On the probabilities of correct or incorrect majority preference relations

While majority cycles may pose a threat to democratic decision making, actual decisions based inadvertently upon an incorrect majority preference relation may be far more expensive to society. We study majority rule both in a statistical sampling and a Bayesian inference framework. Based on any given paired comparison probabilities or ranking probabilities in a population (i.e., culture) of reference, we derive upper and lower bounds on the probability of a correct or incorrect majority social welfare relation in a random sample (with replacement). We also present upper and lower bounds on the probabilities of majority preference relations in the population given a sample, using Bayesian updating. These bounds permit to map quite precisely the entire picture of possible majority preference relations as well as their probabilities. We illustrate our results using survey data.