  A central theme in social choice theory is that of impossibility theorems,
such as Arrow's theorem and the Gibbard-Satterthwaite theorem, which state that
under certain natural constraints, social choice mechanisms are impossible to
construct. In recent years, beginning in Kalai`01, much work has been done in
finding \textit{robust} versions of these theorems, showing "approximate"
impossibility remains even when most, but not all, of the constraints are
satisfied. We study a spectrum of settings between the case where society
chooses a single outcome (\'a-la-Gibbard-Satterthwaite) and the choice of a
complete order (as in Arrow's theorem). We use algebraic techniques,
specifically representation theory of the symmetric group, and also prove
robust versions of the theorems that we state. Our relaxations of the
constraints involve relaxing of a version of "independence of irrelevant
alternatives", rather than relaxing the demand of a transitive outcome, as is
done in most other robustness results.
