## Gibbard–Satterthwaite theorem |

In **Gibbard–Satterthwaite theorem** is a result published independently by philosopher ^{[1]} and economist ^{[2]} It deals with deterministic

- The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
- The rule limits the possible outcomes to two alternatives only; or
- The rule is susceptible to
tactical voting : in certain conditions some voter's sincere ballot may not defend their opinion best.

While the scope of this theorem is limited to ordinal voting,

- informal description
- formal statement
- examples
- corollary
- sketch of proof
- history
- posterity
- see also
- references

Consider some voters named , and , who wish to select an option among four candidates named , , and . Assume that they use the

Assume that preferences are as follows.

This notation means that voter prefers , then , and ; voters and both prefer , then , and . 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.

Voter has strategically upgraded candidate and downgraded candidate . Now, the scores are: . Hence, is elected. Voter is satisfied of her ballot modification, because she prefers the outcome to , 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.