  Our study contributes to the debate on the evolution of cooperation in the
single-shot Prisoner's Dilemma (PD) played on networks. We construct a model in
which individuals are connected with positive and negative ties. Some agents
play sign-dependent strategies that use the sign of the relation as a shorthand
for determining appropriate action toward the opponent. In the context of our
model in which network topology, agent strategic types and relational signs
coevolve, the presence of sign-dependent strategies catalyzes the evolution of
cooperation. We highlight how the success of cooperation depends on a crucial
aspect of implementation: whether we apply parallel or sequential strategy
update. Parallel updating, with averaging of payoffs across interactions in the
social neighborhood, supports cooperation in a much wider set of parameter
values than sequential updating. Our results cast doubts about the realism and
generalizability of models that claim to explain the evolution of cooperation
but implicitly assume parallel updating.
