A quantitative Gibbard-Satterthwaite theorem without neutrality

Abstract

Recently, quantitative versions of the Gibbard-Satterthwaite theorem were proven for k=3 alternatives by Friedgut, Kalai, Keller and Nisan and for neutral functions on k ≥ 4 alternatives by Isaksson, Kindler and Mossel.

We prove a quantitative version of the Gibbard-Satterthwaite theorem for general social choice functions for any number k ≥ 3 of alternatives. In particular we show that for a social choice function f on k ≥ 3 alternatives and n voters, which is ε-far from the family of nonmanipulable functions, a uniformly chosen voter profile is manipulable with probability at least inverse polynomial in n, k, and ε −1.

Ours is a unified proof which in particular covers all previous cases established before. The proof crucially uses reverse hypercontractivity in addition to several ideas from the two previous proofs. Much of the work is devoted to understanding functions of a single voter, and in particular we also prove a quantitative Gibbard-Satterthwaite theorem for one voter.

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Correspondence to Elchanan Mossel.

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Supported by NSF CAREER award (DMS 0548249) and by DOD ONR grant N000141110140.

Supported by a UC Berkeley Graduate Fellowship and by NSF DMS 0548249 (CA-REER).

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Mossel, E., Rácz, M.Z. A quantitative Gibbard-Satterthwaite theorem without neutrality. Combinatorica 35, 317–387 (2015). https://doi.org/10.1007/s00493-014-2979-5

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Mathematics Subject Classification (2000)

  • 05A05