The dynamics of corruption under an optional external supervision service

External supervision services play an important role in combating corruption by detecting potential collusive bribery. This work aims at studying the dynamics of collusive bribery when participants have the option of engaging the external supervision services. To do so, we construct a basic model where collusive bribery can happen between the defecting participants who aim to escape from a punishment by offering a bribe to rule enforcers who monitor interactions among all participants. Among rule enforcers, only the corrupt ones accept the bribe and ignore the violations. The cooperative participants can engage the external supervision service at a certain cost to avoid the risk of potential collusive bribery. Under the framework of evolutionary game theory, we find that a higher initial fraction of honest enforcers is more likely to lead to a trusting cooperating equilibrium. We also find that, when allowing random exploration of available strategies, increasing the exploration rate of rule enforcers is effective in combating corruption for both infinite and finite populations. Lastly, we find that minimizing the cost of external supervision services is not always good. When the system evolves into a cooperating equilibrium, a low cost of external supervision service induces unnecessary costs of seeking external supervision. When the strategy profiles exhibit stable oscillations, there exists an optimal cost of external supervision, considering the trade-off between minimizing the chance of exposing cooperative participants to collusive bribery and strengthening the punishment on the corrupt enforcers. Premised on the results, we discuss practical management suggestions.