  Pro-social punishment and exclusion are common means to elevate the level of
cooperation among unrelated individuals. Indeed, it is worth pointing out that
the combined use of these two strategies is quite common across human
societies. However, it is still not known how a combined strategy where
punishment and exclusion are switched can promote cooperation from the
theoretical perspective. In this paper, we thus propose two different switching
strategies, namely peer switching that is based on peer punishment and peer
exclusion, and pool switching that is based on pool punishment and pool
exclusion. Individuals adopting the switching strategy will punish defectors
when their numbers are below a threshold and exclude them otherwise. We study
how the two switching strategies influence the evolutionary dynamics in the
public goods game. We show that an intermediate value of the threshold leads to
a stable coexistence of cooperators, defectors and players adopting the
switching strategy in a well-mixed population, and this regardless of whether
the pool-based or the peer-based switching strategy is introduced. Moreover, we
show that the pure exclusion strategy alone is able to evoke a limit cycle
attractor in the evolutionary dynamics, such that cooperation can coexist with
other strategies.
