In CSCW applications, users modify shared objects in real-time. Thus, concurrency management protocols are required in order to maintain consistency. Such protocols can be classified as optimistic or pessimistic. Our Pilgrim protocol is pessimistic since it is based on ownership. Our new version of this protocol is optimistic and designed to minimize the delay before writing. This paper presents this new version based on atomization and multi-versioning and compares it to the former one through a probabilistic study. Finally, this study allows us to highlight the parameters that make it possible to choose between the two protocols studied.