Consider some voters named , and , who wish to select an option among four candidates named , , and . Assume that they use the Borda count: each voter communicates 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 no point to the last one. Once all ballots have been counted, the candidate with most points is declared the winner.
Assume that preferences are as follows.
This notation means that voter
. If the voters cast sincere ballots, then the scores are:
. Hence, candidate
get elected, with 7 points.
But voter can figure out that another ballot defend her opinions better. Assume that she modifies her ballot, in order to produce the following situation.
has strategically upgraded candidate
and downgraded candidate
. Now, the scores are:
is elected. Voter
is satisfied of her ballot modification, because she prefers the outcome
, 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.
Unfortunately, the Gibbard–Satterthwaite theorem states that a voting rule must be 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.