Max-Flow Min-Cut Applications:

1
Max-Flow Min-Cut Applications

• Playoff Elimination

2
Quick Review of MLB Playoffs

• Two leagues, American and National independent
• Three divisions per league
• Who makes playoffs?
• Top team in each division
• Team with best record among all second-place
  teams in league (wildcard)

3
End-of-Season Scenario
American League
East

• New York Wins East
• Cleveland Wins Central
• Texas Wins West
• Chicago Wins Wildcard

Won
Lost
Pct.
GB
New York
77
61
0.558
-
Baltimore
73
65
0.529
4
Boston
71
68
0.511
6.5
Toronto
63
76
0.453
14.5
Detroit
50
89
0.360
27.5
Central
Won
Lost
Pct.
GB
Cleveland
81
57
0.587
-
Chicago
75
65
0.536
7
Minnesota
69
70
0.496
12.5
Milwaukee
68
72
0.486
14
Kansas City
65
75
0.464
17
West
Won
Lost
Pct.
GB
Texas
78
60
0.565
-
Seattle
72
66
0.522
6
Oakland
67
74
0.475
12.5
California
64
75
0.460
14.5
4
Can Detroit Win This Division?
8/30/96
W L GB GR 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
5
Remaining Series in the AL East
8/30/96
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
6
Proof of Detroits Elimination

• Try to create a scenario where Detroit wins the
  division.
• Detroit can finish with at most 76 wins.
• New York already has 75 wins.
• If New York wins 2 or more games, Detroit is
  eliminated.

Team W L New York 77 85 Detroit 76 86
7
Scenarios Where Detroit Can Win the Division.

• Case 1 New York loses all its remaining games.
• Case 2 New York wins 1 more game.

8
Case 1 New York Loses All Its Remaining Games.

• New York plays Boston 8 more times.
• Boston already has 69 wins.
• Implies Boston finishes ahead of Detroit.
• Detroit is eliminated in Case 1.

9
Case 2 New York Wins Only 1 More Game

• New York must win 1 game against Boston. 75 1
  76.
• Boston cannot beat any teams other than New York.
  69 (8-1) 76.
• Implies that Detroit, Boston and New York each
  finish with 76 wins.

What about Baltimore ?
10
Baltimore
Team W L Baltimore 76 86 Boston 76 86 Detroit 7
6 86 New York 76 86

• Already has 71 wins.
• Plays New York 3 more times.
• Plays Boston 2 more times.
• Baltimore will finish with at least 76 wins.
• A 4-way tie for first is possible, but only if
  Baltimore only beats New York and Boston.
• What about Toronto ?

11
Toronto

• Already has 63 wins.
• Plays New York 7 more times.
• Plays Baltimore 7 more times.
• Previous assumptions imply Toronto finishes with
  at least 77 wins.
• Detroit cannot win the division in Case 2 either.

12
Literature Review

• B.Schwartz, 1966
• A. Hoffman and J. Rivlin, 1967
• Solve First Place Elimination Problem via Network
  Flows
• L. Robinson, 1991
• Extensions to Other Sports
• D. Gusfield and C. Martel, 1992
• Minimum Number of Games to Avoid Elimination From
  First Place
• K. Wayne, 2001
• First Place Elimination Number via network flows

13
Detroits Elimination Network
Game Nodes
Team Nodes
76-715
2
3
76-697
7
t
s
76-751
8
7
76-6313
Arcs with ? capacity.
14
Detroits Elimination Network
Game Nodes
Team Nodes
5
2
3
7
7
t
s
1
8
7
13
uS,T 26 lt wins remaining 27
15
More Interesting Questions!

• I. Adler, A. Erera, D. Hochbaum and E.
  Olinick, 2002
• http//riot.ieor.berkeley.edu/baseball
• Clinching Questions
• Is my team guaranteed to win division? Make
  playoffs?
• Magic Number Questions
• How many games must my team to win to clinch?
• How many games must they win to avoid
  elimination?