Generalizations of the Choquet theory of integral representation to the setting of vector-valued function spaces goes back to 1980s. However, a satisfactory theory of uniqueness of representing measures which would be analogous to the scalar case is still missing. I will present recent joint results with Jiří Spurný in this direction. In particular, it turns out that there are two natural directions of possible generalization - we call them weak simpliciality and vector simpliciality. In general these two notions are incomparable, but for function spaces containing constants vector simpliciality is strictly stronger. I will focus on similarities and differences from the scalar theory.