When a dyad becomes a triad, the third member can gain tremendous power. which role involves playing one member against the other for his or her own advantage?

The role that involves playing one member against the other for his or her own advantage would be: the third that rejoices

When a team only consist of two members, whenever there is conflict, only those two members will experience the result of the conflict.

When a team consists of three members on the other hand, if two members engage in a conflict, the third member who did not involve usually the one that gain the most benefit. This occurrence is called the third that rejoices.