We study subgame ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}-maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}-maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}. First, we provide necessary and sufficient conditions for a strategy to be a subgame ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}-maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}-maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function ϕ∗\documentclass[12pt]{minimal}
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\begin{document}$$\phi ^*$$\end{document} with the following property: if a player has a subgame ϕ∗\documentclass[12pt]{minimal}
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\begin{document}$$\phi ^*$$\end{document}-maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}-maxmin strategy for every positive tolerance function ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} is equivalent to the existence of a subgame maxmin strategy.
