Generalizing the Gibbard-Satterthwaite theorem: partial preferences, the degree of manipulation, and multi-valuedness

The Gibbard–Satterthwaite (GS) theorem is generalized in three ways: First, it is proved that the theorem is still valid when individual preferences belong to a convenient class of partial preferences; second, it is shown that every non-dictatorial surjective social choice function (SCF) is not only manipulable, but it can be manipulated in such a way that some individual obtains either his best or second best alternative; third, we prove a variant of the theorem where the outcomes of the SCF are subsets of the set of alternatives of an a priori fixed size. In addition, all results are proved not only for finite, but also for countably infinite sets of alternatives