 The game formalism just introduced is not detailed enough to characterize the kind of sequential computation at higher types needed to achieve definability. For this purpose, a richer structure on games is needed, making them closer to dialogue games between Proponent and Opponent exchanging questions and answers. This allows to formulate restrictions on plays by matching answers with the corresponding questions in an appropriate manner. The strategies for this refined game notion, that we study next essentially through examples, will yield a richer notion of morphism between games, allowing to make finer distinctions of a computational nature needed for intensionally fully abstract model of PCF, following essentially the approach of Hyland &amp; Ong (2000) drawing also material from Abramsky &amp; McCusker (1999) and Curien (2006).