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Gibbard's theorem states that a deterministic process of collective decision cannot be strategyproof, except possibly in two cases: if there is a distinguished agent who has a dictatorial power (unilateral), or if the process limits the outcome to two possible options only (duple).
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.
Gibbard's theorem is itself generalized by Gibbard's 1978 theorem [11] and Hylland's theorem, 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. The Gibbard's theorem assumes the collective decision results in exactly one winner ...
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]
Arrow's theorem does not cover rated voting rules, and thus cannot be used to inform their susceptibility to the spoiler effect. However, Gibbard's theorem shows these methods' susceptibility to strategic voting, and generalizations of Arrow's theorem describe cases where rated methods are susceptible to the spoiler effect.
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. Later results show that some methods are more manipulable than others. [2] [12]
Gershgorin circle theorem (matrix theory) Gibbard–Satterthwaite theorem (voting methods) Girsanov's theorem (stochastic processes) Glaisher's theorem (number theory) Gleason's theorem (Hilbert space) Glivenko's theorem (mathematical logic) Glivenko's theorem (probability) Glivenko–Cantelli theorem (probability) Goddard–Thorn theorem ...