Gibbard–Satterthwaite theorem

The Gibbard–Satterthwaite theorem, named after Allan Gibbard and Mark Satterthwaite, is a result about the deterministic voting systems that choose a single winner using only ballots from voters (with a finite number of possible ballot types). The Gibbard–Satterthwaite theorem states that, for three or more candidates, one of the following three things must hold for every voting rule:

Rules that forbid particular eligible candidates from winning or are dictatorial are defective. Hence, every deterministic voting system that selects a single winner either is manipulable or does not meet the preconditions of the theorem.

The theorem does not apply to randomized voting systems, such as the system that chooses a voter randomly and selects the first choice of that voter.

A social-choice-function is a function that maps a set of individual preferences to a social outcome. An example function is the plurality function, which says "choose the outcome that is the preferred outcome of the largest number of voters". We denote a social choice function by ${\displaystyle \operatorname {Soc} }$ and its recommended outcome given a set of preferences by ${\displaystyle \operatorname {Soc} (P)}$.

A social-choice function is called manipulable by player ${\displaystyle i}$ if there is a scenario in which player ${\displaystyle i}$ can gain by reporting untrue preferences (i.e., if the player reports the true preferences then ${\displaystyle \operatorname {Soc} (P)=a}$, if the player reports untrue preferences then ${\displaystyle \operatorname {Soc} (P')=a'}$, and player ${\displaystyle i}$ prefers ${\displaystyle a'}$ to ${\displaystyle a}$). A social-choice function is called incentive-compatible if it is not manipulable by any player.

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