📈 economics

Public Goods Game on Complex Networks: the interplay between conformity and topology

This study investigates how the interplay between individual conformity sensitivity and network topology influences cooperation in the Public Goods Game, revealing that while conformity promotes cooperation on regular networks, it can have null or negative effects on heterogeneous complex networks.

Original authors: Ren Manfredi, Eugenio Vicario, Ennio Bilancini, Rossana Mastrandrea

Published 2026-07-28

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ren Manfredi, Eugenio Vicario, Ennio Bilancini, Rossana Mastrandrea

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 a world where your next move depends entirely on two things: how much money you just made, and what your friends are doing. This is the playground of Evolutionary Game Theory, a branch of science that studies how people (and even animals) learn to cooperate or compete over time. Instead of assuming everyone is a cold, calculating robot, this field asks: How do we actually behave when we are tired, confused, or just want to fit in?

At the heart of this story is a classic puzzle called the Public Goods Game. Picture a group of neighbors deciding whether to chip in money for a community garden. If everyone contributes, the garden blooms and everyone enjoys it. But if you don't pay and just enjoy the flowers, you save your money while still getting the view. This is the "tragedy of the commons": if everyone thinks like that, the garden dies, and no one wins. Scientists have long known that in a chaotic crowd, selfishness usually wins. But what if we arrange these neighbors in specific patterns, like a grid or a web? And what if, instead of just looking at their bank accounts, people also look at their neighbors and think, "Hey, everyone else is doing it, so I should too"? This paper dives into that exact mix: the tug-of-war between the math of profit and the human urge to conform.

The Experiment: A Digital Town Square

The authors set up a massive digital simulation with 1,000 agents (think of them as digital citizens) living on three different types of "towns" or networks. They played a game where they could either Cooperate (pay a cost to help the group) or Defect (cheat and keep their money). The goal was to see who survived and thrived as the game progressed.

But here is the twist: these digital citizens weren't all the same. Each one had a unique personality trait called $\alpha$ (alpha), which determined how much they cared about money versus peer pressure.

Some agents cared mostly about money, but still paid some attention to their neighbors.
Some cared mostly about fitting in, but still checked their bank accounts.
Most were a messy mix of both, constantly weighing their wallet against their social circle.
Crucially, no agent was purely one or the other; every single agent used a combination of both drivers to make decisions.

The researchers tested these agents in three very different neighborhood layouts:

The Circular Lattice: A perfect, orderly grid where everyone has exactly the same number of neighbors (like a chessboard).
The Small-World: A neighborhood that looks mostly orderly but has a few random shortcuts connecting distant houses (like a small town with a few surprise highways).
The Scale-Free Network: A chaotic city where a few super-popular "Hubs" (influencers) have hundreds of friends, while most people only have a few (like a social media platform).

The Findings: When Fitting In Helps (and Hurts)

The results were a rollercoaster, showing that conformity doesn't always help cooperation; it depends entirely on the shape of the neighborhood.

1. The Orderly Grid: Conformity is a Superpower

In the perfect, circular grid, conformity acted like a glue.

Without conformity: Selfish "defectors" could hide in the cracks. Even if they were surrounded by cooperators, they could make more money and refuse to change their ways. They were like uninvited guests who ate all the cake and refused to leave.
With conformity: The glue kicked in. Even if a defector made a little more money, the pressure of seeing everyone else cooperating compromised their immunity to the cooperative strategy. The "social norm" made it harder for them to resist joining the team, though it didn't guarantee an immediate change.
The Result: In this orderly world, conformity helped the group win. It turned the town into a united front of cooperators much faster than money alone ever could, though the paper notes that a "hard core" of defectors can still persist, meaning a total invasion by cooperators is not always guaranteed.

2. The Small-World Town: A Quiet Neutral

In the "small-world" network, where a few random shortcuts connected the town, conformity didn't change much.

The shortcuts were enough to let selfish defectors sneak in and disrupt the cooperation clusters.
Whether the agents were obsessed with money or obsessed with fitting in, the outcome was almost the same. The network structure was just too "leaky" for conformity to fix the problem. It was like trying to hold water in a bucket with a few holes; no matter how much you squeeze the sides (conformity), the water still leaks out.

3. The Scale-Free City: Conformity is a Double-Edged Sword

This was the most surprising and dramatic result. In the city with "Hubs" (the super-connected influencers), conformity had a complex, often negative, effect on cooperation.

The Old Way (Money only): The Hubs were the heroes. Because they played so many games, if a Hub was a cooperator, they made a fortune and stayed a cooperator. They acted as a fortress, protecting the smaller neighbors from the selfish defectors.
The New Way (With Conformity): The Hubs became vulnerable. Imagine a popular influencer who is a generous cooperator. If their hundreds of followers (who are mostly selfish defectors) start acting a certain way, the Hub feels the pressure to "fit in" and stops cooperating.
The Result: The very people who were supposed to lead the charge for cooperation got bullied by the crowd. The Hubs became strategically unstable, and because they were so connected, they dragged the whole city down with them. However, there was a twist: in very bad conditions where cooperation usually fails (low rewards), this same conformity actually helped cooperation survive by forcing agents to stick together despite the lack of profit. But when the rewards were high, the urge to conform made the system less cooperative overall, with cooperation settling at around 50% on average rather than taking over completely.

The Big Picture

The paper concludes that one size does not fit all.

In simple, orderly groups, following the crowd is a great way to build a cooperative society. It stops the selfish few from hiding, though it may not eliminate them entirely.
In complex, "influencer-heavy" groups, following the crowd can be dangerous. It can make the leaders (the Hubs) weak, causing them to abandon cooperation just because their followers are doing it—unless the situation is so dire that sticking together is the only option left.

The authors ran these simulations with 10,000,000 generations of play to be sure, averaging the results over hundreds of different starting scenarios. They didn't just guess; they watched the digital towns evolve. They found that while we often think "fitting in" is good for society, in the wrong kind of network, it can actually break the system. It's a reminder that what works in a small, tight-knit village might be a disaster in a sprawling, chaotic city.

Technical Summary: Public Goods Game on Complex Networks: The Interplay Between Conformity and Topology

Problem Statement
Human cooperation is a persistent social dilemma, often modeled by the Public Goods Game (PGG), where individual incentives to defect conflict with collective benefits. While Evolutionary Game Theory (EGT) has successfully explained cooperation through mechanisms like network reciprocity and payoff-based imitation, these models often rely on the assumption that agents act solely to maximize utility. This paper addresses the gap in understanding how conformity—the tendency to align with the perceived majority—interacts with network topology to shape cooperative dynamics. Specifically, the authors investigate whether conformity acts as a catalyst for cooperation or a hindrance when agents possess heterogeneous sensitivities to payoffs versus social norms across different network structures.

Methodology
The study employs an Agent-Based Model (ABM) of the spatial Public Goods Game on a population of $N=1000$ agents. The core innovation lies in the update rule and the population structure:

Heterogeneous Agents: Unlike previous studies that often divide populations into distinct "payoff-driven" and "conformist" subgroups, this model assigns every agent a unique, fixed sensitivity parameter $\alpha_i \in (0, 1)$ . This parameter, drawn from a Beta distribution, determines the weight an agent places on payoff comparison versus the behavioral norm of their neighborhood.
- $\alpha_i$ : Weight on payoff-based imitation.
- $1 - \alpha_i$ : Weight on conformity-based imitation.
- Three population scenarios were tested: Uniform distribution (Beta(1,1)), conformity-skewed (Beta(1,10)), and payoff-skewed (Beta(10,1)).
Update Mechanism: The probability of an agent $i$ adopting a neighbor's strategy $j$ is a linear combination of two components:
- Payoff-driven: Imitation occurs only if $\pi_j > \pi_i$ , scaled by the payoff difference.
- Conformity-driven: Imitation probability depends on the deviation of the neighbor's strategy from the local social norm ( $SN_i$ ), defined as the fraction of defectors in the local neighborhood.
- The final probability is $P(i \to j) = \alpha_i P_{\pi} + (1-\alpha_i) P_{SN}$ .
Network Topologies: Simulations were run on three distinct topologies with an average degree $k=4$ :
- Regular Circular Lattice: Homogeneous, high clustering, one-dimensional.
- Small-World Network: Generated via the Watts-Strogatz model ( $p=0.05$ ), retaining homogeneity but with shorter path lengths.
- Scale-Free Network: Generated via the Barabási-Albert model ( $m=2$ ), characterized by heterogeneous degree distribution and the presence of hubs.
Simulation Parameters: The enhancement factor $r$ (synergy) varied from 1 to 5. Simulations ran for up to $10^7$ generations, averaged over 300 initializations, with early termination if the system reached a steady state or total dominance by one strategy.

Key Contributions

Unified Update Rule: The paper proposes a model where conformity complements, rather than replaces, payoff-based imitation, allowing for a continuous spectrum of behavioral biases within a single agent.
True Heterogeneity: By assigning individual $\alpha$ values via a distribution rather than using global parameters or discrete agent types, the model captures a more realistic diversity in decision-making heuristics.
Topological Interplay: The study systematically extends the analysis of conformity from regular lattices (common in prior literature) to complex networks (small-world and scale-free), revealing that the effect of conformity is not universal but highly dependent on network structure.

Results
The impact of conformity varies drastically across the three topologies:

Regular Circular Lattice: Conformity acts as a catalyst for cooperation. In standard payoff-driven models, isolated defectors often persist due to payoff advantages. Conformity disrupts this "immunity," causing isolated defectors to adopt the majority cooperative strategy. For $r \ge 3$ , the system converges to total cooperation more robustly. Conformity homogenizes the system, reinforcing the trend dictated by the payoff structure.
Small-World Networks: Conformity has a negligible to neutral effect. While the network structure already disadvantages cooperation compared to lattices (requiring higher $r$ for cooperation to emerge), the introduction of conformity does not significantly alter the final cooperation levels. It merely reinforces existing trends, occasionally inducing bistability (rare invasions by minorities) but failing to fundamentally shift the equilibrium.
Scale-Free Networks: Conformity has a detrimental effect on cooperation. In standard models, hubs (high-degree nodes) drive cooperation by accumulating high payoffs. However, conformity makes these hubs vulnerable to the strategies of their numerous low-degree neighbors. If the majority of a hub's neighbors defect, the hub is likely to conform and defect, even if its payoff would be higher by cooperating. This leads to strong bistability and a general reduction in cooperation levels, particularly for intermediate $r$ values. High conformity makes the system less responsive to increases in the enhancement factor $r$ .

Significance and Claims
The paper claims that the interplay between conformity and topology is non-trivial and context-dependent. The authors assert that:

Structural Sensitivity: Conformity does not have a uniform effect; it reinforces cooperation in regular networks by breaking the immunity of isolated defectors but undermines it in scale-free networks by destabilizing the influential hubs that usually sustain cooperation.
Population Composition: The composition of the population (the distribution of $\alpha$ ) interacts with topology. In heterogeneous networks, high conformity can promote cooperation in adverse conditions ( $r \le 2$ ) but suppress it in favorable ones ( $r \ge 3$ ), a phenomenon not observed in regular topologies.
Limitations and Future Work: The authors modestly note that their model assumes static networks and fixed agent characteristics. They suggest that future work should explore dynamic networks, multiplex structures, and agents capable of endogenously updating their sensitivity parameters, acknowledging that real-world social contexts are fluid and evolving.

In conclusion, the study demonstrates that while conformity can homogenize strategies and drive systems toward ordered phases, its net effect on cooperation is determined by the specific structural features of the interaction network, particularly the role of hubs in heterogeneous topologies.