Complex Collective Decisions and the Probability of Collective Inconsistencies

Many groups have to make decisions over multiple interconnected pro-positions. The doctrinal paradox or discursive dilemma shows that propositionwide majority voting can lead to inconsistent collective outcomes even when individual judgments are all consistent. How likely is the occurrence of this paradox? This paper develops a simple model for determining the probability of the paradoxs occurrence, given various probability distributions over individual judgements. Several convergence results are proved, identifying conditions under which that probability converges to certainty as the number of individuals increases, and conditions under which is vanishes. The model is used for assessing the truth-tracking performance of two escape-routes from the paradox, the premise- and conclusion-based procedures. Finally, the present results are compared with the existing results on the probability of Condorcets paradox. It is suggestd that the doctrinal paradox is likely to occur under plausible conditions.