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This paper describes the mental structures called shared mental models, which are heavily related to group problem solving and cognition, and reports how they are related to theatrical improvisation based on our empirical findings. We have conducted a series of studies on real life improvisers aimed at uncovering the underlying cognition involved in improvisation, with the end goal of having a clear understanding of how to build improvisational synthetic characters. We describe cognitive divergence, when improvisers have conflicting mental models of what is occurring on stage, and cognitive convergence, which is the process of resolving such conflicts within the performance. These findings are supported by examples from our study and are used to make conclusions about improvisational synthetic character design.