A smooth transition from powerlessness to absolute power. http://www.cs.cmu.edu/˜arielpro/papers/ phase.pdf

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 ( p n), where n is the number of voters, then the probability that a random prole is manipulable by the coalition goes to zero as the number of voters goes to innity, whereas if the number of manipulators is! ( p n), then the probability that a random prole is manipulable goes to one. Here we consider the critical window, where a coalition has size c p n, and we show that as c goes from zero to innity, the limiting probability that a random prole 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 not be computationally hard in practice.