Dynamics of collective cooperation under personalised strategy updates

The study addresses how cooperation can emerge and fixate in structured populations where interactions are governed by a network. Classical results show that in homogeneous networks cooperation is favoured when the benefit-to-cost ratio b/c exceeds the average degree k, while heterogeneous networks generally require higher thresholds, implying that heterogeneity hinders cooperation. Prior work typically assumes identical (often synchronous) strategy update rates for all individuals. However, real decision-making is asynchronous and heterogeneous due to differences in cognitive processing speeds and personality traits. The central research question is how personalised (non-identical) strategy update rates interact with structural heterogeneity to influence the fixation of cooperation. The purpose is to relax the identical-rate assumption, derive precise conditions under which cooperation is favoured, and design update-rate policies that promote cooperation. The study is important because heterogeneous networks are ubiquitous in real systems, and understanding the role of behavioural rhythms in conjunction with structure can reconcile conflicting findings and guide the design of cooperative multi-agent systems.

Foundational works on cooperation in structured populations established simple rules for the emergence of cooperation on graphs and highlighted challenges in heterogeneous networks. Classical update rules include death-birth and imitation (pairwise comparison), usually with identical update rates. Empirical and theoretical studies indicate human decision times are heterogeneous due to cognitive and personality differences, and asynchronous updating can alter cooperation dynamics. Previous analyses (e.g., Allen et al.) show that the critical benefit-to-cost ratio C is higher for heterogeneous networks under identical rates, suggesting inhibition of cooperation. Other streams examined temporal networks and public goods games, with mixed findings on heterogeneity’s role. This paper builds on and extends these lines by allowing personalised update rates and by deriving new thresholds and optimisation strategies.

Model: An undirected, unweighted network of N players, each either a cooperator (C) or defector (D). Pairwise donation game on edges: cooperators pay cost c to give benefit b to each neighbour; defectors pay 0. Payoffs are averaged over neighbours. Strategy updating: Each individual i updates according to an independent Poisson process with personalised rate λ_i. When selected to update, i imitates a neighbour j with probability proportional to fitness F_j = 1 + δ f_j, where δ > 0 is the selection intensity (focus on weak selection). Fixation probability p_C is measured starting from a single cooperator placed uniformly at random among defectors; fixation of defection p_D analogously. Cooperation is favoured if p_C > p_D, equivalently p_C > 1/N under weak selection. Analytical framework: Using weak selection and reproductive-value-weighted frequency, derive a closed-form condition for the critical benefit-to-cost ratio C above which cooperation is favoured. The condition involves random-walk transition probabilities p_ij, n-step probabilities, and coalescence times τ_ij, solved via a system of linear equations for η_ij with recurrence relations. A mean-field/approximate expression for C is also derived to reduce computational complexity from O(N^6) to O(N^3). Mechanistic analysis: Pair-approximation reasoning introduces Q = (q_C|C − q_C|D)(k − 1), quantifying the excess number of cooperative neighbours for cooperators relative to defectors; compare cases λ_i = 1, λ_i ∝ 1/k_i, and λ_i ∝ k_i to understand local dispersal advantages. Numerical simulations: Monte Carlo simulations on synthetic networks (lattice, random regular, small-world, Erdős–Rényi, scale-free) and empirical social networks (office and high school contact networks). Typical parameters: N ≈ 98–100, average degree ⟨k⟩ = 6, δ = 0.01; up to 10^7 independent realisations to estimate p_C and C. Robustness checks span different N, ⟨k⟩, and δ. Long-term payoff analysis includes mutation at rate u and averages payoffs over long runs. Optimisation algorithm: OptUpRat uses RMSProp to optimise λ_i = exp(θ_i) to minimise C for a given network. The gradient is computed by differentiating the system for η_ij and solving N(N−1)/2 linear equations each iteration; parameters (learning rate ε, decay ρ, small constant ζ_opt) follow RMSProp practice.

• Personalised update rates profoundly affect cooperation fixation on heterogeneous networks. When λ_i varies inversely with degree (e.g., λ_i = 1/k_i or more generally λ_i ∝ 1/k_i^γ, γ > 0), heterogeneous networks, especially scale-free, can require a lower critical benefit-to-cost ratio C than homogeneous networks, reversing the conventional ordering observed under identical rates.
• Numerical simulations show that under identical rates (λ_i = 1), scale-free networks have the highest C and lattices the lowest; under inverse-degree rates (λ_i = 1/k_i), this ordering reverses, with scale-free becoming most conducive to cooperation and lattice least. This improvement can allow cooperation even when b/c < ⟨k⟩ (i.e., C < ⟨k⟩) in some cases.
• Strengthening the inverse relation (λ_i ∝ 1/k_i^γ with larger γ) further lowers C, while making λ_i positively correlated with degree (λ_i ∝ k_i) increases C and amplifies inhibition on heterogeneous networks.
• Mechanism: Infrequent hub updates create a lock-in effect that allows cooperative clusters to form and spread before hubs revert, increasing Q = (q_C|C − q_C|D)(k − 1) beyond 1; conversely, fast-updating hubs reduce Q below 1, raising C. Time-series on scale-free networks show hubs maintain cooperation longer and have higher q_C|C under λ_i = 1/k_i.
• Analytical results: A closed-form expression for C (via coalescing random walks) matches simulations. An O(N^3) approximation reproduces C accurately on empirical networks and explains why homogeneous networks are relatively insensitive to update-rate heterogeneity.
• Rule-of-thumb: For large heterogeneous networks, cooperation is promoted when for any pair (i, j), (k_i − k_j)(λ_i − λ_j) < 0 (i.e., higher-degree nodes have lower update rates). Identical rates recover C ≈ ⟨k⟩; inverse ordering yields C < ⟨k⟩.
• Long-term payoffs: With mutation (including u = 1), inverse-degree updates yield higher average individual long-term payoffs than identical rates; degree-proportional updates yield lower payoffs. The advantage persists across mutation rates and selection intensities.
• OptUpRat efficiently finds λ_i that minimise C for any given network. Scale-free networks achieve substantially smaller C at the optimum than lattices. Optimised λ_i generally decreases with degree, consistent with the theoretical rule. Even on homogeneous networks, identical rates are not optimal; the algorithm identifies beneficial deviations.
• Generalisation: The condition extends to general two-player social dilemmas; a lower C from update-rate tuning reduces the threshold on R relative to T, S, P for cooperation to be favoured.

Allowing personalised strategy update rates resolves the apparent contradiction between findings based on average cooperation levels and fixation probabilities in heterogeneous networks. Under identical rates, heterogeneity typically raises the threshold C, hindering fixation from a single cooperator. By coupling behavioural rhythms to structure—specifically making hubs update less frequently—heterogeneous networks become fertile ground for cooperation, achieving lower C than homogeneous networks. The mechanism hinges on local cluster formation around hubs and increased excess of cooperative neighbours (Q > 1) that improves cooperators’ relative payoffs in local imitation dynamics. The analytical framework unifies structural and dynamical heterogeneity and yields practical prescriptions: invert the ordering of update rates relative to degrees to promote cooperation. The method performs well on synthetic and empirical networks and scales via an O(N^3) approximation, while the OptUpRat algorithm demonstrates actionable design of update rates for engineered multi-agent systems. These insights also imply improved long-term payoffs due to more time spent in cooperative states, particularly under rare mutation regimes. The findings have broader relevance for other social dilemmas and suggest that behavioural timing and network structure jointly shape collective outcomes.

The paper introduces a general framework for evolutionary games with personalised strategy update rates and derives precise conditions under which cooperation is favoured. Key contributions include: (i) demonstrating that inverse-degree update rates reverse the traditional disadvantage of heterogeneous networks and can lower C below ⟨k⟩; (ii) providing analytical expressions and an efficient approximation for C that match simulations on synthetic and empirical networks; (iii) revealing a simple rule-of-thumb that higher-degree nodes should update less frequently than lower-degree ones; and (iv) developing OptUpRat to optimally design update rates that minimise C for any network. Future research directions include extending to multiple strategies, exploring co-evolution of networks and update rhythms (group formation with similar rhythms), and applying the framework to temporal networks where degrees and interaction patterns vary in time, as well as conducting behavioural experiments that track fixation from a single cooperator to validate the theory.

• The main analytical results rely on weak selection and a specific imitation update rule with fitness-proportional copying; different intensities or update mechanisms might alter thresholds.
• Networks are assumed static, undirected, and unweighted; temporal, directed, or weighted interactions are only discussed as future work.
• Update events follow independent Poisson processes; other timing processes (e.g., bursty or correlated updates) are not modelled explicitly in the main analysis.
• Exact computation of C requires solving large linear systems (O(N^6)), necessitating approximations for scalability; accuracy in extreme topologies or parameter regimes may vary.
• Simulations focus on finite networks with moderate sizes and specific degree distributions; broader classes of real-world structures may introduce additional complexities.
• Human behavioural validation is not provided; translating personalised rates to human decision-making requires experimental confirmation.