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Gibbard's 1978 theorem states that a nondeterministic voting method is only strategyproof if it's a mixture of unilateral and duple rules. For instance, the rule that flips a coin and chooses a random dictator if the coin lands on heads, or chooses the pairwise winner between two random candidates if the coin lands on tails, is strategyproof.
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 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.
On a rated ballot, the voter may rate each choice independently. An approval voting ballot does not require ranking or exclusivity. Rated, evaluative, [1] [2] graded, [1] or cardinal voting rules are a class of voting methods that allow voters to state how strongly they support a candidate, [3] by giving each one a grade on a separate scale.
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]
Social choice theory is a branch of welfare economics that extends the theory of rational choice to collective decision-making. [1] Social choice studies the behavior of different mathematical procedures (social welfare functions) used to combine individual preferences into a coherent whole.
Condorcet's jury theorem shows that so long as p > 1 ⁄ 2, the majority vote of a jury will be a better guide to the relative merits of two candidates than is the opinion of any single member. Peyton Young showed that three further properties apply to votes between arbitrary numbers of candidates, suggesting that Condorcet was aware of the ...