FIFA Foe Fun! Mark Kozek! Whittier College. Tim Chartier! Davidson College. Michael Mossinghoff! Davidson College

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1 FIFA Foe Fun! Tim Chartier! Davidson College Mark Kozek! Whittier College Michael Mossinghoff! Davidson College

2 Group E: Switzerland, Ecuador, France, Honduras. Switzerland is the top seed, based on FIFA s flawed rankings, but might only be the third-best team.

3 FIFA Rankings Oct Basis for World Cup groupings. 1 Spain Germany Argentina Colombia Belgium Uruguay Switzerland Netherlands Italy England Brazil Chile USA Portugal Greece Bosnia-Herz Côte d Ivoire Croatia Russia Ukraine France Ecuador Ghana Mexico 854

4 Nov Portugal beats #25 Sweden twice, jumps from #14 to #5 in November ranking. Belgium loses to #4 Columbia and #44 Japan, drops from #5 to #11. FIFA rankings are volatile! Had groupings been based on November ratings, Portugal would have had better draw. Similar: Switzerland and Italy.

5 FIFA s Method FIFA/Coca-Cola World Ranking. A team is awarded points for winning matches. A team s ranking depends on its average points obtained per year, over four years. Points are based on opponent, type of match, and age of match. Large variation: one win may be worth from 85 to 2400 points, even without aging effect.

6 FIFA Ranking Fix a team X. Let y k = time period starting k years ago and ending k 1 years ago. Let g k = number of games played by X during y k, and let c k = min(1, g k /5). Let a k = c k (average number of points earned per match over y k ). Total points for X = a 1 +.5a 2 +.3a 3 +.2a 4.

7 Points per Match Points = M I T C. M (match outcome): 3 for normal victory, 2 for shootout victory, 1 for shootout loss or tie, 0 for normal loss.

8 Points per Match Points = M I T C. I (importance): 1 for Friendly, 2.5 for World Cup or Confed.-level Qualifier, 3 for Confed. Final or Confederations Cup, 4 for World Cup match.

9 Points per Match Points = M I T C. T (opponent strength): Usually: 200 opponent ranking. Exception 1: Min value for T is 50. Exception 2: Top is worth 200 (not 199).

10 Points per Match Points = M I T C. C (Confederation strength): C = average value of the confederation weight for the two teams. UEFA & CONMEBOL: w = 1. CONCACAF: w = AFC & CAF: w = OFC: w = 0.85.

11 Confederation Weight Compute winning average (1 per win,.5 per draw) in inter-confederation matches in each of last three World Cups. Compute mean m of these three values. E.g., UEFA:.51,.76,.59 produces m =.62. Set m 0 = max m over all confederations. w = max(.85, (m/m 0 ) 1/4 ). CONCACAF: w = max(.85, (.37/.63) 1/4 ) =.88. OFC: w = max(.85, (.17/.63) 1/4 ) =.85.

12 Oddities Sharp drops in age weights. M: Winning penalty shootouts: worth 2? I: Big jump from Friendly weight (1) to WC Qualifier (2.5). Host nation plays no WCQ s! T: No discernment among bottom 60 teams. No team has T = 199. C: fudge factors.

13 New Rankings Several systems: Colley, Massey, and Elo. Similar to FIFA in some respects: Use all matches for past four (or more) years. Weight match based on game type, age. Unlike FIFA: More conservative weights on match type. Smoothed age weights.

14 Colley Method Wesley Colley (2001), astrophysicist. One of the BCS algorithms for college football. Main idea: change winning percentage to account for strength of schedule. N teams; team i has unknown rating r i. Mandate that average rating is always 1/2. At start of season, everyone gets 1 in win column and 1 in loss column, so winning percentage is 50%.

15 Colley Method Assume no ties for now. Suppose team i has W i wins, L i losses, and has played G i games. Let O i denote the set of opponents of team i. Over time, the average rating of the opponents of team i should be near 1/2: 1 G i X j2o i r j 1 2.

16 Colley Method So: r i W i +1 G i +2 = 1+ W i L i + G i 2 2 G i +2 L i 1+ W i + P r 2 j2o i j G i +2. This produces the linear system: X (G i + 2)r i r j =1+ W i j2o i 2 L i. We write Cr = b.

17 Colley Method C is symmetric, and positive definite. The system always has a unique solution. The mean rating is 1/2. Can weigh games by importance, age, Ties: count as half a win and half a loss. Can weigh PSO win anywhere between tie and win.

18 Type Weight Friendly = 1, Continental qualifier = 1.25, Continental tourn. or Confed. Cup = 1.5, World Cup qualifier = 2, World Cup match = 2.25.

19 1.0 Age Weight FIFA Smoothed Total area (nearly) preserved. Keep five years now for smoother aging.

20 Additional Adjustments June 9, 2013: World Cup Qualifier. 0-3 Ignore disqualifications.

21 Additional Adjustments June 28, 2011: World Cup Qualifier. Ignore disqualifications.

22 Weighted Colley 1 Brazil Spain Argentina Germany Colombia Belgium Chile Portugal England USA Netherlands France Uruguay Switzerland Côte d Ivoire Russia Italy Ecuador Ukraine Greece Japan Croatia Bosnia-Herz U.A.E Last column: Change from current FIFA rank.

23 Colley: Group of Death! Gp Third Avg Rk Gap A B C D E F G H

24 Massey Method Ken Massey (1997), undergraduate student. Now consults for the BCS. Main idea: a game outcome is a noisy measurement of one team s superiority over another. Measurement: if team i beats team j by p points then record r i r j = p. Produces inconsistent system. Use least squares.

25 Massey Method Massey matrix: M = C 2I N. Solve Mr = v, where v i = (total points scored by team i) (total points scored on team i). Problem: M is singular. Obvious reason: all equations were for differences of ratings. Alter system: replace one row with NX i=1 r i =0. OK as long as there is a path between any two teams.

26 Adjustments January 29, 2014: Friendly. PSO 4-1 Ignore disqualifications. Count penalty shoot-outs as weak wins.

27 Adjustments October 14, 2010: CONCACAF Qualifier 17-0 Ignore disqualifications. Count penalty shoot-outs as weak wins. Massey: set max score differential to 4.

28 Adjustments July 2 and 9, 2011: World Cup Qualifiers: Only matches for both since 2008! Ignore disqualifications. Count penalty shoot-outs as weak wins. Massey: set max score differential to 4. Ensure connectivity.

29 Omisions!

30 Weighted Massey 1 Brazil Argentina Germany Spain Colombia France Chile Netherlands England Belgium Portugal Ecuador Russia Uruguay Bosnia-Herz Ukraine Serbia Côte d Ivoire USA Italy Switzerland Mexico Croatia Sweden Last column: Change from current FIFA rank.

31 Massey: Group of Death! Gp Third Avg Rk Gap A B C D E F G H

32 Build Your Own! FIFAfoefun.davidson.edu. Build personalized rating of international FIFA teams using your selected parameters. Colley or Massey. Age weights. Number of years to use. Type weights. Age weighting method. Value of win in penalty shoot-out. Max score differential to use in Massey.

33 Press Alex Bellos, The Guardian, June 6. Wall Street Journal blog, June 10 and 12. Galileu, Brazilian science magazine, June 16. Visitors from more than 80 countries.

34 ESPN Bracket Predictor T. Chartier & co.: submits brackets for NCAA basketball for testing predictive power of rankings. ESPN World Cup site: > 1 million entries. Some brackets we generate beat more than 90% of submitted brackets.

35 Elo s Ranking Created for ranking in chess. Adaptation for soccer. Each team has a rating value. After each match, some rating points are exchanged between the two teams. Number of points exchanged depends on outcome of match, weight of match, and disparity in rating points.

36 Suppose team i beats team j. Add to r i and subtract from r j : K(v F(r i r j )). K = weight of the match. v = value of the victory: 0.5 v 1. F(x) = distribution function for logistic distribution

37 World Football Elo Ratings Base K value: (20, 30, 40, 50, 60) for friendly, minor tourn., WC/cont. qual. or major tourn., WC qual., WC match. Magnify K depending on winning margin: 1, 1.5, 1.75, 1.875,. v = 1 for victory (incl. shootouts); 0.5 for tie. Pretend home team is rated 100 higher. Use all FIFA matches back to 1872!

38 Our Elo Rating K = 20 for friendly; magnify by same factors used in earlier systems, e.g., K = 40 for WCQ. v = 1 for victory; 1/2 for tie; 2/3 for shootout victory. Use prior five years of FIFA matches.

39 eloratings.net 1 Brazil Spain Germany Argentina Netherlands England Portugal Colombia Uruguay Chile Italy France USA Belgium Russia Mexico Switzerland Ukraine Ecuador Greece Our Elo ratings 1 Spain Brazil Germany USA Argentina Netherlands Portugal England Côte d Ivoire Chile Uruguay France Colombia Nigeria Iran Japan Belgium Greece Switzerland Egypt

40 Number of Matches µ = 41.6, = 21.0.

41 Elo Summary Elo: seems not well suited to FIFA rankings. World Elo: More than a century of accumulated points. Ours: most games within a confederation. Local powers (USA, Nigeria, Egypt, Côted Ivoire, Iran) have perhaps inflated rankings.

42 FIFA Colley Massey WER Elo SPI 1 Spain Brazil Brazil Brazil Spain Brazil 2 Germany Spain Argentina Spain Brazil Argentina 3 Argentina Argentina Germany Germany Germany Germany 4 Colombia Germany Spain Argentina USA Colombia 5 Belgium Colombia Colombia Netherlands Argentina France 6 Uruguay Belgium France England Netherlands Netherlands 7 Switzerland Chile Chile Portugal Portugal Spain 8 Netherlands Portugal Netherlands Colombia England Belgium 9 Italy England England Uruguay Côte d Ivoire Uruguay 10 England USA Belgium Chile Chile England 11 Brazil Netherlands Portugal Italy Uruguay Bosnia-Herz. 12 Chile France Ecuador France France Ecuador 13 USA Uruguay Russia USA Colombia Mexico 14 Portugal Switzerland Uruguay Belgium Nigeria Switzerland 15 Greece Côte d Ivoire Bosnia-Herz. Russia Iran Portugal 16 Bosnia-Herz. Russia Ukraine Mexico Japan Ghana

43 Thanks!

World Cup draw: quantifying (un)fairness and (im)balance

World Cup draw: quantifying (un)fairness and (im)balance Bloomberg L.P., Quantitative Research Columbia University, Department of Mathematics NYU, Courant Institute of Mathematical Sciences Workshop on Fairness in Sports Ghent University, April 12, 2018 jguyon2@bloomberg.net,