Probabilistic punishment on free riders in threshold public goods games

We focus our attention on the effectiveness of punishing defectors in the evolution of cooperation in rational populations within the threshold public goods game models. We establish two scenarios: defectors will suffer possible punishment whether the game succeeds or not, and defectors will incur punishment only when game fails. A key observation of this paper is that given this assumption, punishing free riders can significantly influence the evolution outcomes, and the results are driven by the specific components of the punishing rule. Particularly, probably punishing defectors always, not only when game fails, can be more effective for maintaining public cooperation and leads to higher total contributions. Intriguingly when defectors face punishment only when game fails, the spread and domination of cooperators can not be ensured even by sufficiently large punishment, which already brings payoff advantages of defectors over cooperators. Further, cooperators are best supported by the large punishment on defectors, and then dominate and stabilize in the population, under the premise that defectors always incur punishment regardless of whether the game ends successfully or not. We hope this work provides some intriguing hints or insights on investigating the collective behaviors by the aid of punishment mechanism.