  Calude et al. have recently shown that parity games can be solved in
quasi-polynomial time, a landmark result that has led to a number of approaches
with quasi-polynomial complexity. Jurdinski and Lasic have further improved the
precise complexity of parity games, especially when the number of priorities is
low (logarithmic in the number of positions). Both of these algorithms belong
to a class of game solving techniques now often called separating automata:
deterministic automata that can be used as witness automata to decide the
winner in parity games up to a given number of states and colours. We suggest a
number of adjustments to the approach of Calude et al. that lead to smaller
statespaces. These include and improve over those earlier introduced by
Fearnley et al.
  We identify two of them that, together, lead to a statespace of exactly the
same size Jurdzinski and Lasic's concise progress measures, which currently
hold the crown as smallest statespace. The remaining improvements, hence, lead
to a further reduction in the size of the statespace, making our approach the
most succinct progress measures available for parity games.
