Particle swarm optimization with historical return decay enhances cooperation in public goods games with investment risks

In the realm of spatial public goods games, heterogeneous investment has emerged as a potent mechanism to enhance cooperative behavior. Savvy investors dynamically adjust their investment levels to optimize returns based on environmental conditions and personal circumstances. This study delves into the evolution of cooperation within a spatial public goods game framework, characterized by heterogeneous investment and associated risks, conducted on a square lattice. We introduce the investors (I) with heterogeneous investment, who updates his investment amount according to the current environment of the game and his own historical experience. We introduce a particle swarm optimization algorithm with decaying historical returns to fine-tune the investment levels of investors. Furthermore, a risk mechanism inspired by the ultimatum game is integrated into the model. This paper investigates the impact of the risk threshold $\theta$ on cooperative behavior and examines the influence of the risk decay factor $\alpha$ on cooperation. Additionally, it analyzes the investment behavior of investors in scenarios where two types of investors coexist. Finally, this study explores the effects of the historical best payoff decay factor $\beta$ and the self-learning rate $c_1$ on cooperative behavior. This research contributes to a nuanced understanding of heterogeneous investment behaviors in public goods games under specified risk mechanisms, providing novel insights into the intricate dynamics of cooperation in complex systems.