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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 ...
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 misrepresents their preferences to obtain a result that is more favorable to ...
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.
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Arrow's impossibility theorem [61] and the Gibbard–Satterthwaite theorem prove that any useful single-winner voting method based on preference ranking is prone to some kind of manipulation. Game theory has been used to search for some kind of "minimally manipulable" (incentive compatibility) voting schemes. Game theory can also be used to ...
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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