engineering management, information, and systems 1 it ain’t over till it’s over: playoff races...

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

Motivating Question: Can The Giants Win the Pennant?

National League West September 8, 1996

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

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?

• 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

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

It ain’t over till

it’s over … unless it’s

over

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 GLNew York 75 59 - 28Baltimore 71 63 4 28Boston 69 66 6.5 27Toronto 63 72 12.5 27Detroit 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 2Baltimore vs. New York 3Baltimore vs. Toronto 7Boston vs. New York 8New 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

Team NodesGame Nodes Balvs. 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


RIOT Site September 8, 2004

10


Remaining Series in the AL West

Teams Games Anaheim vs. Oakland 6Anaheim vs. Seattle 7Anaheim vs. Texas 7Anaheim vs. Other 5Oakland vs. Seattle 7Oakland vs. Texas 7Oakland vs. Other 4Seattle vs. Texas 6Seattle vs. Other 5Texas vs. Other 5

September 8, 2004

11


AL West Scenario Network

Ana Oak Sea Tex

Anavs. Oak

6Anavs. Sea

7Ana.vs. Tex

7Oakvs.Sea

7Oakvs.Tex

7Seavs.Tex

6

D

19

t

-59

Capacity = 5

Capacity = 4

12


How close is Texas to elimination from first place?

and sconstraint flownetwork s.t.

scenario in the winsTexas games of # totalwTex

x tTex ,min

wwwwwwxwxwxwxw

SeaTex

AnaTex

OakTex

tSeaSea

tTexTex

tAnaAna

tOakOak

51757981

,

,

,

,

0 andinteger variablesall

• 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?

scenario in the Anaheim with,or tied of, ahead finishes Texas if 1yAna

wTexmax

1

11

251757981

,

,

,

,

wwMywwMywwMy

yyyxwxwxwxw

SeaTexSea

AnaTexAna

OakTexOak

SeaOakAna

tSeaSea

tTexTex

tAnaAna

tOakOak


• 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?

scenario in the Anaheim with,or tied of, ahead finishes Texas if 1yAna

wTexmax

1

11

251757981

,

,

,

,

wwMywwMywwMy

yyyxwxwxwxw

SeaTexSea

AnaTexAna

OakTexOak

SeaOakAna

tSeaSea

tTexTex

tAnaAna

tOakOak


• In this particular case: o wtex 100o wOak 81o wAna 79 o wsea 51o 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

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

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

2004 American League Final Standings

Playoff Teams:Anaheim wins West DivisionMinnesota wins Central DivisionNew York wins East DivisionBoston 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?

engineering management, information, and systems 1 it ain’t over till it’s over: playoff races...

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