This paper proposes a fully distributed architecture to support massively multiplayer games (MMGs) on a peer-to-peer (P2P) overlay network. P2P networks provide a scalable and robust solution for MMGs. Furthermore, similarly to real world interactions, MMG players interact with their surrounding, i.e., with visible players and objects in the virtual world. Thus, self-organization of peers based on their visibility property is essential for efficient overlay network support in MMGs. Also, the overlay network must be re-organized consistently in the face of players' position changes. In this paper, we achieve these goals by designing a peer-to-peer overlay network based on the Voronoi diagram. Each player has a responsibility region that includes nearby objects, and is responsible for updates' dissemination occurring in this region to provide game state consistency among the players who are concerned by the updates. As a proof of concept, we simulated a simple game application to demonstrate the feasibility of our architecture.