  Introducing strategy complexity into the basic conflict of cooperation and
defection is a natural response to avoid the tragedy of the common state. As an
intermediate approach, quasi-cooperators were recently suggested to address the
original problem. In this study, we test its vitality in structured populations
where players have fixed partners. Naively, the latter condition should support
cooperation unambiguously via enhanced network reciprocity. However, the
opposite is true because the spatial structure may provide a humbler
cooperation level than a well-mixed population. This unexpected behavior can be
understood if we consider that at a certain parameter interval the original
prisoner's dilemma game is transformed into a snow-drift game. If we replace
the original imitating strategy protocol by assuming myopic players, the
spatial population becomes a friendly environment for cooperation. This
observation is valid in a huge region of parameter space. This study highlights
that spatial structure can reveal a new aspect of social dilemmas when strategy
complexity is introduced.
