  According to the evolutionary death-birth protocol, a player is chosen
randomly to die and neighbors compete for the available position proportional
to their fitness. Hence, the status of the focal player is completely ignored
and has no impact on the strategy update. In this work, we revisit and
generalize this rule by introducing a weight factor to compare the payoff
values of the focal and invading neighbors. By means of evolutionary graph
theory, we analyze the model on joint transitive graphs to explore the possible
consequences of the presence of a weight factor. We find that focal weight
always hinders cooperation under weak selection strength. Surprisingly, the
results show a non-trivial tipping point of the weight factor where the
threshold of cooperation success shifts from positive to negative infinity.
Once focal weight exceeds this tipping point, cooperation becomes unreachable.
Our theoretical predictions are confirmed by Monte Carlo simulations on a
square lattice of different sizes. We also verify the robustness of the
conclusions to arbitrary two-player prisoner's dilemmas, to dispersal graphs
with arbitrary edge weights, and to interaction and dispersal graphs
overlapping arbitrarily.
