Fair Standings in Soccer and Other Round-Robin Leagues

Transcription

1 Fair Standings in Soccer and Other Round-Robin Leagues NESSIS, 23 September 2017 David Firth, University of Warwick Heather Turner, University of Warwick and Freelance Consultant alt-3.uk 1 / 29

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4 Retrodictive statistical modelling long history in connection with sports leagues with unbalanced schedules aim is to represent team performance to date, free of any schedule strength effects different from predictive modelling: uses only match results in the current league season objective is fair representation of past performance, not predictive accuracy 4 / 29

5 Fair representation, free from schedule strength effects... Implies that we must: take account of matches played at home and away take account of strength of opponents already played (and, implicitly, opponents still to be played) agree with the actual league table whenever schedule strength differences are known to be absent. The last point here implies a strong restriction on the statistical models that can be used. 5 / 29

6 Football leagues Association football, soccer, calcio, futebol,...: double round robin: every team plays every other team twice (home and away) league points are 3 0 or 0 3 or 1 1 home advantage is a well established phenomenon Exact schedule balance is achieved only at season end. (approximately at halfway too; but slight home advantage effect because #teams is even) Schedule imbalance often becomes a media focus near the end of the season. 6 / 29

7 e.g., the relegation battle in 2016 (Sky Sports, 20 April 2016) 7 / 29

8 Our model: a generalized Bradley-Terry model When home team i plays away team j, in season s, the match outcome probabilities are given by: pr(i wins) : pr(j wins) : pr(draw) = γ s α is : α js : δ s (α is α js ) 1/3 The parameters are: {α is } strength of team i in season s (i = 1,..., n s ) γ s the home advantage effect in season s δ s prevalence of draws in season s. 8 / 29

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10 For a single season: γα i : α j : δ(α i α j ) 1/3 Trinomial model for match outcomes: is a full exponential family sufficient statistics are (3 #wins + 1 #draws)i (i = 1,..., n) total number of home wins total number of draws Because of this, the end-of-season points totals are all reproduced exactly by the MLE. [Compare with the Davidson-Beaver generalization of the Bradley-Terry model, which uses power 1/2 instead of 1/3. Well known to reproduce the end-of-season points table for a scoring system.] 10 / 29

11 Alternative league table During the season: Points total, or points-per-match rate, is affected by schedule imbalance. Use instead the adjusted points per match (ap/m) for team i s whole season, defined as = 1 #matches 1 2(n 1) [E(points when i plays j)] j i ( j i 3γα i + δ(α i α j ) 1/3 γα i + α j + δ(α i α j ) 1/3 + 3α i + δ(α i α j ) 1/3 α i + γα j + δ(α i α j ) 1/3 ) English Premier League: points per match is normally more than 2 for the league winners, and less than 1 for the relegated teams. 11 / 29

12 Estimation 1. Treat the whole-league parameters, γ s (home advantage in season s) δ s (draw prevalence in season s) as structural: take ˆγ s = ˆγ and ˆδ s = ˆδ (constant across seasons), unless there is clear evidence to the contrary. 2. For the team strengths, α 1s,..., α ns : use a shrinkage prior, to deliver plausible ap/m values, even in the early part of the season. 12 / 29

13 Stability of ˆγ s EPL: Estimated home advantage in each of 23 seasons home advantage (log scale) year (end of season) 13 / 29

14 Stability of ˆδ s EPL: Estimated prevalence of draws in each of 23 seasons prevalence of draws (log scale) year (end of season) 14 / 29

15 Estimation 1. Treat the whole-league parameters, γs (home advantage in season s) δ s (draw prevalence in season s) as structural: take ˆγ s = ˆγ and ˆδ s = ˆδ (constant across seasons), unless there is clear evidence to the contrary. 2. For the team strengths, (α 1s,..., α ns ): use a shrinkage prior/penalty, to deliver plausible ap/m values, even in the early part of the season. 15 / 29

16 Estimation 1. Treat the whole-league parameters, γs (home advantage in season s) δ s (draw prevalence in season s) as structural: take ˆγ s = ˆγ and ˆδ s = ˆδ (constant across seasons), unless there is clear evidence to the contrary. 2. For the team strengths, (α 1s,..., α ns ): use a shrinkage prior/penalty, to deliver plausible ap/m values, even in the early part of the season. 16 / 29

17 Typical points per match (top and bottom) EPL: Range of points per match in the last 23 seasons points per match year (end of season) 17 / 29

18 A hyperdirichlet shrinkage prior penalty Consider the likelihood from prior pseudo data : k wins and k losses for each team against a notional team 0, on neutral ground before any actual matches are played. Use this to penalize the likelihood based on actual match results: a shrinkage penalty because all teams are assigned the same strength a priori. The effective prior here is a hyperdirichlet distribution (Hankin, 2010). By design it does not affect the model s end-of-season agreement with the league table. Choose k to yield a plausible ap/m range, even at the very start of the season. k = 1 works well. 18 / 29

19 Results: EPL after 5 matches (from alt-3.uk) 19 / 29

20 Results: EPL after 5 matches (from alt-3.uk) 20 / 29

21 Other leagues? Changes over time? We might expect to need different values, in different leagues, for the structural parameters, i.e., the home-advantage effect the prevalence of draws. And possibly also changes in these parameters over time? A Warwick MSc student, Seulki Chung, has recently looked at this, for the English football leagues and the top leagues in Europe. 21 / 29

22 Home advantage, English leagues: (from the MSc dissertation of Seulki Chung) 22 / 29

23 Home advantage, major European leagues: (from the MSc dissertation of Seulki Chung) 23 / 29

24 Home advantage, major European leagues: (from the MSc dissertation of Seulki Chung) 24 / 29

25 Prevalence of draws, major European leagues: (from the MSc dissertation of Seulki Chung) 25 / 29

26 Summary The end-of-season agreement requirement is crucial...implies a specific type of generalized Bradley-Terry model Readily calibrated from data on past seasons, and through use of a suitably formulated shrinkage prior Interesting (incidental) findings along the way: differences between leagues structural time-trends Uncertainty in the ranking? It s all about (in)sensitivity to the values of γ and δ. All this will become more interesting as the season progresses! 26 / 29

27 [from the M.MORSE dissertation of Jonathan Kent (2017)] 27 / 29

28 Summary The end-of-season agreement requirement is crucial...implies a specific type of generalized Bradley-Terry model Readily calibrated from data on past seasons, and through use of a suitably formulated (conjugate) shrinkage prior Interesting (incidental) findings along the way: differences between leagues structural time-trends Uncertainty in the ranking? It s all about (in)sensitivity to the values of γ and δ. All this will become more interesting as the season progresses! 28 / 29