Gibbard–Satterthwaite theorem

The Gibbard–Satterthwaite theorem states that any strategy-proof, onto deterministic voting rule with at least three alternatives and unrestricted voter preferences is dictatorial.

Strategy-proofness: A property of a mechanism under which no participant can gain by misrepresenting their private information or preferences. The theorem rules out combining this incentive property with its other assumptions except through dictatorship.
Preference relation: A relation that represents how an agent ranks possible outcomes. Voters' preference relations are the inputs whose reports the theorem constrains.
Allan Gibbard: An American philosopher whose work established a major result on strategy-proof social choice. His 1973 paper independently established the theorem's general social-choice result.
Randomized voting rule: A voting procedure that assigns probabilities to possible outcomes rather than selecting one with certainty. Randomization changes the deterministic setting and permits different strategy-proofness results.
Social choice function: A rule that maps profiles of individual preferences to a collective outcome. The theorem classifies which deterministic social choice functions can satisfy its assumptions.
Ordinal utility: A numerical representation that records preference order without measuring preference strength. The theorem relies on rankings, not on comparable utility magnitudes.
Mark Satterthwaite: An American economist whose work established an impossibility result for strategy-proof voting rules. His 1975 dissertation independently proved the theorem for social choice functions.
Single-peaked preferences: Preference rankings in which each voter has one most-preferred point on an ordered spectrum, with alternatives less preferred farther away. Restricting preferences can permit strategy-proof, non-dictatorial rules, such as the median rule.
Onto function: A function whose range includes every element of its codomain. Onto means every alternative can win for some preference profile, excluding rules that effectively discard options.
Voting rule: A procedure that aggregates voters' preferences into a collective choice. The theorem applies to voting rules that select one alternative from each reported profile.

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