The evolution of cooperation in the unidirectional linear division of labour of finite roles

Abstract

Evolution of cooperation is a puzzle in evolutionary biology and social sciences. Previous studies assumed that players are equal and have symmetric relationships. In our society, players are in different roles, have an asymmetric relationship and cooperate together. We focused on the linear division of labour in a unidirectional chain that has finite roles, each of which is assigned to one group with cooperators and defectors. A cooperator in an upstream group produces and modifies a product, paying a cost of cooperation, and hands it to a player in a downstream group who obtains the benefit from the product. If players in all roles cooperate, a final product can be completed. However, if a player in a group chooses defection, the division of labour stops, the final product cannot be completed and all players in all roles suffer damage. By using the replicator equations of the asymmetric game, we investigate which sanction system promotes the evolution of cooperation in the division of labour. We find that not the benefit of the product but the cost of cooperation matters to the evolutionary dynamics and that the probability of finding a defector determines which sanction system promotes the evolution of cooperation.

The evolution of cooperation in the unidirectional division of labour on a tree network

“…When k is the premier, p , the mixU p∪falsefalse{pfalsefalse} equilibrium does not exist, because there is no U p and the case of p being the defector group is included in the equilibrium premierD . When k is a terminal, we can consider Ok∪falsefalse{kfalsefalse} as a linear division of labour [56], where mixDk does not exist.…”

“…Our numerical simulations can also show that the dynamics converge to the mixed equilibrium point (yellow dots in figure 6 b ). While the mixed equilibrium point cannot be locally stable and no numerical simulations converged to the mixed equilibrium point in the linear division of labour when the cost decreases downstream [56], and therefore the existence of mixed equilibrium points when the cost decreases downstream is unique from this study.…”

“…In the beginning, the whole task, which can be a service or development of a product, is assigned to the original group, where a player is randomly selected, who gets a benefit b p . The player receives the benefit for receiving the task and it can be considered as the value of the task [56]. For example, in the multi-layered contract development system, the orderer pays the contract money, b p , to the contractor, which pays the money to the subcontractors and it continues when the terminal sub-subcontractors receive the money and do their tasks.…”

“…Data accessibility. The basic MATLAB code for generating the results of this manuscript are given in the ESM file titled 'Numerical_analysis_code_Nirjhor_and_Nakamaru_tree_graph_2023' [62].…”

“…Though both sanction systems are capable of enforcing cooperation more than the system without sanction, when it is practically impossible to monitor and identify defectors the first role sanction system has been proven to be more successful at ensuring the evolution of cooperation than the defector sanction system. Nirjhor & Nakamaru [56] found out that the premier sanction system is incapable of creating cooperation when the cooperators royalsocietypublishing.org/journal/rsos R. Soc. Open Sci.…”