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

2. April 2004FUTMAX ChileFUTMAX 2/25 Summary Motivation Related work Problem definition Formulation FUTMAX in the WWW FUTMAX in the press Results

3. April 2004FUTMAX ChileFUTMAX 3/25 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! Motivation

4. April 2004FUTMAX ChileFUTMAX 4/25 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

5. April 2004FUTMAX ChileFUTMAX 5/25 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! Flamengo37 Cruzeiro37 Bahia36

6. April 2004FUTMAX ChileFUTMAX 6/25 Motivation: what if 26 teams are playing? São Paulo34Ponte Preta26Fluminense21 São Caetano32Figuerense24Vasco20 Santos32Atlético-PR24Goiás20 Juventude31Grêmio24Portuguesa20 Corinthians31Cruzeiro23Paysandu19 Guarani28Gama22Flamengo19 Vitória27Internacion al 22Palmeiras19 Coritiba27Botafogo22Paraná17 Atlético- MG 27Bahia21

7. April 2004FUTMAX ChileFUTMAX 7/25 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 pointstie: 1 pointloss: 0 point

8. April 2004FUTMAX ChileFUTMAX 8/25 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.

9. April 2004FUTMAX ChileFUTMAX 9/25 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.

10. April 2004FUTMAX ChileFUTMAX 10/25 Related work Berkeleys 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.

11. April 2004FUTMAX ChileFUTMAX 11/25 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)

12. April 2004FUTMAX ChileFUTMAX 12/25 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)

13. April 2004FUTMAX ChileFUTMAX 13/25 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)

14. April 2004FUTMAX ChileFUTMAX 14/25 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

15. April 2004FUTMAX ChileFUTMAX 15/25 Formulation

16. April 2004FUTMAX ChileFUTMAX 16/25 Formulation

17. April 2004FUTMAX ChileFUTMAX 17/25 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

18. April 2004FUTMAX ChileFUTMAX 18/25 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.

19. April 2004FUTMAX ChileFUTMAX 19/25 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: http://www.futmax.org 2006 World Cup

20. April 2004FUTMAX ChileFUTMAX 20/25 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)

21. April 2004FUTMAX ChileFUTMAX 21/25 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.

22. April 2004FUTMAX ChileFUTMAX 22/25 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.

23. April 2004FUTMAX ChileFUTMAX 23/25 Results

24. April 2004FUTMAX ChileFUTMAX 24/25 Results FUTMAX can be used to follow the situation of each team: Possible points Points for guaranteed qualification Points for possible qualification Points accumulated FLUMINE NSE

25. April 2004FUTMAX ChileFUTMAX 25/25 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: http://www.esportemax.org Slides available at: http://www.inf.puc-rio.br/~celso/talks.htm Papers (OR/MS Today 2004; ITOR to appear) available at: http://www.inf.puc- rio.br/~celso/publicacoes.htm