  We study alternating parity good-for-games (GFG) automata, i.e., alternating
parity automata where both conjunctive and disjunctive choices can be resolved
in an online manner, without knowledge of the suffix of the input word still to
be read.
  We show that they can be exponentially more succinct than both their
nondeterministic and universal counterparts. Furthermore, we present a single
exponential determinisation procedure and an Exptime upper bound to the problem
of recognising whether an alternating automaton is GFG. We also study the
complexity of deciding "half-GFGness", a property specific to alternating
automata that only requires nondeterministic choices to be resolved in an
online manner. We show that this problem is PSpace-hard already for alternating
automata on finite words.
