  This paper revisits the well-established relationship between the modal mu
calculus and parity games to show that it is even more robust than previously
known. It addresses the question of whether the descriptive complexity of modal
mu model-checking games, previously known to depend on the syntactic complexity
of a formula, depends in fact on its semantic complexity. It shows that up to
formulas of semantic co-B\"uchi complexity, the descriptive complexity of their
model-checking games coincides exactly with their semantic complexity. Beyond
co-B\"uchi, the descriptive complexity of the model-checking parity games of a
formula is shown to be an upper bound on its semantic complexity; whether it is
also a lower bound remains an open question.
