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?
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Related: https://math.stackexchange.com/q/318570/2206 – endolith Jun 01 '18 at 14:29
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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.
Jonathaniui
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But Gibbard's theorem applies to all conceivable voting systems, right? – endolith Jun 01 '18 at 14:30
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Sorry, I completed the paragraph. And no, Gibbard's theorem again only deals with ordinal voting systems. Cardinal voting systems are not covered. – Jonathaniui Jun 02 '18 at 17:47
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Everything I've read says Gibbard–Satterthwaite applies only to ordinal, but Gibbard applies to everything. – endolith Jun 02 '18 at 20:03
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1I'm sorry, I tought you meant Gibbard-Satterthwhite theorem. Yes, Gibbard's applies to everything. – Jonathaniui Jun 03 '18 at 22:01
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Both are general impossibility theorems that can be presented in the form "No voting system meets all of X criteria." They are very closely related to each other.
If you read Satterthwaite's 1975 paper, you'll see the degree to which the two theorems are closely related: Satterthwaite proves specifically that non-manipulable voting systems meet four of Arrow's criteria and are therefore dictatorships.
http://darp.lse.ac.uk/papersdb/satterthwaite_(jet75).pdf
An important (and sometimes ignored) criterion is that the system needs to be deterministic. One example of violation leads to an important and notable exception that is neither a traditional dictatorship nor a manipulable system: If a random ballot is selected after votes are cast, the system can handle 3+ candidates without violating manipulability (or monotonicity + IIA).
From a certain perspective, the lottery system uses a randomly selected dictator.
The G-S result is quite general and applies to any situation in which voters have preferences and multiple options for how to cast ballots - whether you are rating or ranking candidates, there are situations in which it pays to know how everybody else is voting.
Gibbard and Satterthwaite worked independently and arrived at what was essentially the same result. Gibbard's paper was published in 1973; Satterthwaite arrived at the same result in his dissertation (completed in 1973), but took some time to get his result out into a journal.
