Public and private reputation systems for limiting centralized corruption

Corruption is widely recognized as a major challenge in human institutions. To prevent its spread, measures such as increasing transparency have attracted growing attention. Here, we develop an evolutionary game-theoretic framework with a reputation mechanism to address centralized corruption and sustain cooperation. In our framework, players participate in a donation game monitored by umpires, who enforce centralized punishment on free riders to support cooperation. However, the possibility of bribery by defectors creates incentives for umpires to abuse their authority, thereby undermining cooperation. To counter this, we construct public and private reputation systems that allow players to condition their participation decisions on the reputations of assigned umpires. Theoretical analysis based on our dynamical model, together with simulations, shows that a public reputation system, in which honest umpires receive good reputations and corrupt ones receive bad reputations, effectively reduces corruption. Interestingly, under certain social norms, private reputation systems can outperform public ones, while also offering an advantage in avoiding higher-order corruption problems inherent in centralized systems. Overall, our results highlight the potential of bottom-up transparency mechanisms in centralized institutions and demonstrate that the effectiveness of reputation systems critically depends on the careful design of social norms.