Future wireless systems enable content communication to be offloaded towards the network edges instead of allocating in centralized cloud. In this paper, we investigate the incentives in edge caching networks via tools from game theory by considering heterogeneous edge systems and various caching behaviors. Particularly, we analyze the competitive and cooperative interactions of multiple stakeholders involved in caching systems with the limits of caching capacity and backhaul resources. The participants in the wireless caching network are modeled as rational players due to their selfish nature. The roles of these players in various caching scenarios are thoroughly discussed from the business point of view including their strategies and utilities. The appropriate game models to study the different players are proposed with the solution concepts and algorithms. Given the existing literature, we investigate the potential research challenges concerning wireless edge caching and game models to solve them. A case study is examined showing the effectiveness of applying game theory into wireless cache-enabled networks. The numerical simulations illustrate that our proposed game models result in the win-win solution.