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We present a framework for abstracting two-player turn-based games that preserves any formula of the alternating 驴-calculus (AMC). Unlike traditional conservativ abstractions which can only prove the existence of winning strategies for only one of the players, our framework is based on 3-valued games, and it can be used to prove and disprove formulas of AMC including arbitrarily nested strategy quantifiers. Our main contributions are as follows.We define abstract 3-valued games and an alternating refinement relation on these that preserves winning strategies fo both players.We provide a logical characterization of the alternating refinement relation. We show that our abstractions are as precise as can be via completeness results.We present AMC formulas that solve 3-valued games with w-regular objectives, and we show that such games are determined in a 3-valued sense.We also discuss the complexity of model checking arbitrary AMC formulas on 3-valued games and of checking alternating refinement.