Pairwise Comparison and Ranking: Optimum Properties of the Row Sum Procedure

where ja is a symmetric probability measure on the real line. Then s = (s,, ** *, s8) is a sufficient statistic for the joint distribution of the x j. Under suitable regularity conditions, the converse also holds: if the distributions of the xij are not of this form, then s is not sufficient, and the row sum procedure is not optimal. Thus, the model just described seems to constitute the natural domain of the row sum ranking procedure. Fortunately, this domain contains the important case where the xij (i < j) are independent normal variables with mean t-j and equal variance u2. Another particular case-the case of tournaments without draws-where the xij can take only two values, has been treated in the joint paper [1]. Incidentally, the present paper grew out of an attempt to cover the case of tournaments with draws (see example (iv) in Section 4 below).