A Bayesian approach to the analysis of paired comparison experiments is presented. Two statistical models are considered: the multibinomial and the Bradley-Terry model. For each, the natural conjugate family of priors is used to represent prior beliefs, and the parameters of the prior are given meaningful interpretations. Two different estimators are suggested for the model parameters, the preference probabilities of the multibinomial model and the worth parameters of the Bradley-Terry model. For the Bradley-Terry model the estimated worth parameters can be used to rank the set of objects. When the posterior distribution satisfies a criterion of balance, the rankings determined by the two estimators are identical and are Bayes with respect to a large class of acceptable loss functions. Furthermore, the Bayes ranking can be easily obtained without explicitly evaluating the estimators of the worth parameters.