The conflict between self-interaction and updating passivity in the evolution of cooperation

In social dilemmas under weak selection, the capacity for a player to exhibit updating passivity or interact with its own strategy can lead to conflicting outcomes. The central question is which effect is stronger and how their simultaneous presence influences the evolution of cooperation. We introduce a model that considers both effects using different weight factors. We derive theoretical solutions for the conditions of cooperation success and the cooperation level under weak selection, scanning the complete parameter space. When the weight factors are equally strong, the promoting effect of self-interaction to cooperation surpasses the inhibitory effect of updating passivity. Intriguingly, however, we identify non-monotonous cooperation-supporting effects when the weight of updating passivity increases more rapidly. Our findings are corroborated by Monte Carlo simulations and demonstrate robustness across various game types, including the prisoner's dilemma, stag-hunt, and snowdrift games.