  We study the evolution of cooperation in structured populations within
popular models of social dilemmas, whereby simple coevolutionary rules are
introduced that may enhance players abilities to enforce their strategy on the
opponent. Coevolution thus here refers to an evolutionary process affecting the
teaching activity of players that accompanies the evolution of their
strategies. Particularly, we increase the teaching activity of a player after
it has successfully reproduced, yet we do so depending on the disseminated
strategy. We separately consider coevolution affecting either only the
cooperators or only the defectors, and show that both options promote
cooperation irrespective of the applied game. Opposite to intuitive reasoning,
however, we reveal that the coevolutionary promotion of players spreading
defection is, in the long run, more beneficial for cooperation than the
likewise promotion of cooperators. We explain the contradictory impact of the
two considered coevolutionary rules by examining the differences between
resulting heterogeneities that segregate participating players, and
furthermore, demonstrate that the influential individuals completely determine
the final outcome of the games. Our findings are immune to changes defining the
type of considered social dilemmas and highlight that the heterogeneity of
players, resulting in a positive feedback mechanism, is a fundamental property
promoting cooperation in groups of selfish individuals.
