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The gibbard-saterthwatie theorem states that a social choice function where truthfullness is a dominant strategy must be dictatorial. However, in a median voting system, with single peaked preferences of voters, truthfullness is a dominant strategy yet it is not dictatorial. what is the solution to this contradiction?

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Gibbard and Satterthwaite insist that the social choice function must be defined over all rational preferences over outcomes. That is, if voters' preferences could be anything (subject to the constraint of completeness and transitivity), then we have the Gibbard–Satterthwaite theorem.

On the other hand, if preferences were single-peaked, then the assumptions they make do not hold, and the theorem does not apply.

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