This paper considers an evolving network where each vertex of the network has a target number of edges. If the number of edges connected to a vertex exceeds its target number, the vertex wishes to remove one of the edges in order to reach the optimal number (the vertex in such a position is said to be a Breaker). Similarly, if the number of edges of a vertex is less than its target number, it wishes to build a new edge (a Joiner). If the current number equals the target number, the vertex will do nothing (a Neutral). At each time point a vertex is chosen randomly such that it can modify its connections. The authors introduce a game-theoretical model which allows vertices to break or build edges strategically, instead of randomly. Examples as well as general results are studied with a focus on the types of vertices that are: always Neutral, never Joiners, never Breakers, and the others. A special case where vertices have monotonic decreasing target numbers is studied in detail.