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Empirical studies have shown that individuals' behaviors are largelyinfluenced by social conformity, including punishment. However, acoevolutionary theoretical framework that takes into account effects ofconformity on individuals' punishment behaviors has not been put forward yet.Herein we propose a coevolutionary game model to extend the theory ofcooperation with conformity in spatial public goods game by considering poolpunishment, as well as two converse feedback modes of conformity that stronglyaffect cooperators' punishment behaviors. We focus on how different parametersand spatial structures govern evolutionary dynamics on three different kinds ofnetworks by employing mean-field analysis based on replicator dynamics andMonte Carlo simulations. On regular lattices, defectors are overall extinctedsince cooperators, especially selfish cooperators, have great evolutionaryadvantages due to strong network reciprocity, and at the same time the numberof altruistic cooperators decays. Conversely, abundant shortcuts in regularrandom networks lead to the prevalence of altruistic cooperators, butcooperators suffer from free-riding behaviors of defectors. Of particularinterest, we find that small-world topology can simultaneously help cooperatorssuccessfully outperform defectors by means of strong network reciprocity, andenable rich contacting opportunities with defectors to facilitate the expansionof altruistic cooperators. Therefore, we clarify that small world is theoptimal topology subject to the dominance of altruistic cooperators.

Small world yields optimal public goods in presence of both altruistic and selfish cooperators