Where are the really hard manipulation problems? The phase transition in manipulating the veto rule

TL;DR

This paper investigates the complexity of manipulating voting outcomes under the veto rule, revealing a phase transition from easy to hard manipulation depending on vote correlation and coalition size, with empirical evidence supporting these findings.

Contribution

It provides empirical analysis of manipulation difficulty in voting, identifying conditions under which manipulation becomes computationally hard or easy, especially highlighting the impact of vote correlation.

Findings

Manipulation probability exhibits a smooth phase transition as coalition size increases.

Manipulation is generally easy for independent votes, even with small coalitions.

Highly correlated votes and hung elections can make manipulation computationally hard.

Abstract

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 suggest that this complexity may only be in the worst-case since manipulation is often 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 using the veto rule as the size of the manipulating coalition increases. We show that a rescaled probability curve displays a simple and universal form independent of the size of the problem. We argue that manipulation of the veto rule is asymptotically easy for many independent and identically distributed votes 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.