The “social dilemma” is a problem inherent in forming and maintaining cooperation among selfish individuals, and is of fundamental importance in the biological and social sciences. From the viewpoint of traditional game theory, the existence of the social dilemma necessarily implies degeneration into selfish behavior as the numbers of members in a community increases, unless there exists some external power. In the real world, however, cooperation is often formed and maintained merely through mutual interactions, without the influence of an external power. To answer questions concerning appearance and maintenance of cooperative behavior in societies, we study what we call the “Lumberjacks’ Dilemma (LD) game”, as an application of the dynamical systems (DS) game theory presented in [Physica D 147 (2000) 221], which can naturally deal with the dynamic aspects of games. Dynamical processes that lead to the formation and maintenance of cooperation, which is often observed in ...