Mutual Exclusion of Influence: a New Proof of the Gibbard--Satterthwaite Theorem and the Synergy Obstruction

This paper gives a new proof of the Gibbard–Satterthwaite theorem and develops an information-theoretic framework characterizing dictatorship as the unique entropy profile compatible with strategy-proofness, surjectivity, and three or more alternatives.The proof’s engine is the Mutual Exclusion of Influence: two voters cannot both control the same alternative pair while ranking it differently. Mutual exclusion combined with a transition sequence identifies a decisive voter without constructing a pivotal voter; contagion then extends decisiveness to all pairs and yields dictatorship.The information-theoretic framework produces several new objects: an exact entropy computation, a fiber decomposition of shared control, and the entropy concentration theorem showing dictatorship has the unique influence profile (log |X|, 0, …, 0). Zero overlap — two voters never simultaneously controlling the same pair — emerges as a corollary of concentration rather than a prerequisite. The Gibbard–Satterthwaite theorem is recast as a no-synergy theorem: strategy-proofness with full range forces informationally independent influence. The paper closes by identifying the precise obstruction to a purely entropy-theoretic proof.