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The Gibbard–Satterthwaite theorem is a theorem in social choice theory.It was first conjectured by the philosopher Michael Dummett and the mathematician Robin Farquharson in 1961 [1] and then proved independently by the philosopher Allan Gibbard in 1973 [2] and economist Mark Satterthwaite in 1975. [3]
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.
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 ...
The Gibbard–Satterthwaite theorem implies that the only rule satisfying non-imposition (every alternative can be chosen) and strategyproofness when there are more than two candidates is the dictatorship mechanism. That is, a voter may be able to cast a ballot that misrepresents their preferences to obtain a result that is more favorable to ...
Gibbard and Satterthwaite give an impossibility result similar in spirit to Arrow's impossibility theorem. For a very general class of games, only "dictatorial" social choice functions can be implemented.
If the largest prime factor of 2s 2 +2s+1, p, and its cofactor, k, satisfy 2s+k 2 +1<p, and if p-2*(2s-k+1) is prime; then n=p-(2s+1) outperforms all smaller n of the form 50m 2. Whether such large s other than those give n in the form 50m 2 exist is then the question.
[2]: 224–225 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).
[1] [2] [3] Sen's proof, set in the context of social choice theory, is similar in many respects to Arrow's impossibility theorem and the Gibbard–Satterthwaite theorem. As a mathematical construct, it also has much wider applicability: it is essentially about cyclical majorities between partially ordered sets, of which at least three must ...