Editing Gibbard–Satterthwaite theorem (section)

== Formal statement ==
Let <math>\mathcal{A}</math> 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 <math>\mathcal{N} = \{1, \ldots, n\}</math> the set of ''voters''. Let <math>\mathcal{P}</math> be the set of [[Weak ordering|strict weak orders]] over <math>\mathcal{A}</math>: 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 <math>f: \mathcal{P}^n \to \mathcal{A}</math>. Its input is a profile of preferences <math>(P_1, \ldots, P_n) \in \mathcal{P}^n</math> and it yields the identity of the winning candidate.

We say that <math>f</math> is ''manipulable'' if and only if there exists a profile <math>(P_1, \ldots, P_n) \in \mathcal{P}^n</math> where some voter <math>i</math>, by replacing her ballot <math>P_i</math> with another ballot <math>P_i'</math>, can get an outcome that she prefers (in the sense of <math>P_i</math>).

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

We say that <math>f</math> is ''dictatorial'' if and only if there exists a voter <math>i</math> 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.

{{Math theorem
| math_statement = If an ordinal voting rule has at least 3 possible outcomes and is non-dictatorial, then it is manipulable.
| name = Gibbard–Satterthwaite theorem
}}