We study the evolution of cooperation in group interactions where players are randomly drawn from well-mixed populations of finite size to participate in a public goods game. However, due to the possibility of unforeseen circumstances, each player has a fixed probability of being unable to participate in the game, unlike previous models which assume voluntary participation. We first study how prescribed stochastic opting-out affects cooperation in finite populations, and then generalize for the limiting case of large populations. Because we use a pairwise comparison updating rule, our results apply to both genetic and behavioral evolution mechanisms. Moreover, in the model, cooperation is favored by natural selection over both neutral drift and defection if the return on investment exceeds a threshold value depending on the population size, the game size, and a player’s probability of opting-out. Our analysis further shows that, due to the stochastic nature of the opting-out in finite populations, the threshold of return on investment needed for natural selection to favor cooperation is actually greater than the one corresponding to compulsory games with the equal expected game size. We also use adaptive dynamics to study the co-evolution of cooperation and opting-out behavior. Indeed, given rare mutations minutely different from the resident population, an analysis based on adaptive dynamics suggests that over time the population will tend towards complete defection and non-participation, and subsequently cooperators abstaining from the public goods game will stand a chance to emerge by neutral drift, thereby paving the way for the rise of participating cooperators. Nevertheless, increasing the probability of non-participation decreases the rate at which the population tends towards defection when participating. Our work sheds light on understanding how stochastic opting-out emerges in the first place and on its role in the evolution of cooperation.
The evolution of cooperation in finite populations with synergistic payoffs