  Automata models between determinism and nondeterminism/alternations can
retain some of the algorithmic properties of deterministic automata while
enjoying some of the expressiveness and succinctness of nondeterminism. We
study three closely related such models -- history determinism, good for
gameness and determinisability by pruning -- on quantitative automata. While in
the Boolean setting, history determinism and good for gameness coincide, we
show that this is no longer the case in the quantitative setting: good for
gameness is broader than history determinism, and coincides with a relaxed
version of it, defined with respect to thresholds. We further identify criteria
in which history determinism, which is generally broader than determinisability
by pruning, coincides with it, which we then apply to typical quantitative
automata types. As a key application of good for games and history
deterministic automata is synthesis, we clarify the relationship between the
two notions and various quantitative synthesis problems. We show that
good-for-games automata are central for "global" (classical) synthesis, while
"local" (good-enough) synthesis reduces to deciding whether a nondeterministic
automaton is history deterministic.
