Baseball,

Alan L. Erera

Ilan Adler Department

of Industrial

and

Optimization, World Wide

Engineering

Dorit

and Operations

the

Web S. Hochbaum

Research,

Eli V. Olinick

4135 Etcheverry Hall, 94720

University

Berkeley, California School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia Department of Industrial Engineering and Operations Research, 4135 Etcheverry Hall, University Berkeley, California 94720 Department

of Computer

[email protected]

[email protected]

The

competition closely watched

Science and Engineering, Southern Methodist Texas 75275-0122 Dallas,

30332-0205 of California,

P.O. Box 750122,

University,

[email protected]

[email protected]

one of the most race?is pennant race statistics, such as games back play-off and do not account for the they are overly conservative one can model schedule effects optimization techniques,

for baseball American

of California,

fabled

play-off spots?the sports traditions. While

are informative, of games. Using remaining a team has secured a and determine spot or has been elim explicitly precisely when play-off inated from contention. at the University RIOT Races The Baseball Play-off Web site developed race automatic updates of California, of new, optimization-based Berkeley, provides play-off season. we statistics each day of the major baseball In the found that site, league developing we could determine a status of all teams in a division the first-place elimination using single and magic

number, schedule

since a minimum win threshold formulation, linear-programming a division. We a similar to in teams all identified place applies problem of play-off elimination (Recreation and sports)

Fans their

with

wildcard

of professional sports teams have an insatiable for information about the performance of

desire

teams. Fans of major league baseball are in the United States concerned (MLB) particularly about their teams' prospects for reaching the postsea favorite

son play-offs:

the

race.

fabled

Fans

check

pennant on team sites daily for updates lack (or thereof!). progress As the end of the season nears, teams trailing the

newspapers

current

and Web

division

leader may become mathematically from first place; such teams have no chance even if they were to of finishing first in their division, Bu win all of their remaining The Elias games. Sports eliminated

reau, This

the

paper

was

Interfaces, Vol. 32, No.

official

statistician

for MLB,

refereed.

2,March-April

2002, pp. 12-22

determines

for teams (but weaker)

finishing result

in first for the

teams.

a particular team is eliminated using a simple criterion: if a team trails the first-place team inwins by more than it has remaining, it is eliminated. games whether

That

the San Francisco Giants had suffered on September sirable fate was announced

this unde

10, 1996 in the Giants had 1996); (Gay 59 wins with 20 games left to play, while the first-place San Diego Padres had already won 80 games. The Gi two days had actually been eliminated ants, however, earlier: we had announced the news of their demise on the San Francisco

Chronicle

8 on our Web site (Table 1). September The optimization has long known community to eliminate teams the Elias criterion is sufficient

that from

first place but not necessary (Schwartz 1966). The prob lem is that the criterion the schedule of re ignores the Giants Los games. Continuing maining example,

? 2002 1526-551X

INFORMS electronic

0092-2102/02/3202/0012$05.00 ISSN

ADLER ET AL. Baseball,

Optimization,

NationalLeagueWest Clinch

Avoid Elim

Wins Losses Games Back Games Left 1st Play 1st Play

Team

LosAngeles San Diego Colorado San Francisco

78 78 71 59

21 19 20 22

1 7.5 18.5

1417 17 17 17 * * * *

before

an announcement

was

4 0 7 11 Elim 19

made

in the newspaper.

Since

San

Francisco has 22 games remainingand trails Los Angeles by only 18.5 games, it is not readily apparent from the traditionalstatistics that the team

is eliminated.

at least four of these games and finish with a record of at eliminated 82-80, or better. Thus, the Giants were this point, since they could finish with at most 81 wins. one can techniques, By using simple optimization to determine when model schedule effects explicitly teams are truly mathematically eliminated. is not the fans' only interest. elimination First-place teams may also reach the play-offs In baseball, by se a team with the wild-card the finishes that berth; curing teams in the league is record among second-place is also elimination this berth. Thus, wild-card assigned to track. In addition, teams not yet elimi important

best

of closeness close; a measure elimination would be useful. teams that are performing well

nated may be perilously to first-place or wild-card

fans of Conversely, to first would like know if their teams have clinched berth. A clinch is a guarantee; once place or a wild-card a team has clinched it could first place, for example, lose all of its remaining games and still finish in first. a Web site to provide We developed optimization race statistics to the general public based MLB play-off so that fans can sort out the play-off picture with more information. The Berkeley Baseball precise Play-off Interactive Op Races site, a component of the Remote timization Testbed (RIOT), is up and running during season

at (http: (April through October) The site provides ?baseball). //riot.ieor.berkeley.edu/ and statistics. clinch of elimination daily updates Interfaces

Vol. 32, No. 2, March-April

and

it to achieve an educa also designed entertaining, are useful for tional goal. Sports elimination problems in optimization; they are covered teaching basic ideas we

problem esting. We

by Schr?ge (1984) and Ahuja et al. (1993). (1991) argues that many students relate to the and find the results inter subject intuitively

agree and furthermore suggest that the In the problem and is an ideal place to present broadcast results to attract the interest of both students ternet

and

those who

2002

13

might

otherwise

never

to be exposed site provides links to

concepts. The RIOT sites with an educational

optimization other Web

as the Network

to play each and San Diego were scheduled Angeles 8. Since there other seven more times after September are no ties in baseball, one of these teams would win

the baseball

fans find the site informative

baseball

Although

Robinson

Table 1: The RIOTBaseball Play-off Races Web site declared the San FranciscoGiants eliminated from first place on September 8,1996, two days

Wide Web

in textbooks

?

63 65 71 81

and the World

component,

such

Enabled

(NEOS) Optimization System project sponsored by Argonne National Laboratory and Northwestern The NEOS optimization University. con guide (http://www-fp.mcs.anl.gov/otc/guide/) tains interactive case studies demonstrating the appli to general-interest such problems, RIOT the and diet optimization problem. to Michael Tricks OR page (http: serves as a portal for OR which //mat.gsia.cmu.edu),

cation of OR models as portfolio also links on the web sive

and contains

list of

interactive,

an up-to-date and comprehen Web educational sites (http:

//mat.gsia.cmu.edu/program.html).

Problem

Description

not be aficionados of readers may many we will begin by describ America's national pastime, current struc the baseball ing major play-off league Because

ture. MLB American

teams

are partitioned into two leagues, Each league is further subdi

and National.

into three divisions. Each team in each league a season schedule of 162 games to deter plays regular mine to the play-off the teams that will advance rounds. Four teams from each of the two leagues make vided

the play-offs: the three teams that finish with the best records in their respective divisions, and a fourth team that has the best record among all sec (the wild-card) ond place teams in the league. Ties in the final stand ings for a play-off spot are settled by special one-game playoffs. Each league then conducts a tournament with its four invited play-off teams to determine its pennant winner. the American and National Finally, League

ADLER Baseball,

winners

pennant

in the World

play

championship. Now consider

a particular

Sox, at some point the current win-loss

during records

Series

Optimization,

for the MLB

team, say the Boston Red the regular season. Given of all teams

the re

and

are the Red Sox eliminated

schedule of games, maining in a play-off from finishing and if not, how position, If the Red Sox have not close are they to elimination? been And

eliminated, if they have

a they clinched play-off spot? not, how close are they to clinching? have

Elimination

Questions

The

method

official MLB

elimination

for determining first-place is somewhat naive, and often earlier than the official dec

for a division

teams may be eliminated laration. In the Giants example

presented

was

ing elimination much more difficult.

earlier, prov but it can be

simple by inspection, As an example, consider the case we If examine of the Detroit Tigers on August 30,1996. in the American the standings League East division after the completion of play that night (Table 2), it ap

pears

has a remote

that Detroit New

York

chance

Yankees

since

of catching they have

the 27

first-place and trail New York by only 26 wins. games remaining to show that Detroit is in fact It is possible, however, eliminated from first place using some mathematically the remaining schedule information regarding simple the re Using information (Table 3), the inspired games maining reader should try the elimination proof as an exercise; detail the the following proof. paragraphs of games

teams

between

in the division.

To prove that Detroit is eliminated, we can show that to construct a scenario inwhich Detroit it is impossible

ET AL. and the World

Wide Web

Games

Opponents Baltimore

vs. Boston

Baltimore

vs. New York

Baltimore

vs. Toronto

Boston

vs. New York

Boston

vs. Toronto

2 3 7 8 0

New York vs. Toronto

7

Table 3: Using the remainingschedule of games given here and the stand ings inTable 2, it is possible to show that Detroit cannot finishwith as as New York under any scenario.

many wins

NewYork Baltimore Boston Toronto Detroit

Losses ?2859 428 63 66 6.5 72 86

75 71 69 63 49

Games Back

since

they have

26

fewer wins

would

win

won

If Detroit

its division.

it would finish with games, maining wins and 86 losses. If New York won itwould games, therefore ahead narios

in which First,

games. other game. against New

elim

finish with

all of

its re

a record

of 76

just two more and 85 losses and

77 wins

Thus, we now analyze sce York wins one or no remaining that New York fails to win an

of Detroit. New

suppose Since Boston

has

eight games remaining would finish with at least

Boston

York, in this scenario

finish (69 + 8), and it would to have any chance Thus, for Detroit of finishing first, New York would have to win exactly one of its eight games with Boston and lose all of its 77 wins ahead

of Detroit.

In addition, Boston would have to lose games. all of the games it plays against teams other than New create a three-way tie for first place York. This would (Table 4). Now consider Baltimore and Toronto. Baltimore has other

two games remaining with and therefore would

Boston

York

and three with New

finish with

at least 76 wins

Losses

Games

Back

86?76 ? 8676 ? 8676 ?? ? ?? ?

Detroit Boston New York Baltimore Toronto

27 2712.5 2726.5

but 27 games

has been

Games Left

Table 4: First, suppose Detroitwere to win each of its 27 remaining

Table 2: Can the Detroit Tigers win the pennant? By examining these standings, itappears thatDetroit has a (remote) chance of catchingNew York,

Detroit

Thus,

inated from first place.

WinsTeam

Wins Team

Remaining

remaining.

In fact,

Detroit ismathematically eliminated fromfirst;can you prove itusing the data inTable 3?

games.

Now,

Boston

but were

Boston would

suppose

New York were

to win a single

to lose all of its other

win at least seven

remaining

future games

future game games.

(against

against In this case,

New York).

If Bos

tonwere to lose the rest of its remaininggames, itwould finish tiedwith New Yorkand Detroitwith 76 wins. Butwhat about BaltimoreandToronto?

Interfaces

14

Vol. 32, No. 2, March-April

2002

ADLER Baseball,

in our place

scenario. tie only

Optimization,

could finish in a first Thus, Detroit were to lose all of its re if Baltimore

games to teams other than New York and Bos maining ton. Unfortunately for Detroit, Toronto has seven seven with Baltimore and games remaining remaining to the above logic, Toronto with New York. According 14 games in any scenario in in first place. However, which Detroit finishes if To ronto were to win 14 additional it would finish games, 77 wins with a record of at worst and 85 losses and would

to win

have

therefore

ahead

these

Therefore, Detroit from first place.

eliminated

ematically

ismath

of Detroit.

Clearly, constructing hand can be a tedious

such

elimination

proofs

by

endeavor.

optimi Fortunately, can help. Researchers zation methods have previously addressed the problem of first-place elimination. Schwartz that a maximum-flow calcu (1966) showed can determine lation on a small network precisely

Fans have an insatiable desire about

information

their

for

teams. a team has been eliminated from first necessarily an Robinson such showed that (1991) place. optimi zation approach would teams an av have eliminated

when

Rivlin neces

can be

is eliminated from first place ing when solved as a maximum-flow problem on a bipartite net and Martel work. Gusfield (1992) showed that the min imum avoid solving

a given team must win of games elimination from first place can be found number a

parametric

maximum-flow

problem.

By

to by ex

a result of Gallo et al. (1989) and tending using a binary search procedure, Gusfield and Martel proved a run ning time of 0(n3 + n2\og{nD)), where n is the number of teams and D the number of games the team of interest

has

left

to play,

for

McCormick

(1999) improved this parametric maximum ing

this number. finding the time bound for solv flow problem

Interfaces

Vol. 32, No. 2, March-April

2002

15

or not a team is eliminated whether Determining from first place is only half of the story, since elimi in the wild nated teams might still make the play-offs card berth. Little research has focused on play-off elim ination with

wild-card

the 1994 season the baseball

only

since prior teams, partially the division winners advanced

to to

(1991) briefly dis introduced complications by wild-card in the context of applying his baseball elimina

cussed

Robinson

play-offs.

the

berths

to the National

tion model

Football

a formulation. did not provide For the Baseball Play-off Races Web that the most elimination interesting

(NFL) but

League

site, we decided for information

be statistics that provide ameasure of how team is to elimination, similar to those pro and Martel (1992). Therefore, we by Gusfield a team's number to be the first-place elimination

fans would close

each

posed define

minimum

number

must

to have

win

of remaining games that the team in first place chance of any finishing a num As team's first-place elimination

ber approaches the number of games it has remaining, elimination imminent. In addition, we define becomes a team's number to be the mini elimination play-off mum number of games the team must win to have any chance winner

of earning a play-off or as the wild-card

spot, whether team.

as a division

crite

a team conditions for eliminating sary and sufficient in turn showed from fcth place. McCormick (1987,1999) that determining elimination from fcth place is !NT et Gusfield al. that determin (1987) showed complete. a team

Wide Web

in its division.

favorite

erage of three days earlier than the wins-based rion during the 1987 season. Hoffman and Schwartz's (1970) extended work, developing

ET AL. and the World

to 0(n3).

Clinch Questions Fans of the teams performing well during the regular season have a very different concern: to they want know when their team has clinched first place or a wild-card

the media use magic playoff spot. Currently, to determine clinches. Assume that first-place are ranked in order of increas the teams in a division

numbers

the first-place team has lx ing losses, and suppose losses and g1 games remaining, and the second-place team has Z2losses. The magic number, \i, is given by g1 ? ? combination of wins {l2 l\)- Any by the first-place team and losses team totaling |i by the second-place the first-place guarantees in the division. When spot drops to zero, Unlike the case with

number

team at least a tie for the top the first-place team's magic

the team has clinched

first.

the schedule of re elimination, on a little effect games maining (mathematically) team's ability to clinch first place. However, although has

ADLER Baseball,

numbers

magic tions

give

and

necessary they do not

for clinching, number of future wins

Optimization,

sufficient

of games which, if won, guarantees that the a team finishes in at least tie for first place. Similarly, we define the clinch number for each team to play-off ifwon, guar of games which, that team a position in the play-offs, either as

the division

number

or as the wild-card

winner

Wide Web

NationalLeagueEast

specify the minimum for a team to clinch necessary of other teams' performance,

number

antees

the World

condi

first place independent and they are typically reported only for teams in first place. To address these drawbacks, we define the first for each team to be the minimum place clinch number

be the minimum

ET AL. and

Games

Wins Losses Atlanta Montreal Florida New York

wins,

losses,

Races Web

Site

clinch

Operations of Business

and the Haas School Department at the University of California, Berkeley. focus of the RIOT project has been to pro

Research

The primary vide educational

information

about

industrial

engi

and operations research and to promote in the field via Web pages and easy-to-use, interactive Java applets. Each RIOT component appli cation includes pages describing the details of the un

neering interest

models and algorithms used in optimization derlying the problem solution; once visitors have played with its utility, the application and discovered they can learn about the methods used to produce the results. to fans, the most up-to-date information to be site Web the Baseball Races designed Play-off season. Cre the baseball each updated night during To provide

we

two primary the site required ating development a set of mathematical we activities. First, generated statistics models for calculating the new play-off (Ap a we software system that developed pendix). Second, to produce automated the models nightly up employs dates

of the Web

site. The

system

is scheduled

to run

an in the early morning hours, creating and posting HTML The stand 5). report (Table standings updated by newspaper ings report is similar to those provided teams with sections, sports grouped by league and di to the vision and sorted by win-loss record. In addition information

traditionally

reported,

the report displays

tage

Left

1st Play 1st

0.610 0.553 0.483 0.437 0.406

21 21 19 20 19

13 21 * * *

Play

9 0 0 17 0 8 * 17 9 * Elim 16 * Elim Elim

columns

numbers,

numbers.

each

left to play.

and games

percentage,

current

and first-place the two "Avoid Elim" columns provide the An asterisk for a clinch number indicates that a provide

team's

while

clinch is not currentlypossible, even if the teamwere to win all of its If a team had already

games.

remaining

site is a component of the Berkeley RIOT Internet project, an on-line collection developed and maintained and by the Industrial Engineering The baseball Web

8 18 24.5 29

back, winning

games

two "Clinch"

elimination

Play-off

?

Games

Table 5: Inthis sample standings reportfrom the Baseball Play-offRaces Web site, the first five columns contain traditionalstandings information:

play-off

The Baseball

Percen-

Back

55 63 74 80 85

86 78 69 62 Philadelphia 58

The

team.

Avoid Elim

Clinch

first or a play-off

clinched

spot,

itwould be labeled "In."While New York and Philadelphia are mathe from finishing

eliminated

matically

first, New York has a remote

chance

of securing a wild-card berth (bywinning 16 of the remaining20 games). Also, ifMontrealwere towin its remaining21 games, itwould clinch at least a tie for firstplace.

each

two elimination

team's

numbers

and

two clinch

numbers.

The software

re the standings system that generates as follows. Since the calculations require

ports operates the current win-loss maining maintains results news

records

team and

of each

the re

number

of games between teams, the system database that is simple updated using the of the previous A Internet free day's games. a

service

tomatically

called sends

Infobeat

(www.infobeat.com) an e-mail message

the system the final scores of all MLB

au each

games. The night containing of the system initiates the update pro first component cess by automatically and processing this reading e-mail message, the team win-loss records updating in the database. Next, a program and games remaining to generate text files containing the database the mathematical models that allow calcula optimization uses

tion of the elimination then solves timization the results Finally, numbers HTML

and clinch numbers.

the necessary

models

using

package (www.cplex.com) to determine each team's

a

page-updating and generates format required

The system the CPLEX op and processes

current numbers. uses

the updated program new standings reports in the WWW browser programs by INTERFACES

16

Vol. 32, No.

2,March-April

2002

ADLER ET AL. Baseball,

Optimization,

and the World

Wide Web

Communicator and Microsoft (for example, Netscape runs seam Internet Explorer). The process usually without human intervention. how lessly Occasionally,

Tampa Bay Devil Rays, 1998. To accommodate

ever, the e-mail message game results does containing not arrive as expected, and we must initiate the update We the software sys process manually. implemented tem on a Sun Microsystems SPARCstation 20, and it

League and ican League

completes

typically

its various

tasks

in about

10

minutes.

The bulk of the update

software

iswritten

in the Perl

Perl is specifically programming language. designed for writing Unix script programs, and it is particularly well suited for string manipulation. The update pro text input files (such cedure requires parsing multiple as the nightly e-mail message baseball containing scores

and CPLEX

combining files (such

and files), manipulating output text strings, and then writing out new text as input files for CPLEX or the updated

are gen standings reports). These types of operations to easier in Perl in such much code lan than erally guages as C or C + +. on the site while We began work was in progress, and one of the more

the 1996 season

parts challenging of the project turned out to be determining the number of games remaining between each pair of teams. Al a wealth of MLB data is available on the Web, though we had the necessary information obtaining difficulty in a readily usable format. For our purposes, we would have liked a table or matrix the number of giving games left between pairs of teams. The most common for this type of information, is an ac however, tual schedule of games that lists the games to be played each day of the season. To find out how many games were left between, say, the Boston Red Sox and New format

York Yankees, we had to parse the list and count the number of remaining scheduled the games between we automated task. this Since the sched teams; easily ules we found on the Web contained and inaccuracies, since

different

sources

handled

canceled

and

sus

a correct games in different ways, pended producing schedule of remaining games became an unexpectedly an chore. Eventually, difficult however, we produced accurate schedule. For the 1997 to 2002 seasons, gen erating schedules was much simpler because all of our software was ready before the start of the season. Two new teams, the Arizona Diamondbacks and

Interfaces

Vol. 32, No. 2, March-April

2002

17

Brewers

switched

from

in leagues joined the major these teams, the Milwaukee to the National the American

the Detroit

from the Amer Tigers moved to the American League Central Di

East

we our system to these easily adapted we more to ad make may have changes, significant if structure MLB the is of altered. aptations play-off teams will One possibility is that two additional join vision.

While

two years and teams will be realigned into two leagues with two eight-team If this divisions. team from each league will occurs, a second wild-card and we will need probably be added to the play-offs,

MLB

in the next

new mathematical to develop for wild formulations card elimination and clinching. Since the RIOT Baseball Play-off Races site went on line during the 1996 season, it has been popular with Web surfers. As soon as the site was listed in several Internet

fans started

baseball

directories,

the visiting race is the pennant 100 to 200 hits each day.

when

pages. During September, most heated, the pages attract to its popularity, As further testament tured on a 1996 broadcast Beyond

fea

radio program

Computers.

Mathematical Elimination At

the site was

of the public

for Models and Clinching

the core of the automated

theWeb for updating used for calculating

system that we developed site are the mathematical models

the elimination

and

clinch

num

bers

(Appendix). When we first planned the site, our initial idea was to simply calculate and provide first-place elimination numbers for each team inMLB. Our initial formulation was based on the flow formu parametric maximum lation given by McCormick is an exten (1999), which sion of the original formulation of Schwartz (1966). Us we a separate this created ing modeling methodology, flow formulation for each team to determine its first number. To solve the instances, we place elimination to translate the flow formulation decided into a cor Since we had ac responding integer linear program. cess to a fast, efficient IP solver in CPLEX and since the translation

resulted

in small

problems,

it was

much

ADLER Baseball,

Optimization,

for us

easier

an integer programming to work with from an implementation standpoint. had the models running using real MLB

formulation we

After data, we

an interesting

noticed

ings reports. tion number

Adding together and the current win

elimina

and, suspicion tions can be solved

teams. Again, we formulation for each team

wild-card

as few wins

by considering The primary

a

at least a tie for first and finishes

as

in first place or with the best record among place teams in its league. We again postu exist some threshold that there might vL that elimination the play-off allow us to compute

ishes either

numbers

for all teams

formulation.

in the league by solving a single this was not the case, we were

Although a similar but weaker able to develop us

to compute small integer therefore,

we

(Appendix). At this point, problems approach.

clinch num

(Appendix).

result

k +

by solving for each league. For MLB, programs to solve at most need eight instances it seemed

natural

to address

Races Web

Play-off

on

than

daily, that found

precise as a public forum, we are the improved information without

able to disseminate relying

to fans

statistics

optimization-based information more providing elsewhere. Using the Internet

im

site broadcasts

traditional

to accept

media

the ideas

and

to fans.

the information

modify they normally provide the site provides detailed information Furthermore, are about how the calculations performed, including

about

various

use of interesting through the The Internet can be thought

techniques

optimization real-world

problems. of as a large, distributed, can be used models Optimization

database. public-use amounts to add value to data: by converting unwieldy of data into a usable form, such as an optimal decision or an interesting statistic, they increase the value of the data.

more

As

and

more

data

for more

line, the potential activity only increases.

On

becomes

on

available

value-adding meaningful the RIOT site, we have be further with the develop

this avenue gun to explore ment of an on-line investment-portfolio-design

that allows

at most

the numbers

Conclusions

munity

Itwas

all second would

not necessary

was

model

individuals. copy of this paper for interested In this way, the pages fulfill one of the goals of the of the on-line com RIOT Web site: to educate members

extend

lated

however,

reflection,

an on-line

is to allocate

not too difficult to possible. In this idea to the play-off elimination setting. this case, the idea is to create a feasible end-of-season fin in which the team under consideration scenario

with

remaining for play-off we realized

integer optimization to determine the first-place

proved,

games among teams feasibly to cre the team under scenario in which

attains

consideration

began

separately. elimination models

ate an end-of-season

clinching. After some that solving a formal bers

sce

end-of-season

its remaining wins then find the clinch num

in first. We

finishing

The Baseball

using linear programming (Appen a result, we calculate elimination first-place six small for each team in MLB by solving

one for each division. linear programs, with Our experimentation the integer and linear to for first-place elimination led us naturally programs with of play-off elimination the problem consider

wins

feasible

ber by simply adding one to the maximized a similar formulation wins. We developed

a separate for of utilizing team. We were able to prove this to show that the formula in addition,

in first-place of remaining

a

create

the team maximizes

instead

vk for each division, mulation for each

idea

to

wins

in which

without

total for any team in a specific division k on a given date yielded a constant, could that elimination numbers vk. Thus, we suspected a to determine be calculated formulation using single

dix). As numbers

Wide Web

allocated

nario

in our stand

property the first-place

ET AL. and the World

1

clinching

a similar mathematical-programming To determine clinch numbers, we decided

with

use models that are in some sense reversed. initially to a to determine For example, specific team's first-place an integer program that clinch number, we formulated

system. that automatically tracks the daily closing prices of nearly 100 stocks, the system allows users to solve a portfolio-optimization model. Both the baseball should and portfolio systems give both re

Using

a database

a at the types and practitioners glimpse use to exist research that opportunities operations increase the value of online data. searchers

of to

Acknowledgments All

authors

were

N00014-91-J-1241

of Naval supported by Office this research. Additional

during

Research

contract

funding

for Dorit

Interfaces

18

Vol. 32, No. 2, March-April

2002

ADLER Baseball,

DMI-9713482.

award

dation

was

and Eli Olinick

Hochbaum

the threshold. variable the decision representing team iG Dk wins of future games the number let Xjj represent a of future wins, scenario team / G Dk; let x denote complete to teams in model allocates wins I i, j G Dk}. The following

Let vk be

Foun

Science

by National our gratitude

provided extend

We

to these

ET AL. and the World Wide Web

Optimization,

fund

ing agencies.

We

the mathematical

describe

for

the models

behind

details

Races

the Baseball

we

site.

season

a

Importantly, Play-off can be used elimina to determine program first-place We then show that for all teams in a single division.

linear single tion numbers

min

Notation L be

Let

that can be used

model

a

to calculate

single

team's

in some

the American (for example, Let of divisions. into a number

scenario. we

that follow,

assume

team must

that each with

of games (consistent we a winner. In addition,

its entire

schedule

MLB

rules)

play and that in a

assume

each game has with another tie in the final standings team(s) a such MLB resolves spot; typically play-off game

that finishing to secure is sufficient

ties with

special

one

Elimination

First-Place

the of teams on a given day during single division as follows: the first-place-elimination (FEP) problem the current win-loss records of each team and the remaining given for number the first-place-elimination schedule of games, determine a

consider

season.

each

Define

as

team,

The

defined.

previously formulation

maximum-flow

parametric single-team and Martel (1992) could be

in Gusfield

to solve and (FEP) by creating employed stance for each team. This is unnecessary,

threshold, and may

exists

for each

the minimum

number

there

that

show

sufficient

for any

be found

by

team

solving

a

of wins. Notably, Wayne of a first-place-elimination formulation. istence

First,

in now

a first-place-elimination at season-end necessary in first, and that this threshold

division

of wins

to finish

number

for determining

an appropriate since we however,

solving

the thresh Given single linear program. number for each team in the division

old, the first-place-elimination between is simply the difference

consider

i
Wi + 2

jeDk

j, (1)

*,y v i G Dk, (2)

> Xjj 0 Vi, / G D*, i * j,(3)

the threshold

the team's

current

the ex proves concurrently threshold using a maximum-flow

2002

19

i^

;'(5)

of wins

that the allocation

for all

accounts

each pair of teams in the division. of the remaining games between force vk with the objective Constraints function, (2), in conjunction to be

the minimum

team

at the end

to win

essary

number

ignored;

division Now claim First,

by

Games

played the minimum

to find

nec

of wins

number

a division,

in

scenarios

to consider it is only necessary in division k lose all remaining games

the teams

which

a division-winning teams outside against

attained

of wins

of the season.

are

the division

against

non

opponents. value that the optimal of (PI) is vk. We suppose objective k. for division threshold that vk is the first-place-elimination at that in the optimal solution it is clear from the formulation

team will win Thus, no team winning vk games. exactly can finish atop the division. the To complete than vk games can always be a final can be shown it scenario that standings proof, in which constructed any team / G Dk that can attain at least vkwins least

by with

one

season

integer threshold

linear program k. for division

end

(that vk wins.

is, wx + To do

tx + 2;-eDjk gy so, consider

wins

vk total games. exactly case when in the alternative

^

vk) can win

the division

the optimal allocation of If to (PI) and let vx = wx + xlk. 2;eDfc scenario for / can be attained by

exactly future wins, x, in the solution V\ + tx ^ vk, a division-winning its number (if necessary) increasing

of nondivision

It is also vx +

t?
eL wi + 2 xij - ? i = i.

?Wj

Team /makes the play-offs in this scenario with wi + 2;-eL Xij ^ w? + 2;eL Xij for all i G L\F, where holds from the formulation and the second from x. Thus,

/ necessarily

finishes

the division

except possibly in this new division given by tion of x. The

formulation

x from

allocation team

solving appropriate In the spirit of the we attempt to section, however, previous a avoid solving for each team unnecessarily. instance separate now We show that the play-off-elimination numbers for each team

(11)

is a large integer for example, than the greater (specifically, of games in a season, we 162) and for illustrative purposes, that league L has three divisions. This minimax at model

PEP.

most

integer,

G {1, 2, 3}, i G Dk, (12) af binary V Jfc

rounding = vk [vk~].

scenario

card but

u

the

Elimination Play-off Elimination from the play-offs occurs a team has no only when chance of either in first place in its division or (2) fin (1) finishing the best record all second-place in its teams ishing with among

mine

x,y integer V /, j G L, (10)

affecting

contains

jective function ation obtained

league which

> ziy 0 V ?,; G L, (9)

x may

scenario,

be decreased

teams

? wins,

since

? >

the first

inequality the construction of

at least as many wins as all teams Team / may its actually win since the wins of the division leader

with

leaders.

to ? or fewer wins

during

in F may not be able to finish ? In the optimal solution spot by winning games. only > ? for G F; therefore, we + that possible wf / E;eL xfj exception

(13)

the construc

in a

play-off to (P2), it is

would

need

Interfaces

20

Vol. 32, No. 2, March-April

2002

ADLER ET AL. Baseball,

to decrease f s win

to find a scenario

total ? wins.

offs with

Optimization,

in which

the play / makes in of team / requires a scenario in which /

the wins

exactly Decreasing other teams' wins, which may create creasing does not finish in a play-off Thus, there may be no scenario position. in which the play-offs with ? wins. To address this problem, / makes we

an additional

propose solving team fk. This model, ditional constraint:

denoted

for each exception integer program to is the ad identical (P2), with (P2/fc)

= 0,

4 which

(14)

that team/*, is not an exception team and therefore more wins than the optimal func objective

guarantees finishes

no

with

longer tion value

?k. can now

We

determine numbers for each play-off-elimination team. For i E Dk\ number is minfe, F, the play-off-elimination u\ ? team fk E F, the number is min^, w{. For each exception ?k) a team is not eliminated if it is not elim from the play-offs Wfk. Since inated from first place, the elimination number is the minimum of the first-place-elimination and the wild-card-elimination threshold threshold

minus

elimination

current win

the team's

for a team

number

it is eliminated

from

ing games, an similar argument By straint (10) is unnecessary erate play-off-elimination first solves two integer the American each

numbers linear

and National are

(P2)

for each

formulations Then,

Leagues. and

section,

con

(P2/Jt)). To gen the RIOT system (P2), one for both

(and

team,

of type the exception

teams

solved

for each.

teams grams.

identified,

from

Thus,

thresholds

first-place the play-off-elimination

before

calculation

of

separately It is possible tomod alternatively (P2) and (P2/fc) to calculate min{?fc, ?) and min {vk, omit the details.

ify formulations we

?k) directly; First-Place

are calculated numbers.

Clinch

the First, we consider problems. briefly address clinching a team's on a of clinch number problem determining given first-place the season. For a team i E Dk to clinch first, itmust win day during to guarantee a record that it finishes with games enough remaining at least as good any

prevents

as all other other

games except perhaps bers can be calculated

teams

in its division.

However,

gj =

At/ gj can now

%

be

To briefly justify the above definition, ture games won by team i. To guarantee for i to win f = ;', it is clearly sufficient

let /,- be the number of fu a tie with some other team ? + w{ future games. Wj gj since ihas future games with /, imay need to win fewer However, gf;< case for team i. we assume the worst games. Consider g? lift g?;, that each future win by team / comes against teams other than team ? if f > g{ in the worst case for team i, ;. However, g^, then exactly ~~ + come ?f i*s fu*ure wins must team ;', fi fe against resulting &ij) in the same number of losses for /. Thus, each win for i beyond g2as counts two the for wins, gij effectively justifying expression 0,y.

the number

a team must

the maximum

ing in a play-off To determine

of remaining for each team ; E games clinch number for team /, 0?, first-place

+ 2 (u7/ Si

-

wi + (gi

-

arithmetic

wi + gj 8ij??
Xij 0 V i, j G L, (22)

?fc, a? binary

2002

deter

team

Xij integer V i, j G L, (23)

Interfaces

Vol. 32, No. 2, March-April

of games

we solve such a clinch numbers, play-off problem Consider the following clinch formulation separately. is to maximize the (P3) for some team a in division Da. The objective number of additional wins va accrued by team a such that a finishes with fewer wins than the first-place team in its division, and at least for each

Tmax 9?1. (16) jzDk\{i]

the for number

position.

calculations:

-

model

formulation

number

from winning all of its remaining games team i. Thus, clinch num against first-place as we now using optimization, easily without

+ The tj 2/eD.\(j) gjibe determined via the following

We

However, presented previously. instead of determining the minimum

is reversed:

of games mine

needed to clinch a play-off a mixed this problem with for each team similar to the

of future wins

complicated.

team

the total number

V ; GDk\

Clinch

Play-off

Determining spot is more

nothing

describe. Let

If 0,-^ 0, team i has first place. Alternatively, clinched already if 0j > gi, there is no way for team / to currently at least a guarantee tie with all teams. first-place

rivals.

max

we

Finally,

of 0, guarantees that if team iwere to win 0? games, a record at least as good as all of its division

finish with

linear-programming models play-off-elimination

are of type formulations (P2/*.) numbers for all MLB play-off-elimination are created at most by solving eight small integer linear pro As a final note, in the models that here, we assume presented

division

The definition it would

mulation

in the previous

in formulation

Wide Web

integer

if the play-off total. Again, than its number of remain

is greater the play-offs.

to that made

and the World

V

G Jfc

{1,2,3},

i G Dk. (24)

ADLER

Constraints than

(18) force

teams

all

that each

for which

team a to finish with a* = 0. Constraints at least one

k contains

division

that at least one division

contains

va is the maximum

erty. Therefore, attain without

finishing

in a play-off

at least

team two

number

Baseball,

Optimization,

strictly (20) and

fewer wins

f with

ET AL. and the World

D.

-,-,

(21) ensure = 0, and af

position.

presented 1999. Fast extensions

Theory, Cliffs,

Sympos.

Press, Princeton, NJ. University Technical hard min cut problems. at the TIMS/ORSA New Orleans, Conference, algorithms to parametric

report LA.

come for parametric scheduling flow. Oper. Res. 47(5) maximum

from 744

1988. Integer and Combinatorial G. L., L. A. Wolsey. and Sons, New York. John Wiley

Nemhauser,

Op

timization.

R. E. Tarjan. 1989. A fast parametric G., M. D. Grigoriadis, SIAM J. Comput. and applications. flow algorithm

maximum 18(1)

30-55.

1996. Giants

Chronicle, Gusfield,

'mathematically' on Math.

Princeton

756.

NJ.

Gay, N.

ed. Froc.

for bipartite

algorithms

Princeton Programming. S. T. 1987. Two

-.

Gallo,

J., T. J. Rivlin. H. W. Kuhn,

McCormick,

1993. Network Flows: R. K., T. L. Magnanti, J. B. Orlin. and Prentice-Hall, Applications. Englewood Algorithms,

Fast

16(2) 237-251. Comput. is a team 1970. When

SIAM].

eliminated?

the prop that team a can

References Ahuja,

flow.

A.

Hoffman,

1987.

Fernandez-Baca.

network

teams with

of wins

Wide Web

D.,

officially

September C. Martel.

parametric mica 7(5-6)

minimum 499-519.

leave

the N.L. West

Robinson,

race. San Francisco

10, Dl. for the generalized fast algorithm cut problem and applications. Algorith

L. W.

1991. Baseball

of linear programming. L. 1984. Linear, Schr?ge, LINDO. Schwartz,

1992. A

Scientific

B. L. 1966. Possible SIAM Rev.

K D.

2001. A new

ball

Integer, and Quadratic Palo Alto, CA.

elimination.

An

application

67-74.

Programming

with

Press,

naments. Wayne,

eliminations: playoff Res. Letters 10(2) Oper.

winners

in partially

tour

completed

8(3) 302-308.

and a faster algorithm property SIAM J. Discrete Math. 14(2) 223-229.

for base

Interfaces

22

Vol. 32, No.

2, March-April

2002