1. Engineering Management, Information, and Systems 1 It Ain’t Over Till It’s Over: Playoff Races and Optimization Modeling Exercises Eli Olinick

2. Engineering Management, Information, and Systems Department 2 Motivating Question: Can The Giants Win the Pennant? National League West September 8, 1996 Games TeamWinsLossesBackLeft Dodgers786321 Padres78651.019 Rockies71 7.520 Giants598118.522 It ain’t over till it’s over According to traditional statistics, the Giants are not “mathematically” Eliminated (59+22= 81 > 78).

3. Engineering Management, Information, and Systems Department 3 But What About the Schedule? The Dodgers and Padres will play each other 7 more times There are no ties in baseball One of these two teams will finish the season with at least 82 wins Since they can finish with at most 81 wins, the Giants have already been eliminated from first place Games TeamWinsLossesBackLeft Dodgers786321 Padres78651.019 Rockies71 7.520 Giants598118.522 It ain’t over till it’s over … unless it’s over

4. Engineering Management, Information, and Systems Department 4 Selling Sports Fans on the Science of Better The traditional definition of “mathematical elimination” is based on sufficient, but not necessary conditions (Schwartz [1966]) Giants’ elimination reported in SF Chron. until 9/10/96, but Berkeley RIOT website (http://riot.ieor.berkeley.edu/~baseball) reported it on 9/8/96 [Adler et al. 2002] OR model shows elimination an average of 3 days earlier than traditional methods in 1987 MLB season (Robinson [1991]) –In some sports the traditional calculations are based on methods aren’t even sufficient! Soccer clinches announced prematurely (Ribeiro and Urrutia [2004]) A simple max-flow calculation can correctly determine when a team is really “mathematically eliminated” More interesting questions can be answered by solving straight-forward extensions to the max-flow model

5. Engineering Management, Information, and Systems Department 5 Can Detroit Win This Division? WLGBGL New York7559-28 Baltimore7163428 Boston69666.527 Toronto637212.527 Detroit498626.527 Since Detroit has enough games left to catch New York it’s (remotely) possible.

6. Engineering Management, Information, and Systems Department 6 But What About the Schedule? TeamsGames Baltimore vs. Boston2 Baltimorevs. New York3 Baltimore vs. Toronto7 Boston vs. New York8 New York vs. Toronto7 Assume Detroit wins all of its remaining games to finish the season with 76 wins. Assume the other teams in the division lose all of their games to teams in other divisions. Can we determine winners and losers of the games listed above so that no other team finishes with more than 76 wins?

7. Engineering Management, Information, and Systems Department 7 Proof of Detroit’s Elimination Laborious analysis of possible scenarios –If New York wins two or more games, they will finish with at least 77 wins. Detroit is out. –If New York loses all of their remaining games, then Boston will win at least 8 more games which would give them at least 77 wins. Detroit is out. –If New York wins exactly 1 more game … Detroit is out.

8. Engineering Management, Information, and Systems Department 8 OR Proof: Detroit’s Elimination Network Team Nodes Game Nodes Bal vs. Bos. N.Y. vs. Tor. Bal. vs. N.Y. s 7 2 8 3 Bos. vs. N.Y. 7 Bal Bos N.Y. Tor t 5 7 76-75=1 13 Bal. vs. Tor. u[S,T] = 26 < # wins remaining ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

9. Engineering Management, Information, and Systems Department 9 RIOT Site September 8, 2004

10. Engineering Management, Information, and Systems Department 10 Remaining Series in the AL West TeamsGames Anaheim vs. Oakland6 Anaheim vs. Seattle7 Anaheim vs. Texas7 Anaheim vs. Other5 Oakland vs. Seattle7 Oakland vs. Texas7 Oakland vs. Other4 Seattle vs. Texas6 Seattle vs. Other5 Texas vs. Other5 September 8, 2004

11. Engineering Management, Information, and Systems Department 11 AL West Scenario Network AnaOakSeaTex Ana vs. Oak 6 Ana vs. Sea 7 Ana. vs. Tex 7 Oak vs. Sea 7 Oak vs. Tex 7 Sea vs. Tex 6 D 19 t -59 Capacity = 5 Capacity = 4

12. Engineering Management, Information, and Systems Department 12 How close is Texas to elimination from first place? Find an end-of-season scenario where Texas wins the division with a minimum number of additional wins Texas cannot win the division with fewer additional wins This is the first place elimination number

13. Engineering Management, Information, and Systems Department 13 How close is Texas to clinching first place? Find an end-of-season scenario where Texas wins as many games as possible without winning the division (i.e., at least one other team in the division has a better record) If Texas wins one more game than the optimal value for w Tex, then they are guaranteed at least a tie for first place This is the first-place clinch number

14. Engineering Management, Information, and Systems Department 14 What is an appropriate value for M? In this particular case: ow tex  100 ow Oak  81 ow Ana  79 ow sea  51 oSo, M = (100-51)+1 = 50 is large enough. Since each team plays 162 games, M = 162 + 1 = 163 will always work at any point in the season.

15. Engineering Management, Information, and Systems Department 15 Wild-Card Teams West TeamWLPCT Anaheim92700.568 Oakland91710.562 Texas89730.549 Seattle63990.389 Central TeamWLPCT Minnesota91700.565 Chicago83790.512 Cleveland80810.497 Detroit72900.444 Kansas City581040.358 East TeamWLPCT New York101610.623 Boston98640.605 Baltimore78840.481 Tampa Bay70910.435 Toronto67940.416 2004 American League Final Standings Playoff Teams: Anaheim wins West Division Minnesota wins Central Division New York wins East Division Boston is the Wild-Card Team

16. Engineering Management, Information, and Systems Department 16 Formulation Challenges Elimination and clinch numbers for the Major League Baseball playoffs Formulations for the NBA playoffs –Playoff structure similar to MLB, but with 5 wild-card teams in each conference –Fans interested in questions about clinching home-court advantage in the playoffs

17. Engineering Management, Information, and Systems Department 17 Formulation Challenges Futbol –Standings points determined by the 3-1 system g ij = w ij + w ji + t ij SP i = 3  w ij +  t ij –FutMax project: http://futmax.inf.puc-rio.br/ Top 8 teams (out of 26) make the playoffs Bottom 4 teams demoted to a lower division –Teams wish to avoid elimination from 22 nd place Playoff/Demotion Elimination/Clinch numbers NFL –Standings determined by win-lose-tie percentage: SP i =  w ij + ½  t ij –Complex rules for breaking ties in the final standings NHL –Standings points determined by a 2-1-1 system (wins-ties-overtime losses) –Home-ice advantage

18. Engineering Management, Information, and Systems Department 18 References/Advanced Topics Battista, M. 1993. “Mathematics in Baseball”. Mathematics Teacher. 86:4. 336-342. LP and Integer Programming –Robinson, L. 1991. “Baseball playoff eliminations: An application of linear programming”. OR Letters. 10(2) 67-74. –Alder, I., D. Hochbaum, A. Erera, E. Olinick. 2002, “Baseball, Optimization, and the World Wide Web”. Interfaces. 32(2), 12-22. –Ribeiro, C. and S. Urrutia. 2004. “OR on the Ball”. OR/MS Today. 31:3. 50-54. Network Flows –Schwartz, B. 1966. “Possible winners in partially completed tournaments”. SIAM Rev. 8(3) 302-308. –Gusfield, D., C. Martel, D. Fernandez-Baca. 1987. “Fast algorithms for bipartite maximum flow”. SIAM J. Comp. 16(2) 237-251. –Gusfield, D., C. Martel, D. 1992. “A fast algorithm for the generalized parametric minimum cut problem and applications”. Algorithmica. 7(5-6) 499-519. –Wayne, K. 2001. “A new property and faster algorithm for baseball elimination”. SIAM J. Disc. Math. 14(2) 223-229.

19. Engineering Management, Information, and Systems Department 19 RIOT Site September 8, 2004

20. Engineering Management, Information, and Systems Department 20 References/Advanced Topics Complexity Results –Hoffman, A., T. Rivlin. 1970. “When is a team ‘mathematically eliminated’?”. Proc. Princeton Sympos. On Mat. Programming. –McCormick, S. “Fast algorithms for parametric scheduling come from extensions to parametric maximum flow.” Operations Research. 47(5) 744-756 –Gusfield D., and C. Martel. 2002. “The Structure and Complexity of Sports Elimination Numbers”. Algorithmica. 32(1) 73-86.

21. Engineering Management, Information, and Systems Department 21 The “Magic Number” Definition: the smallest number such that any combination of wins by the first-place team and losses by the second- place team totaling the magic number guarantees that the first-place team will win the division. –Let w 1 = number games the first place team has already won –Let w 2 = number games the first place team has already won –Let g 2 = number games the second place team has left to place –The magic number is w 2 + g 2 – w 1 + 1 –Derivation exercises in Battista [1993] Only given for the first-place team with respect to the second-place team What about teams that aren’t in first place?