  Individuals might abstain from participating in an instance of an
evolutionary game for various reasons, ranging from lack of interest to risk
aversion. In order to understand the consequences of such diverse activity
patterns on the evolution of cooperation, we study a weak prisoner's dilemma
where each player's participation is probabilistic rather than certain. Players
that do not participate get a null payoff and are unable to replicate. We show
that inactivity introduces cascading failures of cooperation, which are
particularly severe on scale-free networks with frequently inactive hubs. The
drops in the fraction of cooperators are sudden, while the spatiotemporal
reorganization of compact cooperative clusters, and thus the recovery, takes
time. Nevertheless, if the activity of players is directly proportional to
their degree, or if the interaction network is not strongly heterogeneous, the
overall evolution of cooperation is not impaired. This is because inactivity
negatively affects the potency of low-degree defectors, who are hence unable to
utilize on their inherent evolutionary advantage. Between cascading failures,
the fraction of cooperators is therefore higher than usual, which lastly
balances out the asymmetric dynamic instabilities that emerge due to
intermittent blackouts of cooperative hubs.
