The preference of controversial impossibility theorems over boring characterisation and classification theorems is not necessarily helpful.

A simple randomised procedure compatible with Arrow's theorem would be to choose an individual randomly, and declare his preference order to be the single overall preference order. Such a procedure could still be unfair, if certain individuals would be chosen with a higher probability than others.

Note however that Arrow's theorem is not sufficient for classifying the possible fair randomised procedures (or at least it is not clear to me how that classification could be derived as a corollary), as highlighted by the randomised quick sort procedure outlined in [this (my) question][1]. (Yes, this answer is motivated by my own question, and especially the opening line and the following last paragraph are based on opinions instead of facts.)

As these two examples of (fair) randomised procedures demonstrate, there cannot be a truly general voting impossibility theorem. Instead of an impossibility theorem, there are various characterisation and classification theorems (like Harsanyi's utilitarian theorem cited in Qiaochu Yuan's answer). But because of the focus on impossibility theorems, not all of them have been worked out yet.


  [1]: https://math.stackexchange.com/questions/2032014/are-there-corollaries-or-generalised-versions-of-arrows-theorem-covering-obvi