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)} · 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 considerably 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.

^* This work is a part of projects CUTACOMBS (Ma. Pilipczuk), BOBR (Mi. Pilipczuk), and VAMOS (K. S. Thejaswini) that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, grant agreements No 714704, 948057, and 101020093, respectively. Ma. Pilipczuk is also partially supported by Polish National Science Centre SONATA BIS-12 grant number 2022/46/E/ST6/00143.

^† The link to the arXiv version of this paper is https://arxiv.org/abs/2310.20433