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Football (soccer) is arguably one of the most unpredictable sports. Countless variables play a role in determining the outcome of a certain football match. Due to the high complexity of the entire set of involved parameters, the results of football matches may seem pretty random at the first glance. However, there are mathematical theories enabling us to model and analyze the behavior of such complex systems (e.g. chaos theory, probability, and dynamical systems).

Also, just like the case of financial markets, there is a great commercial and social motivation behind predicting football matches. For instance, if you are in charge of a company sponsoring a certain football team in an elimination tournament, you might be concerned with the number of matches that your team can play before its elimination so that you can estimate the amount of public attention and profit that you can obtain by sponsoring this certain team.

During my search I came across the following interesting papers in which some methods from dynamical systems and probability theory are used for analyzing football matches.

  • K. Davids, D. Araújo and R. Shuttleworth, Applications of Dynamical Systems Theory to Football. (Link to the paper)

  • L. Bastos; J. da Rosa, Predicting probabilities for the 2010 FIFA World Cup games using a Poisson-Gamma model. J. Appl. Stat. 40 (2013), no. 7, 1533–1544. (MR3286258)

  • G. Angelini and L. De Angelis, PARX model for football match predictions. J. Forecast. 36 (2017), no. 7, 795–807. (MR3714406)

Also, based on the following paper, I guess some chaos theoretic approaches might be relevant to this problem.

  • M. G. Mack, S. Huddleston, K. E. Dutler, and J. K. Mintah, Chaos Theory: A New Science for Sport Behavior? (Link to the article)

I wonder whether there is any known really successful and practically applicable mathematical method for predicting football matches or not.

Question. What are examples of research papers modeling football matches/tournaments using mathematical tools (from chaos theory, dynamical systems or probability) which particularly suggest methods for predicting the results of the football matches/tournaments?

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    $\begingroup$ Football is low scoring which is a big problem, unlike basketball, say. World cup prediction has an extra difficulty. The ratings/models for the teams based on past performance form a graph that's highly connected within each confederation [say Europe] but very lightly connected [other than friendlies] for pairs of teams in different confederations [say Germany v Uruguay] other than possibly at the previous world cup, which is probably relatively old information. $\endgroup$
    – kodlu
    Jun 15, 2018 at 22:30
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    $\begingroup$ such as one team dominating, but other one scoring a lucky goal in injury time ;-) as just happened in the Iran-Morocco match. Disclaimer: Only watched 7 minute long highlights. $\endgroup$
    – kodlu
    Jun 15, 2018 at 22:42
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    $\begingroup$ "Football (soccer) is arguably one of the most complicated sports" What does that even mean? I mean we can make any sport as complicated as we want by simply asking for more and more accurate models. A good enough predictive model of arm wrestling would take sun spots into account. Do you want to cash that out somehow in terms of the change in the variance of outcomes as more factors are added to the statistical model or something? $\endgroup$
    – Peter Gerdes
    Jun 16, 2018 at 0:57
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    $\begingroup$ I made the following observation about soccer once: "A lot of people seem to be interested in a spheroid's semi-random interaction with a rectangle filled with humanoids. Apparently the trajectory of the spheroid is analyzed and two integers are assigned to it. The relative ordering of these two integers is considered significant." $\endgroup$
    – Jim Conant
    Jun 16, 2018 at 3:40
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    $\begingroup$ I wonder whether this would be more suitable for Cross Validated. They have predictive-models tag and also some posts about soccer or football. $\endgroup$
    – Martin Sleziak
    Jun 16, 2018 at 4:00

2 Answers 2

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Typing "mathematics of soccer" into the internet led to J.A. Tenreiro Machado, António M.Lopes, On the mathematical modeling of soccer dynamics, Communications in Nonlinear Science and Numerical Simulation, Volume 53, December 2017, Pages 142-153:

Abstract

This paper addresses the modeling and dynamical analysis of soccer teams. Two modeling perspectives based on the concepts of fractional calculus are adopted. In the first, the power law behavior and fractional-order integration are explored. In the second, a league season is interpreted in the light of a system where the teams are represented by objects (particles) that evolve in time and interact (collide) at successive rounds with dynamics driven by the outcomes of the matches. The two proposed models embed implicitly details of players and coaches, or strategical and tactical maneuvers during the matches. Therefore, the scale of observation focuses on the teams behavior in the scope of the observed variables. Data characterizing two European soccer leagues in the season 2015–2016 are adopted and processed. The model leads to the emergence of patterns that are analyzed and interpreted.

Also, this article, Soccermatics: How maths will change your understanding of football, on the 442 website, reporting on the work of Professor David Sumpter. Sumpter has several math/soccer videos online.

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  • $\begingroup$ (+1) Really interesting references, especially the videos! Thanks for introducing, Gerry! :-) $\endgroup$
    – Morteza Azad
    Jun 16, 2018 at 8:15
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    $\begingroup$ Mathematics may apply to revolutionize soccer gilkalai.wordpress.com/2009/04/20/… (and also gilkalai.wordpress.com/2011/06/09/… ). However this is not in the direction of predictions, and also not quite yet "really successful and practically applicable". $\endgroup$
    – Gil Kalai
    Jun 21, 2018 at 15:30
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There is a great article by Connes (translated in Symmetries, Eur. Math. Soc. Newsl. 54, 11-18 (2004). ZBL1176.00001. - link to free pdf, p. 11-18) motivated by fair pairings in soccer competitions, which quickly leads to nice mathematics.

Recently, Brams et al. Making the rules of sports fairer, SIAM Rev. 60, No. 1, 181-202 (2018). ZBL06837609. analyzed the advantage of starting a penalty shootout, and proposed fairer adjustments.

For some statistical work, see, e.g.,

Fuster-Parra, Pilar; García-Mas, Alex; Cantallops, Jaume; Ponseti, F.Javier; Luo, Yuhua, Ranking features on psychological dynamics of cooperative team work through Bayesian networks, Symmetry 8, No. 5, Article ID 34, 17 p. (2016). ZBL1376.62141.

Seth, Sohan; Eugster, Manuel J.A., Probabilistic archetypal analysis, Mach. Learn. 102, No. 1, 85-113 (2016). ZBL1352.62083.

Emparanza, Ignacio Díaz; Núñez-Antón, Vicente, On the use of simulation methods to compute probabilities: application to the first division Spanish soccer league, SORT 34, No. 2, 181-200 (2010). ZBL1203.62221.

Suzuki, A.K.; Salasar, L.E.B.; Leite, J.G.; Louzada-Neto, F., A Bayesian approach for predicting match outcomes: The 2006 (Association) football world cup, J. Oper. Res. Soc. 61, No. 10, 1530-1539 (2010). ZBL1198.62177.

Yue, Zengyuan; Broich, Holger; Seifriz, Florian; Mester, Joachim, Mathematical analysis of a soccer game. I: Individual and collective behaviors, ZBL05709770.

Yue, Zengyuan; Broich, Holger; Seifriz, Florian; Mester, Joachim, Mathematical analysis of a soccer game. II: Energy, spectral, and correlation analyses, ZBL05709771.

McHale, Ian; Scarf, Phil, Modelling soccer matches using bivariate discrete distributions with general dependence structure, Stat. Neerl. 61, No. 4, 432-445 (2007). ZBL1149.62338.

Kopociński, Bolesław, Components of the game result in a football league, Appl. Math. 28, No. 1, 55-72 (2001). ZBL1008.62548.

Dyte, D.; Clarke, S.R., A ratings based Poisson model for World Cup soccer simulation, J. Oper. Res. Soc. 51, No. 8, 993-998 (2000). ZBL1107.62383.

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  • $\begingroup$ (+1) It is such a nice and comprehensive answer! Thank you, Olaf! $\endgroup$
    – Morteza Azad
    Jun 21, 2018 at 15:27

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