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 (ME) theorem: a six-line result showing that two voters cannot simultaneously control the same alternative pair at the same profile while ranking that pair differently. Shared control forces agreement. Combined with a transition sequence, ME identifies a decisive voter without constructing a pivotal voter in the classical sense—replacing the pivotal-voter construction used in all four existing proof routes (via Arrow, combinatorial induction, Muller–Satterthwaite, and option sets) with a structural exclusion principle. Standard contagion then extends decisiveness to all pairs, yielding dictatorship. The information-theoretic framework built on this proof produces several new objects: An exact computation showing that the conditional entropy of the outcome given all other voters' rankings equals the logarithm of the option-set size. A fiber decomposition of shared control, revealing two regimes: exclusion (at most one voter controls a pair) and forced agreement (both voters control the pair but must rank it identically). Zero overlap—the result that two voters never simultaneously control the same pair—emerges as a corollary of entropy concentration, not a prerequisite. Dictatorship is characterized as the unique entropy profile (log|X|, 0, …, 0). The synergy analysis identifies forced agreement with synergistic influence: it holds if and only if the sum of individual influence entropies exceeds outcome entropy, equivalently, when voters' rankings are outcome-conditionally dependent. For two voters and three alternatives, forced agreement implies range locking (confinement to a two-element subset), and cross-ownership impossibility shows that distinct voters cannot exclusively own different alternative pairs. Together, these yield an alternative proof of the Gibbard–Satterthwaite theorem for the two-voter, three-alternative case that bypasses the transition sequence entirely. The No-Synergy Theorem shows that the Gibbard–Satterthwaite theorem is equivalent to a no-synergy statement: strategy-proofness with full range forces informationally independent influence. The paper closes by identifying the precise obstruction to a purely entropy-theoretic proof—forced agreement is consistent with all standard Shannon-type inequalities, including non-negativity, subadditivity, the data processing inequality, and all polymatroidal constraints—and poses an explicit open problem: find an information-theoretic inequality, specific to strategy-proof social choice functions, that implies non-negative interaction information whenever the range has three or more elements. This paper is part of the Adversarial Aggregation Channel (AAC) research program.