The dynamics of corruption under an optional external supervision service

•We construct a game model to study the influence of external supervision services on combating corruption.•We find introducing the external supervision service can contain the collusive bribery and promote cooperation.•A higher exploration rate of rule enforcers makes players more likely to evolve into the dominance of trusting cooperation.•If a dominance of trusting cooperation is achieved, a higher cost of external supervision services reduces its unnecessary engagements.•If cyclic dominance emerges, the cost of the external supervision service has a tradeoff between combating corruption and protecting trusting cooperators.