© 1971 by Biometrika Trust
On the number of replications of a paired comparison
University of Pennsylvania and Naval Air Development Center Warminster, Pennsylvania
When the better of two objects is to be selected, the choice is frequently the first of the objects to win k comparisons between them. If the better of two objects is defined as the one with the larger probability of winning a single comparison between them, the final choice depends to some extent on chance; the competition could result in a wrong choice and, consequently, a loss to the decision maker. The overall loss function can be expressed as the sum of two terms, one decreasing in k and the other increasing in k. When this function is linear in the expected number of comparisons and either constant for a wrong decision or linear in|p - (1- p), where p is the probability that a particular object wins a single comparison, there is a value of k which minimizes the function. In this paper, a minimizing value of k is shown to exist and is computed for various values of the ratio of the cost of a wrong decision to the cost of making a single comparison. The results are presented graphically along with examples of the use of the graphs.
Key Words: Sample sizo in paired comparisons • Expected duration of play off • Probability of weaker team winning tournament