  One could observe drastically different dynamics of zero-sum and non-zero-sum
games under replicator equations. In zero-sum games, heteroclinic cycles
naturally occur whenever the species of the population supersede each other in
a cyclic fashion (like for the Rock-Paper-Scissors game). In this case, the
highly erratic oscillations may cause the divergence of the time averages. In
contrast, it is a common belief that the most ``reasonable" replicator
equations of non-zero-sum games satisfy ``The Folk Theorem of Evolutionary Game
Theory" which asserts that (i) a Nash equilibrium is a rest point; (ii) a
stable rest point is a Nash equilibrium; (iii) a strictly Nash equilibrium is
asymptotically stable; (iv) any interior convergent orbit evolves to a Nash
equilibrium. In this paper, we propose two distinct vast classes of replicator
equations generated by similar-order preserving mappings which exhibit stable
as well as mean historic behavior. In the latter case, the time averages of the
orbit will slowly oscillate during the evolution of the system and do not
converge to any limit. This will eventually cause the divergence of
higher-order repeated time averages.
