Further Contributions to the Theory of Paired Comparisons

When a pair of objects is presented for comparison and the two are placed in the relationship preferred: not-preferred, we have what is known as a paired comparison. A set of n objects can be compared, a pair at a time, in some or all of the possible n(n 1)/2 ways of choosing a pair, and the set of paired comparisons so derived gives us a picture of the interrelationships of the objects under preference. A pairedcomparison scheme is more general than a ranking; for with the latter A-preferred-to-B and B-preferred-to-C automatically ensures A-preferred-to-C, whereas with paired comparisons it might happen that C was preferred to A. The existence of these departures from the ranking situation may be due to various reasons, such as the fact that 'preference' is a complicated comparison being made with reference to several factors simultaneously; and one reason for using paired comparisons is to give such effects a chance to show themselves. 2. Situations often occur in which a set of m observers express preferences among n objects and we have to select that object, or perhaps that sub-set of objects, which are, in some sense, most preferred. The simplest case is the one where there are only two objects, A and B, and every observer votes for either A or B as president of an institution. If 51 per cent of the votes are cast for A and 49 per cent for B we declare A elected. In doing so we have satisfied 51 per cent of the preferences but have had to proceed contrary to 49 per cent; we may say that 49 per cent of the preferences were violated. More generally, when we have to select a subset of the n objects as elected we shall in general, in the absence of complete unanimity, violate a inumber of preferences. Circumstances force us to do so to some extent. The problem is to do so to the least possible extent. 3. Consider the case in which 8 members of a body have to elect a committee of three from among themselves. We will suppose that no member votes for himself (though this makes no essential difference) and that there are no abstentions (though this too makes no essential