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We model security protocols as a game tree using concepts of game semantics. Using this model we ascribe semantics to protocols written in the standard simple arrow notation. According to the semantics, a protocol is interpreted as a set of strategies over a game tree that represents the type of the protocol. Moreover, in order to specify properties of the model, a logic that deals with games and strategies is developed. A tableau-based proof system is given for the logic, which can serve as a basis for a model checking algorithm. This approach allows us to model a wide range of security protocol types and verify different properties instead of using a variety of methods as is currently the practice. Furthermore, the analyzed protocols are specified using only the simple arrow notation heavily used by protocol designers and by practitioners.