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We investigate the ability-based fitness calculation in an asymmetric setup to the evolution of cooperation on square lattice. In particular, if players can maintain its strategy for a round, then its abilities will increase 1, otherwise, its ability will turn to 0. We also assume that if players' abilities exceed a certain threshold M , then its environment is integrated into its fitness calculation via a simple parameter α. Through extensive numerical simulation, we find that large threshold M decreases cooperation, and large α, players' fitness is mainly decided by its dynamical environment, promotes cooperation. We also provide theoretical analysis and get the same results with simulation. Finally, we test its robustness on homogeneous random network and small-world network. Our work provides a deeper insight to network reciprocity. [ABSTRACT FROM AUTHOR]

• We introduce third parties and disputants on interdependent network. • Disputants play prisoner's dilemma game and bring third parties extra payoffs. • Third parties intervene disputants play snowdrift dilemma game or not. • Conflicts in disputants are relieved though non-interventions exist in third party layer. Interventions from third parties, such as governmental agencies or organizations, play an important role in mitigating conflicts in modern human societies. The goal thereby is to pacify disputants, although this is subject to failure on account of self-interest from all involved. To study how relationships between disputants and third parties evolve, we propose an interdependent network model, where one layer is occupied by disputants and the other layer is occupied by third parties. Disputants play a prisoner's dilemma game, where defection is the dominant strategy, whereas third parties play a snowdrift game, where cooperation and defection coexist more commonly. Moreover, third parties have the ability to mediate a conflict on the other layer by enforcing a snowdrift game onto disputants, for which they can receive a fee. We show that third party interventions improve the evolution of cooperation between disputants, and also, that the improvement of cooperation in turn promotes interventions. Nonetheless, non-intervention does not go extinct, which enables defectors to survive, thus creating a feedback loop between the two networks layers. The evolutionary dynamics is characterized by fascinating spatial pattern formation, which we explore by means of Monte Carlo simulations and via replicator equations. [ABSTRACT FROM AUTHOR]