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A graph interpretation of the least squares ranking method


  • László Csató



The paper aims at analyzing the least squares ranking method for generalized tournaments with possible missing and multiple paired comparisons. The bilateral relationships may reflect the outcomes of a sport competition, product comparisons, or evaluation of political candidates and policies. It is shown that the rating vector can be obtained as a limit point of an iterative process based on the scores in almost all cases. The calculation is interpreted on an undirected graph with loops attached to some nodes, revealing that the procedure takes into account not only the given object’s results but also the strength of objects compared with it. We explore the connection between this method and another procedure defined for ranking the nodes in a digraph, the positional power measure. The decomposition of the least squares solution offers a number of ways to modify the method. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • László Csató, 2015. "A graph interpretation of the least squares ranking method," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 44(1), pages 51-69, January.
  • Handle: RePEc:spr:sochwe:v:44:y:2015:i:1:p:51-69
    DOI: 10.1007/s00355-014-0820-0

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    References listed on IDEAS

    1. Kwiesielewicz, M., 1996. "The logarithmic least squares and the generalized pseudoinverse in estimating ratios," European Journal of Operational Research, Elsevier, vol. 93(3), pages 611-619, September.
    2. P. Herings & Gerard Laan & Dolf Talman, 2005. "The positional power of nodes in digraphs," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 24(3), pages 439-454, June.
    3. Julio González-Díaz & Ruud Hendrickx & Edwin Lohmann, 2014. "Paired comparisons analysis: an axiomatic approach to ranking methods," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(1), pages 139-169, January.
    4. Frederick Mosteller, 1951. "Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations," Psychometrika, Springer;The Psychometric Society, vol. 16(1), pages 3-9, March.
    5. Denis Bouyssou, 2004. "Monotonicity of ‘ranking by choosing’: A progress report," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 23(2), pages 249-273, October.
    6. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
    7. Brozos-Vázquez, Miguel & Campo-Cabana, Marco Antonio & Díaz-Ramos, José Carlos & González-Díaz, Julio, 2008. "Ranking participants in tournaments by means of rating functions," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1246-1256, December.
    8. Peter Borm & René van den Brink & Marco Slikker, 2002. "An Iterative Procedure for Evaluating Digraph Competitions," Annals of Operations Research, Springer, vol. 109(1), pages 61-75, January.
    9. Marco Slikker & Peter Borm & René Brink, 2012. "Internal slackening scoring methods," Theory and Decision, Springer, vol. 72(4), pages 445-462, April.
    10. Shamis, Elena, 1994. "Graph-theoretic interpretation of the generalized row sum method," Mathematical Social Sciences, Elsevier, vol. 27(3), pages 321-333, June.
    11. Harold Gulliksen, 1956. "A least squares solution for paired comparisons with incomplete data," Psychometrika, Springer;The Psychometric Society, vol. 21(2), pages 125-134, June.
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