Public goods game with umpire supervision: Based on complex networks and birandom geometric graph

Using complex networks and the birandom geometric graph (BRGG) model, we first propose a construction algorithm for a two-layer network with umpire supervision (i.e., US-BRGBA). Subsequently, we introduce a reward-punishment mechanism imposed by umpires on players while considering both fair and corrupt umpires, thereby presenting the US-BRGBA game model with umpire supervision. Further, the feasibility and effectiveness of the proposed US-BRGBA model based on Monte Carlo (MC) simulations are verified, as well as the influence on players' cooperative behaviors arising from umpires' spatial distribution, the fraction of fair umpires, the reward value, the fine value, and the bribe value. Abundant simulations demonstrate that the US-BRGBA model promotes cooperation significantly. Additionally, the spatial distribution patterns of umpires do not consistently yield uniform effects on promoting cooperation, as their influences are contingent upon the fraction of fair umpires and the magnitude of the synergy factor. In resource-abundant regions, corruption does not block cooperative behaviors. Meanwhile, the introduction of umpires consistently facilitates cooperation, even in scenarios where all umpires exhibit corrupt behavior. Further, compared with the reward value and fine value, the cooperative behavior in the US-BRGBA game model is more sensitive to the bribe value.