On Permanence of Conservative Replicator Dynamics with Four Strategies
This provides a complete characterization of global dynamics for evolutionary game theory in the four-strategy case, which is incremental as it extends prior work on replicator dynamics.
The paper tackled the problem of characterizing permanence in four-strategy conservative replicator dynamics with constant payoff matrices, establishing necessary and sufficient conditions by linking the payoff matrix to its digraph and identifying five distinct digraph classes that govern global behavior, with the result that all non-equilibrium trajectories are Lyapunov-stable periodic orbits when permanence occurs.
In this paper, we study four-strategy conservative replicator dynamics induced by constant payoff matrices. We establish necessary and sufficient conditions for permanence to occur by associating the payoff matrix with its digraph, revealing exactly five distinct digraph classes governing the global behavior. We further show that, whenever the dynamics is permanent, every non-equilibrium trajectory in the relative interior of the simplex is a Lyapunov-stable periodic orbit. Together with the classification of the boundary phase portraits, these results provide a complete characterization of the global dynamics in the four-strategy case with permanence.