1. 1 913856 OR Applications in Sports Management : The Playoff Elimination Problem IEEM 710300 Topics in Operations Research

2. 2 Results of the MLB case QuestionOptimumRepresentation First-Place-Elimination Number of additional games to win to avoid elimination from first place Play-Off-Elimination Number of additional games to win to avoid elimination from playoffs First-Place-Clinch Number of additional games, if won, guarantees a first-place finish Play-Off-Clinch Number of additional games, if won, guarantees a playoff spot

3. 3 The Selected Case The Brazilian National Football Championship is the most important football tournament in Brazil. The major goal of each team is to be qualified in one of the eight first positions in the standing table at the end of the qualification stage. For the teams that cannot match this objective, their second goal is, at least, not to finish in the last four positions to remain in the competition next year.. The media offers several statistics to help fans evaluate the performance of their favorite teams. However, most often, the information is not correct. Thus, this study aims to solve the GQP (Guaranteed Qualification Problem) and the PQP (Possible Qualification Problem) by finding out the GQS (Guaranteed Qualification Score) and PQS (Possible Qualification Score) for each team..

4. 4 Whats different? 1.The 3-point-rule v.s. the 1-point-rule The regulations to determine whether a team plays better or worse than others 2.Number of teams to be taken into account 3.Quotas for playoff participants

5. 5 The 3-Point-Rule If a team wins against its opponent, it will get 3 points while the other gets none. If theres a tie, both teams will get 1 point.. Team Current points Flamengo37 Cruzeiro37 Bahia36 ElmiminatedN/A * All of these three teams have 2 remaining games to play Comparison of the complexity under different rules Team Current points All possible resulting points under the 1-point-rule Flamengo3738 37 Cruzeiro3738 37 38 37 Bahia363736373637363736 ElmiminatedN/ABB B B B * All of these three teams have 1 remaining game to play Under the 3-point-rule, the number of possible results may be 30,000 times more. Team Current points Some possible resulting points under the 3-point-rule Flamengo3740 Cruzeiro3740 38 37 Bahia36393736393736393736 ElmiminatedN/ABBBCBBC B * All of these three teams have 1 remaining game to play

6. 6 Guaranteed Qualification Problem (GQP) The GQP consists in calculating the minimum number of points of any team has to win (Guaranteed Qualification Score, GQS) to be sure it will be qualified, regardless of any other results. The GQS depends on the current number of points of every team in the league and on the number of remaining games to be played. GQS cannot increase along the competition. A team is mathematically qualified to the playoffs if and only if its number of points won is greater than or equal to its GQS.

7. 7 Possible Qualification Problem (PQP) The PQP consists in computing how many points each team has to win (Possible Qualification Score, PQS) to have any chance to be qualified. The PQS depends on the current number of points of every team in the league and on the number of remaining games to be played. PQS cannot decrease along the competition. A team is mathematically eliminated from the playoffs if and only if the total number of points it has to play plus the current points (Maximum Number of Points, MNP) is less than its PQS. Of course, PQS GQS for any team at any time.

8. 8 Problem Definition: GQP first-eight-place [Restrictions & Assumptions] 1.There are 26 teams in the league. 2.Every team has to finish only one game against each of the other 25 teams; thus, the total number of games for a team is 25. 3.Every game is under the 3-point-rule. 4.A team finishes the qualification stage with the eight most total points will advance to the play-off rounds. 5.Ties in the final standing for a play-off spot are settled by comparing the number of wins of all candidates. [Inputs] Current win-loss records, remaining schedule of games [Outputs] A teams guarantee qualification point (GQP).

9. 9 Notations : GQP first-eight-place Let be the total number of points for team at the end of the qualification stage.Let be the current number of points that team has won. Let be the current number of teams that have no less points than team. Let be the maximum number of points for team such that there exists a valid assignment leading to and at the end of the qualification stage. Therefore, is the minimum number of points that team has to obtain to ensure its qualification among the first teams. Let be the number of teams that can be qualified to the playoffs (among teams).

10. 10 Mathematical Models: GQP first-eight-place (1) (2) (3) (4) Current points 3 points for winning There are at least 8 teams that are ahead of team k. Is a valid upper bound. The maximum number of points foe team k such that it can not be qualified to the playoffs.

11. 11 Problem Definition: PQP first-eight-place [Restrictions & Assumptions] 1.There are 26 teams in the league. 2.Every team has to finish only one game against each of the other 25 teams; thus, the total number of games for a team is 25. 3.Every game is under the 3-point-rule. 4.A team finishes the qualification stage with the eight most total points will advance to the play-off rounds. 5.Ties in the final standing for a play-off spot are settled by comparing the number of wins of all candidates. [Inputs] Current win-loss records, remaining schedule of games [Outputs] A teams guarantee qualification point (GQP).

12. 12 Notations : PQP first-eight-place Let be the total number of points for team at the end of the qualification stage.Let be the current number of points that team has won. Let be the current number of teams that have no less points than team. Let be the minimum number of points for team such that there exists at least one set of valid assignments leading to and at the end of the qualification stage. Let be the number of teams that can be qualified to the playoffs (among teams).

13. 13 Mathematical Models: PQP first-eight-place (1) (2) (3) (4) Current points 3 points for winning There are at most 7 teams that are ahead of team k. Is a valid upper bound. The minimum number of points foe team k such that it has a chancel to be qualified.

14. 14 Results * 2002 Brazilian National Football Championship RankTeamCurrent pointsGames to playPQSGQS 1São Paulo493-- 2São Caetano443-- 3Corínthians423-- 4Juventude413-- 5Atlético MG403-- 6Santos393 40 7Grêmio383 40 8Fluminense3733839 9Coritiba3633740 10Goiás3633940 11Cruzeiro3633940 12Vitória3433738 13Ponte Preta3433738

15. 15 Team Fluminense (2002)

16. 16 Mathematical Models: GQP last-four-place (1) (2) (3) (4) There are at least 18 teams that are ahead of team k.

17. 17 Mathematical Models: PQP last-four-place (1) (2) (3) (4) There are at most 17 teams that are ahead of team k.

18. 18 Conclusions 1.Under a different rule, the playoff elimination problem may be even more complex. 2.This study provides a more general model for solving the playoff elimination problem.

19. 19 Mathematical Models: GQP refined for 1-point-rule (1) (2) (3) (4) At least one team wins, i.e. no ties. 1 point for winning There are no ties.

20. 20 References 1.Ribeiro, C. C. and Urrutia, S. (2004) An Application of Integer Programming to Playoff Elimination in Football Championships, to appear in International Transactions in Operational Research. 2.Footmax, available on the Internet: http://futmax.inf.puc-rio.br/. 3.Bernholt, T., Gulich, A. Hofmeuster, T. and Schmitt, N. (1999) Football Elimination is Hard to Decide Under the 3-Point-Rule, Proceedings of the 24 th International Symposium on Mathematical Foundations of Computer Science, published as Lecture Notes in Computer Science 1672, Springer, pp. 410-418. 4.Adler, I., Erera, A. L., Hochbaum, D.S., and Olinick, E. V. (2002) Baseball, Optimization, and the World Wide Web, Interfaces 32(2), pp. 12-22. 5.Remote Interface Optimization Testbed, available on the Internet: http://riot.ieor.berkeley.edu/.