Evolution of cooperation on interdependent networks: The impact of asymmetric punishment

The role of costly punishment in shaping the evolution of cooperation has long been a subject of debate, given its contested effectiveness due to challenges posed by second-order free-riders and antisocial punishment. However, these conclusions have predominantly arisen from scenarios characterized by symmetry, where individuals possess the ability to mutually impose punishment. In this study, we delve into the impact of asymmetric punishment on the development of cooperation within interdependent networks. Notably, our approach involves the exclusive application of punishment between players situated on different network layers, with no punitive interactions occurring within the same network layer. A critical distinction lies in the fact that punitive authority resides solely with the upper-layer players, while the reverse is not true. Specifically, we investigate two modes of punishment: (i) pro-social punishment, where cooperators penalize defectors, and (ii) anti-social punishment, wherein defectors penalize cooperators. Through extensive numerical simulations, we unveil a synchronization phenomenon of strategies within the two networks when the costs of punishment are comparatively high. This synchronization gives rise to the gradual emergence of cooperative clusters, resulting in higher payoffs for the individuals involved. Conversely, when punishment costs are relatively low, the synchronization phenomenon does not manifest, and the evolution of cooperation takes a divergent course. In summary, our findings enrich the theory of punishment by underscoring the pivotal role played by asymmetric punishment in molding cooperative behavior.