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. Received: 13 November 2000/Accepted: 19 March 2002 This collaborative work was carried out while Regenwetter was a faculty member at the Fuqua School of Business, Duke University. We thank Fuqua for sponsoring our collaboration and the National Science Foundation for grant SBR-97-30076 to Michel Regenwetter. We are indebted to the editor and the referees, as well as to Jim Adams, Bob Clemen, Bernie Grofman, Bob Nau, Saša Pekeč, Jim Smith and Bob Winkler for helpful comments and suggestions.