1. Play-offs elimination in
Celso C. Ribeiro
Sebastián Urrutia
FUTMAX:
Play-offs elimination in
soccer tournaments

2. Summary Motivation Related work Problem definition Formulation
FUTMAX in the WWW
FUTMAX in the press
Results
April 2004
FUTMAX Chile

3. Motivation Guaranteed Qualification Problem:
How many points a team has to make in a tournament to be able to be qualified for the finals (or play-off games)?
Brazilian Soccer Championship:
Press broadcasts already at the first round the “chances of qualification” for the play-offs, based in very obscure statistics and estimates.
Predictions loudly announced are often wrong!
April 2004
FUTMAX Chile

4. Motivation
Interesting application of Operations Research and Integer Programming in sports.
Soccer is the most popular sports in most countries.
Soccer can even make researchers from Argentina and Brazil to work together, instead of trying to prove which of them is the best...
… or arguing about who is the second best player ever! #1
April 2004
FUTMAX Chile

5. Motivation Three teams fight for two places in the play-offs.
Each of them has two games to play: Flamengo against Vasco and Bahia, Cruzeiro against Grêmio and Santos, and Bahia against Flamengo e Fluminense.
How many points Cruzeiro has to make to be sure of being qualified?
4 points!
Flamengo 37
Cruzeiro
Bahia 36
April 2004
FUTMAX Chile

6. Motivation: what if 26 teams are playing?
São Paulo 34
Ponte Preta 26
Fluminense 21
São Caetano 32
Figuerense 24
Vasco 20
Santos
Atlético-PR
Goiás
Juventude 31
Grêmio
Portuguesa
Corinthians
Cruzeiro 23
Paysandu 19
Guarani 28
Gama 22
Flamengo
Vitória 27
Internacional
Palmeiras
Coritiba
Botafogo
Paraná 17
Atlético-MG
Bahia
April 2004
FUTMAX Chile

7. Motivation Brazilian soccer championship (2002 edition) 26 teams
First phase: qualification
Each team plays every other team exactly once.
The eight teams in the first positions are qualified.
The four teams in the last positions do not play next year.
Second phase: play-offs
Eliminatory games
Points:
win: 3 points tie: 1 point loss: 0 point
April 2004
FUTMAX Chile

8. Motivation Brazilian championship: 29 rounds from August to December
The press publishes since the first round (!) the probabilities of qualification of each team.
The press also makes use of very simple statistics from obscure sources to announce very early that a team with a certain number of points (41 in 2002) can be sure to be qualified (often based on estimates or average results).
These predictions are loudly announced and often false!
Trainers also very often make erroneous predictions.
April 2004
FUTMAX Chile

9. Related work
Schwartz 1966: mathematical elimination from play-offs in the Major League Baseball (MLB) solved with maximum flow algorithm
Robinson 1991: integer programming models and further results for the play-offs elimination problem
McCormick 2000: elimination from the p-th position is NP-complete.
Adler et al. 2003: ILP models for MLB
Bernholt et al. 1970: first place elimination is NP-complete under the {(3,0),(1,1)} soccer rule.
April 2004
FUTMAX Chile

10. Related work
Berkeley’s RIOT project posts the number of wins each team needs to have a chance of qualification in the MLB:
No ties
Easier {(1,0)} rule
Small groups with 5 or 6 teams
A team has to finish in the first position within its group to be qualified.
Each team plays up to 160 games!
Problems in each group are easily solvable.
April 2004
FUTMAX Chile

11. Problem definition Play-offs qualification problems:
How many points a team should make to:
… be sure of finishing among the p=8 teams in the first positions? (sufficient condition for play-offs qualification)
… have a chance of finishing among the p=8 teams in the first positions? (necessary condition for play-offs qualification)
April 2004
FUTMAX Chile

12. Problem definition Elimination problems:
How many points a team should make to:
… be sure of finishing among the p=22 teams in the first positions? (sufficient condition for non-elimination)
… have a chance of finishing among the p=22 teams in the first positions? (necessary condition for non-elimination)
April 2004
FUTMAX Chile

13. Problem definition
In this talk, we consider only the Guaranteed Qualification Problem: How many points a team k should make to be sure of finishing among the p=8 in the first positions?
Instead, we compute the maximum number of points a team can make and still not be qualified in the p=8 first positions.
Then, we add 1 to the above number to determine the minimum number of points for guaranteed qualification: GQS(k)
April 2004
FUTMAX Chile

14. Formulation n = number of teams = 26
M = maximum difference of points between any two teams (all wins vs. all losses) = 3*(n-1) = 75
k = team under consideration
April 2004
FUTMAX Chile

15. Formulation 
April 2004
FUTMAX Chile

16. Formulation
April 2004
FUTMAX Chile

17. Formulation
Ties in the number of points are broken in favor of teams with more wins.
In the previous model, add up a very small amount (necessarily smaller than one) to the number of points:
Use e.g.
With a similar model, we can also compute PQS(k): minimum number of points for possible qualification
April 2004
FUTMAX Chile

18. Formulation
When is team k “mathematically qualified” for the play-offs?
Team k is mathematically qualified if the previous problem is infeasible!
When does a team k depend only on itself to be qualifed?
Team k depends only on itself if GQS(k) is less than or equal to the total number of points it can make.
When is team k “mathematically eliminated” from the play-offs?
Team k is mathematically eliminated if PQS(k) is less than or equal to the total number of points it can make.
April 2004
FUTMAX Chile

19. FUTMAX in the WWW FUTMAX project
Results of the games are collected from the web.
Model are generated (four problems for each team).
All problems are solved with CPLEX 9.0
HTML file is automatically built from the results.
Automatic publication in the web:
World Cup
April 2004
FUTMAX Chile

20. FUTMAX in the press September 24, 2002: FUTMAX web site launched
September 29: round table interview to Radio Globo (major sports radio station, program “Enquanto a bola não rola”)
September 30: article in the Internet section of Jornal do Brasil
October 24: interview for TV Campus
December 18: article in Jornal da PUC
October 2003: talks with SPORTV (TV Globo) and computations of “chances” of qualification (live)
April 2004
FUTMAX Chile

21. Results
Already at the 11th round, some teams did not depend only on themselves to be qualified.
At this time, Vasco was in a difficult situation and fans were complaining.
The trainer of Vasco met the press and said that “if we win the next ten games, then we will be qualified”.
FUTMAX was able to prove to the press that this was not true.
April 2004
FUTMAX Chile

22. Results
October 31, 2002: São Paulo won Ponte Preta and made a total of 43 points.
The press (Folha de São Paulo) announced that “São Paulo was mathematically qualified for the play-offs”.
FUTMAX was able to prove to the press that this was not true.
November 3, 2002: São Caetano made a total of 42 points and the press also announced it was qualified.
Again, FUTMAX was able to prove to the press that this was not true.
April 2004
FUTMAX Chile

23. Results
April 2004
FUTMAX Chile

24. Results FUTMAX can be used to follow the situation of each team:
Possible points
FLUMINENSE
Points for guaranteed
qualification
Points for possible
qualification
Points accumulated
April 2004
FUTMAX Chile

25. Results
Spin-offs: followed by HockeyPlex project (same idea for National Hockey League, USA) based on FUTMAX
Motivation for students
Activities of our research group made public
New web page:
Slides available at:
Papers (OR/MS Today 2004; ITOR to appear) available at:
April 2004
FUTMAX Chile