Gibbard-Satterthwaite theorem
Encyclopedia
The Gibbard–Satterthwaite theorem, named after Allan Gibbard
and Mark Satterthwaite
, is a result about the deterministic voting system
s 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:
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 shows that the result holds even if ties are allowed in the ballots: the winner is then chosen from the candidates tied at the top of the dictator's ballot. 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.
published influential articles on the theory of voting: In particular, in an article with Michael Dummett
, Farquharson conjectured that deterministic voting rules with at least three issues faced endemic strategic voting.
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.
Allan Gibbard
Allan Gibbard is the Richard B. Brandt Distinguished University Professor of Philosophy at the University of Michigan, Ann Arbor. Allan Gibbard has made several contributions to contemporary ethical theory, in particular metaethics...
and Mark Satterthwaite
Mark Satterthwaite
Mark A. Satterthwaite is an economist at the Kellogg School of Management at Northwestern University in Evanston, Illinois. He is currently A.C. Buehler Professor in Hospital & Health Services Management, Professor of Strategic Management & Managerial Economics, and chair of the Management &...
, is a result about the deterministic voting system
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....
s 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:
- The rule is dictatorial (i.e., there is a single individual who can choose the winner), or
- There is some candidate who can never win, under the rule, or
- The rule is susceptible to tactical votingTactical votingIn voting systems, tactical voting occurs, in elections with more than two viable candidates, when a voter supports a candidate other than his or her sincere preference in order to prevent an undesirable outcome.It has been shown by the Gibbard-Satterthwaite theorem that any voting method which is...
, 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 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 shows that the result holds even if ties are allowed in the ballots: the winner is then chosen from the candidates tied at the top of the dictator's ballot. Arrow's impossibility theorem
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...
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 FarquharsonRobin Farquharson
Reginald "Robin" Farquharson was an academic whose interest in mathematics and politics led him to work on game theory, which he wrote an influential analysis of voting systems in his doctoral thesis, later published as Theory of Voting.Farquharson diagnosed himself as suffering from bipolar...
published influential articles on the theory of voting: In particular, in an article with Michael Dummett
Michael Dummett
Sir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...
, Farquharson conjectured that deterministic voting rules with at least three issues faced endemic strategic voting.
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.