This paper reports our study of how to gain consistency of states in a ball-game typed Distributed Virtual Environment (DVE) with lag, in peer-to-peer (P2P) architecture. That is, we are studying how to reduce in real-time the difference of states between the participating terminals in a virtual ball game caused by transmission lag or update interval. We are also studying how to control shared objects in real-time in a server-less network architecture. Specifically, a priority field called Allocated Topographical Zone (AtoZ) is used in P2P for DVE. By implementing this function, each terminal can compute which avatar holds the ownership of a shared object by calculating mutually the state of the local avatar predicted by the remote terminals. The region for ownership determined by AtoZ allows an avatar to access and control an object dominantly, so that a geometrical property is dynamically changed depending upon the relative arrangement between the object and avatars. Moreover considering the critical case, defined as inconsistent phenomena between the peers, caused by the network latency, a stricter ownership determination algorithm, called dead zone is introduced. By using these protocols in combination, a robust and effective scheme is achieved for a virtual ball game. As an example of the application, a real-time networked doubles air-hockey is implemented for evaluation of the influence of these protocols on interactivity and on consistency.