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Performance Rating Equilibrium

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  • Mehmet S. Ismail

Abstract

In this note, I introduce a novel performance rating system called Performance Rating Equilibrium (PRE). A PRE is a vector of hypothetical ratings for each player, such that if these ratings were each player's initial rating at the start of a tournament, scoring the same points against the same opponents would leave each player's initial rating unchanged. In other words, all players' initial ratings perfectly predict their actual scores in the tournament. This property, however, does not hold for the well-known Tournament Performance Rating. PRE is defined as a fixed point of a multidimensional rating function. I show that such a fixed point, and hence a PRE, exists under mild conditions. I provide an implementation of PRE along with several empirical applications. PREs have broad applicability, from sports competitions to the evaluation of large language models.

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  • Mehmet S. Ismail, 2024. "Performance Rating Equilibrium," Papers 2410.19006, arXiv.org, revised Oct 2024.
    • Handle: RePEc:arx:papers:2410.19006
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    References listed on IDEAS

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    1. Mehmet S. Ismail, 2023. "Performance rating in chess, tennis, and other contexts," Papers 2312.12700, arXiv.org.
    2. Brams, Steven J. & Ismail, Mehmet S. & Kilgour, D. Marc & Stromquist, Walter, 2016. "Catch-Up: A Rule That Makes Service Sports More Competitive," MPRA Paper 75650, University Library of Munich, Germany.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
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