The very notion of a fully Peer-to-Peer (P2P) Virtual Environment (VE) places exacting demands on its underlying network. Subset applications such as online games are notorious for their sensitivity to latency and high bandwidth demands. By omitting the centralised mechanisms that underpin current commercial implementations, the task of managing a disparate and dynamic peer population is made ever more daunting.For any VE, arbitrating the interactions between players is an inescapable need. Online games are equal parts collaboration and competition, requiring robust conflict resolution mechanisms to govern game-play. With centralised systems, arbitration duties are simply assigned to a provisioned server infrastructure. In a P2P system however, the issue looms large. As such, managing arbitration loads across the peer population is the focus here. Being a game-play arbitrator entails added bandwidth and processing demands. Thus, great care is needed to avoid overloading peers whilst providing a responsive and uninterrupted experience.The work here exploits 3D Voronoi Diagrams (3D-VD) as a scalable, flexible and fault-tolerant P2P overlay that is able to automatically balance arbitration loads amongst peers. Simulation results indicate how 3D-VD, with the right arbitrator-selection policy, can appropriately distribute loads and reduce load fluctuations by up to 90%. This is then augmented with algorithms based on classical Newtonian gravity laws. Doing so provides an autonomous method to detect and respond to high-demand areas within the VE. Further experimentation demonstrates an ability to reduce the instances of failed arbitration attempts by 50%.Index Terms-Peer-to-peer overlays, distributed virtual environments, load management.