We introduce a winner-weaken-loser-strengthen rule and study its effects on how cooperation evolves on interdependent networks. The new rule lowers the learning ability of a player if its payoff is larger than the average payoff of its neighbors, thus enhancing its chance to hold onto its current strategy. Conversely, when a player gaining less than the average payoff of its neighborhood, its learning ability is increased, thus weakening the player by increasing the chance of strategy change.

Furthermore, considering the nature of human pursue fairness, we let a loser, someone who has larger learning ability, can benefit from another network, whereas a winner cannot. Our results show that moderate values of the threshold lead to a high cooperation plateau, while too high or too small values of the threshold inhibit cooperation. At moderate thresholds, the flourishing cooperation is attributed to species diversity and equality, whereas a lacking of species diversity determines the vanishing of cooperation. We thus demonstrate that a simple winner-weaken-loser-strengthen rule significantly expands the scope of cooperation on structured populations.

L. Shi, […] M. Perc, et al., Winner-weaken-loser-strengthen rule leads to optimally cooperative interdependent networks, Nonlinear Dyn 96 (2019) 49–56