Key players in bullying networks

Individuals are embedded in a network of relationships and they can be victims, bystanders, or perpetrators of bullying and harassment. Each individual decides noncooperatively how much effort to exert in preventing misbehavior. Each individual's optimal effort depends on the contextual effect, the social multiplier effect and the social conformity effect. We characterize the Nash equilibrium and we derive an inter-centrality measure for finding the key player who once isolated increases the most the aggregate effort. An individual is more likely to be the key player if she is influencing many other individuals, she is exerting a low effort because of her characteristics, and her neighbors are strongly influenced by her. The key player policy increases substantially the aggregate effort and the targeted player should never be selected randomly. The key player is likely to remain the key player in presence of social workers except if she is becoming much less influential due to her closeness to social workers. Finally, we consider alternative policies (e.g. training bystanders for helping victims) and compare them to the policy of isolating the key player.