Catalysing cooperation: the power of collective beliefs in structured populations

The study investigates whether and how collective, non-moralising beliefs can catalyse the emergence of cooperation (trust) in populations where cooperation is individually risky but collectively beneficial. Using the stag hunt (trust dilemma), the authors focus on coordination-dependent cooperation, where stag hunting is profitable only if enough partners also cooperate. They argue that real social interactions are structured (not well-mixed), and that beliefs and actions may propagate on different timescales. The research question is how population structure (network topology), heterogeneous group sizes, and distinct propagation rates of beliefs versus actions shape the evolution, speed, and robustness of trust. The work is motivated by evidence that networks often promote cooperation in other dilemmas, and by observations that social norms and shared narratives can coordinate behaviour even without explicit moral content (e.g., money as a collectively believed coordination device). The purpose is to quantify how network properties (degree heterogeneity, clustering, diameter) and belief dynamics alter takeover times of cooperative strategies and the size of social tipping points required for population-wide change.

Prior work shows that spatial and network structures can sustain cooperation in social dilemmas (e.g., prisoner’s dilemma, snowdrift), but the stag hunt—centred on coordination and trust—has been less explored on networks. Existing studies indicate that structure can promote coordination in stag hunt dynamics on random networks. Research in cultural evolution and social norms highlights how beliefs, narratives, and tags can bias decisions and spread via social processes, enabling coordination without direct payoff changes. Social diversity and network properties like degree distribution and clustering have been linked to cooperative outcomes. The role of hubs in diffusion is nuanced: they can accelerate spread or act as firewalls, and targeting mid-degree nodes can sometimes be more effective. Literature on critical mass and tipping points suggests a committed minority (10–40%) can induce social change, with thresholds depending on network structure. Building on these strands, the present work examines non-moralising collective beliefs as coordination devices in structured populations and quantifies their effects across ER, BA, and NWS networks.

Game and strategies: The trust dilemma is modelled via an N-player stag hunt. A hare yields payoff PH (risk-free). A stag hunt succeeds only if at least M stag hunters participate; success yields payoff PS for stag hunters, otherwise 0. Parameters used in the main simulations: PH = 1, PS = 4, M = 4. The two-player case is extended to N-player groups while preserving that successful cooperation is more rewarding than defection. Collective narratives: Following Gokhale et al., each individual holds a personal belief (narrative 1 or 2). Groups select a narrative via a frequency-dependent mechanism (probability proportional to the fraction of believers in the group). Each individual conditions actions on the group’s chosen narrative, yielding 8 possible strategies (action under narrative 1, action under narrative 2, personal belief). Beliefs do not directly affect payoffs; they act as coordination devices. Population structure: Three random network classes represent structured populations of size Z = 32 with comparable average degree (~7.375) and edges (~118): (i) Erdős–Rényi (ER) with ne = 118 edges; (ii) Newman–Watts–Strogatz (NWS) small-world with initial lattice dimension d = 6 and edge-addition probability p = 0.24 (high clustering, small path length); (iii) Barabási–Albert (BA) scale-free with minimal connectivity m = 4 (preferential attachment, power-law degree). Degree distributions and key metrics differ: average global clustering ≈ 0.23 (ER), 0.44 (NWS), 0.31 (BA); average diameter ≈ 3.22 (ER), 3.61 (NWS), 3.02 (BA). Mutation processes (distinct timescales): Two uncorrelated mutation rates are used: μA for actions and μg for beliefs. Action mutations occur upon reproduction (at most once per lifetime), drawing new actions from {HH, SH, HS, SS} while inheriting the parent’s belief. Belief mutations occur independently of reproduction—at each time step with probability μg the belief of a randomly chosen individual is reset to one of the two beliefs with equal probability. The ratio μg/μA proxies the relative speed of belief versus action propagation. Targeted belief mutations: To study targeted introduction of new beliefs, the node i for belief mutation can be chosen with probability Pi ∝ e^{α k_i}, where k_i is node degree. α > 0 targets hubs; α = 0 corresponds to random placement; α < 0 targets periphery (less-connected nodes). Evolutionary dynamics: A synchronous birth–death Moran process operates over generations of Z time steps. Each time step: (1) belief mutation with probability μg (as above); (2) payoffs computed from one hunting interaction with all neighbours; fitness ψ_i = 1 + ω Π_i with selection intensity ω (main text uses ω = 1); (3) one individual reproduces with probability proportional to fitness; its offspring replaces a randomly chosen neighbour (or itself) and undergoes action mutation with probability μA; belief is inherited unchanged. Simulation design: Initial population is egoistic, all (H,H,1). Networks: ER (ne = 118), NWS (d = 6, p = 0.24), BA (m = 4). Also a well-mixed comparison where individuals play with 8 random partners; reproduction–death occurs between any two random individuals. Main parameters: Z = 32, μA = 0.001, PH = 1, PS = 4, M = 4, ω = 1. For each network type and parameter set: 10 network realizations × 100 runs each = 1000 runs. Maximum duration per simulation: 5 × 10^6 generations. Group-size heterogeneity and selection intensity variations, as well as network size sensitivity (Z from 32 to 252), are explored in Supplementary Material. Outcome metrics: Because mutations ensure eventual appearance of hare hunters post-takeover, the main metric is takeover time: first generation at which all individuals are stag hunters. Additional analyses include relative takeover time under targeting (α) and estimation of social tipping points: the minimum fraction of stag hunters after first crossing that threshold such that the fraction never falls below it thereafter. Averages and standard deviations are reported across 1000 runs per condition.

• Structured populations accelerate cooperation: With one belief (μg = 0), the average stag-hunter takeover time in a well-mixed population is 7.2× longer than on NWS and 3.4× longer than on ER and BA networks.
• Belief diversity speeds cooperation: Increasing μg from 0 to 0.01 reduces takeover time by ~65% on ER and BA and ~50% on NWS. Introducing a second belief (μg > 0) consistently accelerates takeover across all networks.
• Network topology matters: NWS networks (higher clustering coefficient ≈ 0.44) yield faster takeovers than ER (≈ 0.23) and BA (≈ 0.31) for all parameter sets. Lower diameter alone (≈ 3.02–3.22 for BA/ER vs 3.61 for NWS) is less predictive of faster cooperation than high clustering.
• Targeted belief mutation effects depend on topology and degree heterogeneity: In BA and ER networks, targeting hubs (α > 0) slows takeover; targeting periphery (α < 0) generally decreases takeover time or matches random placement (α = 0). In NWS (more homogeneous degree), targeting has negligible or small effects; for low μg, most non-zero α slightly lowers takeover time, but effects dissipate as μg increases.
• Mechanism for periphery advantage: In scale-free BA networks, random placement already tends to select periphery nodes (they are numerous). Explicitly targeting periphery can thus be as good as or better than random, while targeting hubs can impede spread (acting as firewalls).
• Social tipping points: In the absence of strong competing mutations, a committed minority of ~15–30% can trigger population-wide cooperation. For small μg and α < 0, average tipping points are ≈ 19% (NWS), 25% (ER), 28% (BA). Higher μg increases both mean tipping point and its standard deviation, as competing mutants are more likely to appear before takeover, hindering or reversing cooperative growth.
• Overall, high clustering and belief diversity most strongly catalyse coordination-based cooperation; degree heterogeneity shifts optimal targeting toward less-connected nodes rather than hubs.

The findings show that collective, non-moralising beliefs can coordinate actions and catalyse the transition from individually safe but socially suboptimal behaviours (hare hunting) to cooperative, high-payoff behaviours (stag hunting), particularly in structured populations. High clustering facilitates local reinforcement and rapid consolidation of cooperative clusters, thereby shortening takeover times relative to well-mixed populations and lower-clustering networks. Belief diversity (higher μg) enhances the availability of narratives that enable coordination, thereby lowering the effective barrier to stag-hunter dominance. Targeted introduction of beliefs reveals that hubs are not universally optimal seeds for behavioural change. In degree-heterogeneous networks (BA, ER), targeting hubs can slow the spread, consistent with the notion of hubs as potential firewalls. In contrast, targeting the periphery—where most individuals reside—can accelerate coordination, highlighting the power of commoners and suggesting grassroots-focused strategies for social change. The tipping point analysis demonstrates that structured populations with high clustering require a smaller committed minority to achieve irreversible cooperative adoption, whereas lower clustering and higher mutation rates raise the threshold, reflecting greater vulnerability to competing innovations or beliefs. These outcomes directly address the research question by quantifying how network topology and the relative timescales of belief versus action changes modulate the speed and robustness of cooperation. The work underscores the strategic importance of leveraging network structure—particularly clustering—and designing belief dissemination to align with degree heterogeneity to promote trust and cooperative conventions.

This study demonstrates that non-moralising collective beliefs can act as powerful coordination devices that catalyse cooperation in structured populations. Compared with well-mixed settings, random network structures—especially those with higher clustering—significantly reduce the time to cooperative takeover. Increasing belief diversity further accelerates trust formation. Contrary to common intuition, seeding beliefs in hubs can hinder spread in degree-heterogeneous networks; targeting periphery nodes is more effective. Social tipping points depend on topology and mutation rates, with high clustering supporting lower thresholds (~19%) and higher belief mutation raising both the mean threshold and variability. The main contributions include: (i) extending stag hunt with narratives to structured populations and heterogeneous group sizes; (ii) identifying clustering as a key accelerator of coordination; (iii) revealing that periphery targeting can outperform hub targeting; and (iv) quantifying topology-dependent tipping points. Future research should examine alternative consensus mechanisms (e.g., majority vote, charismatic leadership, conformity), the interplay of beliefs with different evolutionary update rules, larger and empirical network structures, varying selection intensities, and richer cultural dynamics beyond two narratives.

• Network realism: The study uses synthetic random networks (ER, NWS, BA) to isolate structural effects; real social networks may differ (e.g., community structure, assortativity, temporal dynamics).
• Population size and parameters: Main results use small populations (Z = 32) with fixed payoffs (PH = 1, PS = 4), threshold (M = 4), and selection intensity (ω = 1). Although sensitivity analyses are provided in Supplementary Material, generalizability to larger or differently parameterized systems may vary.
• Belief processes: Only two narratives are considered, and beliefs have no direct payoff effects; the group narrative arises via a frequency-dependent (groupthink) rule. Other consensus mechanisms or belief-payoff couplings could change dynamics.
• Mutation modelling: Belief mutations are exogenous and uncorrelated with reproduction; targeted placement via degree-based probabilities abstracts away from richer socio-cognitive determinants of belief change and diffusion.
• Stochasticity and competing mutants: High mutation rates increase stochastic interference between strategies, affecting takeover and tipping point estimates; results are averages over finite runs and small networks.
• Occasional small hunting groups in ER networks (due to degree variability) did not alter averages but represent a constraint of the model’s one-shot neighbourhood interaction assumption.