  Cooperation is a very common, yet not fully-understood phenomenon in natural
and human systems. The introduction of a network within the population is known
to affect the outcome of cooperative dynamics, allowing for the survival of
cooperation in adverse scenarios. Recently, the introduction of multiplex
networks has yet again modified the expectations for the outcome of the
Prisoner's Dilemma game, compared to the monoplex case. However, much remains
unstudied regarding other social dilemmas on multiplex, as well as the
unexplored microscopic underpinnings of it. In this paper, we systematically
study the evolution of cooperation in all four games in the $T-S$ plane on
multiplex. More importantly, we find some remarkable and previously unknown
features in the microscopic organization of the strategies, that are
responsible for the important differences between cooperative dynamics in
monoplex and multiplex. Specifically, we find that in the stationary state,
there are individuals that play the same strategy in all layers (coherent), and
others that don't (incoherent). This second group of players is responsible for
the surprising fact of a non full-cooperation in the Harmony Game on multiplex,
never observed before, as well as a higher-than-expected cooperation rates in
some regions of the other three social dilemmas.
