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,
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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
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Programming
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winners
in partially
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and a faster algorithm property SIAM J. Discrete Math. 14(2) 223-229.
for base
Interfaces
22
Vol. 32, No.
2, March-April
2002