The Cooperation Ladder: Scale-dependent payoffs and population dynamics create surges, stalls and reversals

This paper presents a theoretical model extending the Stag Hunt game to demonstrate how endogenous population growth and scale-dependent cooperative thresholds create a "cooperation ladder" that drives historical surges in social complexity while explaining why societies often stall or revert due to intermediate incentives for free-riding.

The Gist

The Big Idea: Why Do Some Societies Grow Huge While Others Stay Small?

Imagine human history not as a straight line of progress, but as a giant, multi-story ladder.

• The Rungs: Each rung represents a massive reward (like a huge pile of food, a new energy source like coal, or a complex city).
• The Climb: To reach a higher rung, you need more people working together than the rung below it.
• The Catch: You can't just jump to the top. You have to climb step-by-step. And if you don't have enough people, you can't even reach the next step.

This paper argues that cooperation isn't just about being "nice." It's a mathematical game where the size of the reward depends on how many people are cooperating, and the size of the population depends on how big the reward is. It's a cycle of "Cooperation $\rightarrow$ Big Rewards $\rightarrow$ More People $\rightarrow$ Even Bigger Cooperation."

---

The Game: Hunting Stags vs. Catching Hares

To understand the model, let's use the classic story of the Stag Hunt:

• The Hare: A lone hunter can catch a small rabbit (a hare) anytime. It's easy, safe, and guaranteed. But it's not very filling.
• The Stag: A massive deer (a stag) is worth a feast for the whole village. But one person can't catch it. It takes a whole team working perfectly together. If even one person quits, the stag escapes, and everyone gets nothing.

The Paper's Twist:
Most games only have one Stag and one Hare. This paper says: What if there are Stags of all sizes?

• Rung 1: A small group can catch a small stag (like gathering firewood).
• Rung 2: A medium group can catch a bison (like early farming).
• Rung 3: A massive group can catch a "dragon" (like mining coal or building a nuclear power plant).

The Three Rules of the Ladder

1. The "Surge" (When the Ladder is Within Reach)

Imagine you are part of a team trying to catch a bison. You have 99 hunters, but you need 100 to succeed.

• What happens? Suddenly, everyone is super motivated to cooperate! The person on the fence thinks, "If I just join in, we get the feast!"
• The Result: Cooperation spikes. The group catches the bison, everyone eats, and the population booms because there is so much food.

2. The "Stall" (When the Ladder is Too High)

Now imagine you have 50 hunters, but the next big prize (a bison) requires 100.

• What happens? People get discouraged. "Why bother trying to catch a bison? We'll never get there. I'll just go catch my own rabbit."
• The Result: Cooperation drops. People stop trying to work together because the goal feels impossible. They settle for the small, safe rewards. This is why some societies get "stuck" at a small scale.

3. The "Free-Rider" Problem (The Lazy Guest)

Once the group does catch the bison, a funny thing happens.

• The Situation: The feast is ready. The hunters who did the work are tired. The people who didn't help (the free-riders) are just as hungry.
• The Result: The group tolerates the lazy people. Why? Because if the lazy people start helping, they might accidentally mess up the coordination, or if they quit, the whole group loses the feast. So, the group stays stable, even with some lazy members, as long as the big reward is secured.

---

The Feedback Loop: The "Virtuous Cycle" vs. The "Trap"

This is the most important part of the paper. It explains how history moves forward (or backward).

The Virtuous Cycle (The Climb):

1. A group manages to catch a small stag.
2. They have extra food, so the population grows (more babies survive).
3. Because there are more people, they can now try to catch a bigger stag that was previously impossible.
4. They catch the bison, get even more food, and the population explodes.
5. History Example: This is how we went from small tribes to farming villages to industrial cities. Each step unlocked the next.

The Trap (The Stuck Society):

1. A group starts with low cooperation or a small population.
2. They can't reach the threshold for the next big reward.
3. Because the big reward is out of reach, people stop cooperating and go back to hunting hares.
4. The population stays small, and they never get the chance to try for the big prize.
5. Key Insight: It's not that these people are "less moral" or "less smart." They are just stuck on a lower rung because they didn't have the initial push to get to the next level.

What About Technology and Resources?

The paper adds two more ingredients to the mix:

• Technology (The Magic Ladder): Technology is like a magic booster that makes the ladder easier to climb. If you invent a better spear or a tractor, you need fewer people to catch the same amount of food. This helps societies escape the "Trap."
• Resource Limits (The Slipping Ladder): Imagine the ladder is made of ice. As more people climb, the ice gets thinner. If resources (like oil or fish) run out, the reward gets smaller. Now, you need even more people to cooperate just to get the same amount of food. This forces societies to keep climbing or risk falling.

Why Does This Matter Today?

The authors suggest this model helps us understand modern problems like Climate Change.

• The Problem: We are currently on a high rung of the ladder (industrial society). But our "food" (fossil fuels) is running out or getting harder to get.
• The Danger: If we don't find a new, massive reward (like cheap fusion energy or a breakthrough in renewables) that requires global cooperation, we might hit a "stall."
• The Warning: If the next rung looks too far away, or if the cost of cooperation is too high, countries might stop cooperating and start fighting over the remaining scraps.

The Takeaway

Human history isn't just about us getting "nicer" over time. It's a game of critical mass.

• We need enough people and enough motivation to reach the next "big prize."
• Once we reach it, the prize supports a bigger population, which helps us reach the next one.
• But if we lose our momentum, or if the next prize seems too far away, we can slide back down the ladder.

To solve big global problems, we don't just need to ask people to "be good." We need to make sure the next rung of the ladder is visible, reachable, and worth the climb.

Technical Summary

1. Problem Statement

The paper addresses a central puzzle in evolutionary biology and social science: How do human societies transition from small-scale cooperation (based on kinship and reciprocity) to large-scale cooperation (involving millions of anonymous strangers)?

While mechanisms sustaining cooperation at specific scales are well-understood, the transition between scales remains unclear. The authors note that historical evidence shows a dramatic expansion in cooperative scale (e.g., from hunter-gatherers to nation-states), yet this trajectory is not universal; some societies stagnate or regress. The core question is what drives these expansions and why some societies get "stuck" at smaller scales despite the availability of high-value resources.

2. Methodology

The authors develop a theoretical model extending Evolutionary Game Theory and Cultural Evolution frameworks.

• Core Framework: The model generalizes the classic Stag Hunt game (a coordination game) into an N-person Stag Hunt with multiple, escalating thresholds.
  • Multiple Rewards ("Stags"): Instead of a single high-value reward, the model posits a "ladder" of rewards. Each rung represents a resource or energy source (e.g., firewood $\to$ agriculture $\to$ fossil fuels) that offers exponentially higher payoffs but requires a larger threshold of cooperators ($k$) to unlock.
  • Endogenous Population Dynamics: Unlike static models, population size ($N$) is dynamic. It follows logistic growth, where the carrying capacity ($K$) is determined by the average utility (payoff) of the group. Successful cooperation increases resource availability, raising $K$ and allowing population growth.
  • Replicator Dynamics: Individuals adopt strategies (cooperate vs. defect) based on relative fitness. If cooperation yields a higher expected payoff than defection, the cooperation rate ($r$) increases.
  • Modulating Factors:
    • Technology: Modeled as a factor that reduces the number of cooperators required to reach a threshold as population grows.
    • Resource Decay: Modeled as a factor where per-capita returns diminish as population increases (simulating resource scarcity/depletion), effectively lowering the carrying capacity unless a new, higher threshold is reached.

3. Key Contributions

The paper introduces the concept of the "Cooperation Ladder," a mechanistic framework explaining the non-linear, path-dependent nature of social scaling.

• Scale-Dependent Payoffs: It formalizes the idea that larger rewards (like industrial energy) require larger cooperative thresholds, creating discrete "steps" in social complexity.
• Coupled Dynamics: It explicitly couples cooperation rates with population growth, showing how they form a feedback loop: cooperation $\to$ higher resources $\to$ population growth $\to$ ability to reach higher thresholds $\to$ more cooperation.
• Path Dependence: The model demonstrates that historical outcomes are highly sensitive to initial conditions (initial cooperation rates and population size), leading to divergent trajectories (high-cooperation growth vs. low-cooperation stagnation).

4. Key Results

The model yields six primary insights regarding the dynamics of cooperation:

1. Surges at Thresholds: Cooperation rates spike dramatically when a group is just one or a few cooperators away from reaching a new threshold. The marginal incentive to cooperate is highest at these "tipping points."
2. Stalls and Reversals Between Thresholds: When the next reward is far out of reach, cooperation tends to stagnate or decline. Individuals revert to defection (free-riding) because their individual effort cannot bridge the gap to the next reward. Free-riding is tolerated and stable at intermediate levels.
3. Positive Feedback Loops: Successful threshold crossings trigger population booms (due to increased carrying capacity). This larger population provides more potential cooperators, making the next threshold attainable, creating a "ratcheting" effect of growth.
4. Suboptimal Equilibria (Traps): Societies can become trapped in stable, low-cooperation equilibria. If a group starts with low cooperation or a small population, it may never reach the first critical threshold, even if high-value resources are physically present.
5. Role of Technology: Technology lowers the effective threshold (fewer people needed to succeed). While this helps groups escape traps, highly efficient technology can paradoxically lower the equilibrium proportion of cooperators needed, as tasks become easier to complete with fewer contributors.
6. Role of Resource Limits: Resource decay (scarcity) increases the pressure to cooperate. As per-capita returns drop, groups must innovate to reach higher thresholds just to maintain current living standards, preventing unbounded growth but driving continuous cooperative innovation.

Visual Dynamics:

• Figure 4 & 5 illustrate path dependence: Two societies with identical parameters but different initial cooperation rates diverge. One climbs the ladder (high growth), while the other stalls or regresses (low growth).
• Figure 3 shows that without resource decay, the model predicts unbounded population growth; with decay, growth is capped, forcing the need for new thresholds.

5. Significance and Implications

• Historical Explanation: The model provides a materialist explanation for historical transitions (e.g., Agricultural and Industrial Revolutions). These were not just cultural shifts but necessary steps to unlock exponentially higher energy returns (EROI) that supported larger populations.
• Explaining Inequality: It suggests that the lack of large-scale cooperation in some societies is not due to a lack of "moral" development but rather initial conditions and material constraints that prevented them from crossing the first critical thresholds.
• Modern Challenges (Climate Change): The framework offers a cautionary lens for global coordination. If the current global society fails to find new high-EROI energy sources (a new "stag"), the model predicts a plateau or reversal in cooperation. Without a material payoff to justify the cost of cooperation, nations may revert to short-term self-interest (defection).
• Policy Insight: To sustain global cooperation, incentives must be aligned such that the next threshold of collective benefit is "within reach" and offers a tangible, high payoff. Moral appeals alone are insufficient without the underlying material reality of increased returns.

In summary, the paper argues that large-scale cooperation is a self-reinforcing cycle driven by the interplay of population size, resource thresholds, and technology. It succeeds when groups can cross critical thresholds to unlock new resources, but it is fragile and prone to stalling if those thresholds become too distant or if initial conditions are unfavorable.