Consider concurrent, infinite duration, two-player win/lose games played on graphs. If the winning condition satisfies some simple requirement, the existence of Player 1 winning (finitememory) strategies is equivalent to the existence of winning (finite-memory) strategies in finitely many derived one-player games. Several classical winning conditions satisfy this simple requirement. Under an additional requirement on the winning condition, the non-existence of Player 1 winning strategies from

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... l vertices is equivalent to the existence of Player 2 stochastic strategies almost-sure winning from all vertices. Only few classical winning conditions satisfy this additional requirement, but a fairness variant of omega-regular languages does. ACM Subject Classification Theory of computation → Formal languages and automata theory, Theory of computation → Verification by model checking, Software and its engineering → Software verification, Software and its engineering → Model checking Keywords and phrases Two-player win/lose, graph, infinite duration, abstract winning condition Digital Object Identifier 10.4230/LIPIcs.MFCS.2018.40 Acknowledgements Referees (of several conferences) and Sasha Rubin made helpful comments. A simplification of the proof of Lemma 11 was triggered by a conversation with Arno Pauly.