# Gibbard–Satterthwaite theorem

The Gibbard–Satterthwaite theorem, named after Allan Gibbard and Mark Satterthwaite, is a result about the deterministic voting systems that choose a single winner using only the preferences of the voters, where each voter ranks all candidates in order of preference. The Gibbard–Satterthwaite theorem states that, for three or more candidates, one of the following three things must hold for every voting rule:

1. The rule is dictatorial (i.e., there is a single individual who can choose the winner), or
2. There is some candidate who can never win, under the rule, or
3. The rule is susceptible to tactical voting, in the sense that there are conditions under which a voter with full knowledge of how the other voters are to vote and of the rule being used would have an incentive to vote in a manner that does not reflect his or her preferences.

Rules that forbid particular eligible candidates from winning or are dictatorial are defective. Hence, every voting system that selects a single winner either is manipulable or does not meet the preconditions of the theorem. Taylor (2002, Theorem 5.1) shows that the result holds even if ties are allowed in the ballots (but a single winner must nevertheless be chosen): for such elections, a dictatorial rule is one in which the winner is always chosen from the candidates tied at the top of the dictator's ballot, and with this modification the same theorem is true. Arrow's impossibility theorem is a similar theorem that deals with voting systems designed to yield a complete preference order of the candidates, rather than simply choosing a winner. Similarly, the Duggan–Schwartz theorem deals with voting systems that choose a (nonempty) set of winners rather than a single winner.

## Conjecture by Dummett and Farquharson

Robin Farquharson published influential articles on the theory of voting; in an article with Michael Dummett, he conjectured that deterministic voting rules with at least three issues faced endemic tactical voting.[1]

After the establishment of the Farquarson-Dummett conjecture by Gibbard and Sattherthwaite, Michael Dummett contributed three proofs of the Gibbard–Satterthwaite theorem in his monograph on voting.[2]

## Notes

1. ^ Dummett, Michael (2005). "The work and life of Robin Farquharson". Social Choice and Welfare 25 (2): 475–483. doi:10.1007/s00355-005-0014-x.
2. ^ Michael Dummett Voting Procedures (Oxford, 1984)