Mathematical  Culture and Thought

Mathematical Culture and Thought

An Application of Linear Algebra ‎in ‎the ‎Ranking‎ Problem of Sports Teams

Document Type : Review

Authors
Faculty of ‎Mathematics‎, University of Kashan‎, ‎Iran
Abstract
In this paper, a simple application of linear algebra to the problem of ranking players in a tournament is given. We will see that in the case where every player plays exactly one match against each of the other players, one can use a certain eigenvector of the score matrix, known as Perron eigenvalue, to rank players.
Keywords

Subjects


[1]    Bryce, R. A., Smythe, N. F., A note on ranking tournaments, Austral. Math. Soc. Gaz., 7 (1980), no. 1, 5-10.
[2]    Daniels, H. E., Round-robin tournament scores, Biometrica, 56 (1969), 295-299.
[3]    David, H. A., Ranking the players in a round robin tournament, Revue de l’Institut International de Statistique, 39 (1971), no. 2, 137-147.
[4]    David, H. A., Ranking from unbalanced paired-comparison data, Biometrika, 74 (1987), no. 2, 432436.
[5]    David, H. A., The Method of Paired Comparisons, 2nd ed., Charles Griffin & Co., Ltd., London, 1988.
[6]    Eschenbach, C., Hall, F., Hemasinha, R., Kirkland, S., Li, Z., Shader, B., Stuart, J., Weaver, J., Properties of tournaments among well-matched players, Amer. Math. Monthly, 107 (2000), no. 10, 881-892.
[7]    Goddard, S. T., Ranking in tournaments and group decisionmaking, Management Sci., 29 (1983), no. 12, 1384-1392.
[8]    Horn, R. A., Johnson, C. R., Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge, 2013.
[9]    Kendall, M. G., Further contributions to the theory of paired comparisons, Biometrics, 11 (1955), 43-62.
[10]    Kirkland, S., A note on Perron vectors for almost regular tournament matrices, Linear Algebra Appl., 266 (1997), 43-47.
[11]    Landau, E., Zur relativen Wertbemessung der Turnierresultate, Deutsches Wochenschach, 11 (1895), 366-369.
[12]    Landau, E., Über Preisverteilung bei Spieltunrnieren, Zeitschrift für Mathematik und Physik, 63 (1915), 192-202.
[13]    Maybee, J. S., Pullman, Norman J., Tournament matrices and their generalizations, I, Linear Multilinear Algebra, 28 (1990), no. 1-2, 57-70.
[14]    Moon, J. W., Topics on Tournaments, Holt, Rinehart and Winston, London, 1968.
[15]    Ramanujacharyulu, C., Analysis of preferential experiments, Psychometrika, 29 (1964), 257-261.
[16]    Seeley, J. R., The net of reciprocal influence: A problem in treating sociometric data, Canadian Journal of Psychology, 3 (1949), no. 4, 234-240.
[17]    Stob, M., Rankings from round-robin tournaments., With a reply by Stephen Goddard, Management Sci., 31 (1985), no. 9, 1191-1195.
[18]    Vigna, S., Spectral ranking, Network Science, 4 (2016), no. 4, 433-445.
[19]    Wei, T. H., The algebraic foundations of ranking theory, Ph.D. thesis, Cambridge University, 1952.

  • Receive Date 21 January 2023
  • Revise Date 13 May 2023
  • Accept Date 27 May 2023
  • Publish Date 22 July 2024