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913856 盧俊銘

OR Applications in Sports Management : The Playoff Elimination Problem

IEEM 710300 Topics in Operations Research

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Results of the MLB case

Question Optimum Representation

First-Place-EliminationNumber of additional games to win to avoid elimination from first place

Play-Off-Elimination Number of additional games to win

to avoid elimination from playoffs

First-Place-ClinchNumber of additional games, if won, guarantees a first-place finish

Play-Off-ClinchNumber of additional games, if won, guarantees a playoff spot

k iv w

min ,k iv u w min ,

kk k fv u w

\max .

ki ij

j D i

1av

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The Selected Case

The Brazilian National Football Championship is the most importan

t football tournament in Brazil. The major goal of each team is to be qu

alified in one of the eight first positions in the standing table at the end

of the qualification stage. For the teams that cannot match this objectiv

e, their second goal is, at least, not to finish in the last four positions to

remain in the competition next year. .

The media offers several statistics to help fans evaluate the perfor

mance of their favorite teams. However, most often, the information is

not correct. Thus, this study aims to solve the GQP (Guaranteed Quali

fication Problem) and the PQP (Possible Qualification Problem) by fin

ding out the GQS (Guaranteed Qualification Score) and PQS (Possibl

e Qualification Score) for each team. .

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What’s different?

1. The 3-point-rule v.s. the 1-point-rule

The regulations to determine whether a team plays better or worse than others

2. Number of teams to be taken into account

3. Quotas for playoff participants

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The 3-Point-Rule

If a team wins against its opponent, it will get 3 points while the

other gets none. If there’s a tie, both teams will get 1 point. .

TeamCurrent points

Flamengo 37

Cruzeiro 37

Bahia 36

Elmiminated N/A

* All of these three teams have 2 remaining games to play

Comparison of the complexity under different rules

TeamCurrent points All possible resulting points under the 1-point-rule

Flamengo 37 38 38 38 38 37 37 37 37

Cruzeiro 37 38 38 37 37 38 38 37 37

Bahia 36 37 36 37 36 37 36 37 36

Elmiminated N/A B B N/A B N/A B N/A B

* All of these three teams have 1 remaining game to play

Under the 3-point-rule, the number of possible results may be 30,000 times more.

TeamCurrent points Some possible resulting points under the 3-point-rule

Flamengo 37 40 40 40 40 40 40 40 40 40

Cruzeiro 37 40 40 40 38 38 38 37 37 37

Bahia 36 39 37 36 39 37 36 39 37 36

Elmiminated N/A B B B C B B C N/A B

* All of these three teams have 1 remaining game to play

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Guaranteed Qualification Problem (GQP)

The GQP consists in calculating the minimum number of points of any team

has to win (Guaranteed Qualification Score, GQS) to be sure it will be

qualified, regardless of any other results.

The GQS depends on the current number of points of every team in the league

and on the number of remaining games to be played.

GQS cannot increase along the competition.

A team is mathematically qualified to the playoffs if and only if its number of

points won is greater than or equal to its GQS.

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Possible Qualification Problem (PQP)

The PQP consists in computing how many points each team has to win

(Possible Qualification Score, PQS) to have any chance to be qualified.

The PQS depends on the current number of points of every team in the league

and on the number of remaining games to be played.

PQS cannot decrease along the competition.

A team is mathematically eliminated from the playoffs if and only if the total

number of points it has to play plus the current points (Maximum Number of

Points, MNP) is less than its PQS.

Of course, PQS GQS for any team at any time.

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Problem Definition: GQP first-eight-place

[Restrictions & Assumptions]

1. There are 26 teams in the league.2. Every team has to finish only one game against each of the other 25 teams; thus, t

he total number of games for a team is 25.3. Every game is under the 3-point-rule.4. A team finishes the qualification stage with the eight most total points will advance t

o the play-off rounds.5. Ties in the final standing for a play-off spot are settled by comparing the number of

wins of all candidates.

[Inputs]

Current win-loss records, remaining schedule of games

[Outputs]

A team’s guarantee qualification point (GQP).

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Notations : GQP first-eight-place

Let be the total number of points for team at the end of the qualification stage.it i

1, if team wins over team

0, otherwiseij

i jx

1, if (team is not ahead of team )

0, otherwise

j k

j

t t k jy

Let be the current number of points that team has won.ip i

Let be the current number of teams that have no less points than team .Pi i

Let be the maximum number of points for team such that there exists a valid as

signment leading to and at the end of the qualification stage.

Therefore, is the minimum number of points that team has to obt

ain to ensure its qualification among the first teams.

kGQS kk

kt GQSkP m

1kkGQS GQS km

Let be the number of teams that can be qualified to the playoffs (among teams).m n

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Mathematical Models: GQP first-eight-place

max

subject to

1 1

3 1 ( )

( ) 1

kk

ij ji

j j ji ji iji j i j

GQS t

x x i j n

t p x x x

GQP k j n

(1 ) 1 ,

8

0,1 1 ,1 ,

0,1 1 ,

k j j

jj k

ij

j

t t M y j n j k

y

x i n j n i j

y j n j k

(1)

(2)

(3)

(4)

0, if and are tied

1, if either or winsij ji

i jx x

i j

Current points

, if and

0, if either

1

o

are ti

r win

ed

s

i j

i j

3 points for

winning

There are at least 8 teams that are ahead of team k.

M 3 25 75 Is a valid upper bound.

The maximum number of points foe team k such that it can not be qualified to the playoffs.

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Problem Definition: PQP first-eight-place

[Restrictions & Assumptions]

1. There are 26 teams in the league.2. Every team has to finish only one game against each of the other 25 teams; thus, t

he total number of games for a team is 25.3. Every game is under the 3-point-rule.4. A team finishes the qualification stage with the eight most total points will advance t

o the play-off rounds.5. Ties in the final standing for a play-off spot are settled by comparing the number of

wins of all candidates.

[Inputs]

Current win-loss records, remaining schedule of games

[Outputs]

A team’s guarantee qualification point (GQP).

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Notations : PQP first-eight-place

Let be the total number of points for team at the end of the qualification stage.it i

1, if team wins over team

0, otherwiseij

i jx

1, if (team is ahead of team )

0, otherwise

j k

j

t t j kz

Let be the current number of points that team has won.ip i

Let be the current number of teams that have no less points than team .Pi i

Let be the minimum number of points for team such that there exists at least

one set of valid assignments leading to and at the end of the

qualification stage.

kPQS kk

kt PQS 1kP m

Let be the number of teams that can be qualified to the playoffs (among teams).m n

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Mathematical Models: PQP first-eight-place

min

subject to

1 1

3 1 ( )

( ) 1

kk

ij ji

j j ji ji iji j i j

PQS t

x x i j n

t p x x x

PQP k j n

(1 ) 1 ,

7

0,1 1 ,1 ,

0,1 1 ,

j k j

jj k

ij

j

t t M y j n j k

z

x i n j n i j

z j n j k

(1)

(2)

(3)

(4)

0, if and are tied

1, if either or winsij ji

i jx x

i j

Current points

, if and

0, if either

1

o

are ti

r win

ed

s

i j

i j

3 points for

winning

There are at most 7 teams that are ahead of team k.

M 3 25 75 Is a valid upper bound.

The minimum number of points foe team k such that it has a chancel to be qualified.

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Results

* 2002 Brazilian National Football Championship

Rank Team Current points Games to play PQS GQS

1 São Paulo 49 3 - -

2 São Caetano 44 3 - -

3 Corínthians 42 3 - -

4 Juventude 41 3 - -

5 Atlético MG 40 3 - -

6 Santos 39 3 39 40

7 Grêmio 38 3 38 40

8 Fluminense 37 3 38 39

9 Coritiba 36 3 37 40

10 Goiás 36 3 39 40

11 Cruzeiro 36 3 39 40

12 Vitória 34 3 37 38

13 Ponte Preta 34 3 37 38

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Team Fluminense (2002)

MNP GQS

P PQS

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Mathematical Models: GQP last-four-place

max

subject to

1 1

3 1 ( )

( ) 1

kk

ij ji

j j ji ji iji j i j

GQS t

x x i j n

t p x x x

GQP k j n

(1 ) 1 ,

0,1 1 ,1 ,

0,1

1

,

8

1

k j j

jj k

ij

j

t t M y j n j k

y

x i n j n i j

y j n j k

(1)

(2)

(3)

(4)There are at least 18 teams that are ahead of team k.

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Mathematical Models: PQP last-four-place

min

subject to

1 1

3 1 ( )

( ) 1

kk

ij ji

j j ji ji iji j i j

PQS t

x x i j n

t p x x x

PQP k j n

(1 ) 1 ,

0,1 1 ,1 ,

0,1

1

,

7

1

j k j

jj k

ij

j

t t M y j n j k

z

x i n j n i j

z j n j k

(1)

(2)

(3)

(4)There are at most 17 teams that are ahead of team k.

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Conclusions

1. Under a different rule, the playoff elimination problem may be even more complex.

2. This study provides a more general model for solving the playoff elimination problem.

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Mathematical Models: GQP refined for 1-point-rule

max

subject to

1 1

3

( )

kk

ij ji

j j

GQS t

x x i j n

t p

GQP k

1 ( )ji ijii j

ji j

x xx

1

(1 ) 1 ,

8

0,1 1 ,1 ,

k j j

jj k

ij

j n

t t M y j n j k

y

x i n j n i j

0,1 1 ,jy j n j k

(1)

(2)

(3)

(4)

At least one team wins, i.e. no ties.

1 point for winning

There are no ties.

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References

1. Ribeiro, C. C. and Urrutia, S. (2004) An Application of Integer Programming to Playoff Elimination in Football Championships, to appear in International Transactions in Operational Research.

2. Footmax, available on the Internet: http://futmax.inf.puc-rio.br/.

3. Bernholt, T., Gulich, A. Hofmeuster, T. and Schmitt, N. (1999) Football Elimination is Hard to Decide Under the 3-Point-Rule, Proceedings of the 24th International Symposium on Mathematical Foundations of Computer Science, published as Lecture Notes in Computer Science 1672, Springer, pp. 410-418.

4. Adler, I., Erera, A. L., Hochbaum, D.S., and Olinick, E. V. (2002) Baseball, Optimization, and the World Wide Web, Interfaces 32(2), pp. 12-22.

5. Remote Interface Optimization Testbed, available on the Internet: http://riot.ieor.berkeley.edu/.