The conflict of social norms may cause the collapse of cooperation: Indirect reciprocity with opposing attitudes towards in-group favoritism

Indirect reciprocity is a cooperation maintaining mechanism based on the social evaluation of players. Here, we consider the case of a group in which two social norms with opposing attitudes towards in-group favoritism are mixed. One norm, called Bushido (the way of warriors), regards cooperation with outsiders as betrayal, whereas the second norm, called Shonindo (the way of merchants), regards cooperation with outsiders as desirable. Each member of the group, irrespective of being a Bushido or a Shonindo player, is evaluated in two different ways and assigned two different labels: "ally" or "enemy" according to the Bushido evaluation; "good" or "bad" according to the Shonindo evaluation. These labels change in response to the action taken (cooperation or defection) when acting as a donor, as well as the label attached to the recipient. In addition to Bushido players, who cooperate with an ally and defect from an enemy, and Shonindo players, who cooperate with a good recipient and defect from a bad recipient, the group contains a third kind of players - unconditional defectors. The fractions of the three types of players follow the replicator dynamics. If the probability of interacting with outsiders is small, and if the cost-to-benefit ratio of cooperation is low, we observe several important patterns. Each social norm is able to maintain a high level of cooperation when dominant. Bushido and Shonindo players evaluate each other unfavorably and engage in a severe conflict. In the end, only one norm permeates the whole group driving the other to the extinction. When both social norms are equally effective, a rare occurrence of unconditional defectors may lead to a successful invasion.