Where are the really hard manipulation problems ? The manipulation phase transition

Voting is a simple mechanism to aggregate the preferences of agents. Many voting rules have been shown to be NP-hard to manipulate. However, a number of recent theoretical results have suggested that this complexity may only be in the worst-case and manipulation may be easy in practice. In this paper, we show that empirical studies are useful in improving our understanding of this issue. We demonstrate that there is a smooth transition in the probability that a coalition can elect a desired candidate as the size of the manipulating coalition is varied. We show that a rescaled probability curve displays a simple and universal form independent of the size of the problem. We argue that for many independent and identically distributed votes, manipulation will be computationally easy even when the coalition of manipulators is critical in size. Based on this argument, we identify a situation in which manipulation is computationally hard. This is when votes are highly correlated and the election is “hung”. We show, however, that even a single uncorrelated voter is enough to make manipulation easy again.