Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Quantitative Analysis in Sports

An official journal of the American Statistical Association

Editor-in-Chief: Steve Rigdon, PhD

CiteScore 2018: 1.67

SCImago Journal Rank (SJR) 2018: 0.587
Source Normalized Impact per Paper (SNIP) 2018: 1.970

See all formats and pricing
More options …
Volume 9, Issue 1


Volume 1 (2005)

Determining the level of ability of football teams by dynamic ratings based on the relative discrepancies in scores between adversaries

Anthony Costa Constantinou
  • Corresponding author
  • Electronic Engineering and Computer Science, Queen Mary, University of London, CS332, RIM GROUP, EECS, Mile End, London E1 4NS, UK
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Norman Elliott Fenton
  • Electronic Engineering and Computer Science, Queen Mary, University of London, CS435, RIM GROUP, EECS, Mile End, London E1 4NS, UK
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2013-03-30 | DOI: https://doi.org/10.1515/jqas-2012-0036


A rating system provides relative measures of superiority between adversaries. We propose a novel and simple approach, which we call pi-rating, for dynamically rating Association Football teams solely on the basis of the relative discrepancies in scores through relevant match instances. The pi-rating system is applicable to any other sport where the score is considered as a good indicator for prediction purposes, as well as determining the relative performances between adversaries. In an attempt to examine how well the ratings capture a team’s performance, we have a) assessed them against two recently proposed football ELO rating variants and b) used them as the basis of a football betting strategy against published market odds. The results show that the pi-ratings outperform considerably the widely accepted ELO ratings and, perhaps more importantly, demonstrate profitability over a period of five English Premier League seasons (2007/2008–2011/2012), even allowing for the bookmakers’ built-in profit margin. This is the first academic study to demonstrate profitability against market odds using such a relatively simple technique, and the resulting pi-ratings can be incorporated as parameters into other more sophisticated models in an attempt to further enhance forecasting capability.

Keywords: dynamic sports rating; ELO rating; football betting; football prediction; football ranking


  • Baio, G., and M. Blangiardo. 2010. “Bayesian Hierarchical Model for the Prediction of Football Results.” Journal of Applied Statistics 37(2):253–264.CrossrefGoogle Scholar

  • Buchner, A., W. Dubitzky, A. Schuster, P. Lopes, P. O’Doneghue, J. Hughes, D. A. Bell, K. Adamson, J. A. White, J. M. C. C. Anderson and M. D. Mulvenna. 1997. Corporate Evidential Decision Making in Performance Prediction Domains. Proceedings of the Thirteenth Conference on Uncertainty in Artificial Intelligence (UAI ’97). Providence, Rhode Island, USA: Brown University.Google Scholar

  • Clarke, S. R. and J. M. Norman. 1995. “Home Ground Advantage of Individual Clubs in English Soccer.” The Statistician 44:509–521.CrossrefGoogle Scholar

  • Constantinou, A. C. and N. E. Fenton. 2012. Evidence of an (Intended) Inefficient Association Football Gambling Market.Under Review. Draft available at: http://constantinou.info/downloads/papers/evidenceofinefficiency.pdf.

  • Constantinou, A. C., N. E. Fenton, and M. Neil. 2012a. “pi-football: A Bayesian Network Model for Forecasting Association Football Match Outcomes”. Knowledge-Based Systems, 322–339. Draft available at: http://www.constantinou.info/downloads/papers/pi-model11.pdf.

  • Constantinou, A. C., N. E. Fenton, and M. Neil. 2012b. Profiting from an Inefficient Association Football Gambling Market: Prediction, Risk and Uncertainty Using Bayesian Networks.Under Review. Draft available at: http://www.constantinou.info/downloads/papers/pi-model12.pdf.

  • Crowder, M., M. Dixon, A. Ledford and M. Robinson. 2002. “Dynamic Modelling and Prediction of English Football League Matches for Betting.” The Statistician 51:157–168.Google Scholar

  • Dixon, M., and S. Coles. 1997. “Modelling Association Football Scores and Inefficiencies in the Football Betting Market.” Applied Statistics 46:265–280.Google Scholar

  • Dixon, M., and P. Pope. 2004. “The Value of Statistical Forecasts in the UK Association Football Betting Market.” International Journal of Forecasting 20:697–711.CrossrefGoogle Scholar

  • Dunning, E. 1999. Sport Matters: Sociological Studies of Sport, Violence and Civilisation. London: Routledge.Google Scholar

  • Dunning, E. G., A Joseph and R.E. Maguire. 1993. The Sports Process: A Comparative and Developmental Approach. p. 129. Champaign: Human Kinetics.Google Scholar

  • Elo, A. E. 1978. The Rating of Chess Players, Past and Present. New York: Arco Publishing.Google Scholar

  • Fenton, N. E. and M. Neil. 2012. Risk Assessment and Decision Analysis with Bayesian Networks. London: Chapman and Hall.Google Scholar

  • FIFA. 2012. FIFA. Retrieved March 27, 2012, from FIFA/Coca-Cola World Ranking Procedure: http://www.fifa.com/worldranking/procedureandschedule/menprocedure/index.html.

  • Football-Data. 2012. Football-Data.co.uk. Retrieved August 2, 2012, from Football Results, Statistics & Soccer Betting Odds Data: http://www.football-data.co.uk/englandm.php.

  • Forrest, D., J. Goddard and R. Simmons. 2005. “Odds-Setters as Forecasters: The Case of English Football.” International Journal of Forecasting 21:551–564.CrossrefGoogle Scholar

  • Goddard, J. 2005. “Regression Models for Forecasting Goals and Match Results in Association Football.” International Journal of Forecasting 21:331–340.CrossrefGoogle Scholar

  • Goddard, J. and I. Asimakopoulos. 2004. “Forecasting Football Results and the Efficiency of Fixed-odds Betting.” Journal of Forecasting 23:51–66.Google Scholar

  • Halicioglu, F. 2005a. “Can We Predict the Outcome of the International Football Tournaments?: The Case of Euro 2000.” Doğuş Üniversitesi Dergisi 6:112–122.Google Scholar

  • Halicioglu, F. 2005b. Forecasting the Professional Team Sporting Events: Evidence from Euro 2000 and 2004 Football Tournaments. 5th International Conference on Sports and Culture: Economic, Management and Marketing Aspects. Athens, Greece, pp. 30–31.Google Scholar

  • Harville, D. A. 1977. “The Use of Linear-model Methodology to Rate High School or College Football Teams.” Journal of American Statistical Association 72:278–289.Google Scholar

  • Hirotsu, N. and M. Wright. 2003. “An Evaluation of Characteristics of Teams in Association Football by Using a Markov Process Model.” The Statistician 52(4):591–602.Google Scholar

  • Hvattum, L. M. and H. Arntzen. 2010. “Using ELO Ratings for Match Result Prediction in Association Football.” International Journal of Forecasting 26:460–470.CrossrefGoogle Scholar

  • Joseph, A., N. Fenton and M. Neil. 2006. “Predicting Football Results Using Bayesian Nets and Other Machine Learning Techniques.” Knowledge-Based Systems 7:544–553.Google Scholar

  • Karlis, D. and I. Ntzoufras. 2000. “On Modelling Soccer Data.” Student 229–244.Google Scholar

  • Karlis, D. and I. Ntzoufras. 2003. “Analysis of Sports Data by Using Bivariate Poisson Models.” The Statistician 52(3): 381–393.Google Scholar

  • Knorr-Held, L. 1997. Hierarchical Modelling of Discrete Longitudinal Data, Applications of Markov Chain Monte Carlo. Munich: Utz.Google Scholar

  • Knorr-Held, L. 2000. “Dynamic Rating of Sports Teams.” The Statistician 49(2):261–276.Google Scholar

  • Koning, R. 2000. “Balance in Competition in Dutch Soccer.” The Statistician 49(3):419–431.Google Scholar

  • Koning, R., H. Koolhaas, M. Renes and G. Ridder. 2003. “A Simulation Model for Football Championships.” European Journal of Operational Research 148:268–276.Google Scholar

  • Kuonen, D. 1996. Statistical Models for Knock-Out Soccer Tournaments. Technical Report, Department of Mathematics, Ècole Polytechnique Federale de Lausanne.Google Scholar

  • Kuypers, T. 2000. “Information and Efficiency: an Empirical Study of a Fixed Odds Betting Market.” Applied Economics 32: 1353–1363.CrossrefGoogle Scholar

  • Lee, A. J. 1997. “Modeling Scores in the Premier League: is Manchester United Really the Best?” Chance 10:15–19.Google Scholar

  • Leitner, C., A. Zeileis and K. Hornik. 2010. “Forecasting Sports Tournaments by Ratings of (prob)abilities: A Comparison for the EURO 2008.” International Journal of Forecasting 26:471–481.CrossrefGoogle Scholar

  • Maher, M. J. 1982. “Modelling Association Football Scores.” Statististica Neerlandica 36:109–118.Google Scholar

  • Min, B., J. Kim, C. Choe, H. Eom, and R. B. McKay. 2008. “A Compound Framework for Sports Results Prediction: A Football Case Study.” Knowledge-Based Systems 21:551–562.Google Scholar

  • Mueller, F. O., R. C. Cantu and S. P. Camp. 1996. Catastrophic Injuries in High School and College Sports. Champaign: Human Kinetics, p. 57.Google Scholar

  • Murali, V. (2011, October 28) Bleacher Report. Retrieved March 28, 2012, from World Football: 40 Biggest Scandals in Football History: http://bleacherreport.com/articles/909932-world-football-40-biggest-scandals-in-football-history.

  • Poulter, D. R. 2009. “Home Advantage and Player Nationality in International Club Football.” Journal of Sports Sciences 27(8):797–805.Google Scholar

  • Reid, D. A. and M. S. Nixon. 2011. “Using Comparative Human Descriptions for Soft Biometrics.” International Joint Conference on Biometrics (IJCB) 2011.Google Scholar

  • Rotshtein, A., M. Posner and A. Rakytyanska. 2005. “Football Predictions Based on a Fuzzy Model with Genetic and Neural Tuning.” Cybernetics and Systems Analysis 41(4):619–630.CrossrefGoogle Scholar

  • Rue, H. and O. Salvesen. 2000. “Prediction and Retrospective Analysis of Soccer Matches in a League.” The Statistician 3:339–418.Google Scholar

  • Tsakonas, A., G. Dounias, S. Shtovba and V. Vivdyuk. 2002. Soft Computing-Based Result Prediction of Football Games. The First International Conference on Inductive Modelling (ICIM 2002). Lviv, Ukraine.Google Scholar

About the article

Corresponding author: Anthony Costa Constantinou, Electronic Engineering and Computer Science, Queen Mary, University of London, CS332, RIM GROUP, EECS, Mile End, London E1 4NS, UK

Published Online: 2013-03-30

It might also worth mentioning that the ELO rating algorithm was featured prominently in the popular movie The Social Network (also known as the Facebook movie), whereby during a scene Eduardo Saverin writes the mathematical formula for the ELO rating system on Zuckerberg’s dorm room window.

If the rating is applied to a single league competition, the average team in that league will have a rating of 0. If the rating is applied to more than one league in which adversaries between the different leagues (or cup competitions) play against each other, the average team over all leagues will have a rating of 0.

If the prediction is +4 in favour of the home side then an actual result of 5–0 will give you an error of approximately 1. But if the prediction is 0 in favour of the home side and the actual result is 1–0, then this also gives you the same error as above.

The learning parameters could have been optimised based on predictions of type {H, D, A} (corresponding to home win, draw and away win), based on profitability, based on scoring rules, or based on many other different accuracy measurements and metrics. We have chosen score difference for optimising the learning parameters since the pi-ratings themselves are exclusively determined by that information.

The first five EPL seasons (1992/1993 to 1996/1997) are solely considered for generating the initial ratings for the competing teams. This is important because training the model on ignorant team ratings (i.e., starting from 0) will negatively affect the training procedure. Thus, learning parameters λ and γ are trained during the subsequent ten seasons; 1997/1998 to 2006/2007 inclusive.

For the pi-rating system the ratings are segregated into intervals of 0.10 (from ≤–1.1 to >1.6), for ELOb the ratings are segregated into intervals of 25 (from ≤–330 to >345), and for ELOg the ratings are segregated into intervals of 35 (from ≤–475 to >470).

Assumes a profit margin of 5%.

For the newly promoted team Wolves the development of the ratings start at match instance 760 since no performances have been recorded relative to the residual EPL teams during the two preceding seasons.

Where the pi-ratings of the home and away team follow ∼Normal (x, y) distributions for capturing rating uncertainty, where x is the pi-rating value (RαH or RβA) and y is the pi-rating variance, which can be measured over n preceding match instances.

Citation Information: Journal of Quantitative Analysis in Sports, Volume 9, Issue 1, Pages 37–50, ISSN (Online) 1559-0410, ISSN (Print) 2194-6388, DOI: https://doi.org/10.1515/jqas-2012-0036.

Export Citation

©2013 by Walter de Gruyter Berlin Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Werner Dubitzky, Philippe Lopes, Jesse Davis, and Daniel Berrar
Machine Learning, 2018
Ondřej Hubáček, Gustav Šourek, and Filip Železný
Machine Learning, 2018
Olav Drivenes Sæbø and Lars Magnus Hvattum
Journal of Sports Analytics, 2018, Page 1
Marcel Ausloos, Adam Gadomski, and Nikolay K Vitanov
Physica Scripta, 2014, Volume 89, Number 10, Page 108002

Comments (0)

Please log in or register to comment.
Log in