Coalitional Structure of the Muller-Satterthwaite Theorem

The Muller-Satterthwaite theorem states that social choice functions that satisfy unanimity and monotonicity are also dictatorial. Unlike Arrow’s theorem, it does not assume that the function produces a transitive social ordering. Wilson showed that a voting process under Arrow’s conditions can be interpreted as a strong and proper simple game—as defined by von Neumann and Morgenstern. We show this to be the case also under the Muller-Satterthwaite conditions. Our main theorem, which we prove using two very different approaches—one partially automated and one manual—is that a winning coalition coincides with a blocking coalition under unanimity and monotonicity. This might be of independent interest. We also show that this can be used to generate a short proof of the Muller-Satterthwaite theorem.