Effect of the group size on the evolution of cooperation when an exit option is present

The evolution of cooperation has been one of the main topics in evolutionary biology. If cooperators maintain interaction with cooperators and halt interaction with defectors, then cooperation can pay and can be favored by natural selection. This is called an exit option. Here, not only cooperation in dyadic interactions but also cooperation in sizable groups can be observed. Rivalry is about whether usage of the benefit by one individual reduces its availability to others or not. A common good is a rivalrous good, whereas a public good is a non-rivalrous good. In this paper, by analyzing n-player prisoner's dilemma games, we examine whether the effect of the group size on cooperation is positive or negative in the context of exit option. When goods are common goods, defectors always dominate cooperators when the group size is infinitely large. Thus, the group size has only negative effects on the evolution of cooperation when goods are common goods. In contrast, when goods are public goods, an increase in group size has positive effects as well as negative effects on the evolution of cooperation. In addition, we reveal that it has both positive and negative effects on the evolution of cooperation for cooperators to tolerate some defection and hope to keep the interaction.