  We study the complexity of problems related to subgame-perfect equilibria
(SPEs) in infinite duration non zero-sum multiplayer games played on finite
graphs with parity objectives. We present new complexity results that close
gaps in the literature. Our techniques are based on a recent characterization
of SPEs in prefix-independent games that is grounded on the notions of
requirements and negotiation, and according to which the plays supported by
SPEs are exactly the plays consistent with the requirement that is the least
fixed point of the negotiation function. The new results are as follows. First,
checking that a given requirement is a fixed point of the negotiation function
is an NP-complete problem. Second, we show that the SPE constrained existence
problem is NP-complete, this problem was previously known to be ExpTime-easy
and NP-hard. Third, the SPE constrained existence problem is fixed-parameter
tractable when the number of players and of colors are parameters. Fourth,
deciding whether some requirement is the least fixed point of the negotiation
function is complete for the second level of the Boolean hierarchy. Finally,
the SPE-verification problem -- that is, the problem of deciding whether there
exists a play supported by a SPE that satisfies some LTL formula -- is
PSpace-complete, this problem was known to be ExpTime-easy and PSpace-hard.
