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An Explanation for Individually Rational, Voluntary Investment in Communal Capital in Large Groups:Toward a Population Pressure – Production Theory of Social Order

This paper argues that in large groups facing fixed land resources, increased population pressure creates conditions where individually rational agents voluntarily invest in communal capital to maximize free time, thereby explaining how high levels of cooperation and social order can emerge without coercion.

Original authors: Noah P. Mark

Published 2026-08-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Noah P. Mark

Original paper licensed under CC BY 4.0 (https://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

The Great Human Puzzle: Why We Stick Together When It Gets Crowded

Imagine you are trying to solve the biggest mystery of human history: How did we go from living in tiny, scattered tribes to building massive cities where millions of strangers cooperate every single day? For a long time, scientists have been puzzled by this. In the world of game theory—a branch of math that studies how people make decisions—there is a famous rule that says big groups are actually bad for cooperation. Think of it like a potluck dinner: if you invite just three friends, everyone brings a dish. But if you invite three hundred strangers, everyone thinks, "Someone else will bring the cake," and nobody brings anything. This is the "free-rider problem," and for decades, it seemed to explain why large-scale cooperation should be impossible without a boss, a police force, or strict laws to force people to play nice.

But here is the twist: humans did figure it out long before we had governments or contracts. We started cooperating in huge numbers while still living as foragers and farmers. So, how did we break the rules of the game? The question matters because understanding the roots of our ability to work together helps us understand the very foundation of society itself. If we can figure out why we started cooperating when things were tough, we might learn something profound about how human civilization actually grew.

The Crowded Room and the Magic Tool

In this paper, sociologist Noah Mark suggests a surprising answer: we didn't start cooperating because we became nicer or smarter. We started cooperating because we got crowded, and the math of survival changed.

Imagine a group of early humans living on a fixed patch of land, like an island. Let's say the island has a set amount of soil (land) that doesn't grow, but the number of people on the island keeps increasing. At first, everyone has plenty of space. But as the crowd grows, each person gets a smaller and smaller slice of that land.

In this crowded scenario, the paper argues that people have a choice. They can spend all their time just trying to scratch out a living on their tiny slice of land (private work), or they can spend some of their time building a "magic tool" that everyone shares. This tool could be a better hoe, a fence, or an irrigation system. The paper calls this "communal capital."

Here is the magic part: In the old game theory models, adding more people to a group usually makes everyone contribute less because everyone thinks, "I'll let the others do it." But Mark's model suggests that when land is scarce and the number of people is high, the opposite happens. The more crowded the group gets, the more each person is willing to voluntarily chip in to build that shared tool.

The "Land vs. Sweat" Tug-of-War

Why does crowding make people more generous? It comes down to a tug-of-war between two things: Land and Labor (your sweat and time).

The paper uses a mathematical recipe (called a Cobb-Douglas production function) to show how food is made. It turns out that in farming and foraging, the amount of food you get depends much more on how much land you have than on how hard you work. If you have a huge field, you can grow a lot of food even if you work at a normal pace. But if you have a tiny, crumb-sized piece of land, no amount of sweating will make it grow much food.

As the population grows, the land per person shrinks. This makes the land the "bottleneck." To survive on that tiny piece of land, you need to make it super productive. The only way to do that is to add "capital"—better tools that everyone shares.

Mark's model shows that when land is the limiting factor (which it was for early farmers), the pressure to survive actually incentivizes cooperation.

The Logic: If you are on a tiny plot of land, you are in big trouble. But if you and your neighbors all pitch in to build a shared irrigation system, that tiny plot becomes much more productive.
The Result: The more people there are (and the smaller the land per person gets), the more valuable that shared tool becomes. So, it becomes "rational" for each individual to spend more of their free time building that tool, even though they are giving up their own leisure.

The Numbers Don't Lie

The paper runs a simulation to prove this point. Imagine a group of people where the land is fixed, but the population grows.

In a group of 1 person, they might contribute 1 unit of time to building tools.
In a group of 10 people, each person contributes 10 units of time.
In a group of 15 people, each person contributes 15 units of time.

The paper explicitly states that as the group gets bigger, the amount of time each person voluntarily spends helping the group goes up. This is the opposite of what traditional theories predicted. Usually, bigger groups mean less individual effort. Here, bigger groups mean more individual effort.

However, the paper is careful to note that this doesn't mean life gets better. In fact, the simulation shows that as the group grows and everyone works harder to build tools, the total "free time" for each person actually goes down.

The Trade-off: In a group of 1, a person might have 238 hours of free time (in the model's units). In a group of 15, that free time drops to 0.
The Takeaway: People aren't cooperating because they are happy; they are cooperating because they are desperate. They are trading their free time for a slightly better chance of survival. The cooperation slows down the slide into starvation, but it doesn't stop it completely.

What This Means for Social Order

The paper suggests that this "Population Pressure – Production" theory offers a new way to understand how social order began. It argues that we didn't need a king or a law to force us to cooperate. Instead, the sheer pressure of having too many people on too little land created a situation where building shared tools was the only logical move for a rational person.

This idea challenges the old view that cooperation only happens when people are forced to or when the group is small. It suggests that the very things that make life hard—like overcrowding and resource scarcity—might have been the engine that drove us to build the complex, cooperative societies we live in today. The paper doesn't claim this is the only reason we cooperate, but it suggests it's a powerful, previously overlooked reason that fits the math of early human survival.

In short, the paper proposes that we learned to work together not because we were saints, but because the math of a crowded world made it the smartest thing to do.

Technical Summary: An Explanation for Individually Rational, Voluntary Investment in Communal Capital in Large Groups

Problem Statement
The paper addresses a fundamental puzzle in social theory: how high levels of cooperation in large groups are possible among individually rational actors. Traditional collective action theory (e.g., Olson 1965; Hardin 1982) suggests that per capita contributions to collective goods decrease as group size increases, particularly when the good is rivalrous and non-excludable. While prior research has identified conditions under which large groups might cooperate (e.g., heterogeneous interests or perfect jointness of supply), there remains a theoretical gap regarding homogeneous groups producing rivalrous collective goods where prior models predict a decline in cooperation as the group grows. The paper seeks to explain how individually rational, voluntary investment in communal capital could increase on a per capita basis as group size expands, specifically within the context of early human foraging or agricultural societies facing fixed land constraints.

Methodology
The author employs a mathematical modeling approach using a one-shot game framework to analyze the allocation of time by $m$ actors subsisting on a fixed quantity of land ($TA$).

Agents and Objectives: Actors seek to maximize "free time" ( $n_i$ ), defined as their initial endowment of waking time ($IE$) minus time spent on private subsistence work ( $pw_i$ ) and time voluntarily contributed to the group ( $gw_i$ ) for building communal capital.
Production Function: The model utilizes a three-factor Cobb-Douglas production function to determine the time required for subsistence. Output ( $Y$ ) is produced using Labor ( $pw_i$ ), Land ( $x_i$ ), and Capital ( $k_i$ ). The function is defined as $y_i = A \cdot (pw_i)^{b_w} \cdot (x_i)^{b_x} \cdot (k_i)^{(1-b_x-b_w)}$ , where $A$ is total factor productivity, and $b_w$ and $b_x$ represent the elasticity of production with respect to labor and land, respectively.
Constraints and Assumptions:
- Land is fixed and divided equally among members ( $x_i = TA/m$ ).
- Capital is a collective good created by the aggregation of voluntary individual investments ( $k_i = \sum gw_i / m$ ).
- The good is divisible, rival, and non-excludable (shared equally).
- The analysis focuses on a one-shot game to isolate the logical source of the relationship between group size and cooperation, rather than relying on iterated game dynamics.
Equilibrium Derivation: The author derives the Nash equilibrium by taking the partial derivative of the payoff function with respect to individual contribution ( $gw_i$ ), setting it to zero, and solving for the aggregate contribution ( $\sum gw$ ) that simultaneously maximizes individual payoffs for all $m$ members.

Key Results
The analysis yields a counter-intuitive result regarding the relationship between group size and per capita contribution:

Condition for Increasing Cooperation: The sign of the relationship between per capita contributing ( $\sum gw / m$ $\sum g w / m$ ) and group size ( $m$ $m$ ) is determined by the relative magnitudes of the production elasticities of land ( $b_x$ $b_{x}$ ) and labor ( $b_w$ $b_{w}$ ).
- If $b_x < b_w$ : Per capita contribution decreases as group size increases.
- If $b_x = b_w$ : Per capita contribution remains constant.
- If $b_x > b_w$ : Per capita contribution increases as group size increases.
Relevance to Early Societies: The condition $b_x > b_w$ (where production is more responsive to land than labor) is identified as characteristic of foraging and farming economies. Under this condition, as population grows on fixed land, the scarcity of land increases the marginal value of capital. Consequently, it becomes individually rational for actors to invest more time in building communal capital (e.g., irrigation, tools) to offset the diminishing returns of labor on shrinking land plots.
Numerical Illustration: Using specific parameters ( $b_x=3/5, b_w=1/5$ ), the model demonstrates that as group size ( $m$ ) increases from 1 to 15, per capita contribution increases linearly (from 1 to 15 units), while aggregate contribution increases quadratically ( $m^2$ ).
Welfare Implications: Despite the increase in cooperation, the model shows that mean individual payoff (free time) decreases as group size increases. The increase in cooperation serves to slow the rate at which material conditions deteriorate due to population pressure, rather than reversing the decline. The "cost" of the worsening environment (increased private work time) outpaces the "benefit" of increased cooperation, but the voluntary investment remains rational because it mitigates an otherwise even more dire outcome.

Significance and Claims
The paper claims to offer a "Population Pressure – Production Theory of Social Order" with several theoretical implications:

Rational Voluntary Cooperation: It provides a mechanism for individually rational, voluntary unilateral contributions to a collective good that increases per capita with group size, challenging the consensus that rivalry in collective goods necessitates a decline in per capita cooperation as groups grow.
Origin of the State: The theory suggests a potential rational and voluntary pathway for the emergence of the state. If resources pooled for communal capital can be channeled toward establishing property rights or taxation powers, this model offers a starting point for explaining state formation without relying solely on coercion or pre-existing institutions.
Historical Correlation: The model aligns with the historical trend where both human population size and levels of social order have increased over time. It posits that population growth on fixed land resources creates the specific production conditions ( $b_x > b_w$ ) that drive the intensification of cooperation.
Hardship-Driven Cooperation: Unlike Prisoner's Dilemma models where cooperation improves collective payoff, this model posits that cooperation arises because material conditions are deteriorating. Actors cooperate not to maximize absolute gains, but to ameliorate the severity of land scarcity and population pressure.
Complementary Pathways: The theory complements existing literature on population pressure and war, suggesting that both intensification of land use (via communal capital) and inter-group conflict are rational responses to the same underlying demographic and resource constraints.

The author concludes that this framework offers a promising approach to understanding the foundations of social order, specifically how large-scale cooperation could emerge in the absence of modern coercive institutions, driven by the economic necessities of subsistence production under population pressure.

The Great Human Puzzle: Why We Stick Together When It Gets Crowded

The Crowded Room and the Magic Tool

The "Land vs. Sweat" Tug-of-War

The Numbers Don't Lie

What This Means for Social Order

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