Two-layer network model of public goods games with intervention and corruption

Public goods games are widely used to model social dilemmas involving multiple agents. Though defection is the only rational choice for an individual in a public goods game, cooperative behavior is observed in a variety of social dilemmas, which is the subject of our research. Punishing defectors has been shown to be an effective mechanism for promoting cooperation, but it relies on the third-party umpire being fair. In this article, an umpire intervention model with corruption is proposed to explore the impact of corruption on punishment mechanisms. In our model, players and umpires operate in a multilayer network. The players play public goods games, which are overseen by umpires. Fair umpires punish defectors, whereas corrupt umpires take bribes from defectors rather than meting out a punishment. We separately explore the effects of the fraction of fair umpires _ρ, the spatial distribution, and the fine cost α and bribe cost β. Our Monte Carlo simulation shows that the above factors have a significant impact on cooperation. Intervention by an umpire always improves social efficiency, even for an entirely corrupt system. Moreover, relatively developed systems can resist corruption. Staggered and centralized distributions always have opposite effects on cooperative behavior, and these effects depend on _ρ and r. We also find that whether cooperators fully occupy the player layer depends only on whether β reaches a certain threshold.