913856 盧俊銘

OR Applications in Sports Management : The Playoff Elimination Problem

IEEM 710300 Topics in Operations Research

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. .

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.

The Selected 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. .

Regulations of the MLB Playoffs

Chicago Minnesota Cleveland Detroit Kansas City

Central Division

Toronto Baltimore Boston Tampa Bay New York

East Division

Los Angeles Texas Oakland Seattle

West Division

Atlanta Florida New York Washington Philadelphia

East Division

St. Louis Chicago Cincinnati Houston Milwaukee

Central Division

Pittsburgh

Los Angeles

Minnesota New York

Boston

Wild Card

v.s.

v.s.

Atlanta

St. Louis Los Angeles

v.s.

v.s.

Houston

Wild Card

Los Angeles Arizona San Francisco San Diego

West Division

Colorado

Wild Card

Wild Card

An Example: The Case of Detroit Tigers

Current win-loss records

Remaining schedule of games

For Detroit Tigers, there’s a remote chance of catching the first-place:

If Detroit wins all of its remaining games, it will end up with 49 + 27 = 76 wins. However, New York will meet this record by easily adding just one more win of the remaining 28 games.

Therefore, is it reasonable to say that the Detroit Tigers has been eliminated from the first-place? (The answer is “yes.”)

Considering the scenario that New York wins no more than one games in the remaining 28 games.

1) New York fails to win another game [75-87]

Since Boston has 8 more games against New York, it will win them all. Thus, Boston may end up the season with at least (69 + 8 = 77) wins. That is to say, there’s no chance for Detroit to catch the first place.

2) New York wins only one more game [76-86]

According to the previous scenario, the only one win for New York must be against Boston, so that Boston may not end up with 77 wins. In addition, Boston have to lose all of the remaining games except the 7 wins against New York. Therefore,the final record for Boston would be also 76-86.

Estimates in a specific scenario

Estimates in a specific scenario

An Example: The Case of Detroit Tigers

Current win-loss records

Remaining schedule of games

Therefore, is it reasonable to say that the Detroit Tigers has been eliminated from the first-place? (The answer is “yes.”) [continued]

2) New York wins only one more game [76-86] [continued]

Now consider Baltimore and Toronto. Since Boston wins only the 7 (of 8) games against New York. It will undoubtedly lose when encountering all other teams. Thus, Baltimore will win all of its 2 games against Boston. Similarly, New York can not win any game except the only one game against Boston. Hence, Baltimore will win all of its 3 games against New York. Based on these facts, Baltimore’s final wins will be at least (71 + 2 + 3 = 76), which results in a four-way tie for first place.

In order not to finish ahead of Detroit, Baltimore must lose all other games except the 5 additional wins. That is to say, it will lose all of its 7 games against Toronto. In addition, since New York can not win any game against teams except Boston, Toronto will also win its 7 games against New York. This will make Toronto finish with (63 + 7 + 7 = 77) wins.

In this scenario,the final records for the division is shown as the table in the bottom. Toronto will catch the first-place, and then clinch for the play-off.

In summary, there’s no chance for Detroit to catch the first-place; it has been eliminated from the first-place.

Estimates in a specific scenario

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

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

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)

Mathematical Models: First-Place-Elimination

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

Mathematical Models: First-Place-Elimination

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)

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)

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

Problem Definition: Clinching 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. 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]

Current win-loss records, remaining schedule of games

[Outputs]

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

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

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

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

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

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

Results

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

Results

Conclusion & Discussion

1. The method is simple and useful.

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

3. Problems for kth-place-elimination or kth-place-clinching need to be discovered.

References

1. 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. Wayne, K.D. (2001) A New Property and A Faster Algorithm For Baseball Elimination, SIAM J. Discrete Math 14(2), pp. 223-229.

8. Ribeiro, C. C. and Urrutia, S. (2004) OR Applications In Sports Scheduling and Management, OR/MS Today 31(3), pp. 50-54.

9. 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.