Stochastic punishment by authorized third parties in a public goods game: the role of reputation-based migration

This paper examines how delegating a third-party punishment can sustain cooperation in large-scale, mobile societies facing public goods dilemmas. Specifically, we explore how the effectiveness of delegated punishment changes in more realistic scenarios (e.g., free-riders are not equally likely to be punished)—when the sanctions are neither certain nor immediate. Additionally, we examine whether informal sanctions, such as reputation-based "voting with one's feet", can complement or substitute formal punishment mechanisms in promoting and maintaining cooperation, especially in the presence of mutant defectors.

We propose an agent-based model of authorized third-party punishment that is non-deterministic and non-immediate in unstructured populations (e.g., during the Great Migration). Three types of players—unconditional cooperators, unconditional defectors, and conditional cooperators—are matched in a one-shot public goods game, after which point their contributions are mapped onto reputation scores that accumulate continuously up to a fixed upper bound. The reputation scores guide players’ movements and determine the intensity of the punishment they receive. Agents update their strategies via payoff-based imitation to capture social learning dynamics. We assess the cooperation across scenarios defined by three factors—punishment type (none/deterministic/stochastic), migration rule (random vs. voting with one’s feet), and mutation status (presence vs. absence).

The experimental results demonstrate that reputation-based “voting with one’s feet” migration significantly enhances the cooperators’ spatial clustering and survival probability. This spatial clustering reduces the enforcement burden on third-party punishers, making it easier to sustain cooperation than under a random migration scenario. However, in the absence of any punishment, cooperation collapses rapidly regardless of migration mode. Without mutation, deterministic punishment maintains higher average contributions under both random and reputation-based migration. But under the reputation-based “voting with one’s feet” scenario, once mutation is introduced, deterministic punishment exhibits a lower tolerance for internal variations within the core cluster than that under stochastic punishment, causing the cooperation to collapse to a low level.