 The moves of the refined game notion will be either questions or answers played by Proponent or by the Opponent. We have then four classes of moves each represented by a kind of (round or square) bracket: Proponent’s questions ‘(’; Opponent’s answers ‘)’; Opponent’s questions ‘[’; and Proponent’s answer ‘]’. This labeling of the moves subsumes under the usual well-formedness criterion for bracket sequences, at one time: the alternation between Proponent and Opponent, the fact that Opponent is the first to move and that each answer of a player answers a unique question of the partner. This is not enough, however: a further justification structure on questions and answers is needed to discipline the nesting of (sub)dialogues in the evaluation of higher-order functions, allowing to characterize the well-bracketed strategies. Consider now the strategy in \((G^{11}_\texttt{bool}\to G^{12}_\texttt{bool}\to G^1_\texttt{bool}) \to G_\texttt{bool}\), described informally using a labeling of the copies of \(G_\texttt{bool}\) as shown: