futmax: play-offs elimination in soccer tournaments celso c. ribeiro sebastián urrutia

FUTMAX: Play-offs elimination in

soccer tournaments

Celso C. RibeiroSebastián Urrutia

April 2004 FUTMAX Chile FUTMAX 2/25

Summary

Motivation Related work Problem definition Formulation FUTMAX in the WWW FUTMAX in the press Results


Guaranteed Qualification Problem: How many points a team has to make in a tournament to be able to be qualified for the finals (or play-off games)?

Brazilian Soccer Championship: Press broadcasts already at the first round the “chances of qualification” for the play-offs, based in very obscure statistics and estimates.

Predictions loudly announced are often wrong!

Motivation


Motivation Interesting application of Operations Research and

Integer Programming in sports. Soccer is the most popular sports in most

countries. Soccer can even make researchers from Argentina

and Brazil to work together, instead of trying to prove which of them is the best...

… or arguing about who is the second best player ever!

#1


Motivation

Three teams fight for two places in the play-offs. Each of them has two games to play: Flamengo

against Vasco and Bahia, Cruzeiro against Grêmio and Santos, and Bahia against Flamengo e Fluminense.

How many points Cruzeiro has to make to be sure of being qualified?

4 points! Flamengo 37

Cruzeiro 37

Bahia 36


Motivation: what if 26 teams are playing?

São Paulo 34 Ponte Preta

26 Fluminense

21

São Caetano

32 Figuerense 24 Vasco 20

Santos 32 Atlético-PR 24 Goiás 20

Juventude 31 Grêmio 24 Portuguesa

20

Corinthians 31 Cruzeiro 23 Paysandu 19

Guarani 28 Gama 22 Flamengo 19

Vitória 27 Internacional

22 Palmeiras 19

Coritiba 27 Botafogo 22 Paraná 17

Atlético-MG

27 Bahia 21


Motivation Brazilian soccer championship (2002 edition) 26 teams First phase: qualification

• Each team plays every other team exactly once.• The eight teams in the first positions are qualified.• The four teams in the last positions do not play

next year. Second phase: play-offs

• Eliminatory games Points:

• win: 3 points tie: 1 point loss: 0 point


Motivation

Brazilian championship: • 29 rounds from August to December• The press publishes since the first round (!) the

probabilities of qualification of each team.• The press also makes use of very simple statistics

from obscure sources to announce very early that a team with a certain number of points (41 in 2002) can be sure to be qualified (often based on estimates or average results).

• These predictions are loudly announced and often false!

• Trainers also very often make erroneous predictions.


Related work

Schwartz 1966: mathematical elimination from play-offs in the Major League Baseball (MLB) solved with maximum flow algorithm

Robinson 1991: integer programming models and further results for the play-offs elimination problem

McCormick 2000: elimination from the p-th position is NP-complete.

Adler et al. 2003: ILP models for MLB Bernholt et al. 1970: first place elimination

is NP-complete under the {(3,0),(1,1)} soccer rule.


Related work

Berkeley’s RIOT project posts the number of wins each team needs to have a chance of qualification in the MLB:• No ties• Easier {(1,0)} rule• Small groups with 5 or 6 teams• A team has to finish in the first position

within its group to be qualified.• Each team plays up to 160 games!• Problems in each group are easily solvable.


Problem definition

Play-offs qualification problems:How many points a team should make to: … be sure of finishing among the p=8

teams in the first positions? (sufficient condition for play-offs qualification)

… have a chance of finishing among the p=8 teams in the first positions? (necessary condition for play-offs qualification)


Problem definition

Elimination problems:How many points a team should make to: … be sure of finishing among the p=22

teams in the first positions? (sufficient condition for non-elimination)

… have a chance of finishing among the p=22 teams in the first positions? (necessary condition for non-elimination)


Problem definition

In this talk, we consider only the Guaranteed Qualification Problem: How many points a team k should make to be sure of finishing among the p=8 in the first positions?

Instead, we compute the maximum number of points a team can make and still not be qualified in the p=8 first positions.

Then, we add 1 to the above number to determine the minimum number of points for guaranteed qualification: GQS(k)


Formulation

n = number of teams = 26 M = maximum difference of points

between any two teams (all wins vs. all losses) = 3*(n-1) = 75

k = team under consideration


Formulation

xi j =½

1; if team i wins over team j0; otherwise

pj = points already accumulated by team j

tj = points won by team j at theend of the tournament

yj =½

1; if tj ¸ tk (i.e. if team k is not ahead j )0; otherwise


Formulation

( ) 1 max

subject to:

1 , to be played

3 1 ( )

(1 )

8

{0,1} , to be played

{0,1}

k

ij ji

j j ji ij jii j i j

k j j

jj k

ij

j

GQS k t

x x i j

t p x x x j

t t M y j

y

x i j

y j

0j k jt t y


Formulation

Ties in the number of points are broken in favor of teams with more wins.

In the previous model, add up a very small amount (necessarily smaller than one) to the number of points:

Use e.g. With a similar model, we can also compute

PQS(k): minimum number of points for possible qualification

(3 ) [1 ( )]j j ji ij jii j i jt p x x x

0.01


Formulation

When is team k “mathematically qualified” for the play-offs?Team k is mathematically qualified if the previous problem is infeasible!

When does a team k depend only on itself to be qualifed?Team k depends only on itself if GQS(k) is less than or equal to the total number of points it can make.

When is team k “mathematically eliminated” from the play-offs?Team k is mathematically eliminated if PQS(k) is less than or equal to the total number of points it can make.


FUTMAX in the WWW

FUTMAX project Results of the games are collected from the

web. Model are generated (four problems for each

team). All problems are solved with CPLEX 9.0 HTML file is automatically built from the

results. Automatic publication in the web:

http://www.futmax.org 2006 World Cup


FUTMAX in the press

September 24, 2002: FUTMAX web site launched

September 29: round table interview to Radio Globo (major sports radio station, program “Enquanto a bola não rola”)

September 30: article in the Internet section of Jornal do Brasil

October 24: interview for TV Campus December 18: article in Jornal da PUC October 2003: talks with SPORTV (TV Globo)

and computations of “chances” of qualification (live)


Results

Already at the 11th round, some teams did not depend only on themselves to be qualified.

At this time, Vasco was in a difficult situation and fans were complaining.

The trainer of Vasco met the press and said that “if we win the next ten games, then we will be qualified”.

FUTMAX was able to prove to the press that this was not true.


Results

October 31, 2002: São Paulo won Ponte Preta and made a total of 43 points.

The press (Folha de São Paulo) announced that “São Paulo was mathematically qualified for the play-offs”.

FUTMAX was able to prove to the press that this was not true.

November 3, 2002: São Caetano made a total of 42 points and the press also announced it was qualified.

Again, FUTMAX was able to prove to the press that this was not true.


Results


Results

FUTMAX can be used to follow the situation of each team:

Possible points

Points for guaranteed qualification

Points for possible qualification

Points accumulated

FLUMINENSE


Results

Spin-offs: followed by HockeyPlex project (same idea for National Hockey League, USA) based on FUTMAX

Motivation for students Activities of our research group made public New web page: http://www.esportemax.org Slides available at:

http://www.inf.puc-rio.br/~celso/talks.htm Papers (OR/MS Today 2004; ITOR to appear)

available at:http://www.inf.puc-rio.br/~celso/publicacoes.htm

futmax: play-offs elimination in soccer tournaments celso c. ribeiro sebastián urrutia

Documents

futmax chilefutmax

motivation slide

www futmax

27bahia21 slide

point slide

playoffs elimination

soccer rule

press results