Two New Proofs of the Gibbard--Satterthwaite Theorem

We present two new proofs of the Gibbard–Satterthwaite theorem, the foundational result in social choice theory establishing that every surjective, strategy-proof social choice function on three or more alternatives is dictatorial. Both proofs share a common engine—the Mutual Exclusion of Influence (a six-line theorem showing that two voters cannot both control the same alternative pair at a shared profile while ranking the pair differently)—but diverge in how they derive dictatorship from this principle.

The first proof is purely combinatorial: mutual exclusion combined with a transition sequence identifies a uniquely decisive voter without constructing a classical pivotal voter. The second proof is information-theoretic: under the uniform distribution on preference profiles, strategy-proofness yields an exact identity relating conditional outcome entropy to option-set size. The zero-overlap theorem—a measure-theoretic consequence of mutual exclusion—forces influence entropy to concentrate entirely in a single voter, characterizing dictatorship as the unique entropy profile (log |X|, 0, …, 0) compatible with strategy-proofness and surjectivity.

To our knowledge, the second proof is the first to establish the Gibbard–Satterthwaite theorem via Shannon-type information-theoretic quantities. The Mutual Exclusion Theorem itself is new and replaces the pivotal-voter construction across all four established proof routes with a single structural principle. Both proofs connect to the Adversarial Aggregation Channel (AAC) framework, in which the influence entropy corresponds to adversarial sub-channel capacity and the mutual exclusion principle instantiates a channel-capacity conservation law.