Symmetry vs. complexity in proving the Muller-Satterthwaite theorem

In this short note, we first provide two rather straightforward proofs for the Muller - Satterthwaite theorem in the baseline cases of 2 person 3 alternatives, and 2 person n ≥ 3 alternatives. We also show that it suffices to prove the result in the special case of 3 alternatives (with arbitrary N individuals) as it then can easily be extended to the general case. We then prove the result in the decisive case of 3 alternatives (with arbitrary N individuals) by induction on N.