Editing Gibbard–Satterthwaite theorem (section)

== Informal description ==
Consider three voters named Alice, Bob and Carol, who wish to select a winner among four candidates named <math>a</math>, <math>b</math>, <math>c</math> and <math>d</math>. Assume that they use the [[Borda count]]: each voter communicates his or her preference order over the candidates. For each ballot, 3 points are assigned to the top candidate, 2 points to the second candidate, 1 point to the third one and 0 points to the last one. Once all ballots have been counted, the candidate with the most points is declared the winner.

Assume that their preferences are as follows.
{| class="wikitable"
|-
! Voter !! Choice 1 !! Choice 2 !! Choice 3 !! Choice 4
|-
| Alice || <math>a</math> || <math>b</math> || <math>c</math> || <math>d</math>
|-
| Bob || <math>c</math> || <math>b</math> || <math>d</math> || <math>a</math>
|-
| Carol || <math>c</math> || <math>b</math> || <math>d</math> || <math>a</math>
|}

If the voters cast sincere ballots, then the scores are: <math>(a : 3, b : 6, c : 7, d : 2)</math>. Hence, candidate <math>c</math> will be elected, with 7 points.

But Alice can vote strategically and change the result. Assume that she modifies her ballot, in order to produce the following situation.
{| class="wikitable"
|-
! Voter !! Choice 1 !! Choice 2 !! Choice 3 !! Choice 4
|-
| Alice || <math>b</math> || <math>a</math> || <math>d</math> || <math>c</math>
|-
| Bob || <math>c</math> || <math>b</math> || <math>d</math> || <math>a</math>
|-
| Carol || <math>c</math> || <math>b</math> || <math>d</math> || <math>a</math>
|}
Alice has strategically upgraded candidate <math>b</math> and downgraded candidate <math>c</math>. Now, the scores are: <math>(a : 2, b : 7, c : 6, d : 3)</math>. Hence, <math>b</math> is elected. Alice is satisfied by her ballot modification, because she prefers the outcome <math>b</math> to <math>c</math>, which is the outcome she would obtain if she voted sincerely.

We say that the Borda count is ''manipulable'': there exists situations where a sincere ballot does not defend a voter's preferences best.

The Gibbard–Satterthwaite theorem states that every [[Ranked voting|ranked-choice voting]] is manipulable, except possibly in two cases: if there is a distinguished voter who has a dictatorial power, or if the rule limits the possible outcomes to two options only.