interests of rulers and the ruled.
Preventing Civilizational Collapse: The ultimate goal is to avoid the trap of "advanced technology but outdated governance." By balancing the "spear" (force/military) and the "shield" (law/justice), civilizations can achieve a dynamic equilibrium where internal conflicts drive progress rather than collapse.

Conclusion:
The paper postulates that the path to an "optimal civilization" lies in the quantitative control of wealth distribution (via lognormal equity incentives) and the structural adaptation of power (via mixed selection and cavalry-like efficiency). It calls for a shift from passive historical adaptation to active, mathematically informed evolution.