Enhancing cooperative evolution in spatial public goods game by particle swarm optimization based on exploration and q-learning

In evolutionary game theory, the emergence and maintenance of cooperative behavior in a population often face challenges posed by the temptation of free-riding behavior, which offers high individual payoff. Recently, apart from a range of mechanisms that promote the formation of cooperation, individual learning abilities under limited information have emerged as a key factor in adjusting agents' strategies. This paper introduces q-learning and particle swarm optimization into the realm of evolutionary dynamics. The primary focus is on investigating the impact of Exploration-based Particle Swarm Optimization (EPSO) and Q-learning-based Particle Swarm Optimization (QPSO) on the evolution of cooperation in a continuous version of the spatial public goods game (SPGG) with punishment. EPSO defines a rule for updating agents' strategies based on individual and limited population information. It also integrates an exploration mechanism to increase the diversity and directionality of the strategies. Additionally, QPSO serves to adaptively optimize the parameters of EPSO, addressing the issue of parameter control limiting the EPSO's performance. Leveraging experiential learning and iterative adjustment, QPSO progressively refines system parameters, thus rationally assimilating knowledge and updating individual strategies to attain optimal payoff. Through extensive simulation studies, it has been observed that employing QPSO's adaptively optimized parameters in EPSO significantly promotes the cooperative evolution in the SPGG with punishment. Furthermore, individual learning coefficients, when too low or too high, both facilitate the occurrence of cooperation. Simultaneously, higher inertia weight coefficients strengthen the system's cooperation level, while lower punishment intensity coefficients and higher gain intensity coefficients effectively promote the cooperation emergence and exert a significant influence on the overall cooperation level of the system. This research provides a new perspective for designing real-world schemes that encourage cooperation and offers insights into the intricate dynamics of cooperation in complex systems.