We present a dynamic game framework to model and design defense strategies for advanced persistent threats (APTs). The model is based on a sequence of nested finite two-person zero-sum games, in which the APT is modeled as the attempt to get through multiple protective shells of a system towards conquering the target located in the center of the infrastructure. In each stage, a sub-game captures the attack and defense interactions between two players, and its outcome determines the security level and the resilience against penetrations as well as the structure of the game in the next stage. By construction, interdependencies between protections at multiple stages are automatically accounted for by the dynamic game. The game model provides an analysis and design framework to develop effective protective layers and strategic defense-indepth strategies against APTs. We discuss a few closed form solutions of our sequential APT-games, upon which design problems can be formulated to optimize the quality of security (QoS) across several layers. Numerical experiments are conducted in this work to corroborate our results.