The long-run average cost control problem for discrete time Markov chains on a countable state space is studied in a very general framework. Necessary and sufficient conditions for optimality in terms of the dynamic programming equations are given when an optimal stable stationary strategy is known to exist (e.g., for the situations studied in [5]). A characterization of the desired solution of the dynamic programming equations is given in a special case. Also included is a novel convex analytic argument for deducing the existence of an optimal stable stationary strategy when that of a randomized one is known.