Proposed Statistical model for Scoring and Ranking Sport Tournaments
Racquetball, Squash and Badminton
Abstract
A class of modification is proposed for calculating a score for each Player/team in Unbalanced Incomplete paired Comparisons Sports Tournaments. Many papers dealing with Balanced Incomplete Paired Comparison Sports 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 to 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.(1988) 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 suggest (modification) another way of quantifying the outcomes of the games/tournaments, in particular, ratings on a scales, 0 to 5, 1 to 10 .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 + SjRj, 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.
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References
Antoine, M. J. (2011). The Method of Pairwise Comparisons-Lecture Note. Retrieved fromOnline: http://www.eajournals.org/.../Statistical-model-forUnbalanced-Incomplete-paired-Comparis. [Last accessed on 2018Retrieved on September 12, 2018].
Arnold, B. C., & Strauss, D. (1987). Bivariate distributions with Pareto conditionals. Statistics and Probability Letters, 5, 263-266.
Bradley, R. A. (1954). Incomplete block rank analysis: On the appropriateness of the model for a method of paired comparisons. Biometric, 375, 10.
Bradley, R. A., & Terry, M. E. (1952). The rank analysis of incomplete book designs. Biometrics, 39, 324.
Buhlmann, H., & Huber, P. J. (1963). Pair wise comparison and ranking in tournaments. Annals of Mathematical Statistics, 34, 501.
Chung, F. R. D., & Hwang, F. K. (1978). Do stronger players win more knock-out tournaments? Journal of the American Statistical Association, 593, 76.
Cowden, D. J. (1975). A method of evaluating contestants. The American Statistician, 29, 82.
Csat́o, L. (2013). Ranking by pairwise comparisons for Swiss-system tournaments. Central European Journal of Operations Research, 21, 783-803.
David, H. A. (1987). Ranking from unbalanced paired comparison data. Biometrika, 74, 432-436.
David, H. A. (1988). The Method of Paired Comparisons. London: Griffin Company.
David, H., & Andrews, M. (1993). Nonparametric Methods of Ranking from Paired Comparisons. In: Fligner, M. A., & Verducci, J. S. (Eds.), Probability Models and Statistical Analyses for Ranking Data. New York: Springer, pp.20-36.
Dykstra, O. (1990). Rank analysis of incomplete block deigns: A method of paired comparisons employing unequal repetitions on pairs. Biometrics, 176, 16.
Glenn, W. A., & David, H. A. (1960). Ties in paired comparison experiments using a modified thurston-mosteller model. Biometrics, 86, 53.
Gonzalez, D., Hendrickx, R., & Lohmann, E. (2013). Paired comparisons
analysis: An axiomatic approach to ranking methods, social choice and welfare. The Society Social Choice and Welfare, 42, 139-169R.
John, M. (2006). The Paired T-test and Hypothesis Testing. Baltimore, Maryland: Johns Hopkins-University.
Laslier, J. (1997). Tournament Solutions and Majority Voting. Berlin: Springer, Berlin.
Ramanujacharyulu, C. (1964). Analysis of preferential experiments. Psychometrika, 29, 257-261.
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