We consider a zero-sum average cost stochastic game with an unbounded lower semi-continuous cost function, and by using the contraction property for the average case we give sufficient conditions for which there exists a minimax stationary strategy. Also, we formulate a minimax inventory model as a stochastic game and show that for any \(\epsilon >0\) there exists an \(\epsilon\)-minimax random (s,S) ordering policy, which is a modification of (s,S) ordering policy, under some weak conditions.