Gibbard's theorem

It has been suggested that this article be merged into Gibbard–Satterthwaite theorem. (Discuss) Proposed since November 2023.

In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973.^[1] 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 is a single voter whose vote chooses the outcome.
2. The process limits the possible outcomes to two options only.
3. The process is not straightforward; the optimal ballot for a voter depends on their beliefs about other voters' ballots.

A corollary of this theorem is the Gibbard–Satterthwaite theorem about voting rules. The key difference between the two theorems is that Gibbard–Satterthwaite applies only to ranked voting. Because of its broader scope, Gibbard's theorem makes no claim about whether voters need to reverse their ranking of candidates, only that their optimal ballots depend on the other voters' ballots.^{[note 1]}

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 or otherwise rate candidates (cardinal voting). Gibbard's theorem can be proven using Arrow's impossibility theorem.^{[citation needed]}

Gibbard's theorem is itself generalized by Gibbard's 1978 theorem^[3] and Hylland's theorem,^[4] which extend these results to non-deterministic processes, i.e. where the outcome may not only depend on the agents' actions but may also involve an element of chance.

Gibbard's theorem assumes the collective decision results in exactly one winner and does not apply to multi-winner voting. A similar result for multi-winner voting is the Duggan–Schwartz theorem.

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