Tax thresholds yield multiple optimal cooperation levels in the spatial public goods game.

Income redistribution, which involves transferring income from certain individuals to others, plays a crucial role in human societies. Previous research has indicated that tax-based redistribution can promote cooperation by enhancing incentives for cooperators. In such a tax system, all individuals, irrespective of their income levels, contribute to the tax system, and the tax revenue is subsequently redistributed to everyone. In this study, we relax this assumption by introducing a tax threshold, signifying that only individuals with incomes exceeding the threshold will be subject to taxation. In particular, we employ the spatial public goods game to investigate the influence of tax rates-the percentage of income allocated to tax-and tax thresholds, which determine the income level at which individuals become taxable, on the evolution of cooperation. Our extensive numerical simulations disclose that tax thresholds produce complex outcomes for the evolution of cooperation, depending on tax rates. Notably, at low tax rates (i.e., below 0.41), as the tax threshold increases, discontinuous phase transitions in cooperation performance suggest the presence of multiple intervals of effective tax thresholds that promote peak cooperation levels. Nevertheless, irrespective of the chosen tax rate, once the tax threshold surpasses a critical threshold, the redistribution mechanism fails, causing the collapse of cooperation. Evolutionary snapshots show that self-organized redistribution forms an intermediary layer on the peripheries of cooperative clusters, effectively shielding cooperators from potential defectors. Quantitative analyses shed light on how self-organized redistribution narrows the income gap between cooperators and defectors through precise identification of tax-exempt entities, thereby amplifying the cooperative advantage. Collectively, these findings enhance our comprehension of how income redistribution influences cooperation, highlighting the pivotal role of tax thresholds.