This paper studies a special kind of equilibrium termed as "balanced equilibrium" which arises in the power allocation game defined in \cite{allocation}. In equilibrium, each country in antagonism has to use all of its own power to counteract received threats, and the "threats" made to each adversary just balance out the threats received from that adversary. This paper establishes conditions on different types of networked international environments in order for this equilibrium to exist. The paper also links the existence of this type of equilibrium on structurally balanced graphs to the Hall's Maximum Matching problem and the Max Flow problem.