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Otherwise, the other voters use a classic voting rule, for example the Borda count. This game form is clearly dictatorial, because voter 1 can impose the result. However, it is not strategyproof: the other voters face the same issue of strategic voting as in the usual Borda count. Thus, Gibbard's theorem is an implication and not an equivalence.
Gibbard's proof of the theorem is more general and covers processes of collective decision that may not be ordinal, such as cardinal voting. [note 1] Gibbard's 1978 theorem and Hylland's theorem are even more general and extend these results to non-deterministic processes, where the outcome may depend partly on chance; the Duggan–Schwartz ...
A strategic vote which improves a voter's satisfaction under one method could have no effect or be outright self-defeating under another method. Gibbard's theorem shows that no deterministic single-winner voting method can be completely immune to strategy, but makes no claims about the severity of strategy or how often strategy succeeds.
Gibbard's theorem, built upon the earlier Arrow's theorem and the Gibbard–Satterthwaite theorem, to prove that for any single-winner deterministic voting methods, at least one of the following three properties must hold: The process is dictatorial, i.e. there is a single voter whose vote chooses the outcome.
Methods like anti-plurality voting and Coombs' method have the opposite effect, being dominated by a voter's bottom rankings and so tending to elect the "least offensive" candidates. First-preference votes are used by psephologists and the print and broadcast media to broadly describe the state of the parties at elections and the swing between ...
Gibbard's theorem shows that any strategyproof game form (i.e. one with a dominant strategy) with more than two outcomes is dictatorial. The Gibbard–Satterthwaite theorem is a special case showing that no deterministic voting system can be fully invulnerable to strategic voting in all circumstances, regardless of how others vote.
The pandemic-era 2020 election had the highest ever early vote totals, at more than 101 million, or 63% of all votes cast. But beside 2020, the share of early votes has remained steady since 2012 ...
The revelation principle shows that, while Gibbard's theorem proves it is impossible to design a system that will always be fully invulnerable to strategy (if we do not know how players will behave), it is possible to design a system that encourages honesty given a solution concept (if the corresponding equilibrium is unique). [3] [4]