Altruism in multiplayer snowdrift games with threshold and punishment

The puzzle of cooperation attracts broader concerns of the scientific community nowadays. Here we adopt an extra mechanism of punishment in the framework of a threshold multiple-player snowdrift game employed as the scenario for the cooperation problem. Two scenarios are considered: defectors will suffer punishment regardless of the game results, and defectors will incur punishment only when the game fails. We show by analysis 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, punishing defectors always, not only when the game fails, can be more effective for maintaining public cooperation in multi-player systems. Intriguingly larger thresholds of the game provide a more favorable scenario for the coexistence of the cooperators and defectors under a broad value range of parameters. 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