We study the complexity of computing or approximating refinements of Nash
equilibrium for a given finite n-player extensive form game of perfect recall
(EFGPR), where n >= 3. Our results apply to a number of well-studied
refinements, including sequential (SE), extensive-form perfect (PE), and
quasi-perfect equilibrium (QPE). These refine Nash and subgame-perfect
equilibrium. Of these, the most refined notions are PE and QPE. By classic
results, all these equilibria exist in any EFGPR. We show that, for all these
notions of equilibrium, approximating an equilibrium for a given EFGPR, to
within a given desired precision, is FIXP_a-complete. We also consider the
complexity of corresponding "almost" equilibrium notions, and show that they
are PPAD-complete. In particular, we define "delta-almost
epsilon-(quasi-)perfect" equilibrium, and show computing one is PPAD-complete.
We show these notions refine "delta-almost subgame-perfect equilibrium" for
EFGPRs, which is PPAD-complete. Thus, approximating one such (delta-almost)
equilibrium for n-player EFGPRs, n >= 3, is P-time equivalent to approximating
a (delta-almost) NE for a normal form game (NFG) with 3 or more players. NFGs
are trivially encodable as EFGPRs without blowup in size. Thus our results
extend the celebrated complexity results for NFGs to refinements of equilibrium
in the more general setting of EFGPRs. For 2-player EFGPRs, analogous
complexity results follow from the algorithms of Koller, Megiddo, and von
Stengel (1996), von Stengel, van den Elzen, and Talman (2002), and Miltersen
and Soerensen (2010). For n-player EFGPRs, an analogous result for Nash and
subgame-perfect equilibrium was given by Daskalakis, Fabrikant, and
Papadimitriou (2006). However, no analogous results were known for the more
refined notions of equilibrium for EFGPRs with 3 or more players
