A Quantitative Version of the Gibbard-Satterthwaite Theorem for Three Alternatives

TL;DR

This paper provides a quantitative analysis of the Gibbard-Satterthwaite theorem, showing that non-dictatorial voting rules among three options are susceptible to strategic manipulation with significant probability.

Contribution

It establishes a quantitative version of the theorem, quantifying the likelihood of successful manipulation for rules far from dictatorships.

Findings

Random manipulation succeeds with non-negligible probability

Results apply to election rules with three alternatives

Extends the classical theorem with quantitative bounds

Abstract

The Gibbard-Satterthwaite theorem states that every non-dictatorial election rule among at least three alternatives can be strategically manipulated. We prove a quantitative version of the Gibbard-Satterthwaite theorem: a random manipulation by a single random voter will succeed with a non-negligible probability for any election rule among three alternatives that is far from being a dictatorship and from having only two alternatives in its range.