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

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..

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 Schwartz’s 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.

The Selected 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..

Regulations of the MLB Playoffs Chicago Minnesota ClevelandDetroit Kansas City Central Division Toronto Baltimore BostonTampa Bay New York East Division Los Angeles Texas OaklandSeattle West Division Atlanta Florida New YorkWashington Philadelphia East Division St. Louis Chicago CincinnatiHouston Milwaukee Central Division Pittsburgh Los Angeles Minnesota New York Boston Wild Card v.s. Atlanta St. Louis Los Angeles v.s. Houston Wild Card Los Angeles Arizona San FranciscoSan Diego West Division Colorado Wild Card

An Example: The Case of Detroit Tigers Current win-loss records Remaining schedule of games For Detroit Tigers, there’s a remote chance of catching the first-place: If Detroit wins all of its remaining games, it will end up with = 76 wins. However, New York will meet this record by easily adding just one more win of the remaining 28 games. Therefore, is it reasonable to say that the Detroit Tigers has been eliminated from the first-place? (The answer is “yes.”) Considering the scenario that New York wins no more than one games in the remaining 28 games. 1) New York fails to win another game [75-87] Since Boston has 8 more games against New York, it will win them all. Thus, Boston may end up the season with at least ( = 77) wins. That is to say, there’s no chance for Detroit to catch the first place. 2) New York wins only one more game [76-86] According to the previous scenario, the only one win for New York must be against Boston, so that Boston may not end up with 77 wins. In addition, Boston have to lose all of the remaining games except the 7 wins against New York. Therefore,the final record for Boston would be also Estimates in a specific scenario

An Example: The Case of Detroit Tigers Current win-loss records Remaining schedule of games Therefore, is it reasonable to say that the Detroit Tigers has been eliminated from the first-place? (The answer is “yes.”) [continued] 2) New York wins only one more game [76-86] [continued] Now consider Baltimore and Toronto. Since Boston wins only the 7 (of 8) games against New York. It will undoubtedly lose when encountering all other teams. Thus, Baltimore will win all of its 2 games against Boston. Similarly, New York can not win any game except the only one game against Boston. Hence, Baltimore will win all of its 3 games against New York. Based on these facts, Baltimore’s final wins will be at least ( = 76), which results in a four-way tie for first place. In order not to finish ahead of Detroit, Baltimore must lose all other games except the 5 additional wins. That is to say, it will lose all of its 7 games against Toronto. In addition, since New York can not win any game against teams except Boston, Toronto will also win its 7 games against New York. This will make Toronto finish with ( = 77) wins. In this scenario,the final records for the division is shown as the table in the bottom. Toronto will catch the first-place, and then clinch for the play-off. In summary, there’s no chance for Detroit to catch the first-place; it has been eliminated from the first-place. Estimates in a specific scenario

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.There’s 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 team’s first-place-elimination number and play-off-elimination number

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 season’s 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.

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)

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.

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)

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)

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

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.There’s 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 team’s first-place-clinch number and play-off-clinch number

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.

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,

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.

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

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 =.

Results 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

Results

Conclusion & Discussion 1.The method is simple and useful. 2.The applications are very attractive, which encourages students to study optimization problems in Operations Research. 3.Problems for k th -place-elimination or k th -place-clinching need to be discovered.

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 Wayne, K.D. (2001) A New Property and A Faster Algorithm For Baseball Elimination, SIAM J. Discrete Math 14(2), pp Ribeiro, C. C. and Urrutia, S. (2004) OR Applications In Sports Scheduling and Management, OR/MS Today 31(3), 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.