Products are often paired with additional services to satisfy customers' needs, differentiate product offerings, and remain competitive in today's market. This research is motivated by the need to provide guidelines and methods to support the design of such services, addressing the lack of knowledge on customized service design as well as methods for designing and evaluating services for mass customization. We extend concepts from module-based product family design to create a method for designing families of services. In particular, we introduce a strategic platform design method for developing customized families of services using game theory to model situations involving dynamic market environments. A module-based service model is proposed to facilitate customized service design and represent the relationships between functions and processes that constitute a service offering. A module selection problem for platform design is considered as a strategic module sharing problem under collaboration, and we use a coalitional game to model module sharing and decide which modules provide more benefit when in the platform based on marginal contribution of each module. To demonstrate implementation of the proposed method, we use a case study involving a family of banking services.