Statistical model for Unbalanced Incomplete paired Comparisons Tournaments


A class of modification is proposed for calculating a score for each Player/team in Unbalanced Incomplete paired Comparisons Tournaments. Many papers dealing with Balanced Incomplete Paired Comparison Tournaments with at most one comparison per pair have appeared since 1950. However, little has been written about unbalanced situations in which the player /the team (object)  ( j ) plays unequal number of games against the player/the team( m ) in a tournament, and the results of all games can be summarized in a Win-Lose matrix Y = { Yjm } ,  where   Yjm = 1,0,1/2, respectively , according as the player or the team ( j ) wins, losses or draws against the player or the team (m ).  Published papers by Ramanujacharyulu (1964), Cowden,D.J. (1975), and David,H. A.  have concentrated on the problem of converting the results of unbalanced incomplete paired comparison tournaments into rank with little consideration of the main relative ability on each player or team. We seggest ( modification )  other ways of quantifying the outcomes of the games / tournaments , in particular , ratings on a scale, 0 to 5 ,   0 to10,..ect. It is important to consider not only the vector Vj(d) or the vectors Sj , in scoring and ranking the k teams in such tournaments , but also the vector Zj,  where Zj = Sj + Sj Rj ,to take into account the ratio of the relative ability of each team ( Rj ). The proposed modification helps to introduce these methods for use in comparisons/games (tournaments), where the player/team are quantified on a special scale.  e.g.0-5, 1-10, ..etc.  We conclude the following:- The scores stabilized to three decimal places at iteration 2 in Cowden’s method Vj(d) .see table(1.4).   The scores stabilized to three decimal places at iteration 2 in David’s method Sj ,and it’s modification Zj. The proposed modification (Zj) has the advantage of removing ties from David’s method (Sj), and hence it is the best method.

Keywords: Unbalanced Incomplete design; paired Comparisons; Scores; Ranking; Rating on a scale.

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