  Understanding cooperative behavior in biological and social systems
constitutes a scientific challenge, being the object of intense research over
the past decades. Many mechanisms have been proposed to explain the presence
and persistence of cooperation in those systems, showing that there is no
unique explanation, as different scenarios have different possible driving
forces. In this paper, we propose a model to study situations in which
voluntary participation involves an access cost to the cooperative interaction,
besides the cost associated with cooperation. The proposed Costly-Access
Prisoner's Dilemma, a symmetric donation game with voluntary and costly
participation, breaks the symmetry between abstainers and participants of the
Voluntary Prisoner's Dilemma. A mean-field approach shows that, in well-mixed
populations, the dynamic always leads the system to abstention. However,
depending on the return parameter, numerical simulations in structured
populations display an alternating behavior between mono-strategic,
multi-stable, and coexistence phases. This behavior is fully explained through
a theoretical analysis of the strategic motifs, the transitions being
determined by the change in stability of those motifs.
