  In the evolutionary Prisoner's Dilemma (PD) game, agents play with each other
and update their strategies in every generation according to some microscopic
dynamical rule. In its spatial version, agents do not play with every other
but, instead, interact only with their neighbors, thus mimicking the existing
of a social or contact network that defines who interacts with whom. In this
work, we explore evolutionary, spatial PD systems consisting of two types of
agents, each with a certain update (reproduction, learning) rule. We
investigate two different scenarios: in the first case, update rules remain
fixed for the entire evolution of the system; in the second case, agents update
both strategy and update rule in every generation. We show that in a well-mixed
population the evolutionary outcome is always full defection. We subsequently
focus on two-strategy competition with nearest-neighbor interactions on the
contact network and synchronized update of strategies. Our results show that,
for an important range of the parameters of the game, the final state of the
system is largely different from that arising from the usual setup of a single,
fixed dynamical rule. Furthermore, the results are also very different if
update rules are fixed or evolve with the strategies. In these respect, we have
studied representative update rules, finding that some of them may become
extinct while others prevail. We describe the new and rich variety of final
outcomes that arise from this co-evolutionary dynamics. We include examples of
other neighborhoods and asynchronous updating that confirm the robustness of
our conclusions. Our results pave the way to an evolutionary rationale for
modelling social interactions through game theory with a preferred set of
update rules.
