Scheideler has shown that peer-to-peer overlays networks can only survive Byzantine attacks if malicious nodes are not able to predict what is going to be the topology of the network for a given sequence of join and leave operations. In this paper we investigate adversarial strategies by following specific games. Our analysis demonstrates first that an adversary can very quickly subvert DHT-based overlays by simply never triggering leave operations. We then show that when all nodes (honest and malicious ones) are imposed on a limited lifetime, the system eventually reaches a stationary regime where the ratio of polluted clusters is bounded, independently from the initial amount of corruption in the system.