How Deterministic are Good-For-Games Automata? 
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In {\em good for games\/} (GFG) automata~\cite{HP06}, it is possible to resolve nondeterminism in a way that only depends on the past and still accepts all the words in the language. Formally, a nondeterministic automaton $\A$ over an alphabet $\Sigma$ is GFG if there is a strategy $f$ that maps each finite word $u \in \Sigma^*$ to the transition to be taken after $u$ is read; and following $f$ results in accepting runs for all the words in the language of $\A$. 
We continue the ongoing effort of studying the power of nondeterminism in GFG automata. 
Initial indications have hinted that every GFG automaton embodies a deterministic one. Today we know that this is not the case, and in fact GFG automata may be exponentially more succinct than deterministic ones. We focus on {\em typeness} in automata on infinite words, namely the question of whether an automaton with a certain acceptance condition has an equivalent automaton with a weaker acceptance condition on the same structure. For deterministic automata, typeness holds: when a deterministic Rabin automaton recognizes a language that can be defined with a deterministic B\"uchi automaton, then it is guaranteed that the Rabin condition can be replaced by a B\"uchi condition, yielding an equivalent automaton on the same structure; similarly for the dual case of deterministic Streett automata that recognize a language that can be defined with a deterministic co-B\"uchi automaton. For nondeterministic automata, typeness does not hold. We study typeness for GFG automata and show that the picture is similar to the one of deterministic automata. In particular, our results imply that when Rabin or Streett GFG automata have equivalent B\"uchi or Co-B\"uchi GFG automata, respectively, then such equivalent automata can be defined on a substructure of the original automata. We also strengthen the relation between tightness and determinism, and study further practical applications of the tightness results.