Uncertainty driven decision making and perturbation dynamics in evolutionary games on small-world networks

The efficiency of models that aim to interpret real world scenarios largely depends on how well they can reproduce empirical data. However, this task is challenged by uncertainties arising from dynamic environments. Recent studies on feedback-evolving games which characterize the interplay between the evolution of strategies and environmental changes, provide a theoretical framework to address this uncertainty problem. While previous studies assumed uncertainty caused by dynamic environments would impact all decisions equally, the proposed model reveals that decision-specific uncertainty influences the behavioral dynamics of a population differently. By introducing uncertainty as perturbations to a 2 × 2 payoff structure, the proposed model represents different aspects of real world uncertainty. This study reveals that perturbations applied to the off diagonal elements promote coexistence among strategies, whereas perturbations spanning the entire payoff matrix enhance cooperative behaviors. In contrast, perturbations introduced through the main diagonal and cost-benefit terms drive the system towards defective behavior. It is also evident that those findings remain consistently across various network structures. Expanding upon these findings, the study was extended to small-world networks, investigating the impact of key parameters such as the average degree (number of neighbors) and rewiring probability. Our results uncover intricate dependencies between these structural parameters and behavioral dynamics over time under various perturbation mechanisms. Overall, this research provides a comprehensive understanding of how external dynamics and network structures collectively shape evolutionary dynamics while highlighting the role of game transitions in evolutionary dynamics. • Introduces a systematic framework to integrate environmental uncertainty into evolutionary games through four novel perturbation mechanisms (MDP, ODP, FP, CBP), each targeting distinct regions of the payoff matrix. • Reveals that the mode of perturbation integration, fundamentally alters the evolutionary trajectory, with FP promoting cooperation and ODP sustaining coexistence across time. • Demonstrates the robustness of behavioral dynamics across diverse network topologies (line, cycle, lattice, regular, small-world, and random), emphasizing that perturbation effects transcend spatial structure. • Uncovers how small-world network parameters—local connectivity ( k ) and rewiring probability ( p )—nonlinearly modulate the convergence and coexistence patterns induced by different perturbation schemes. • Establishes that inter-game transitions—reflecting temporally fluctuating external conditions—consistently yield more favorable behavioral outcomes than intra-game state transitions, underscoring their evolutionary advantage.