Large group size promotes the evolution of cooperation in the mutual-aid game.

Our society is based on group cooperation, but this remains an unsolved problem from the viewpoint of evolutionary biology and the social sciences. Group cooperation is often studied through the public goods game, and it has been shown that large group size never promotes the evolution of cooperation. We consider the mutual-aid game, in which one member of the group is chosen randomly as the aid recipient, and other members decide whether to help the recipient. This game can describe the early stage of insurance provision in England, for example. With reference to existing indirect reciprocity studies, we investigate what promotes the evolution of cooperation in the mutual-aid game by means of replicator equations and agent-based simulations. Our key findings are as follows. In a multilateral relationship in which members play the mutual-aid game in a group whose size is greater than two, once two or more defectors are in bad standing, they remain bad under the rule that a donor's bad reputation remains bad, whether or not the donor helps a recipient with a bad reputation. Then, cooperators never help them, and mutation helps the invasion of rare cooperators more when the group size is larger in the finite population, even when implementation and perception errors occur. Meanwhile, this rule never helps the invasion in the bilateral relationship in which the group size is two. Our results suggest that large group cooperation can be sustainable if social institutions are equipped with systems such as those in the mutual-aid game.