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The Gibbard–Satterthwaite theorem is a theorem in social choice theory. It was first conjectured by the philosopher Michael Dummett and the mathematician Robin Farquharson in 1961 [ 1 ] and then proved independently by the philosopher Allan Gibbard in 1973 [ 2 ] and economist Mark Satterthwaite in 1975. [ 3 ]
A corollary of this theorem is the Gibbard–Satterthwaite theorem about voting rules. The key difference between the two theorems is that Gibbard–Satterthwaite applies only to ranked voting. Because of its broader scope, Gibbard's theorem makes no claim about whether voters need to reverse their ranking of candidates, only that their optimal ...
There are several famous theorems concerning social choice functions. The Gibbard–Satterthwaite theorem implies that the only rule satisfying non-imposition (every alternative can be chosen) and strategyproofness when there are more than two candidates is the dictatorship mechanism. That is, a voter may be able to cast a ballot that ...
There are several different degrees of incentive-compatibility: [4] The stronger degree is dominant-strategy incentive-compatibility (DSIC). [1]: 415 It means that truth-telling is a weakly-dominant strategy, i.e. you fare best or at least not worse by being truthful, regardless of what the others do.
Gibbard's theorem shows that any strategyproof game form (i.e. one with a dominant strategy) with more than two outcomes is dictatorial. The Gibbard–Satterthwaite theorem is a special case showing that no deterministic voting system can be fully invulnerable to strategic voting in all circumstances, regardless of how others vote.
This means that, for 3 or more outcomes, the only RFTT mechanisms are dictatorships (by the Gibbard–Satterthwaite impossibility theorem); and for 2 outcomes, a mechanism is RFTT if and only if it is an extended majority rule. As an example, to see that plurality voting is not RFTT for 3 outcomes, suppose an agent's preference ranking is z>y>x.
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