I have a question related to two very important theorems from Social Choice Theory. What is the difference between Arrow Theorem and Gibbard-Saterthwaite theorem? I mean, the obvious one is that in G-S theorem we have non-manipulable (strategyproof) voting system and there is no assumption about it in Arrows' theorem. So are there any other differences?


The difference is that Arrow's theorem says that if you want to have two desirable conditions in your voting system (unanimity and independence of irrelevant alternatives) then necessarily your voting system is a dictatorship. It gives consequences for having "nice" properties. In this case if you do not want a dictatorship you have to sacrifice a property or relax the notion of unanimity or independence of irrelevant alternatives.

In the other hand, Gibbard-Sattertawhite theorem says that in any voting system you only have three options and they are mutually exclusive. Either you get a dictatorship, the election is only between two choices or the voting system can be manipulated. This theorem says that if you do not want one of the options in the list, then you are left with the other two options and only those. It gives you the alternatives you get when you reject one of the three options.

Note that these theorems only apply to ordinal voting systems.

  • $\begingroup$ But Gibbard's theorem applies to all conceivable voting systems, right? $\endgroup$ – endolith Jun 1 '18 at 14:30
  • $\begingroup$ Sorry, I completed the paragraph. And no, Gibbard's theorem again only deals with ordinal voting systems. Cardinal voting systems are not covered. $\endgroup$ – Jonathaniui Jun 2 '18 at 17:47
  • $\begingroup$ Everything I've read says Gibbard–Satterthwaite applies only to ordinal, but Gibbard applies to everything. $\endgroup$ – endolith Jun 2 '18 at 20:03
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    $\begingroup$ I'm sorry, I tought you meant Gibbard-Satterthwhite theorem. Yes, Gibbard's applies to everything. $\endgroup$ – Jonathaniui Jun 3 '18 at 22:01

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