A Smooth Transition from Powerlessness to Absolute Power
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.
 Publication:

arXiv eprints
 Pub Date:
 May 2012
 arXiv:
 arXiv:1205.2074
 Bibcode:
 2012arXiv1205.2074M
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Mathematics  Probability
 EPrint:
 22 pages