1 913856 OR Applications in Sports Management : The Playoff Elimination Problem IEEM 710300 Topics in Operations Research

1

913856 盧俊銘

OR Applications in Sports Management : The Playoff Elimination Problem

IEEM 710300 Topics in Operations Research

2

Introduction

Sports management is a very attractive area for Operations

Research. Deciding playoff elimination and Timetabling are the two

problems discussed most frequently. The former helps the fans to be

aware of the status of their favorite teams, either

qualified to or eliminated from the playoffs. This

information is also very useful for

team managers to decide whether to

spend time in planning the future or

to struggle for the current season.

The latter can be used to devise a

fairer and more cost-effective

schedule for the league. .

3

The Playoff Elimination Problem

1. Schwartz (1966) showed that a maximum-flow calculation on a

small network can determine precisely when a team has been

necessarily eliminated from the first place. .

2. Hoffman and Rivlin (1970) extended Schwartz’s work, developing

necessary and sufficient conditions for eliminating a team from kth

place. McCormick (1987, 1999) in turn showed that determining

elimination from kth place is NP-complete. .

3. Robinson (1991) applied linear programming in solving baseball

playoff eliminations, which resulted in eliminating team three days

earlier than the wins-based criterion during the 1987 MLB season.

4

The MLB Case

The Elias Sports Bureau, the official statistician for MLB, determine

s whether a particular team is eliminated using a simple criterion: if a t

eam trails the first-place team in wins by more games than it has remai

ning, it is eliminated. However, according to this study, a team had act

ually been eliminated few days earlier than it was announced by MLB.

.

First-place elimination is not the fans’ only interest. In baseball, te

ams may also reach the play-off by securing a wild-card berth; the tea

m that finishes with the best record among second-place teams in the l

eague is assigned this berth. Based on the MLB statistics and the mod

els provided, fans can sort out the play-off picture with more precise inf

ormation. .

5

Problem Definition: Elimination Questions

[Restrictions & Assumptions]

1. There are three divisions for each of the two leagues.2. Every team has to finish 162 games per season.3. There’s neither rain-outs nor ties. (Every game has a winner.)4. A team finishes the season with the best record of the division will advance to the pl

ay-off rounds.5. Ties in the final standing for a play-off spot are settled by special one-game playoffs.6. A team with the best record among all second-pace teams in the league will advanc

e to the play-off rounds as the “wild card.”7. To find the minimum number of wins necessary to win a division, it is only necessar

y to consider scenarios in which the teams in the division lose all remaining games against non-division opponents.

[Inputs]

Current win-loss records, remaining schedule of games

[Outputs]

A team’s first-place-elimination number and play-off-elimination number

6

Notations : Elimination Questions

Let be the decision variable representing the first-place-elimination threshold for division .

L : the set of teams in a league

kD : the set of teams in a division kk kD L

For each team in division , let be its number of current wins, the number the number of games remaining against team , and the number of games remaining against nondivision opponents.

i k iw ijgj

it

Finally, let be the total number of wins attained by team by season’s end in some scenario.

iW i

kvk

Further, let represent the number of future games that team wins against team ; let denote a complete scenario of future wins, .

ijx ki Dkj D x ,ij kx x i j D

Let be the decision variable representing the play-off-elimination threshold for league .

uL

7

Mathematical Models: First-Place-Elimination

min

subject tokv

, , ,

0 , , ,

integer,

integer , ,

k

ij ji ij k

k i ij kj D

ij k

k

ij k

x x g i j D i j

v w x i D

x i j D i j

v

x i j D i j

(1)

(2)

(3)

(4)

(5)

team winsi

team winsj

team against teami j

─(1)

12g 21gis the same as

→ Every game has a winner.

k

k

k

a aj kj D

b bjj D

c cjj D

w x v

w x

w x

1

2

3

Ranking Number of wins

─(2)

8


Suppose that the optimal objective value is , the first-place-elimination threshold for division .

kvk

Any team that can attain at least wins by season end will win the division.kl D kv

k

l l lj kj D

w t g v

Let ,

If , a division-winning scenario can be attained for team by increasing its number of non-division wins such that wins exactly total games.

If , a division-winning scenario can be attained for team by winning all of its non-division games( ) and an additional ( ) division games.

k

l l ljj D

v w x

l l kv t v ll kv

l l kv t v llt k l lv v t

9


It is clear that a team is eliminated from first-place if and only ifki D

k

i i ij kj D

w t g v

Further, if a team is not eliminated, .

Therefore, its first-place-elimination number is ( ), the minimum number of future wins that team needs to reach the threshold.

In addition, as mentioned above, a team is eliminated from the first-place, if its first-place number is greater than the number of its remaining games, i.e.

ki Dk

i i ij kj D

w t g v

k iv wi

k

k i i ijj D

v w t g

(first-place-elimination number) (number of remaining games)

10

Mathematical Models: Play-Off-Elimination min

subject to

u

, ,

u M 1,2,3 , ,

1 1, 2,3 ,

0 , ,

integer , ,

k

ij ji ij

ki ij i k

j L

ki

i D

ij

ij

x x g i j L i j

w x k i D

k

x i j L

x i j L

integer,

binary 1,2,3 ,

ki k

u

k i D

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Every game has a winner.

1 if team i wins division k,

0 else.ki

The variable u will not be affected by the number of wins for the first-place team if the three divisions in the league .L

The variable u is at least as large as the number of wins by all teams except first-place teams of the three divisions.

kiIf =0, u 0i ij i ij

j L j L

w x M w x

kiIf =1, u , where M is larger than 0i ij i ij

j L j L

w x M w x u

─(2)

11

Mathematical Models: Play-Off-Elimination

Suppose that the optimal objective value is , the play-off-elimination threshold for league .

The play-off-elimination number for each team with

Is

The play-off-elimination number for each team that wins the division

Is

uL

ki D 0ki

min ,k iv u w

kf

min ,kk k fv u w min

subject toku

k

, ,

u M 1,2,3 , ,

1 1,2,3 ,

0 , ,

integer , ,

k

ij ji ij

ki ij i k

j L

ki

i D

ij

ij

x x g i j L i j

w x k i D

k

x i j L

x i j L

integer,

binary 1,2,3 , ,

0 k

ki k

kf

u

k i D

12

Problem Definition: Clinching Questions


1. There are three divisions for each of the two leagues.2. Every team has to finish 162 games per season.3. There’s neither rain-outs nor ties. (Every game has a winner.)4. A team finishes the season with the best record of the division will advance to the pl

ay-off rounds.5. A team with the best record among all second-pace teams in the league will advanc

e to the play-off rounds as the “wild card.”6. Ties in the final standing for a play-off spot are settled by special one-game playoffs.

[Inputs]


[Outputs]

A team’s first-place-clinch number and play-off-clinch number

13

Notations : Clinching Questions

Let be the number of games for team to win to tie up with team .

L : the set of teams in a league

kD : the set of teams in a division kk kD L

For each team in division , let be its number of current wins, the number the number of games remaining against team , the number of games remaining against nondivision opponents, and the number of its future wins.

i k iw ijgj

it

ij

Further, let represent the number of future games that team wins against team ; let denote a complete scenario of future wins, .

ijxki D

kj D x ,ij kx x i j D

Let be the total wins accrued by team such that finishes with fewer wins than the first-place team in its division, and at least one division contains two teams with better records. Thus, ( ) is the play-off clinch number for team .

av

if

i j

Let be the number of games for team to win to tie up with all teams in the division, i.e. the first-place-clinching number for team .

i ii

a a

a1a

14

Mathematical Models: First-Place-Clinching

\

1min ,

2

\

max .k

ij j j i i ij j j i

k

i ijj D i

w g w g g w g w

j D i

(1)

(2)

If ,i i ijf g g

team must win some games against .

As team wins one game against team , the number of games that trails by will decrease by two, however.

Therefore, the number of games that has to win against is .

In addition, team may win at most games against teams other than .

To guarantee a tie with team , .

Thus, in this case,

ji

j ji i

i j

2i i ijf g g

j i ijg gi

j i j j if w g w

1

2 2 2i i ij i i ij

ij i ij j j i i ij

f g g f g gg g w g w g g

15

Mathematical Models: First-Place-Clinching

\

1min ,

2

\

max .k

ij j j i i ij j j i

k

i ijj D i

w g w g g w g w

j D i

(1)

(2)

If ,i i ijf g g

we assume that each future win by team comes against teams other than .

To guarantee a tie with team , .

Thus, in this case,

ji

j i j j if w g w

ij j j iw g w

The first-place-clinch number for team can be calculated as , without optimization.i i

[Remarks] Magic Number is calculated as , where denotes current numbers of wins for the first and second place teams respectively and denotes the number of remaining games for the second-place team. If either the 1st-place team wins one more game or the 2nd-place team loses one more game, the magic number decreases by 1. As the magic number approaches 0, the first-place team wins the division.

2 2 1 1w g w 1 2,w w

2g

16

Mathematical Models: Play-Off-Clinching max

subject toav

a

3

1

, ,

M -1 1, 2,3 , , ,

,

1 1, 2,3 ,

k

ij ji ij

ki ij i k

j L

a a ajj L

k k ki

i D

k

k

x x g i j L i j

v w x k i D i a

v w x

N k

1,

0 , ,

integer , ,

, binary 1,2,3 ,

ij

ij

k ki k

x i j L

x i j L

k i D

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Every game has a winner.

0 if team i clinches a play-off spot

1 else.ki

1 if the wild-card team is in division k,

0 else.k

kN denotes the number of teams in division k

1k kN denotes the number of teams without play-off positions in division k

17

Mathematical Models: Play-Off-Clinching max

subject toav

a

3

1

, ,

M -1 1, 2,3 , , ,

,

1 1, 2,3 ,

k

ij ji ij

ki ij i k

j L

a a ajj L

k k ki

i D

k

k

x x g i j L i j

v w x k i D i a

v w x

N k

1,

0 , ,

integer , ,

, binary 1,2,3 ,

ij

ij

k ki k

x i j L

x i j L

k i D

ki aIf =0, v 1i ij

j L

w x

All teams that finish in a play-off position will have more wins than does. a

ki aIf =1, v M

All teams that fail to finish in a play-off position will not be taken into consideration.

The play-off-clinch number for team = .a 1av

18

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

19

Results of the MLB case

20

The CBF 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. .

21

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

22

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 1 remaining game to play

Comparison of the complexity under different rules

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

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


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


23

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.

24

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.

25

Problem Definition: GQP first-eight-place


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]


[Outputs]

A team’s guarantee qualification score (GQS).

26

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 kkj

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

27

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 ,

kk j j

kj

j k

ij

kj

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.

28

Problem Definition: PQP first-eight-place


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]


[Outputs]

A team’s possible qualification score (PQS).

29

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 kkj

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

30

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 ,

kj k j

kj

j k

ij

kj

t t M z 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.

31

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

2

1

2

,

kk j j

kj

j k

ij

kj

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 22 teams ahead of team k.

32

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

21

,

kj k j

kj

j k

ij

kj

t t M z 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 21 teams ahead of team k.

33

Results of the CBF case

* 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

34

Team Fluminense (2002)

MNP GQS

P PQS

Results of the CBF case

35

Conclusions

1. The applications are very attractive, which encourages students to study optimization problems in Operations Research.

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

3. The CBF case provides a more general model for solving the playoff elimination problem.

4. Probabilistic models may describe more exactly how likely a team is able to be clinched to or eliminated from the playoffs.

36

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 ,

kk j j

kj

j k

ij

j n

t t M y j n j k

y

x i n j n i j

0,1 1 ,kjy 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.

37

References1. 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.

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

3. Schwartz, B. L. (1966) Possible Winners In Partially Completed Tournaments, SIAM Rev. 8(3), pp. 302-308.

4. McCormick, S. T. (1987) Two Hard Min Cut Problems, Technical report presented at the TMS/ORSA Conference, New Orleans, L.A.

5. McCormick, S. T. (1999) Fast Algorithms for parametric Scheduling Come From Extensions To Parametric Maximum Flow, Oper. Res. 47(5), pp.744-756.

6. Robinson, L.W. (1991) Baseball Playoff Eliminations: An Application of Linear Programming, Operations Research Letters 10, pp. 67-74.

7. 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.

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

9. 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.

Documents

1 913856 OR Applications in Sports Management : The Playoff Elimination Problem IEEM 710300 Topics in Operations Research