Evolutionary dynamics in public goods games with general frequency-dependent returns

The public goods game serves as a significant paradigm for investigating the emergence and maintenance of cooperation in conflicting situations. In the traditional public goods game, the multiplication factor characterizing the synergy effect of common efforts is typically assumed to be constant. In real-world scenarios, however, investment returns are often dynamic and vary with the strategic composition of the interaction group. To date, the evolutionary dynamics of the public goods game with such frequency-dependent returns have remained not fully understood. In this work, we introduce a general frequency-dependent multiplication factor that depends on the strategy composition within the game group. Through theoretical analysis, we derive the mathematical conditions under which cooperation is favored. Our results show that whether cooperation has an evolutionary advantage over defection depends on the investment return rate in the full-contribution state of the game group, irrespective of the return rates in other states. An increase in this rate leads to a higher abundance of cooperators. Furthermore, we introduce a general frequency-dependent multiplication factor into the public goods game with peer punishment, and systematically explore its effects on the cooperation dilemma and the second-order free-rider problem. Our results highlight that the abundance of cooperators or punishers is governed solely by the investment return values in the full-cooperation and full-punishment compositions of the group. A higher return rate in the full-cooperation state facilitates the promotion of cooperation, whereas a higher return rate in the full-punishment state favors the emergence of punishment. Our theoretical findings are verified by individual-based simulations.