Ranking by pairwise comparisons for Swiss-system tournaments


Pairwise comparison matrices are widely used in multicriteria decision making. This article applies incomplete pairwise comparison matrices in the area of sport tournaments, namely proposing alternative rankings for the 2010 Chess Olympiad Open tournament. It is shown that results are robust regarding scaling technique. In order to compare different rankings, a distance function is introduced with the aim of taking into account the subjective nature of human perception. Analysis of the weight vectors implies that methods based on pairwise comparisons have common roots. Visualization of the results is provided by multidimensional scaling on the basis of the defined distance. The proposed rankings give in some cases intuitively better outcome than currently used lexicographical orders.

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  1. Arsigny V, Fillard P, Pennec X, Ayache N (2006) Fast and simple calculus on tensors in the log-Euclidean framework. Magn Reson Med 56(2): 411–421

    Article  Google Scholar 

  2. Bozóki S, Fülöp J, Rónyai L (2010) On optimal completion of incomplete pairwise comparison matrices. Math Comput Model 52(1–2): 318–333

    Article  Google Scholar 

  3. Crawford G, Williams C (1980) Analysis of subjective judgment matrices. The Rand Corporation, Office of the Secretary of Defense USA, R-2572-AF

  4. Crawford G, Williams C (1985) A note on the analysis of subjective judgment matrices. J Math Psychol 29: 387–405

    Article  Google Scholar 

  5. De Graan JG (July 22–25 1980) Extensions of the multiple criteria analysis method of T.L. Saaty. Presented at EURO IV Conference, Cambridge

  6. Fedrizzi M, Giove S (2007) Incomplete pairwise comparison and consistency optimization. Eur J Oper Res 183(1): 303–313

    Article  Google Scholar 

  7. Gass SI (1998) Tournaments, transitivity and pairwise comparison matrices. J Oper Res Soc 49: 616–624

    Google Scholar 

  8. Handbook of FIDE. D.II.02 (2010) Regulations for specific competitions, Chess Olympiad, Olympiad pairing rules. Approved by the 1994 and 1998 Executive Councils, amended by the 2006 Presidential Board and the 2007 and 2009 Executive Council. http://www.fide.com/fide/handbook.html?id=95&view=article

  9. Harker PT (1987) Incomplete pairwise comparisons in the analytic hierarchy process. Math Model 9(11): 837–848

    Article  Google Scholar 

  10. Kéri G (2011) On qualitatively consistent, transitive and contradictory judgment matrices emerging from multiattribute decision procedures. Central Eur J Oper Res 19(2): 215–224

    Article  Google Scholar 

  11. Kruskal JB, Wish M (1978) Multidimensional scaling. Sage, Beverly Hills

    Google Scholar 

  12. Norris AN (2005) The isotropic material closest to a given anisotropic material. J Mech Mater Struct 1(2): 231–246

    Google Scholar 

  13. Perron O (1907) Zur Theorie der Matrizen. Mathematische Annalen 64: 248–263

    Article  Google Scholar 

  14. Results of the 39th Olympiad Khanty-Mansiysk 2010 Open tournament (2010) http://chess-results.com/tnr36795.aspx?lan=1

  15. Saaty TL (1980) The analytic hierarchy process. McGraw-Hill, New York

    Google Scholar 

  16. Spearman C (1904) The proof and measurement of association between two things. Am J Psychol 15: 72–101

    Article  Google Scholar 

  17. Tarsitano A (2008) Nonlinear rank correlations. Working Paper. Agostino Tarsitano, Italy (in press). http://eprints.bice.rm.cnr.it/586/

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Correspondence to László Csató.

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Csató, L. Ranking by pairwise comparisons for Swiss-system tournaments. Cent Eur J Oper Res 21, 783–803 (2013). https://doi.org/10.1007/s10100-012-0261-8

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  • Multicriteria decision making
  • Incomplete pairwise comparison matrix
  • Ranking for Swiss-system tournaments
  • Multidimensional scaling