User:RobLa/Gibbard

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This is my attempt to summarize both Gibbard's theorem and the Gibbard–Satterthwaite theorem, using the prose from the #Copied source text section below:

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Allen Gibbard would become well-known in the field of social choice theory through his work on Gibbard's theorem, published in 1973. Mark Satterthwaite's later worked on a similar theorem which he published in 1975. Satterthwaite and Jean Marie Brin published a paper in 1978 describing Gibbard's and Satterthwaite's mathematical proofs as the "Gibbard–Satterthwaite theorem" and described it's relationship to Arrow's impossibility theorem.

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Gibbard's theorem

Further information: Gibbard's theorem

Below is a copy of a portion of the Gibbard's theorem article, specifically oldid=1033066316 of the "Gibbard's theorem" article:

In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973. It states that for any deterministic process of collective decision, at least one of the following three properties must hold:

1. The process is dictatorial, i.e. there exists a distinguished agent who can impose the outcome;
2. The process limits the possible outcomes to two options only;
3. The process is open to strategic voting: once an agent has identified their preferences, it is possible that they have no action at their disposal that best defends these preferences irrespective of the other agents' actions.

A corollary of this theorem is Gibbard–Satterthwaite theorem about voting rules. The main difference between the two is that Gibbard–Satterthwaite theorem is limited to ranked (ordinal) voting rules: a voter's action consists in giving a preference ranking over the available options. Gibbard's theorem is more general and considers processes of collective decision that may not be ordinal: for example, voting systems where voters assign grades to candidates. Gibbard's theorem can be proven using Arrow's impossibility theorem.

Gibbard–Satterthwaite theorem

Further information: Gibbard–Satterthwaite theorem

This portion is copied from Gibbard–Satterthwaite theorem, specifically https://en.wikipedia.org/w/index.php?title=Gibbard%E2%80%93Satterthwaite_theorem&oldid=1034639892 :

In social choice theory, the Gibbard–Satterthwaite theorem is a result published independently by philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic ordinal electoral systems that choose a single winner. It states that for every voting rule, one of the following three things must hold:

1. The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
2. The rule limits the possible outcomes to two alternatives only; or
3. The rule is susceptible to tactical voting: in certain conditions, a voter's sincere ballot may not best defend their opinion.

While the scope of this theorem is limited to ordinal voting, Gibbard's theorem is more general, in that it deals with processes of collective decision that may not be ordinal: for example, voting systems where voters assign grades to candidates. Gibbard's 1978 theorem and Hylland's theorem are even more general and extend these results to non-deterministic processes, i.e. where the outcome may not only depend on the voters' actions but may also involve a part of chance.