Public goods cooperation via group interaction with opponent selection

Understanding how cooperation emerges in biological or social systems remains a significant scientific conundrum. The public goods game is one of the most frequently used paradigms for studying the evolution of cooperative behavior in structured populations. Here we consider the group interaction with opponent selection and assume that individuals randomly choose some neighbors from the whole neighborhood to form an interaction group both in infinite and finite structured populations. Using the pair-approximation approach in the weak selection regime, we first derive the dynamical equations of the frequency of cooperators in infinitely large populations. We consider four different strategy update rules, including death-birth, imitation, birth-death, and pairwise comparison. Our findings demonstrate that death-birth and imitation updating can promote the emergence of cooperation, whereas cooperation fails to emerge under birth-death and pairwise comparison. We then extend our theoretical analysis to finite populations and respectively calculate the fixation probabilities of cooperation and defection. We identify the mathematical condition under which selection favors cooperators over defectors. Our results indicate that cooperators dominate the population more often than defectors under death-birth and imitation updating, while birth-death and pairwise comparison confer an evolutionary advantage to defection. To complete our study, the theoretical predictions are confirmed numerically and by Monte-Carlo simulations.