The evolution of cooperation in the division of labor on generalized networks with a single output

A general division of labor has an arbitrary number of subtasks, and asymmetric interactions among players in the subtask-holder upstream groups and the downstream groups. Cooperation among the subtask-holders is essential in division of labor. We construct a general division of labor network model with a single output, where subtask-holder groups are located, and the process never stops for a defection. We consider that cooperators and defectors exist in each subtask-holder group initially; cooperators in a group pay a cost of cooperation to upgrade the product or service quality, defectors do not pay any cost, reducing the quality. More cooperator groups can upgrade the quality more, thus all the players in all groups can obtain a better reputation defined as a bonus. We found that the division of labor with a bonus meets the social dilemma condition, and our model can be studied from the viewpoint of evolution of cooperation. Our analysis of replicator equations shows the benefits are canceled out in the equations; thus, benefits never influence the evolution of cooperation. As a result, the network structure of the division of labor never influences the dynamics. We also show that sanction on the defectors induces the evolution of cooperation and co-existences of cooperator groups and defector groups, and that the bonus functions determine whether the number of the subtask-holder groups in the division of labor influences on the evolution of cooperation. Our results will contribute to the theoretical understanding of division of labor on the network structure.