  The framework of iterated Prisoner's Dilemma (IPD) is commonly used to study
direct reciprocity and cooperation, with a focus on the assessment of the
generosity and reciprocal fairness of an IPD strategy in one-on-one settings.
In order to understand the persistence and resilience of reciprocal
cooperation, here we study long-term population dynamics of IPD strategies
using the Moran process where stochastic dynamics of strategy competition can
lead to the rise and fall of cooperation. Although prior work has included a
handful of typical IPD strategies in the consideration, it remains largely
unclear which type of IPD strategies is pivotal in steering the population away
from defection and providing an escape hatch for establishing cooperation. We
use a network-based approach to analyze and characterize networks of
evolutionary pathways that bridge transient episodes of evolution dominated by
depressing defection and ultimately catalyze the evolution of reciprocal
cooperation in the long run. We group IPD strategies into three types according
to their stationary cooperativity with an unconditional cooperator: the good
(fully cooperative), the bad (fully exploitive), and the ugly (in between the
former two types). We consider the mutation-selection equilibrium with rare
mutations and quantify the impact of the presence versus absence of any given
IPD strategy on the resulting population equilibrium. We identify catalysts
(certain IPD strategies) as well as bridges (particular evolutionary pathways)
that are most crucial for boosting the abundance of good types and suppressing
that of bad types or having the highest betweenness centrality. Our work has
practical implications and broad applicability to real-world cooperation
problems by leveraging catalysts and bridges that are capable of strengthening
persistence and resilience.
