We present a new multi-valued monitoring approach for linear-time temporal logic that classifies trace prefixes not only according to the existence of correct and erroneous continuations, but also according to the strategic power of the system and its environment to avoid or enforce a violation of the specification. We classify the monitoring status into four levels: (1) the worst case is a
, where no continuation satisfies the specification any more; (2)
means that the environment can force the system to violate the specification; (3)
means that the system can enforce that the specification is satisfied; (4) the best case,
, indicates that all possible continuations satisfy the specification. Because our approach recognizes situations where the system cannot avoid a violation even though there may still be continuations in which the specification is satisfied, our approach detects errors earlier, and it detects errors that are missed by less detailed classifications. We give an asymptotically optimal construction of multi-valued monitoring automata based on parity games.