Gibbard–Satterthwaite theorem
Informal description

Consider three voters named Alice, Bob and Carol, who wish to select a winner among four candidates named a {\displaystyle a}, b {\displaystyle b}, c {\displaystyle c} and d {\displaystyle d}. 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.

If the voters cast sincere ballots, then the scores are: (a: 3, b: 6, c: 7, d: 2) {\displaystyle (a:3,b:6,c:7,d:2)}. Hence, candidate c {\displaystyle c} 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.

Alice has strategically upgraded candidate b {\displaystyle b} and downgraded candidate c {\displaystyle c}. Now, the scores are: (a: 2, b: 7, c: 6, d: 3) {\displaystyle (a:2,b:7,c:6,d:3)}. Hence, b {\displaystyle b} is elected. Alice is satisfied by her ballot modification, because she prefers the outcome b {\displaystyle b} to c {\displaystyle c}, 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-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.
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