stylistic:style_transfer
Formal statement

Let A {\displaystyle {\mathcal {A}}} be the set of alternatives (which is assumed finite), also called candidates, even if they are not necessarily persons: they can also be several possible decisions about a given issue. We denote by N = { 1, …, n } {\displaystyle {\mathcal {N}}=\{1,\ldots,n\}} the set of voters. Let P {\displaystyle {\mathcal {P}}} be the set of strict weak orders over A {\displaystyle {\mathcal {A}}}: an element of this set can represent the preferences of a voter, where a voter may be indifferent regarding the ordering of some alternatives. A voting rule is a function f: P n → A {\displaystyle f:{\mathcal {P}}^{n}\to {\mathcal {A}}}. Its input is a profile of preferences (P 1, …, P n) ∈ P n {\displaystyle (P_{1},\ldots,P_{n})\in {\mathcal {P}}^{n}} and it yields the identity of the winning candidate.

We say that f {\displaystyle f} is manipulable if and only if there exists a profile (P 1, …, P n) ∈ P n {\displaystyle (P_{1},\ldots,P_{n})\in {\mathcal {P}}^{n}} where some voter i {\displaystyle i}, by replacing her ballot P i {\displaystyle P_{i}} with another ballot P i ′ {\displaystyle P_{i}'}, can get an outcome that she prefers (in the sense of P i {\displaystyle P_{i}}).

We denote by f (P n) {\displaystyle f({\mathcal {P}}^{n})} the image of f {\displaystyle f}, i.e. the set of possible outcomes for the election. For example, we say that f {\displaystyle f} has at least three possible outcomes if and only if the cardinality of f (P n) {\displaystyle f({\mathcal {P}}^{n})} is 3 or more.

We say that f {\displaystyle f} is dictatorial if and only if there exists a voter i {\displaystyle i} who is a dictator, in the sense that the winning alternative is always her most-liked one among the possible outcomes regardless of the preferences of other voters. If the dictator has several equally most-liked alternatives among the possible outcomes, then the winning alternative is simply one of them.

Gibbard–Satterthwaite theorem — If an ordinal voting rule has at least 3 possible outcomes and is non-dictatorial, then it is manipulable.