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Gibbard–Satterthwaite theorem

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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 ballots from voters (with a finite number of possible ballot types). The Gibbard–Satterthwaite theorem states that, for three or more candidates, one of the following three things must hold for every voting rule:

Contents

  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 deterministic voting system that selects a single winner either is manipulable or does not meet the preconditions of the theorem.

The theorem does not apply to randomized voting systems, such as the system that chooses a voter randomly and selects the first choice of that voter.

Definitions

A social-choice-function is a function that maps a set of individual preferences to a social outcome. An example function is the plurality function, which says "choose the outcome that is the preferred outcome of the largest number of voters". We denote a social choice function by Soc and its recommended outcome given a set of preferences by Soc ⁡ ( P ) .

A social-choice function is called manipulable by player i if there is a scenario in which player i can gain by reporting untrue preferences (i.e., if the player reports the true preferences then Soc ⁡ ( P ) = a , if the player reports untrue preferences then Soc ⁡ ( P ′ ) = a ′ , and player i prefers a ′ to a ). A social-choice function is called incentive-compatible if it is not manipulable by any player.

A social-choice function is called monotone if, whenever the following is true:

  • When i has some preferences P , Soc ⁡ ( P ) = a ;
  • When i has other preferences P ′ , Soc ⁡ ( P ′ ) = a ′ ;

    • Then, under the preferences P , player i prefers outcome a , and under the preferences P ′ , player i prefers outcome a ′ . It can be demonstrated that incentive-compatibility and monotonicity are equivalent.

    For example, when there are only two possible outcomes, the majority rule is incentive-compatible and monotone: when a player switches his preference from one option to the other, this can only be better for the other option.

    A player i is called a dictator in a social-choice function Soc if Soc always selects the outcome that player i prefers over all other outcomes. Soc is called a dictatorship if there is a player i who is a dictator in it.

    Formal statement

    If Soc is incentive-compatible and returns at least three different outcomes, then Soc is a dictatorship.

    Proof

    The GS theorem can be proved based on Arrow's impossibility theorem. Arrow's impossibility theorem is a similar theorem that deals with social ranking functions - voting systems designed to yield a complete preference order of the candidates, rather than simply choosing a winner.

    Given a social choice function Soc , it is possible to build a social ranking function Rank , as follows: in order to decide whether a ≺ b , the Rank function creates new preferences in which a and b are moved to the top of all voters' preferences. Then, Rank examines whether Soc chooses a or b .

    It is possible to prove that, if Soc is incentive-compatible and not a dictatorship, then Rank satisfies the properties: unanimity and independence-of-irrelevant-alternatives, and it is not a dictatorship. Arrow's impossibility theorem says that, when there are three or more alternatives, such a Rank function cannot exist. Hence, such a Soc function also cannot exist.

    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.

    The Duggan–Schwartz theorem deals with voting systems that choose a (nonempty) set of winners rather than a single winner.

    Noam Nisan describes the relation between the GS theorem and mechanism design:

    The main idea of these "escape routes" is that they deal only with restricted classes of preferences (in contrast to GS, which deals with arbitrary preferences). For example, suppose that all agents have quasi-linear preferences. This means that their utility function depends linearly on money. This means that monetary transfers can be used to induce them to act truthfully. This is the idea behind the successful Vickrey–Clarke–Groves auction.

    History

    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.

    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.

    The theorem is also covered by Hervé Moulin.

    References

    Gibbard–Satterthwaite theorem Wikipedia
    (Text) CC BY-SA


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