The biased interaction game: Its dynamics and application in modelling social systems

This paper presents the biased interaction game as a versatile modeling tool for social systems, demonstrating how boundedly rational interactions under scarcity and bias naturally generate emergent hierarchies, inequality, and non-linear stability patterns while offering a framework to evaluate competing wealth redistribution philosophies like social welfare and universal basic income.

The Gist

Imagine a giant, invisible playground where everyone is playing a game to get resources (like food, money, or power). This paper introduces a new way of understanding how that game works, called the "Biased Interaction Game."

Here is the simple breakdown of what the authors discovered, using everyday analogies.

1. The Core Idea: The "Weighted" Playground

In most games, everyone starts with the same rules. But in the real world, the rules aren't fair. If you are rich and powerful, you have an advantage. If you are poor, you have a disadvantage.

The authors created a computer simulation to test this. They gave every "player" (agent) two things:

• Incumbent Value: Their current "weight" or status (wealth, power, reputation).
• Scarcity: How hard it is to get resources. Is there plenty of food, or is everyone starving?

The Big Discovery: When resources are scarce, the game naturally creates a hierarchy (a ladder of rich and poor). It's not because someone planned it; it just happens because the "heavier" players (the rich) tend to win more often when things are tight.

2. The Two Ways to Play

Every player has to choose one of two strategies when they meet someone else:

• Cultivate (The Hustler): You actively work, fight, or compete to get resources.
• Utilise (The Leech): You sit back and try to take what the environment or others provide without doing much work.

The paper tested what happens if everyone picks the same strategy:

• Scenario A: The Utopia (Everyone is Equal)
  If everyone starts with exactly the same amount of money and plays fair, they stay equal forever. It's a stable, boring, but happy world.
• Scenario B: Hyper-Capitalism (Everyone Hustles)
  If everyone tries to "Cultivate" (fight for everything), chaos ensues. A tiny few people end up with almost everything, and the vast majority end up with nothing. It's like a gladiator arena where only two people survive.
• Scenario C: Social Egalitarianism (Everyone Leeches)
  If everyone just sits back and "Utilises" (takes without giving), the system collapses into total equality. Everyone ends up with the same tiny amount. It's like a potluck where everyone eats the same meager leftovers.

The Lesson: Real life is a mix. We need a balance between hustling and resting to keep society stable.

3. The "Earthquake" Effect (Social Mobility)

You might think that once you are born poor or rich, you stay there. But the simulation showed something surprising: Social mobility is real, but it's weird.

Imagine a ladder with rungs (bands of wealth). Most people stay on their rung for a long time. But occasionally, the system has a "glitch" or an "earthquake."

• A rich person might have a string of bad luck and fall down a rung.
• A poor person might have a string of lucky breaks and jump up a rung.

Sometimes, these jumps happen in a cascade. One person falls, which knocks the person above them down, who knocks the next one down, and suddenly, a whole group of people swap places in a short burst. The authors call this a "Mobility Cascade." It explains why societies can look stable for years, then suddenly shake up and change.

4. The Tax Experiment: Welfare vs. Universal Basic Income

The authors used their game to test two famous ways governments try to fix inequality: Social Welfare and Universal Basic Income (UBI).

• Social Welfare (Helping only the poor):
  • How it works: The government taxes everyone and gives money only to the poorest people.
  • The Result: It helps the poor, but it creates a mess in the middle. The "middle class" gets squeezed because they pay taxes but don't get the help. In the simulation, this blurred the lines between the poor and the middle class, making it hard to tell who actually needs help. It's like trying to fix a leaky boat by only patching the bottom hole while the middle is still flooding.
• Universal Basic Income (Helping everyone):
  • How it works: The government taxes everyone and gives a small check to every single person, rich or poor.
  • The Result: This kept the social ladder (the rich are still richer than the poor) but made the rungs closer together. It reduced inequality without breaking the structure of society. However, it requires higher taxes to work.

The Bottom Line

This paper suggests that inequality isn't just a mistake; it's a natural result of how we interact when resources are scarce. However, the system is alive and moving.

• Stability: Hierarchies (rich/poor classes) naturally form.
• Movement: People can and do move up and down, sometimes in sudden waves.
• Policy: If you want to fix inequality, how you do it matters. Giving money only to the poor might break the middle class, while giving money to everyone (even the rich) keeps the system stable but costs more.

In short: Society is like a complex, self-organizing dance. Sometimes the dancers move in perfect lines, and sometimes they trip and swap partners in a chaotic burst. Understanding the music (the rules of the game) helps us understand why the dance looks the way it does.

Technical Summary

Here is a detailed technical summary of the paper "The biased interaction game: Its dynamics and application in modelling social systems" by Phil Mercy and Martin Neil.

1. Problem Statement

The paper addresses the challenge of modeling complex social systems where hierarchy, inequality, and social mobility emerge from the interactions of boundedly rational agents under conditions of scarcity. Traditional game theory often assumes perfect rationality or static environments, while standard agent-based models may require complex rules to generate emergent hierarchy.

The authors seek to answer:

• How do bias (derived from incumbent value/wealth) and environmental scarcity interact to shape social structures?
• Can a simple set of interaction rules naturally produce emergent hierarchies, inequality, and social mobility without explicit programming of these outcomes?
• How can this framework be applied to evaluate competing real-world economic policies, specifically wealth redistribution strategies?

2. Methodology

The authors utilize the Biased Interaction Game (BIG), a novel game-theoretic framework introduced in a prior publication (Mercy & Neil, 2025). The methodology involves Agent-Based Modeling (ABM) using the Python Mesa library.

Core Mechanics of the Game:

• Agents: A population of $N$ agents (typically 50) with an Incumbent Value ($M$), representing accumulated wealth, power, or influence.
• Environment: Defined by a Scarcity parameter ($S$, where $0 \le S \le 1$).
• Strategies: Agents choose between two strategies during interactions:
  • Cultivate (Active): Engaging actively with the environment.
  • Utilise (Passive): Relying on the environment or the other agent.
• Interaction Dynamics:
  • Agents interact in pairs. The "pot" of value is distributed based on relative incumbent values and the scarcity level.
  • Bias Mechanism: The payout equation (e.g., for a passive-active interaction) favors the agent with lower incumbent value in low-scarcity environments but shifts advantage to the agent with higher incumbent value (or the active cultivator) as scarcity increases.
  • Equilibrium: Agents iteratively adjust their strategies to maximize benefit, leading to an equilibrium where the payout matches the contribution.
• Experimental Design:
  • Edge-Case Scenarios: Experiments testing "Utopian Equality" (equal start), "Hyper-Capitalism" (all cultivate), and "Social Egalitarianism" (all utilise).
  • Social Mobility: Long-term simulations (50,000–200,000 iterations) tracking agent movement between hierarchical bands.
  • Policy Simulation: A taxation and redistribution model comparing Social Welfare (redistributing only to the poorest band) vs. Universal Basic Income (UBI) (redistributing equally to all).
• Metrics:
  • Gini Coefficient: To measure wealth inequality.
  • Hierarchical Bands: Agents stratify into bands based on similar incumbent values.
  • Transition Density: The frequency of agents moving between bands over time.

3. Key Contributions

1. Emergent Hierarchy from Minimal Rules: The paper demonstrates that complex social stratification (hierarchical bands) emerges naturally from simple rules of bounded rationality and scarcity, without pre-defined social classes.
2. Non-Linear Dynamics and "Mobility Cascades": The authors identify that the system exhibits non-linear behavior. While the overall hierarchical structure is robust, individual agents experience "mobility cascades"—periods of rapid, chaotic transition between bands triggered by rare sequences of favorable/unfavorable interactions.
3. Stability of Edge Cases:
   • Absolute Equality: If agents start with equal value, the system remains egalitarian even with scarcity.
   • Strategy Constraints: If all agents are forced to "Cultivate," the system collapses into an oligarchy (extreme inequality). If all "Utilise," the system collapses into equality.
4. Policy Modeling Framework: The paper provides a novel simulation framework to visualize the systemic impacts of wealth redistribution, revealing unintended consequences of specific tax policies on social structure stability.

4. Key Results

A. Edge-Case Scenarios:

• Utopian Equality (Equal Start): The system remains stable with a Gini coefficient of 0.00.
• Hyper-Capitalism (All Cultivate): Results in extreme inequality (Gini $\approx$ 1.0). Most agents end with near-zero value, while a tiny fraction (1–2 agents) capture the majority of wealth.
• Social Egalitarianism (All Utilise): Results in total equality (Gini = 0.00) regardless of starting conditions, as the passive strategy erodes differences.

B. Social Mobility and Non-Linearity:

• Mobility is Frequent: In long-term runs, agents transition between hierarchical bands frequently (average of 23.2 transitions per agent in 50k iterations).
• Mobility Cascades: Transitions are not uniform. The system exhibits "abrupt non-continuous change," where long periods of stability are punctuated by short bursts where a large proportion of agents shift bands simultaneously. This mimics real-world social mobility patterns.
• Structural Resilience: Despite individual mobility, the overall hierarchical structure (the existence of bands and their relative ratios) remains robust and repeatable.

C. Wealth Redistribution (Social Welfare vs. UBI):

• Social Welfare Model:
  • Redistributes tax only to the lowest band.
  • Result: Even low tax rates (1–5%) blur the distinction between the lowest and middle bands. High tax rates (50%) cause the middle band to collapse entirely, creating a large underclass and a dominant oligarchy.
  • Implication: While it reduces the Gini coefficient, it destabilizes the social hierarchy and creates bureaucratic inefficiencies due to the difficulty of defining eligibility in a fluid system.
• Universal Basic Income (UBI) Model:
  • Redistributes tax equally to all agents.
  • Result: Reduces inequality (lowers Gini) while preserving the hierarchical structure. The distance between bands shrinks, but the bands themselves remain distinct.
  • Implication: UBI maintains social stability and structure but requires higher tax rates to achieve significant inequality reduction. At extreme tax rates (50%), the hierarchy collapses into total equality.

5. Significance

• Theoretical Insight: The paper challenges the notion that social mobility requires complex evolutionary mechanisms. It suggests that mobility is an inherent, non-linear property of systems driven by scarcity and biased interactions.
• Policy Relevance: The simulation offers a counter-intuitive insight: Social Welfare policies, while well-intentioned, may inadvertently destabilize the middle class and create a rigid two-tier society (oligarchy vs. underclass) in the long run. Conversely, UBI, often criticized for being inefficient, appears to reduce inequality while maintaining the natural social hierarchy, provided tax rates are moderate.
• Modeling Paradigm: The Biased Interaction Game provides a new tool for social scientists to model systems where "power" and "scarcity" are dynamic variables, offering a bridge between game theory, complexity science, and sociology.

In conclusion, the paper establishes that the Biased Interaction Game is a powerful, parsimonious model capable of reproducing complex social phenomena—hierarchy, inequality, and mobility—and serves as a valuable sandbox for stress-testing economic policies before real-world implementation.