Talk:Gibbard-Satterthwaite theorem

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I redirected Gibbard's theorem here though I've heard they're actually different? (Gibbard had multiple theorems?) But I'm not sure. — Psephomancy (talk) 03:46, 16 April 2020 (UTC)

Yes. G-S is about ranked methods. Gibbard's theorem shows that you can't escape the conclusions simply by making your method cardinal. See the introductory paragraph at w:Gibbard–Satterthwaite_theorem. Kristomun (talk) 09:17, 16 April 2020 (UTC)
"Whereas Satterthwaite's version only applies to ordinal voting systems, Gibbard's version applies to all deterministic voting systems, including non-ordinal ones. Combining Gibbard's version and a remark made by Satterthwaite" ... gives us G-S? Or Gibbard's theorem is a separate one from G-S? https://politics.stackexchange.com/a/14245Psephomancy (talk) 19:32, 16 April 2020 (UTC)
From Gibbard's paper:[1] "I shall prove in this paper that any non-dictatorial voting scheme with at least three possible outcomes is subject to individual manipulation" ... "The result on voting schemes follows from a theorem I shall prove which covers schemes of a more general kind". So in his paper, Gibbard proves the result for (ordinal) voting methods as a special case of a more general result. The special result is G-S, since it was also independently discovered by Satterthwaite. However, the general result, which generalizes beyond ordinal voting methods, is not covered by Satterthwaite.
Thus (at least it seems to me), Gibbard-Satterthwaite is used to specifically refer to the result that both found independent of each other, which covers ordinal voting systems, and Gibbard's theorem refers to the more general result that Gibbard showed G-S is a corollary of. Kristomun (talk) 19:47, 16 April 2020 (UTC)

References[edit source]

  1. Gibbard, Allan (1973). "Manipulation of voting schemes: A general result". Econometrica. 41 (4): 587–601. doi:10.2307/1914083. JSTOR 1914083.