  We study alternating good-for-games (GFG) automata, i.e., alternating
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 lift many results
from nondeterministic parity GFG automata to alternating ones: a single
exponential determinisation procedure, an Exptime upper bound to the GFGness
problem, a PTime algorithm for the GFGness problem of weak automata, and a
reduction from a positive solution to the $G_2$ conjecture to a PTime algorithm
for the GFGness problem of parity automata with a fixed index. The $G_2$
conjecture states that a nondeterministic parity automaton A is GFG if and only
if a token game, known as the $G_2$ game, played on A is won by the first
player. So far, it had only been proved for B\"uchi automata; we provide
further evidence for it by proving it for coB\"uchi automata. 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 strictly more difficult than
GFGness check, already for alternating automata on finite words.
