In this paper, we use computer simulation to better understand mixed-initiative dialogues. We compare two models of mixed-initiative: unrestricted initiative, where either participant can take over control at any point; and restricted initiative where one participant keeps control and the other plays a secondary role, but greater than what single-initiative allows. We find that restricted initiative results in similar solution quality as unrestricted, less communication effort, and similar or less reasoning effort. These results agree with our empirical studies on human-human dialogues, in which we find that participants seem to follow the restricted initiative model.