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.
Federgruen, Awi. "On the functional equations in undiscounted and sensitive discounted stochastic games." Mathematical Methods of Operations Research 24, no. 7 (December 1980): 243-262.
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