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AbstractThis article provides sufficient conditions for the existence of solutions for two‐person zero‐sum games with inf/sup‐compact payoff functions and with possibly noncompact decision sets for both players. Payoff functions may be unbounded, and we do not assume any convexity/concavity‐type conditions. For such games expected payoff may not exist for some pairs of strategies. The results of this article imply several classic facts. The article also provides sufficient conditions for the existence of a value and solutions for each player. The results of this article are illustrated with the number guessing game.