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.

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 $Soc(Prefs)$.

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 $Soc(Prefs) = a$, if the player reports untrue preferences then $Soc(Prefs') = a'$, 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 Prefs, $Soc(Prefs) = a$;
• When $i$ has other preferences Prefs', $Soc(Prefs') = a'$;

Then, under the preferences Prefs, player $i$ prefers outcome $a$, and under the preferences Prefs', player $i$ prefers outcome $a'$. It is easy to see 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 $a\prec b$, 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.

Related results

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 GS theorem seems to quash any hope of designing incentive-compatible social-choice functions. The whole field of Mechanism Design attempts escaping from this impossibility result using various modifications in the model."

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 Hervi Moulin.

External links

• The Proof of the Gibbard–Satterthwaite Theorem Revisited
• Arrow’s Theorem and the Gibbard–Satterthwaite Theorem: A Unified Approach
• The Gibbard-Satterthwaite theorem about honest & strategic voting - in the RangeVoting website.
• An example of an election situation in which, by all common voting methods, it pays to vote strategically.