  We give a simple proof that assuming the Exponential Time Hypothesis (ETH),
determining the winner of a Rabin game cannot be done in time $2^{o(k \log k)}
\cdot n^{O(1)}$, where $k$ is the number of pairs of vertex subsets involved in
the winning condition and $n$ is the vertex count of the game graph. While this
result follows from the lower bounds provided by Calude et al [SIAM J. Comp.
2022], our reduction is simpler and arguably provides more insight into the
complexity of the problem. In fact, the analogous lower bounds discussed by
Calude et al, for solving Muller games and multidimensional parity games,
follow as simple corollaries of our approach. Our reduction also highlights the
usefulness of a certain pivot problem -- Permutation SAT -- which may be of
independent interest.
