I think I have seen papers in which participants performed simple forced-choice tasks, where "don't know" was one of the possible choices. So for example, a signal detection task in which the response alternatives are "no", "don't know" and "yes". It seems to me that it should be fairly easy to construct an alternative signal detection theory (SDT) model, and therefore derived, for this kind of task.
Can anyone point to any papers that have derived such a model, or failing that, any papers that report experiments with "don't know" options? Thanks!