Threshold-initiated spatial public goods games

The Public Goods Game (PGG) encounters hurdles when donations are scarce, resulting in failed game initiations. To investigate such phenomena, we propose a threshold -based spatial PGG model operating on a lattice with periodic boundaries. Players strategically choose between cooperation (C) and defection (D), with PGG initiation determined by a cooperative contribution threshold. Additionally, we introduce an adjustment factor for player reputation, reflecting how individual strategy choices and responses from interacting partners influence reputation changes. We hypothesize that within non -initiated PGG, defectors' reputations decrease, while within initiated PGG, defectors are evaluated and penalized by cooperators. Our findings reveal that higher initiation thresholds can enhance final cooperation levels. Moreover, the inclusion of the reputation adjustment factor acts as a catalyst for cooperative behavior. Interestingly, greater uncertainty in strategy adoption is associated with increased cooperative levels under higher initiation thresholds. This study adds new insights into the evolution of cooperation in the context of spatial structure.