This paper considers two-person zero-sum sequential games with finite state and action spaces. We consider the pair of functional equations (f.e.) that arises in the undiscounted infinite stage model, and show that a certain class of successive approximation schemes is guaranteed to converge to a solution pair whenever an equilibrium policy with respect to the average return per unit time criterion (AEP) exists. Existence of the latter thus implies the existence of a solution to this pair of f.e. whereas the converse implication is shown only to hold under special circumstances. In addition to this pair of f.e., a complete sequence of f.e. has to be considered when analyzing more sensitive optimality criteria that make further selections within the class of AEPs. A number of characterizations and interdependences between the existence of solutions to the f.e. and existence of stationary sensitive optimal equilibrium policies are obtained.