On the comparison of group performance with categorical data

There are many different evaluation problems that involve several groups (societies, firms or institutions) whose members can be classified into ordered categories, pursuant to their characteristics or their achievements. This paper addresses these types of problems and provides an evaluation criterion based on the distribution of the agents across categories. The starting point is that of dominance relations in pair-wise comparisons. We say that group i dominates group j when the expected category of a member of i is higher than the expected category of a member of j. We introduce the notion of relative advantage of a group to extend this principle to multi-group comparisons and show that there is a unique evaluation function that ranks all groups consistently in terms of this criterion. This function associates to each evaluation problem the (unique) dominant eigenvector of a matrix whose entries describe the dominance relations between groups in pair-wise comparisons. The working of the model is illustrated by means of three different applications.