A Smooth Transition from Powerlessness to Absolute Power

TL;DR

This paper analyzes the phase transition in the probability of coalitional manipulation in voting systems, showing a smooth transition from unlikely to likely manipulability as coalition size increases.

Contribution

It establishes a precise, smooth phase transition in manipulability probability for generalized scoring rules, extending previous zero-one results.

Findings

Probability of manipulability transitions smoothly from 0 to 1

Results validate empirical observations of manipulation likelihood

Suggests limited computational hardness in practical manipulation scenarios

Abstract

We study the phase transition of the coalitional manipulation problem for generalized scoring rules. Previously it has been shown that, under some conditions on the distribution of votes, if the number of manipulators is $o(\sqrt{n})$, where $n$ is the number of voters, then the probability that a random profile is manipulable by the coalition goes to zero as the number of voters goes to infinity, whereas if the number of manipulators is $\omega(\sqrt{n})$, then the probability that a random profile is manipulable goes to one. Here we consider the critical window, where a coalition has size $c\sqrt{n}$, and we show that as $c$ goes from zero to infinity, the limiting probability that a random profile is manipulable goes from zero to one in a smooth fashion, i.e., there is a smooth phase transition between the two regimes. This result analytically validates recent empirical results, and suggests that deciding the coalitional manipulation problem may be of limited computational hardness in practice.