Abstract
This work is concerned with the interpretation of the results produced by the well known Elo algorithm applied in various sport ratings. The interpretation consists in defining the probabilities of the game outcomes conditioned on the ratings of the players and should be based on the probabilistic rating-outcome model. Such a model is known in the binary games (win/loss), allowing us to interpret the rating results in terms of the win/loss probability. On the other hand, the model for the ternary outcomes (win/loss/draw) has not been yet shown even if the Elo algorithm has been used in ternary games from the very moment it was devised. Using the draw model proposed by Davidson in 1970, we derive a new Elo-Davidson algorithm, and show that the Elo algorithm is its particular instance. The parameters of the Elo-Davidson are then related to the frequency of draws which indicates that the Elo algorithm silently assumes games with 50% of draws. To remove this assumption, often unrealistic, the Elo-Davidson algorithm should be used as it improves the fit to the data. The behaviour of the algorithms is illustrated using the results from English Premier League.
Acknowledgement
Many thanks to J.-C. Gregoire (INRS, Canada) and E. V. Kuhn (Federal University of Santa Catarina, Brazil) for critical reading.
funding: The work was supported by NSERC, Canada.
References
Aldous, D. 2017. “Elo Ratings and the Sports Model: A Neglected Topic in Applied Probability?” Statistical Science 32:616–629. https://doi.org/10.1214/17-STS628.10.1214/17-STS628Search in Google Scholar
Bishop, C. 2006. Pattern Recognition and Machine Learning. Springer.Search in Google Scholar
Bradley, R. A. and M. E. Terry. 1952. “Rank Analysis of Incomplete Block Designs: 1 the Method of Paired Comparisons.” Biometrika 39:324–345.10.1093/biomet/39.3-4.324Search in Google Scholar
Caron, F. and A. Doucet. 2012. “Efficient Bayesian Inference for Generalized Bradley–Terry Models.” Journal of Computational and Graphical Statistics 21:174–196. https://doi.org/10.1080/10618600.2012.638220.10.1080/10618600.2012.638220Search in Google Scholar
Cattelan, M. 2012. “Models for Paired Comparison Data: A Review with Emphasis on Dependent Data.” Statistical Science 27:412–433.10.1214/12-STS396Search in Google Scholar
David, H. 1963. The Method of Paired Comparison. Charles Griffin & Co. Ltd.Search in Google Scholar
Davidson, R. R. 1970. “On Extending the Bradley-Terry Model to Accommodate Ties in Paired Comparison Experiments.” Journal of the American Statistical Association 65:317–328. http://www.jstor.org/stable/2283595.10.1080/01621459.1970.10481082Search in Google Scholar
Davidson, R. R. and R. J. Beaver. 1977. “On Extending the Bradley-Terry model to Incorporate Within-Pair Order Effects.” Biometrics 33:693–702.10.2307/2529467Search in Google Scholar
Elo, A. E. 2008. The Rating of Chess Players, Past and Present. Ishi Press International.Search in Google Scholar
Fahrmeir, L. and G. Tutz. 1994. “Dynamic Stochastic Models for Time-Dependent Ordered Paired Comparison Systems.” Journal of the American Statistical Association 89:1438–1449. http://dx.doi.org/10.1093/biomet/39.3-4.324.10.1080/01621459.1994.10476882Search in Google Scholar
FIFA. 2018. “Fédération International de Football Association: Men’s Ranking Procedure.” https://www.fifa.com/fifa-world-ranking/procedure/.Search in Google Scholar
Football-data.co.uk. 2019. “Historical Football Results and Betting Odds Data.” https://www.football-data.co.uk/data.php.Search in Google Scholar
Gelman, A., J. Hwang, and A. Vehtari. 2014. “Understanding Predictive Information Criteria for Bayesian Models.” Statistics and Computing 24:997–1016. https://doi.org/10.1007/s11222-013-9416-2.10.1007/s11222-013-9416-2Search in Google Scholar
Glickman, M. E. 1999. “Parameter Estimation in Large Dynamic Paired Comparison Experiments.” Journal of the Royal Statistical Society: Series C (Applied Statistics) 48:377–394. http://dx.doi.org/10.1111/1467-9876.00159.10.1111/1467-9876.00159Search in Google Scholar
Glickman, M. 2018. “Paired Comparison Models with tie Probabilities and Order Effects as a Function of Strength.” http://www.fields.utoronto.ca/talks/Paired-Comparison-Models-Tie-Probabilities-and-Order-Effects-Function-Strength.Search in Google Scholar
Herbrich, R. and T. Graepel. 2006. “Trueskill(TM): A Bayesian Skill Rating System.” Technical Report. https://www.microsoft.com/en-us/research/publication/trueskilltm-a-bayesian-skill-rating-system-2/.10.7551/mitpress/7503.003.0076Search in Google Scholar
Joe, H. 1990. “Extended Use of Paired Comparison Models, with Application to Chess Rankings.” Journal of the Royal Statistical Society Series C (Applied Statistics) 39:85–93. http://www.jstor.org/stable/2347814.10.2307/2347814Search in Google Scholar
Király, F. J. and Z. Qian. 2017. “Modelling Competitive Sports: Bradley-Terry-Elo Models for Supervised and On-Line Learning of Paired Competition Outcomes.” arXiv e-prints arXiv:1701.08055.Search in Google Scholar
Koning, R. H. 2000. “Balance in Competition in Dutch Soccer.” Journal of the Royal Statistical Society: Series D (The Statistician) 49:419–431. https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/1467-9884.00244.10.1111/1467-9884.00244Search in Google Scholar
Langville, A. N. and C. D. Meyer. 2012. Who’s #1, The Science of Rating and Ranking. Princeton University Press.10.1515/9781400841677Search in Google Scholar
Lasek, J., Z. Szlávik, and S. Bhulai. 2013. “The Predictive Power of Ranking Systems in Association Football.” International Journal of Applied Pattern Recognition 1:27–46. https://www.inderscienceonline.com/doi/abs/10.1504/IJAPR.2013.052339, pMID: 52339.10.1504/IJAPR.2013.052339Search in Google Scholar
Rao, P. V. and L. L. Kupper. 1967. “Ties in Paired-Comparison Experiments: A Generalization of the Bradley-Terry Model.” Journal of the American Statistical Association 62:194–204. https://amstat.tandfonline.com/doi/abs/10.1080/01621459.1967.10482901.10.1080/01621459.1967.10482901Search in Google Scholar
Thurston, L. L. 1927. “A law of Comparative Judgement.” Psychological Review 34:273–286.10.1037/h0070288Search in Google Scholar
Wikipedia contributors. 2019. “Wikipedia: Elo Rating System.” https://en.wikipedia.org/wiki/Elo_rating_system.Search in Google Scholar
©2020 Walter de Gruyter GmbH, Berlin/Boston
