OR Applications in Sports Management : The Playoff Elimination Problem IEEM Topics in Operations Research

2 Introduction Sports management is a very attractive area for Operations Research. Deciding playoff elimination and Timetabling are the two problems discussed most frequently. The former helps the fans to be aware of the status of their favorite teams, either qualified to or eliminated from the playoffs. This information is also very useful for team managers to decide whether to spend time in planning the future or to struggle for the current season. The latter can be used to devise a fairer and more cost-effective schedule for the league..

3 The Playoff Elimination Problem 1.Schwartz (1966) showed that a maximum-flow calculation on a small network can determine precisely when a team has been necessarily eliminated from the first place.. 2.Hoffman and Rivlin (1970) extended Schwartzs work, developing necessary and sufficient conditions for eliminating a team from k th place. McCormick (1987, 1999) in turn showed that determining elimination from k th place is NP-complete.. 3.Robinson (1991) applied linear programming in solving baseball playoff eliminations, which resulted in eliminating team three days earlier than the wins-based criterion during the 1987 MLB season.

4 The MLB Case The Elias Sports Bureau, the official statistician for MLB, determines whether a particular team is eliminated using a simple criterion: if a team trails the first-place team in wins by more games than it has remaining, it is eliminated. However, according to this study, a team had actually been eliminated few days earlier than it was announced by MLB.. First-place elimination is not the fans only interest. In baseball, teams may also reach the play-off by securing a wild-card berth; the team that finishes with the best record among second-place teams in the league is assigned this berth. Based on the MLB statistics and the models provided, fans can sort out the play-off picture with more precise information..

5 Problem Definition: Elimination Questions [Restrictions & Assumptions] 1.There are three divisions for each of the two leagues. 2.Every team has to finish 162 games per season. 3.Theres neither rain-outs nor ties. (Every game has a winner.) 4.A team finishes the season with the best record of the division will advance to the play-off rounds. 5.Ties in the final standing for a play-off spot are settled by special one-game playoffs. 6.A team with the best record among all second-pace teams in the league will advance to the play-off rounds as the wild card. 7.To find the minimum number of wins necessary to win a division, it is only necessary to consider scenarios in which the teams in the division lose all remaining games against non-division opponents. [Inputs] Current win-loss records, remaining schedule of games [Outputs] A teams first-place-elimination number and play-off-elimination number

6 Notations : Elimination Questions Let be the decision variable representing the first-place-elimination threshold for division. : the set of teams in a league : the set of teams in a division k For each team in division, let be its number of current wins, the number the number of games remaining against team, and the number of games remaining against nondivision opponents. Finally, let be the total number of wins attained by team by seasons end in some scenario. Further, let represent the number of future games that team wins against team ; let denote a complete scenario of future wins,. Let be the decision variable representing the play-off-elimination threshold for league.

7 Mathematical Models: First-Place-Elimination (1) (2) (3) (4) (5) team wins team against team (1) is the same as Every game has a winner RankingNumber of wins (2)

8 Mathematical Models: First-Place-Elimination Suppose that the optimal objective value is, the first-place-elimination threshold for division. Any team that can attain at least wins by season end will win the division. Let, If, a division-winning scenario can be attained for team by increasing its number of non-division wins such that wins exactly total games. If, a division-winning scenario can be attained for team by winning all of its non-division games( ) and an additional ( ) division games.

9 Mathematical Models: First-Place-Elimination It is clear that a team is eliminated from first-place if and only if Further, if a team is not eliminated,. Therefore, its first-place-elimination number is ( ), the minimum number of future wins that team needs to reach the threshold. In addition, as mentioned above, a team is eliminated from the first-place, if its first- place number is greater than the number of its remaining games, i.e. (first-place-elimination number) (number of remaining games)

10 Mathematical Models: Play-Off-Elimination (1) (2) (3) (4) (5) (6) (7) Every game has a winner. The variable u will not be affected by the number of wins for the first- place team if the three divisions in the league. The variable u is at least as large as the number of wins by all teams except first- place teams of the three divisions. (2)

11 Mathematical Models: Play-Off-Elimination Suppose that the optimal objective value is, the play-off-elimination threshold for league. The play-off-elimination number for each team with Is The play-off-elimination number for each team that wins the division Is

12 Problem Definition: Clinching Questions [Restrictions & Assumptions] 1.There are three divisions for each of the two leagues. 2.Every team has to finish 162 games per season. 3.Theres neither rain-outs nor ties. (Every game has a winner.) 4.A team finishes the season with the best record of the division will advance to the play-off rounds. 5.A team with the best record among all second-pace teams in the league will advance to the play-off rounds as the wild card. 6.Ties in the final standing for a play-off spot are settled by special one-game playoffs. [Inputs] Current win-loss records, remaining schedule of games [Outputs] A teams first-place-clinch number and play-off-clinch number

13 Notations : Clinching Questions Let be the number of games for team to win to tie up with team. : the set of teams in a league : the set of teams in a division k For each team in division, let be its number of current wins, the number the number of games remaining against team, the number of games remaining against nondivision opponents, and the number of its future wins. Further, let represent the number of future games that team wins against team ; let denote a complete scenario of future wins,. Let be the total wins accrued by team such that finishes with fewer wins than the first-place team in its division, and at least one division contains two teams with better records. Thus, ( ) is the play-off clinch number for team. Let be the number of games for team to win to tie up with all teams in the division, i.e. the first-place-clinching number for team.

14 Mathematical Models: First-Place-Clinching (1) (2) team must win some games against. As team wins one game against team, the number of games that trails by will decrease by two, however. Therefore, the number of games that has to win against is. In addition, team may win at most games against teams other than. To guarantee a tie with team,. Thus, in this case,

15 Mathematical Models: First-Place-Clinching (1) (2) we assume that each future win by team comes against teams other than. To guarantee a tie with team,. Thus, in this case, The first-place-clinch number for team can be calculated as, without optimization. [Remarks] Magic Number is calculated as, where denotes current numbers of wins for the first and second place teams respectively and denotes the number of remaining games for the second-place team. If either the 1 st -place team wins one more game or the 2 nd -place team loses one more game, the magic number decreases by 1. As the magic number approaches 0, the first-place team wins the division.

16 Mathematical Models: Play-Off-Clinching (1) (2) (3) (4) (5) (6) (7) (8) Every game has a winner. denotes the number of teams in division k denotes the number of teams without play-off positions in division k

17 Mathematical Models: Play-Off-Clinching All teams that finish in a play-off position will have more wins than does. All teams that fail to finish in a play-off position will not be taken into consideration. The play-off-clinch number for team =.

18 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

19 Results of the MLB case

20 The CBF 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..

21 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

22 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 1 remaining game to play Comparison of the complexity under different rules Under the 3-point-rule, the number of possible results may be 30,000 times more. Team Current points All possible resulting points under the 1-point-rule Flamengo Cruzeiro Bahia ElmiminatedN/ABB B B B * All of these three teams have 1 remaining game to play Team Current points Some possible resulting points under the 3-point-rule Flamengo3740 Cruzeiro Bahia ElmiminatedN/ABBBCBBC B * All of these three teams have 1 remaining game to play

23 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.

24 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.

25 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 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 score (GQS).

26 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).

27 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.

28 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 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 possible qualification score (PQS).

29 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).

30 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.

31 Mathematical Models: GQP last-four-place (1) (2) (3) (4) There are at least 22 teams ahead of team k.

32 Mathematical Models: PQP last-four-place (1) (2) (3) (4) There are at most 21 teams ahead of team k.

33 Results of the CBF case * 2002 Brazilian National Football Championship RankTeamCurrent pointsGames to playPQSGQS 1São Paulo São Caetano Corínthians Juventude Atlético MG Santos Grêmio Fluminense Coritiba Goiás Cruzeiro Vitória Ponte Preta

34 Team Fluminense (2002) Results of the CBF case

35 Conclusions 1.The applications are very attractive, which encourages students to study optimization problems in Operations Research. 2.Under a different rule, the playoff elimination problem may be even more complex. 3.The CBF case provides a more general model for solving the playoff elimination problem. 4.Probabilistic models may describe more exactly how likely a team is able to be clinched to or eliminated from the playoffs.

36 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.

37 References 1.Adler, I., Erera, A. L., Hochbaum, D.S., and Olinick, E. V. (2002) Baseball, Optimization, and the World Wide Web, Interfaces 32(2), pp Remote Interface Optimization Testbed, available on the Internet: 3.Schwartz, B. L. (1966) Possible Winners In Partially Completed Tournaments, SIAM Rev. 8(3), pp McCormick, S. T. (1987) Two Hard Min Cut Problems, Technical report presented at the TMS/ORSA Conference, New Orleans, L.A. 5.McCormick, S. T. (1999) Fast Algorithms for parametric Scheduling Come From Extensions To Parametric Maximum Flow, Oper. Res. 47(5), pp Robinson, L.W. (1991) Baseball Playoff Eliminations: An Application of Linear Programming, Operations Research Letters 10, pp 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. 8.Footmax, available on the Internet: 9.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