It Ain’t Over Till It’s Over: Playoff Races and Optimization Modeling Exercises

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

2. Motivating Question: Can The Giants Win the Pennant? It ain’t over till it’s over According to traditional statistics, the Giants are not “mathematically” Eliminated (59+22= 81 > 78).

3. But What About the Schedule? It ain’t over till it’s over … unless it’s over • 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

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. Can Detroit Win This Division? W L GB GL New York 75 59 - 28 Baltimore 71 63 4 28 Boston 69 66 6.5 27 Toronto 63 72 12.5 27 Detroit 49 86 26.5 27 Since Detroit has enough games left to catch New York it’s (remotely) possible.

6. But What About the Schedule? Teams Games Baltimore vs. Boston 2 Baltimore vs. New York 3 Baltimore vs. Toronto 7 Boston vs. New York 8 New York vs. Toronto 7 • 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. 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. OR Proof: Detroit’s Elimination Network Bal vs. Bos. Bal Bal. vs. Tor. Bal. vs. N.Y. Bos s Bos. vs. N.Y. N.Y. N.Y. vs. Tor. Tor u[S,T] = 26 < # wins remaining Team Nodes Game Nodes ∞ ∞ 5 2 ∞ ∞ 7 3 t ∞ 7 76-75=1 ∞ 8 13 ∞ 7 ∞ ∞ ∞

9. RIOT Site September 8, 2004

10. Remaining Series in the AL West September 8, 2004 Teams Games Anaheim vs. Oakland 6 Anaheim vs. Seattle 7 Anaheim vs. Texas 7 Anaheim vs. Other 5 Oakland vs. Seattle 7 Oakland vs. Texas 7 Oakland vs. Other 4 Seattle vs. Texas 6 Seattle vs. Other 5 Texas vs. Other 5

11. AL West Scenario Network Ana vs. Oak Ana vs. Sea Ana. vs. Tex Oak vs. Sea Sea vs. Tex Oak vs. Tex D Ana Oak Tex Sea 6 7 7 7 7 6 19 Capacity = 5 Capacity = 4 t -59

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. 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 wTex, then they are guaranteed at least a tie for first place • This is the first-place clinch number

14. What is an appropriate value for M? • In this particular case: • wtex 100 • wOak 81 • wAna 79 • wsea 51 • So, 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. Wild-Card Teams 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. 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. Formulation Challenges • Futbol • Standings points determined by the 3-1 system • gij = wij + wji + tij • SPi = 3 wij +  tij • 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 22nd place • Playoff/Demotion Elimination/Clinch numbers • NFL • Standings determined by win-lose-tie percentage: SPi = wij + ½  tij • 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. 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. RIOT Site September 8, 2004

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. 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 w1 = number games the first place team has already won • Let w2 = number games the first place team has already won • Let g2= number games the second place team has left to place • The magic number is w2 + g2 – w1 + 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?