  Empirical studies have shown that individuals' behaviors are largely
influenced by social conformity, including punishment. However, a
coevolutionary theoretical framework that takes into account effects of
conformity on individuals' punishment behaviors has not been put forward yet.
Herein we propose a coevolutionary game model to extend the theory of
cooperation with conformity in spatial public goods game by considering pool
punishment, as well as two converse feedback modes of conformity that strongly
affect cooperators' punishment behaviors. We focus on how different parameters
and spatial structures govern evolutionary dynamics on three different kinds of
networks by employing mean-field analysis based on replicator dynamics and
Monte Carlo simulations. On regular lattices, defectors are overall extincted
since cooperators, especially selfish cooperators, have great evolutionary
advantages due to strong network reciprocity, and at the same time the number
of altruistic cooperators decays. Conversely, abundant shortcuts in regular
random networks lead to the prevalence of altruistic cooperators, but
cooperators suffer from free-riding behaviors of defectors. Of particular
interest, we find that small-world topology can simultaneously help cooperators
successfully outperform defectors by means of strong network reciprocity, and
enable rich contacting opportunities with defectors to facilitate the expansion
of altruistic cooperators. Therefore, we clarify that small world is the
optimal topology subject to the dominance of altruistic cooperators.
