The Geometry of Manipulation - a Quantitative Proof of the Gibbard Satterthwaite Theorem

TL;DR

This paper provides a quantitative proof of the Gibbard-Satterthwaite theorem, showing that nearly all voting functions with four or more options are susceptible to manipulation, extending previous results to more alternatives.

Contribution

It introduces a geometric, isoperimetric approach to quantify manipulation probability in social choice functions with four or more options, extending prior work from three options.

Findings

Manipulation probability is at least proportional to the minimal distance from dictators.

The measure of profiles on the interface of multiple outcomes is large.

First isoperimetric result for interfaces involving more than two outcomes.

Abstract

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 $q \ge 4$ alternatives and n voters will be manipulable with probability at least $10^{-4} \eps^2 n^{-3} q^{-30}$, where $\eps$ is the minimal statistical distance between f and the family of dictator functions. Our results extend those of FrKaNi:08, which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in BarthOrline:91, ConitzerS03b, ElkindL05, ProcacciaR06 and ConitzerS06) 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.