Non-signalized intersection is a typical and common scenario for connected and automated vehicles (CAVs). How to balance safety and efficiency remains difficult for researchers. To improve the original Responsibility Sensitive Safety (RSS) driving strategy on the non-signalized intersection, we propose a new strategy in this paper, based on right-of-way assignment (RWA). The performances of RSS strategy, cooperative driving strategy, and RWA based strategy are tested and compared. Testing results indicate that our strategy yields better traffic efficiency than RSS strategy, but not satisfying as the cooperative driving strategy due to the limited range of communication and the lack of long-term planning. However, our new strategy requires much fewer communication costs among vehicles.