Three Body Problems in Evolutionary Game Dynamics: Convergence, Periodicity and Limit Cycles

We study the asymptotic behavior of replicator dynamics in settings of network interaction. We focus on three agent graphical games where each edge/game is either a 2x2 zero-sum or a 2x2 coordination game. Using tools from dynamical systems such as Lyapunov functions and invariant functions we establish that this simple family of games can exhibit an interesting range of behaviors such as global convergence, periodicity for all initial conditions as well as limit cycles. In contrast, we do not observe more complex behavior such as toroids or chaos while it is possible to reproduce them in slightly more complicated settings.