  We introduce a natural notion of limit-deterministic parity automata and
present a method that uses such automata to construct satisfiability games for
the weakly aconjunctive fragment of the $\mu$-calculus. To this end we devise a
method that determinizes limit-deterministic parity automata of size $n$ with
$k$ priorities through limit-deterministic B\"uchi automata to deterministic
parity automata of size $\mathcal{O}((nk)!)$ and with $\mathcal{O}(nk)$
priorities. The construction relies on limit-determinism to avoid the full
complexity of the Safra/Piterman-construction by using partial permutations of
states in place of Safra-Trees. By showing that limit-deterministic parity
automata can be used to recognize unsuccessful branches in pre-tableaux for the
weakly aconjunctive $\mu$-calculus, we obtain satisfiability games of size
$\mathcal{O}((nk)!)$ with $\mathcal{O}(nk)$ priorities for weakly aconjunctive
input formulas of size $n$ and alternation-depth $k$. A prototypical
implementation that employs a tableau-based global caching algorithm to solve
these games on-the-fly shows promising initial results.
