Editing Gibbard–Satterthwaite theorem (section)

== Corollary ==
We now consider the case where by assumption, a voter cannot be indifferent between two candidates. We denote by <math>\mathcal{L}</math> the set of [[Total order|strict total orders]] over <math>\mathcal{A}</math> and we define a ''strict voting rule'' as a function <math>f: \mathcal{L}^n \to \mathcal{A}</math>. The definitions of ''possible outcomes'', ''manipulable'', ''dictatorial'' have natural adaptations to this framework.

For a strict voting rule, the converse of the Gibbard–Satterthwaite theorem is true. Indeed, a strict voting rule is dictatorial if and only if it always selects the most-liked candidate of the dictator among the possible outcomes; in particular, it does not depend on the other voters' ballots. As a consequence, it is not manipulable: the dictator is perfectly defended by her sincere ballot, and the other voters have no impact on the outcome, hence they have no incentive to deviate from sincere voting. Thus, we obtain the following equivalence.
{{Math theorem
| math_statement = If a strict voting rule has at least 3 possible outcomes, it is non-manipulable if and only if it is dictatorial.
}}
In the theorem, as well as in the corollary, it is not needed to assume that any alternative can be elected. It is only assumed that at least three of them can win, i.e. are ''possible outcomes'' of the voting rule. It is possible that some other alternatives can be elected in no circumstances: the theorem and the corollary still apply. However, the corollary is sometimes presented under a less general form:<ref name="weber">{{cite journal|first1=Tjark|last1=Weber|title=Alternatives vs. Outcomes: A Note on the Gibbard-Satterthwaite Theorem|journal=Technical Report (University Library of Munich)|year=2009|url=http://econpapers.repec.org/paper/pramprapa/17836.htm}}</ref> instead of assuming that the rule has at least three possible outcomes, it is sometimes assumed that <math>\mathcal{A}</math> contains at least three elements and that the voting rule is ''onto'', i.e. every alternative is a possible outcome.<ref name="reny">{{cite journal|first1=Philip J.|last1=Reny|title=Arrow's Theorem and the Gibbard-Satterthwaite Theorem: A Unified Approach|journal=Economics Letters|volume=70|issue=1|year=2001|pages=99–105|doi=10.1016/S0165-1765(00)00332-3|citeseerx=10.1.1.130.1704}}</ref> The assumption of being onto is sometimes even replaced with the assumption that the rule is ''unanimous'', in the sense that if all voters prefer the same candidate, then she must be elected.<ref name="benoit">{{cite journal|first1=Jean-Pierre|last1=Benoît|title=The Gibbard-Satterthwaite Theorem: A Simple Proof|journal=Economics Letters|volume=69|issue=3|date=2000|issn=0165-1765|pages=319–322|doi=10.1016/S0165-1765(00)00312-8}}</ref><ref name="sen">{{cite journal|first1=Arunava|last1=Sen|title=Another Direct Proof of the Gibbard-Satterthwaite Theorem|journal=Economics Letters|volume=70|issue=3|date=2001|issn=0165-1765|url=http://econdse.org/wp-content/uploads/2012/02/arunavags.pdf|pages=381–385|doi=10.1016/S0165-1765(00)00362-1}}</ref>