The Smoothed Likelihood of Doctrinal Paradox

When aggregating logically interconnected judgments from n agents, the result might be inconsistent with the logical connection. This inconsistency is known as the doctrinal paradox, which plays a central role in the field of judgment aggregation. Despite a large body of literature on the worst-case analysis of the doctrinal paradox, little is known about its likelihood under natural statistical models, except for a few i.i.d. distributions [List, 2005]. In this paper, we characterize the likelihood of the doctrinal paradox under a much more general and realistic model called the smoothed social choice framework [Xia, 2020b], where agents' ground truth judgments are arbitrarily correlated while the noises are independent. Our main theorem states that under mild conditions, the smoothed likelihood of the doctrinal paradox is either 0, exp(-Theta(n)), Theta(n-1/2) or Theta(1). This not only answers open questions by List [2005] for i.i.d. distributions but also draws clear lines between situations with frequent and with vanishing paradoxes.