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Many two-sided matching situations involve multiperiod interaction. Traditional cooperative solutions, such as pairwise stability or the core, often identify unintuitive outcomes (or are empty) when applied to such markets. As an alternative, this study proposes the criterion of perfect a-stability. An outcome is perfect a-stable if no coalition prefers an alternative assignment in any period that is superior for all plausible market continuations. The solution posits that agents have foresight, but cautiously evaluate possible future outcomes. A perfect a-stable matching exists, even when assignments are intertemporal complements. The perfect a-core, a stronger solution, is nonempty under standard regularity conditions, such as history-independence. Our analysis extends to markets with arrivals and departures, transfers, and many-to-one assignments.