Compulsory persistent cooperation in continuous public goods games

The public goods game (PGG), where players either contribute an amount to the common pool or do nothing, is a paradigm for exploring cooperative behaviors in biological systems, economic communities and other social systems. In many situations, including climate game and charity donations, any contribution, however large or small, should be welcome. Consequently, the conventional PGG is extended to a PGG with continuous strategy space, which still cannot escape the tragedy of commons without any enforcing mechanisms. Here we propose the persistent cooperation investment mechanisms based on continuous PGG, including single-group games, multi-group games with even investment, non-even investment and non-even investment with preference. We aim to reveal how these investment styles promote the average cooperation level in the absence of any other enforcing mechanisms. Simulations indicate that the multi-group game outperforms the single-group game. Among the multi-group game, non-even investment is superior to even investment, but inferior to non-even investment with preference. Our results may provide an explanation to the emergence of cooperative actions in continuous phenotypic traits based on inner competition and self-management without extrinsic enforcing mechanisms.