With a finite set of at least three alternatives and unrestricted strict preference orders, every onto deterministic strategyproof social choice function is a dictatorship in social choice. Restricting to two alternatives or to a restricted preference domain removes essential hypotheses.
For an onto strategyproof rule on unrestricted strict preference orders, rank-raising monotonicity implies unanimity and respect for unanimous pairwise preference. Promote each alternative pair to the top two positions in all reports. The selected member depends only on each voter's comparison of that pair. These binary choices form a strict transitive social order: a directed three-cycle would contradict the rule's choice when that triple is placed above all other alternatives, because the chosen member must win both of its binary comparisons. The original outcome is the top of this order. This creates a social ordering with pairwise independence without presuming it as an axiom of the original choice rule.
For two voters and at least three alternatives, pairwise unanimity, pairwise independence and transitivity force the binary social ordering from strategyproof choice to follow one fixed voter. If voter one wins a conflict against , use profiles and to show it also wins against and against . Repeating this implication yields both directions of every pair. If the initial conflict instead follows voter two, exchange the roles. The social choice, being the top of that ordering, is a dictatorship in social choice. This proves the two-voter case of the Gibbard-Satterthwaite theorem.
For two agents and a strategyproof social choice function, if an alternative is selected when one agent ranks it first and the other ranks it last, it is selected whenever the first agent ranks it first, whatever the other agent reports. Changing the first agent's order preserves its attainable top choice; changing the second agent's order cannot offer it an improvement over a selected bottom alternative.
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The Gibbard–Satterthwaite theorem is a fundamental result in social choice theory and mechanism design that addresses the limitations of voting systems. It states that any voting rule (or voting mechanism) that satisfies certain reasonable conditions is susceptible to strategic manipulation, meaning that voters can gain by misrepresenting their true preferences.