It Ain

1
It Aint Over Till Its Over Playoff Races and
Optimization Modeling Exercises

• Eli Olinick

2
Motivating Question Can The Giants Win the
Pennant?
National League West September 8, 1996 National League West September 8, 1996 National League West September 8, 1996 National League West September 8, 1996 National League West September 8, 1996
Games Games
Team Wins Losses Back Left
Dodgers 78 63 21
Padres 78 65 1.0 19
Rockies 71 71 7.5 20
Giants 59 81 18.5 22
According to traditional statistics, the Giants
are not mathematically Eliminated (5922 81 gt
78).
3
But What About the Schedule?
Games Games
Team Wins Losses Back Left
Dodgers 78 63 21
Padres 78 65 1.0 19
Rockies 71 71 7.5 20
Giants 59 81 18.5 22

• 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 arent 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 its (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 Detroits 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 Detroits Elimination Network
Team Nodes
Game Nodes
8
8
5
2
8
8
7
3
t
8
7
76-751
8
8
13
8
7
8
8
8
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
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
East
Team W L PCT
New York 101 61 0.623
Boston 98 64 0.605
Baltimore 78 84 0.481
Tampa Bay 70 91 0.435
Toronto 67 94 0.416
West
Team W L PCT
Anaheim 92 70 0.568
Oakland 91 71 0.562
Texas 89 73 0.549
Seattle 63 99 0.389
Central
Team W L PCT
Minnesota 91 70 0.565
Chicago 83 79 0.512
Cleveland 80 81 0.497
Detroit 72 90 0.444
Kansas City 58 104 0.358
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. 864. 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. 313. 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 arent in first place?