  Parity games play a central role in model checking and satisfiability
checking. Solving parity games is computationally expensive, among others due
to the size of the games, which, for model checking problems, can easily
contain $10^9$ vertices or beyond. Equivalence relations can be used to reduce
the size of a parity game, thereby potentially alleviating part of the
computational burden. We reconsider (governed) bisimulation and (governed)
stuttering bisimulation, and we give detailed proofs that these relations are
equivalences, have unique quotients and they approximate the winning regions of
parity games. Furthermore, we present game-based characterisations of these
relations. Using these characterisations our equivalences are compared to
relations for parity games that can be found in the literature, such as direct
simulation equivalence and delayed simulation equivalence. To complete the
overview we develop coinductive characterisations of direct- and delayed
simulation equivalence and we establish a lattice of equivalences for parity
games.
