The geometry of manipulation - A quantitative proof of the Gibbard-Satterthwaite theorem

Marcus Isaksson*, Guy Kindler, Elchanan Mossel

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ≥ 4 alternatives and n voters will be manipulable with probability at least 10 -42n -3q -30, where ∈ is the minimal statistical distance between f and the family of dictator functions. Our results extend those of [11], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [4,6,9,15,7]) cannot hide manipulations completely. Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies.

Original languageAmerican English
Pages (from-to)221-250
Number of pages30
Issue number2
StatePublished - Mar 2012

Bibliographical note

Funding Information:
∗ Supported by the Israel Science Foundation and by the Binational Science Foundation. † Supported by DMS 0548249 (CAREER) award, by ISF grant 1300/08, by a Minerva Foundation grant and by an ERC Marie Curie Grant 2008 239317.


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