  In a social dilemma situation, where individual and collective interests are
in conflict, it sounds a reasonable assumption that the presence of super or
smart players, who simultaneously punish defection and reward cooperation
without allowing exploitation, could solve the basic problem. The behavior of
such a multi-strategy system, however, is more subtle than it is firstly
anticipated. When exploring the complete parameter space, we find that the
emergence of cyclic dominance among strategies is rather common, which results
in several counter-intuitive phenomena. For example, the defection level can be
lowered at higher temptation, or weaker punishment provides better conditions
for smart players. Our study indicates that smart cooperators can unexpectedly
thrive under high temptation, emphasizing the complexity of strategic
interactions. This study suggests that the principles governing these
interactions can be applied to other moral behaviors, such as truth-telling and
honesty, providing valuable insights for future research in multi-agent
systems.
