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 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.