FIFA Foe Fun!

Michael Mossinghoff!Davidson College

Tim Chartier!Davidson College

Mark Kozek!Whittier College

• 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.”

1 Spain 15132 Germany 13113 Argentina 12664 Colombia 11785 Belgium 11756 Uruguay 11647 Switzerland 11388 Netherlands 11368 Italy 1136

10 England 108011 Brazil 107812 Chile 1051

13 USA 104014 Portugal 103615 Greece 98316 Bosnia-Herz. 92517 Côte d’Ivoire 91718 Croatia 90119 Russia 87420 Ukraine 87121 France 87022 Ecuador 86223 Ghana 86024 Mexico 854

FIFA Rankings Oct. 2013

• Basis for World Cup groupings.

Nov. 2013• 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.

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.

FIFA Ranking• Fix a team X.

• Let yk = time period starting k years ago and ending k ⎼ 1 years ago.

• Let gk = number of games played by X during yk, and let ck = min(1, gk/5).

• Let ak = ck・(average number of points earned per match over yk).

• Total points for X = a1 + .5a2 + .3a3 + .2a4.

• 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.

Points per Match

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.

• 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).

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 = 0.88.

• AFC & CAF: w = 0.86.

• OFC: w = 0.85.

Points per Match

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 m0 = max m over all confederations.

• w = max(.85, (m/m0)1/4).

• CONCACAF: w = max(.85, (.37/.63)1/4) = .88.

• OFC: w = max(.85, (.17/.63)1/4) = .85.

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.

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.

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 ri.

• 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%.

• Assume no ties for now.

• Suppose team i has Wi wins, Li losses, and has played Gi games.

• Let Oi 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

Gi

X

j2Oi

rj ⇡1

2.

Colley Method

ri ⇡Wi + 1

Gi + 2

=1 + Wi�Li2 +

Gi2

Gi + 2

⇡1 + Wi�Li2 +

Pj2Oi rj

Gi + 2.

(Gi + 2)ri �X

j2Oi

rj = 1 +Wi � Li

2.

So:

This produces the linear system:

We write Cr = b.

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.

Colley Method

Type Weight

• Friendly = 1,

• Continental qualifier = 1.25,

• Continental tourn. or Confed. Cup = 1.5,

• World Cup qualifier = 2,

• World Cup match = 2.25.

1 2 3 4 5

0.2

0.4

0.6

0.8

1.0

Age Weight

FIFA

.20.30

.50

Smoothed

.717.283.054.008 .946

1

0

• Total area (nearly) preserved. • Keep five years now for smoother aging.

June 9, 2013: World Cup Qualifier.

0-3

• Ignore disqualifications.

Additional Adjustments

June 28, 2011: World Cup Qualifier.

• Ignore disqualifications.

Additional Adjustments

1 Brazil 1.058 ⬆️ 22 Spain 1.008 ⬇️ 13 Argentina 0.975 ⬆️ 24 Germany 0.951 ⬇️ 25 Colombia 0.934 ⬆️ 36 Belgium 0.929 ⬆️ 57 Chile 0.883 ⬆️ 78 Portugal 0.876 ⬇️ 49 England 0.872 ⬆️ 110 USA 0.869 ⬆️ 311 Netherlands 0.859 ⬆️ 412 France 0.859 ⬆️ 5

13 Uruguay 0.849 ⬇️ 614 Switzerland 0.842 ⬇️ 815 Côte d’Ivoire 0.823 ⬆️ 816 Russia 0.823 ⬆️ 317 Italy 0.814 ⬇️ 818 Ecuador 0.813 ⬆️ 819 Ukraine 0.810 ⬇️ 320 Greece 0.810 ⬇️ 821 Japan 0.780 ⬆️ 2522 Croatia 0.776 ⬇️ 423 Bosnia-Herz. 0.776 ⬇️ 224 U.A.E. 0.775 ⬆️ 48

Weighted Colley

• Last column: Change from current FIFA rank.

Colley: Group of Death!

Gp 1 2 3 4 Third Avg Rk Gap

A 1 22 25 48 25 24.0 25.0B 2 7 11 37 11 14.3 19.5C 5 15 20 21 20 15.3 13.0D 9 13 17 34 17 18.3 15.0E 12 14 18 46 18 22.5 16.0F 3 23 27 29 27 20.5 17.0G 4 8 10 3 0 10 13 15.5H 6 16 26 4 1 26 22.3 19.5

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 ri – rj = p.

• Produces inconsistent system.

• Use least squares.

Massey Method• Massey matrix: M = C – 2IN.

• Solve Mr = v, where vi = (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

• OK as long as there is a path between any two teams.

NX

i=1

ri = 0.

January 29, 2014: Friendly.

• Ignore disqualifications.

• Count penalty shoot-outs as weak wins.

PSO 4-1

Adjustments

October 14, 2010: CONCACAF Qualifier

• Ignore disqualifications.

• Count penalty shoot-outs as weak wins.

17-0

• Massey: set max score differential to 4.

Adjustments

July 2 and 9, 2011: World Cup Qualifiers:

• Ignore disqualifications.

• Count penalty shoot-outs as weak wins.

Only matches for both since 2008!

• Massey: set max score differential to 4.• Ensure connectivity.

Adjustments

Omisions!

1 Brazil 3.633 ⬆️ 22 Argentina 3.126 ⬆️ 33 Germany 3.031 ⬇️ 14 Spain 2.940 ⬇️ 35 Colombia 2.875 ⬆️ 36 France 2.670 ⬆️ 117 Chile 2.620 ⬆️ 78 Netherlands 2.616 ⬆️ 79 England 2.541 ⬆️ 110 Belgium 2.528 ⬆️ 111 Portugal 2.354 ⬇️ 712 Ecuador 2.306 ⬆️ 14

13 Russia 2.304 ⬆️ 614 Uruguay 2.202 ⬇️ 715 Bosnia-Herz. 2.187 ⬆️ 616 Ukraine 2.123 017 Serbia 2.038 ⬆️ 1318 Côte d’Ivoire 2.019 ⬆️ 519 USA 1.991 ⬇️ 620 Italy 1.984 ⬇️ 1121 Switzerland 1.942 ⬇️ 1522 Mexico 1.866 ⬇️ 223 Croatia 1.836 ⬇️ 524 Sweden 1.712 ⬆️ 8

Weighted Massey

• Last column: Change from current FIFA rank.

Massey: Group of Death!

Gp 1 2 3 4 Third Avg Rk Gap

A 1 22 23 45 23 22.8 22.5B 4 7 8 54 8 18.3 25.5C 5 18 27 32 27 20.5 18.0D 9 14 20 44 20 21.8 20.5E 6 12 21 56 21 23.8 29.5F 2 15 30 47 30 23.5 30.0G 3 11 19 2 6 19 14.8 15.5H 10 13 28 5 7 28 27.0 31.0

Build Your Own!• FIFAfoefun.davidson.edu.

• Build personalized rating of international FIFA teams using your selected parameters.

• Colley or Massey.

• Number of years to use.

• Type weights.

• Age weighting method.

• Age weights.

• Value of win in penalty shoot-out.

• Max score differential to use in Massey.

http://FIFAfoefun.davidson.edu

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.

• 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.

ESPN Bracket Predictor

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.

• Suppose team i beats team j.

• Add to ri and subtract from rj: K(v ⎼ F(ri ⎼ rj)).

• K = weight of the match.

• v = value of the victory: 0.5 ≤ v ≤ 1.

• F(x) = distribution function for logistic distribution.

-1000 -500 500 1000

0.2

0.4

0.6

0.8

1.0

World Football Elo Ratings• www.eloratings.net

• 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!

http://www.eloratings.net

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.

1 Brazil 2113 ⬆️ 22 Spain 2086 ⬇️ 13 Germany 2046 ⬇️ 14 Argentina 1989 ⬆️ 15 Netherlands 1959 ⬆️ 106 England 1914 ⬆️ 47 Portugal 1902 ⬇️ 38 Colombia 1897 09 Uruguay 1895 ⬇️ 210 Chile 1895 ⬆️ 411 Italy 1879 ⬇️ 212 France 1869 ⬆️ 513 USA 1832 014 Belgium 1824 ⬇️ 315 Russia 1821 ⬆️ 416 Mexico 1820 ⬆️ 417 Switzerland 1820 ⬇️ 1118 Ukraine 1815 ⬇️ 219 Ecuador 1813 ⬆️ 720 Greece 1796 ⬇️ 8

eloratings.net Our Elo ratings1 Spain 316 02 Brazil 307 ⬆️ 13 Germany 265 ⬇️ 14 USA 234 ⬆️ 95 Argentina 217 06 Netherlands 216 ⬆️ 97 Portugal 213 ⬇️ 38 England 186 ⬆️ 29 Côte d’Ivoire 182 ⬆️ 1410 Chile 174 ⬆️ 411 Uruguay 173 ⬆️ 512 France 168 ⬆️ 513 Colombia 167 ⬇️ 514 Nigeria 163 ⬆️ 3015 Iran 159 ⬆️ 2816 Japan 158 ⬆️ 3017 Belgium 157 ⬇️ 618 Greece 156 ⬇️ 619 Switzerland 151 ⬇️ 1320 Egypt 151 ⬆️ 16

http://eloratings.net

Number of Matches

20 40 60 80 100

10

20

30

40

µ = 41.6, � = 21.0.

• 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, Côte-d’Ivoire, Iran) have perhaps inflated rankings.

Elo Summary

FIFA Colley Massey WER Elo SPI1 Spain Brazil Brazil Brazil Spain Brazil2 Germany Spain Argentina Spain Brazil Argentina3 Argentina Argentina Germany Germany Germany Germany4 Colombia Germany Spain Argentina USA Colombia5 Belgium Colombia Colombia Netherlands Argentina France6 Uruguay Belgium France England Netherlands Netherlands7 Switzerland Chile Chile Portugal Portugal Spain8 Netherlands Portugal Netherlands Colombia England Belgium9 Italy England England Uruguay Côte d’Ivoire Uruguay10 England USA Belgium Chile Chile England11 Brazil Netherlands Portugal Italy Uruguay Bosnia-Herz.12 Chile France Ecuador France France Ecuador13 USA Uruguay Russia USA Colombia Mexico14 Portugal Switzerland Uruguay Belgium Nigeria Switzerland15 Greece Côte d’Ivoire Bosnia-Herz. Russia Iran Portugal16 Bosnia-Herz. Russia Ukraine Mexico Japan Ghana

http://espnfc.com/world-cup/story/_/id/4447078/ce/us/guide-espn-spi-ratings?cc=5901&ver=us

Thanks!