  Even though existence of non-convergent evolution of the states of
populations in ecological and evolutionary contexts is an undeniable fact,
insightful game-theoretic interpretations of such outcomes are scarce in the
literature of evolutionary game theory. As a proof-of-concept, we tap into the
information-theoretic concept of relative entropy in order to construct a
game-theoretic interpretation for periodic orbits in a wide class of
deterministic discrete-time evolutionary game dynamics, primarily investigating
the two-player two-strategy case. Effectively, we present a consistent
generalization of the evolutionarily stable strategy -- the cornerstone of the
evolutionary game theory -- and aptly term the generalized concept: information
stable orbit. The information stable orbit captures the essence of the
evolutionarily stable strategy in that it compares the total payoff obtained
against an evolving mutant with the total payoff that the mutant gets while
playing against itself. Furthermore, we discuss the connection of the
information stable orbit with the dynamical stability of the corresponding
periodic orbit.
