Effect of Population Size on Random Punisher Cooperator in the Optional Spatial Public Goods Game

This work examines how population size affects the behavior of cooperation in a spatial optional public goods game with a random punisher mechanism where player can choose between cooperate, defect or abstain(loner). For small populations, the system is highly sensitive to random fluctuations, frequently collapsing into total defection or total cooperation without sustaining any stable cyclic dynamic. As population size increases, a stable coexistence regime emerges where cooperators, defectors, and loner players cycle. Beyond a determined threshold, loners become extinct and defectors dominate the system, after that threshold both small and large populations converge toward a cooperation-dominated regime, where defectors are completely eliminated regardless of grid size. The results show that larger populations sustain more stable cooperative regimes, the extinction of the loners converts the game into a compulsory scenario, forcing a
binary outcome between total cooperation or defection, while smaller population remain sensitive due to the random nature of the punisher selection.