ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE

Statistical Models for Knock-out Soccer Tournaments

Diego Kuonen Department of Mathematics Chair of Applied Statistics
Prof. S. Morgenthaler, DMA, EPF Lausanne Assistant: E. Chavez, DMA, EPF Lausanne Winter 1996/97

Table of Contents
1 Introduction
1.1 Preliminary . . . . . . . . . 1.2 European Cups . . . . . . . 1.2.1 Champions Cup . . . 1.2.2 Cup Winners Cup . . 1.2.3 UEFA Cup . . . . . 1.2.4 UEFA Intertoto Cup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1 1 1 2 2 4

2 Descriptive Data Analysis
2.1 Data Structure . . . . . . . . . 2.2 Tree Structure . . . . . . . . . . 2.3 Seeding Coe cients . . . . . . . 2.3.1 First Approach . . . . . 2.3.2 Improvement . . . . . . 2.4 Tournament Analysis . . . . . . 2.4.1 Possible Sets of Winners 2.4.2 Probability of Winning . 2.4.3 Potential Opponents . .

5
5 6 8 9 11 13 13 14 16

3 Statistical Modelisation
3.1 Logistic Regression Model . . . . . . . 3.2 Constancy of Team Strength . . . . . . 3.2.1 Model Chosen . . . . . . . . . . 3.2.2 The Program . . . . . . . . . . 3.2.3 Number of Correct Predictions 3.3 Variation of Team Strength . . . . . . 3.3.1 Model Chosen . . . . . . . . . . 3.3.2 Using Probable Opponents . . . i

17
17 17 18 20 23 25 25 26

Table of Contents

3.3.3 Using All Potential Opponents . . . . . . . . . . . . . . . . . . . . . 29 3.3.4 Number of Correct Predictions . . . . . . . . . . . . . . . . . . . . 32 3.3.5 Evaluation of the Coe cients . . . . . . . . . . . . . . . . . . . . . 34

4 Comparison of the Methods 5 Conclusion A Appendix
A.1 The Bradley-Terry Model . . . . . . . A.2 Tournament Trees . . . . . . . . . . . . A.2.1 Cup Winners Cup . . . . . . . . A.2.2 UEFA Cup . . . . . . . . . . . A.3 Evaluation of the Coe cients . . . . . A.3.1 Using Probable Opponents . . . A.3.2 Using All Potential Opponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 39 40
40 41 41 48 55 55 59

B References

63

ii

Introduction

1 Introduction
1.1 Preliminary
Sports events and tournament competitions provide excellent opportunities for model building and using basic statistical methodology in an interesting way. In this paper, a logistic regression model using seed positions (conceived through a seeding coe cient) is applied to European soccer Cups tournament data in order to predict the probability of winning the tournament for each one of the participating teams, and the predicted probabilities of each team reaching a certain leg such as the quarter nal.

1.2 European Cups
This section was coined by M. Protzen (1996).

There are four di erent European Cups the Champions Cup (CC), the Cup Winners Cup (CWC), the UEFA Cup (UC), originally intended for League runners up, and the UEFA Intertoto Cup (UIC). The quali cation for these competitions depends on the performance in National Leagues Cup competitions respective. The Champion usually enters the CC, the Cup Winner (losing nalist in case of a double respective) enters the CWC, while a variable number of League runners up enter the UC (in some countries winners of a League Cup enter the UC as well). Teams nishing below those qualifying for the UC may enter the UIC.

1.2.1 Champions Cup
Originally this Cup was intended for the league winners of the member FAs of UEFA. A while ago the quali cation process for the CC has been modi ed such that only the CC defender together with the top 23 champions enters the CC while all other champions will enter the UC. The CC defender and the 7 best champions advance directly to the league stage while the champions ranked 8th to 15th have to play a preliminary tie against one of the champions ranked 16th to 23rd. To determine the top champions (and the number of UC berths allocated to a country) UEFA maintains a ranking list.

1

Introduction

New format for 1997/98 For the upcoming season UEFA has once again changed the format to allow the vice champions of the 8 top ranked countries as well as the champions of up to 48 countries into the competition. The league stage will be played in six groups of four teams. From there the six group winners together with two best runners-up will advance to the quarter nals. The two best runners-up will be determined by points achieved, goal di erence, goals scored and goals scored away. To cut down the number of teams the champions of countries ranked 17 to 48 in the most recent ranking table play a preliminary round in late July. For the losers of this tie their European campaign is over. Winners play in the quali cation round together with champions of countries ranked 9 to 16 and vice champions of countries ranked 1 to 8. Winners of this round advance to the league stage where they will be joined by the champions of the top ranked countries. Losers will play in the rst round of the UEFA Cup. The defending Cup Winner gets an automatic place in the league stage regardless where he nishes in his national competition. This may mean that the champions of the countries ranked 8th, 16th (and 48th) are 'downgraded'.

1.2.2 Cup Winners Cup
The Cup Winners of the FA Cups of the member FAs of UEFA get to play in this Cup unless they also win the domestic championship (in which case they will be replaced by the losing cup nalists). The CWC defender is also eligible to play (unless he becomes national champion). UEFA is also considering to allow a second team from the eight top ranked countries to play the CWC. How the second team will be determined has not yet been xed. This modi cation will not e ect for 96/97 and 97/98 competitions.

1.2.3 UEFA Cup
The UEFA Cup started (under the name "Fairs Cup") as a competition between city teams from cities which host a trade fair. Later it was a competition for the runners up of the various leagues, some countries enter also their League Cup winner. The number of participants from each country is determined by the UEFA ranking list. Since the Champions League was introduced the champions not eligible for the Champions League are entered into the UEFA Cup as well, another three berths are reserved for the Intertoto Cup semi nalists and three are given to countries who top the UEFA Fair Play competition. 2

Introduction

79 participants are determined by the UEFA coe cient table: { 3 countries with four berths { 5 countries with three berths { 13 countries with two berths { 26 countries with one berth 24 champions not entering the CC 3 Intertoto Cup participants 3 Fair Play competition winners The three countries which top the UEFA Fair Play competition each get one additional UEFA cup berth in the next competition. The 1995/96 berths were allocated to Norway, England and Luxembourg, the 1996/97 berths have been awarded to Sweden, Russia and Finland (one berth originally awarded to England has been revoked because of elding youth and reserve teams in the 1995 UIC) and in 1997/98 Norway, England and Sweden will get the berths. 1 defending UEFA Cup Holder (only if they do not qualify for any European competition otherwise) 8 preliminary round losers from the Champions Cup (since 1996/97) Altogether, there are 118 participants. For the 96/97 competition the following format will be used: Preliminary round in July, involving 54 teams from countries with the lowest country coe cient in the 95/96 ranking Qualifying round in August, involving 50 teams (27 winners from the qualifying round, 23 teams from the countries with low coe cients which did not play in the preliminary round) + 6 Intertoto teams. First round in September, involving 64 teams (25 winners from the preliminary round, 8 teams eliminated from the Champions Cup, 28 teams from countries with high rankings and 3 teams qualifying through the UIC). Barring participation of teams from San Marino, Andorra and Bosnia the 97/98 competition will have 16 participants fewer: 24 champions of low ranked countries will not enter the UC but 16 instead of 8 CC quali cation round losers will play in the rst round. From 1997/98 on the nal will be decided in a single game on neutral ground. 3

Introduction

1.2.4 UEFA Intertoto Cup
The Intertoto Cup started as a way to guarantee income to the Toto-industry in the early sixties (in fact, this generated a signi cant sum of money for clubs in several countries, one of them the Netherlands). The rst one or two seasons the group winners played out a nal stage to determine an overall winner (the rst being Ajax). After a few years the competition degenerated into summer practice for mid table teams of North, Central and East European countries. Teams from the British Islands, Spain or Italy rarely participated. No nal stage was played after the initial (one or two) season(s) and nobody really cared for it. This new directive by UEFA seems like a nal attempt of attracting some attention to it. The 'new' UEFA Intertoto-Cup has started in the summer of 1995. The cup will be played during the summer, and the teams allowed to participate should have been the best nishing teams in the respective leagues the previous season not already quali ed for any of the other three European Cups. The number of teams from each country is determined by the UEFA ranking of countries just as in the UC (with the exception that also the bottom clubs are entitled to one berth). 60 teams are divided into 12 groups with the winners of each group qualifying for a knock-out round, from where the best 3 teams qualify for the rst round proper of the UEFA Cup the following autumn. From the 1995 UIC two French teams, Racing Strasbourg and Girondins de Bordeaux, emerged successful and participated in the UC, Bordeaux even reached the UC nal. A number of changes have been announced for the 1996 UIC: The number of UC berths available through this competition has been raised to 3, Italian teams will join the competition and teams are urged to take the competition serious, i.e. play with their rst team. Some countries do not take part in the competition: England, Portugal, Scotland and Luxembourg. Italy sends only three teams, Spain and Greece only one. Italy and Spain later renounced their remaining berths for the UIC, they were heavily ned for that. Also Greece and Albania did not take their berths. Those six berths went to Estonia, Lithuania, Austria, Turkey, Denmark and Sweden. Also the format of the cup has changed slightly: Again there will be 12 groups with 5 teams each. But fortunately the concept of "best seconds" has been cancelled: The 12 winner will play one elimination round with the 6 winners playing another round as part of the UC preliminary round. In 1996 Silkeborg IF, Karlsruher SC and EA Guingamp quali ed for the UC.

4

Descriptive Data Analysis

2 Descriptive Data Analysis
2.1 Data Structure
As said in the introduction European soccer Cups tournament data is used. We only considered the data from the Champions Cup (CC), from the Cup Winners Cup (CWC) and from the UEFA Cup (UC). The typical tournament structure does not begin until after the preliminary round(s), so we did not consider them. In this context the structure of an ordinary data looks like this: leg team.a country.a team.b country.b s.a.1 s.b.1 s.a.2 s.b.2 Round1 FKAustria(Wien) Aut DinamoMinsk Bls 1 2 0 1 Round1 Milan Ita ZaglebieLubin Pol 4 0 4 1 ... Round1 SpartakVladikavkaz Rus Liverpool Eng 1 2 0 0 Round1 ZimbruChisinau Mol RAFYelgava Lat 1 0 2 1 Round1 RodaJC(Kerkrade) Ned SCTOlimpija(Ljubljana) Slo 5 0 0 2 Round2 BrondbyIF Den Liverpool Eng 0 0 1 0 ... Round2 SpartaPraha Tch ZimbruChisinau Mol 4 3 2 0 Round2 GirondinsBordeaux Fra RotorVolgograd Rus 2 1 1 0 Round3 PSV(Eindhoven) Ned WerderBremen Ger 2 1 0 0 Round3 BayernMuenchen Ger SLBenfica Por 4 1 3 1 ... Round3 Sevilla Esp FCBarcelona Esp 1 1 1 3 Round3 SlaviaPraha Tch RCLens Fra 0 0 1 0 Quarter FCBarcelona Esp PSV(Eindhoven) Ned 2 2 3 2 Quarter SlaviaPraha Tch Roma Ita 2 0 1 3 Quarter Milan Ita GirondinsBordeaux Fra 2 0 0 3 Quarter BayernMuenchen Ger NottinghamForest Eng 2 1 5 1 Semi BayernMuenchen Ger FCBarcelona Esp 2 2 2 1 Semi SlaviaPraha Tch GirondinsBordeaux Fra 0 1 0 1 Final BayernMuenchen Ger GirondinsBordeaux Fra 2 0 3 1

where s.a.1:s.b.1 is for example the score of the 1st leg. This list represents the data from the UEFA Cup 1995/96.

5

Descriptive Data Analysis

2.2 Tree Structure
An example of a typical tournament structure is represented by gure 1. The tree structure is cognizable. In this section we want to describe how we obtained the tournament tree through the data. So let us have a look at gure 1. We always start with the winner of the tournament, in this case it is A. Then we will look for the team A played in the previous round, that would be in this case C . In the future C is marked. Now we are looking for the team A played against before playing against C , that would be B . We realize that the game A against B had be played in the rst round of the tournament, so we are going back to the last marked team, that would be C , and so on.
A A B A C C D

Figure 1 Example of a typical tournament structure.

For this reason we have programmed a Splus routine titled pairings.tree. In applying this routine to the Cup Winners Cup of the year 1989/90 we obtain gure 2.

6

Descriptive Data Analysis

cwc.8990
Sampdoria Sampdoria SKBrann Sampdoria BorussiaDortmund BorussiaDortmund Besiktas Sampdoria GrasshopperClubZuerich GrasshopperClubZuerich SlovanBratislava GrasshopperClubZuerich TorpedoMoscow TorpedoMoscow CorkCity ASMonaco ASMonaco CFOsBelenenses ASMonaco BerlinerFCDynamo BerlinerFCDynamo Valur(Reykjavik) ASMonaco RealValladolid RealValladolid HamrunSpartans RealValladolid DjurgardensIF DjurgardensIF USLuxembourg RSCAnderlecht RSCAnderlecht BallymenaUnited RSCAnderlecht FCBarcelona FCBarcelona LegiaWarsaw RSCAnderlecht AdmiraWacker AdmiraWacker AEL(Limassol) AdmiraWacker Ferencvarosi Ferencvarosi ValkeakoskenHaka DinamoBucharest DinamoBucharest KSTirane DinamoBucharest Panathinaikos Panathinaikos SwanseaCity DinamoBucharest FKPartizan(Belgrade) FKPartizan(Belgrade) Celtic FKPartizan(Belgrade) FCGroningen FCGroningen IkastFS

Sampdoria

Sampdoria

RSCAnderlecht

Figure 2 The Cup Winners Cup 1989/90.

In the appendix you will nd the tournament trees of all European Cups considered.

7

Descriptive Data Analysis

2.3 Seeding Coe cients
To obtain the seed positions we calculated the seeding coe cients with the help of a ranking list. The basis for this ranking is the performance of teams in the three major European Cups during a three year period. Each team gets two points for a win and one point for a draw. One bonus point is allocated for reaching the quarter nal, the semi nal and the nal. This procedure is similar to the one the UEFA uses for their ranking list. In the rst time we calculated a coe cient which will be replaced later on, by a more appropriated coe cient: A weighted coe cient. Both coe cients are based on a list containing the performances during an European Cup. An example is the following list which shows the points achieved for the participants during the Champions Cup of the year 1995/96:
THE TABLE FOR cc.9596 ---------------------------------team country win draw lost scored Ajax Ned 8 3 1 22 Juventus Ita 6 3 3 22 Panathinaikos Gre 5 3 2 11 SpartakMoscow Rus 6 1 1 17 FCNantes Fra 4 4 2 14 RealMadrid Esp 4 1 3 12 BorussiaDortmund Ger 2 3 3 8 LegiaWarsaw Pol 2 2 4 5 FCPorto Por 1 4 1 6 SteauaBucharest Rom 1 2 3 1 RosenborgBK Nor 2 0 4 11 Rangers Sco 1 2 3 6 Ferencvarosi Hun 1 2 3 9 BlackburnRovers Eng 1 1 4 5 AaB(Aalborg) Den 1 1 4 5 GrasshopperClubZuerich Sui 0 2 4 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

against points 3 22 9 18 6 15 8 14 12 14 7 10 11 8 11 7 5 6 5 4 16 4 13 4 19 4 8 3 11 3 13 2

8

Descriptive Data Analysis

2.3.1 First Approach
In this approach we calculated the performances of the teams over a three year period and summarized them into a list. The following list shows the points achieved for all teams who participated in an European Cup from 1992 to 1995.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 ... 211 212 213 team country win draw lost scored Milan Ita 23 8 4 55 Juventus Ita 22 5 5 68 Parma Ita 18 8 6 37 Ajax Ned 17 6 3 47 ParisSaintGermain Fra 15 6 7 39 BorussiaDortmund Ger 16 4 10 43 SLBenfica Por 13 7 3 43 FCPorto Por 14 5 8 43 Arsenal Eng 11 7 2 35 FCBarcelona Esp 11 7 6 42 SpartakMoscow Rus 12 6 5 46 RealMadrid Esp 11 4 5 40 EintrachtFrankfurt Ger 10 4 6 35 BayerLeverkusen Ger 9 4 3 41 IFKGothenburg Swe 10 3 5 26 AJAuxerre Fra 8 5 5 37 OlympiqueMarseille Fra 10 3 2 26 Internazionale Ita 10 1 3 23 RSCAnderlecht Bel 7 9 5 31 RealZaragoza Esp 9 1 6 31 ClubBruggeKV Bel 9 2 5 19 Boavista Por 7 5 4 22 Galatasaray Tur 6 7 9 17 Lazio Ita 8 2 2 15 KSVWaregem DACDunajskaStreda Motherwell Bel Tch Sco 0 0 0 0 0 0 2 2 2 1 0 0 against points 10 60 22 56 15 53 14 45 19 42 30 42 23 37 20 35 15 35 27 32 31 32 23 28 14 26 19 25 18 24 21 24 9 24 11 24 24 23 17 22 15 21 11 20 29 19 7 19 6 4 3 0 0 0

This ranking list over a three year period is now used to calculate the seeding coe cients. We de ned the coe cient in the following way: during the three year period Coe cient = Points achievedduring the three year period Games played This is di erent from the UEFA coe cients, because the UEFA uses the sum of the ratio points achieved over games played for each of the ve past years. An example of our calculations is the following list which shows the seeding coe cients for the UEFA Cup of the year 1993/4. 9

Descriptive Data Analysis team country coeff Juventus Ita 2.4000000 AtleticoMadrid Esp 2.0625000 BayernMuenchen Ger 1.9166667 AJAuxerre Fra 1.8571429 BorussiaDortmund Ger 1.7222222 SportingCP(Lisbon) Por 1.6428571 Internazionale Ita 1.6428571 BrondbyIF Den 1.5000000 AstonVilla Eng 1.5000000 RoyalAntwerp Bel 1.4166667 Olympiakos(Pireus) Gre 1.4000000 AdmiraWacker Aut 1.4000000 Vitesse(Arnhem) Ned 1.2500000 Trabzonspor Tur 1.1666667 GirondinsBordeaux Fra 1.1666667 EintrachtFrankfurt Ger 1.1000000 SlaviaPraha Tch 1.0000000 DinamoMoscow Rus 1.0000000 BSCYoungBoys AaB(Aalborg) Sui 0.0000000 Den 0.0000000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ... 63 64

This approach was not satis able because there is a problem with teams which were playing in the Champions Cup. In the Champions Cup less games are played and so the teams will achieve less points. That is why their seeding coe cient would be smaller, even if they played in the most important European Cup. By calculating the probability for a team of winning the tournament for example (in using the program prob.cte (cf. section 3.2)) for the UEFA Cup 1995/96 we noticed that Girondins Bordeaux (Fra) had more chances to win the tournament than Barcelona (Esp). For every soccer fan this seems to be impossible, so for us. For this reason we did not use this approach to calculate the seeding coe cients.

10

Descriptive Data Analysis

2.3.2 Improvement
The improvement consists in calculating a weighted mean of the ratio points achieved over games played for each of the three past years, let be Cx this ratio for the year x. Hence we de ned the coe cient for the year x as Coe cient = Cx? + Cx? + Cx? .
3 6 1 2 6 2 1 6 3

With this choice of the weights we give most importance on the ratio of the previous year. If for example a team plays in a given year's UEFA Cup, but it did not play any European Cup for the last two years, the team will get a seeding coe cient (in using the previous method (see 2.3.1)) which seems to be too high because they played for the last time in the European Cup three years before - that is nonsense. This problem is absorbed by the improved coe cient because there will be the weight 1/6. That is why from now on this weighted seeding coe cient (brie y coe cient) is used. The following list shows the coe cient for the Cup Winners Cup of the year 1992/93: team country coeff 1 ASMonaco Fra 1.44722222 2 AtleticoMadrid Esp 1.33333333 3 SpartakMoscow Rus 1.25000000 4 WerderBremen Ger 1.21666667 5 Trabzonspor Tur 1.00000000 6 SteauaBucharest Rom 1.00000000 7 Feyenoord Ned 0.91666667 8 AdmiraWacker Aut 0.72222222 9 SpartaPraha Tch 0.70833333 10 Glenavon Nir 0.66666667 11 Boavista Por 0.58333333 12 Liverpool Eng 0.56250000 13 Parma Ita 0.50000000 14 Olympiakos(Pireus) Gre 0.50000000 15 RoyalAntwerp Bel 0.37500000 16 FCLucerne Sui 0.33333333 17 AvenirBeggen Lux 0.33333333 18 Apollon(Limassol) Cyp 0.33333333 19 Valur(Reykjavik) Isl 0.25000000 20 ChernomoretsOdessa Ukr 0.25000000 21 FCLevski(Sofia) Bul 0.08333333 22 UjpestiTE Hun 0.00000000 23 TPS(Turku) Fin 0.00000000 24 MiedzLegnica Pol 0.00000000 25 MariborBranik Slo 0.00000000 26 HapoelPetachTikva Isr 0.00000000 27 Hannover96 Ger 0.00000000

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Descriptive Data Analysis
28 29 30 31 32 CardiffCity Bohemians Airdrieonians AIK(Stockholm) AGF(Arhus) Wal Irl Sco Swe Den 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000

Figure 3 shows the coe cient for di erent three year periods for some chosen teams. Let us have a look at the evolution of the coe cient of Juventus (Ita): Between the periods 1989/92 and 1990/93 the coe cient increased because they won the UEFA Cup in 1992/93 (yes, Juventus also wins the UEFA Cup not only the Champions Cup) in achieving 24 points. Between the periods 1990/93 and 1991/94 the coe cient decreased because in the 1993/94 edition of the UEFA Cup Juventus reached the quarter nal in achieving in 8 games only 10 points (4 wins, 1 draw and 3 lost). And between the periods 1991/94 and 1992/95 he increased again because they won the UEFA Cup in 1994/95 again before becoming Italian champion. We also remark the constancy of the coe cient of Barcelona (Esp) who was the dominating team in European Cups during this periods.
Coefficients for different periods for chosen teams:
• •
1.6

•
1.4

• •

•

•

•
1.2

• •

•

•

1.0

•

•

• • 89-92 90-93 91-94

Juventus Milan FCBarcelona BorussiaDortmund

0.8

92-95

Figure 3 The coe

cient for di erent periods for some chosen teams.

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Descriptive Data Analysis

2.4 Tournament Analysis
2.4.1 Possible Sets of Winners
Predicting the probability of each seed winning the tournament requires the consideration of all possible paths and opponents. An example of a tournament is shown by gure 4.
Round 1
1

Round 2

2

3

4

Figure 4 Example of a tournament.

Let us determine the teams that must be played for a team to become the champion. In Round 1 there are 2 possible sets of winners and in Round 2 there are 2 possible outcomes. Hence there are 2 2 = 2 possible outcomes for the 3 games.
2 2 3

By generalizing, we obtain the following table: Round number Games per round Possible sets of winners 1 n=2 2n= 2 n=4 2n= ... ... ... n=(2m) 2n= m m
2 4 (2 )

where n is the number of teams participating in the tournament. From this it follows that in a tournament with n teams (and so n ? 1 games) there are m Y n= k 2
(2 )

k=1

possible sets of winners for the tournament, where m is the number of legs to play for reaching the nal game.
Remark: In fact m = log2 n, because of the following relationship: n = 2m.

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Descriptive Data Analysis

2.4.2 Probability of Winning
Let Pk (i; j ) be the probability that team i wins against team j in the kth leg (8i 6= j ). As in a tournament there is always a winner (i.e. there are any draws), it follows that Pk (j; i) = 1 ? Pk (i; j ). In the rst round each team has only one possible opponent, but in the second round there are two possible opponents as shown in the previous section. That is why the probability analysis must include not only the probability of defeating each potential opponent, but also the probability of each potential opponent advancing to a particular game. To illustrate, let us have a look at gure 5.
A Game 1 B Game 3 C Game 2 D

Figure 5 Example of a tournament.

Suppose that team A wins the tournament by defeating team B in game 1 and the winner of game 2 (C or D) in game 3, then the probability that team A wins the tournament would be :

P (A wins the tournament) = = = =

P (A wins in game 3) P (A; B )P (A; winner of game 2) P (A; B )fP (C; D)P (A; C ) + P (D; C )P (A; D)g P (A wins in leg 1)fP (C wins in leg 1)P (A; C ) + P (D wins in leg 1)PX D)g (A; = P (A wins in leg 1) P (j wins in leg 1)P (A; j )
1 2 1 1 2 1 2 2 2

j 2fC;Dg
2

2

This probability is the probability of A winning the tournament and as well the proba= 2. bility of A winning in game 3, i.e. in leg m = log n = log 4 =
2 log 4 log 2

14

Descriptive Data Analysis

By generalizing, we obtain:

P (i wins in leg k) = P (i wins in leg k ? 1) P P (j wins in leg k ? 1)Pk (i; j ) j 2J

(1)

for k = 2; : : : ; log n. J is the set of all potential opponents of i for leg k.
2

Hence:

P (i wins the tournament ) = P (i wins in leg k = log n)
2

(2)

We now must nd probability models for determining Pk (i; j ).

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Descriptive Data Analysis

2.4.3 Potential Opponents
By equation (1) we are forced to determinate the set of all potential opponents for a chosen team. For this reason we have programmed a function titled potential.opponents. The work of this function is illustrated by gure 6.
Quarter
1 2 3

Semi

Final

Upper half Lower half

4 5 6 7 8

Figure 6 Illustration of potential.opponents.

In gure 6 there are eight teams participating in the tournament. If we want to determine all potential opponents for the nal game of a team we only have to look if the team chosen is in the upper or in the lower half of the tournament tree. If the team is in the upper half the potential opponents will be all teams playing in the lower half, i.e. the following four teams: 5, 6, 7 and 8. In the same way it is possible to determine all potential opponents for the semi nal: If the team is among the teams 3 and 4 the potential opponents are the teams playing in the upper quarter of the tournament tree, i.e. 1 and 2, and so on.

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Statistical Modelisation

3 Statistical Modelisation
3.1 Logistic Regression Model
Schwertman et al. (1996) suggested in their work on probability models for the NCAA Regional Basketball Tournaments 11 di erent models for assigning probabilities of winning for each team in each individual game. These models were compared in three ways by using a chi-square statistic as a measure of the relative t of the models. The chi-square values provided a measure of the relative accuracy of the various models. Hence the logistic model seemed to be the most satisfactory if the objective is to predict the winner of the tournament. That is the reason why we choose a logistic regression model. The model we assume is closely related to the Bradley-Terry model for paired comparisons (Bradley and Terry 1952). This relationship is shown in the appendix of this paper. Therefore the formula relating the seeding coe cients to winning probabilities is: + (S (i) ? S (j )) Pk (i; j ) = e + (kSk (i) ?kSk (j )) 1+e (3)

where Sk (i) is the seeding coe cient of team i depending on leg k. Clearly we have 0 Pk (i; j ) 1 and Pk (j; i) = 1 ? Pk (i; j ), 8k and 8i = j . 6 These seeding coe cients represent the team strength.

3.2 Constancy of Team Strength
In the rst time, we assume that the games are independent and that the seeding coefcients remain constant throughout the tournament. Hence Sk (i) = S (i) and therefore Pk (i; j ) = P (i; j ). As each game has only two outcomes: A win or a loss, we are opposed to an independent Bernoulli trial. Let be Yij one Bernoulli trial, therefore: ( Yij = 1 if i defeats j 0 else. It follows that P (Yij = 1) = P (i; j ). Hence the model (3) becomes: e S i ?S j E Yij ] = 1 + e S i ?S j where and are the parameters to estimate.
+ ( ( ) ( )) + ( ( ) ( ))

17

Statistical Modelisation

3.2.1 Model Chosen
The model is t to a data set consisting of the game outcomes from season 1992/93 to 1995/6 in all three European Cups, i.e. the UEFA Cup, the Cup Winners Cup and the Champions Cup. It is important to notice that in the Champions Cup we took only the games of type knock-out, i.e. we did not take into consideration the league stage of the Champions Cup. In all we considered 442 game outcomes. In this regard it is important to say that even if two teams played twice against each other in a leg, we considered only the outcome of the leg, and not the outcomes of each game of the leg. As response variable for the regression we took the following variable, let it be y: 3 2 yuc: 7 6y 7 6 uc: 7 6 7 6 yuc: 7 6 7 6 yuc: 7 y=6 6 7 6 ycwc: 7 7 6 7 6 ::: 5 4 ycc: where for example yuc: is a vector containing 1's and 0's. Let us have a look at two games of the UEFA Cup 1995/96:
9596 9495 9394 9293 9596 9293 9596

leg 20 Round1 21 Round1

team.a country.a Glenavon Nil BayernMuenchen Ger
9596

team.b country.b WerderBremen Ger LokomotivMoscow Rus

s.a.1 s.b.1 s.a.2 s.b.2 0 2 0 5 0 1 5 0

The 20th element of yuc: would be 0 because WerderBremen (i.e. team.b) won the game, and the 21th element of yuc: would be 1 because BayernMuenchen (i.e. team.a) won the game.
9596

By tting the model (3) we obtained the following estimations: Estimations Value Std. Error t value ^ -0.1305 0.1039 -1.2559 ^ 1.2610 0.1608 7.8420 It is a good t. This statement is validated by gure 7. One notices that this regression rejects the necessity of the intercept . Therefore (3) becomes: ^(S (i) ? S (j )) P (i; j ) = e ^ e (S (i) ? S (j ))
1+

(4)

18

Statistical Modelisation

Fitting of the logistic regression model
1.0

•
Mean of the response variable on each interval 0.8

With intercept Without intercept

• • •

0.4

0.6

• •

0.2

• •

0.0

-1

0

1

Mean of the seeding coefficient difference on each interval

Fitting of the logistic regression model. We noticed that the seeding coe cient di erences lied between -1.8 and 1.8, therefore we divided this interval in 8 subintervals. In each of this subintervals we calculated the mean of the response variable and the mean of the seeding coe cient di erences. One remarks that the curve without the intercept ts the observations best.
Figure 7

19

Statistical Modelisation

3.2.2 The Program
With the help of equation (4) it is now possible to calculate the probability of each team winning against any other team (i.e. P (i; j )). In introducing these probabilities into equation (1) we are able to know the probability of each team winning any chosen leg and so, using equation (2), the tournament. For this reason we have programmed a function titled prob.cte. An issue of this function is shown by the following table who shows the probabilities for the Cup Winners Cup 1995/96:
ParisSaintGermain MoldeFK Celtic DinamoBatumi Parma KSTeuta(Durres) HalmstadsBK LokomotivSofia RCDeportivoLaCoruna Apoel(Lefkosia) Trabzonspor ZalgirisVilnius RealZaragoza InterZTSBratislava ClubBruggeKV ShakhtarDonetsk SKRapid(Wien) PetrolulPloiesti SportingCP(Lisbon) MaccabiHaifa DinamoMoscow AraratErevan SKHradecKralove FCKobenhavn Feyenoord DAGLiepaya Everton KR(Reykjavik) BorussiaMonchengladbach SileksKratovo AEK(Athens) FCSion Round1 Round2 Quarter Semi Final 87.60 71.81 39.57 27.36 18.98 12.40 4.97 0.95 0.24 0.06 65.26 17.30 4.93 1.88 0.73 34.74 5.93 1.13 0.28 0.07 89.01 79.22 49.19 35.38 25.51 10.99 5.50 1.12 0.28 0.07 50.00 7.64 1.56 0.39 0.10 50.00 7.64 1.56 0.39 0.10 68.37 32.21 15.36 4.69 2.21 31.63 9.49 2.88 0.51 0.15 74.09 47.98 27.31 10.53 5.93 25.91 10.33 3.36 0.65 0.21 76.19 40.65 21.63 7.39 3.81 23.81 6.63 1.80 0.27 0.07 78.28 46.19 25.88 9.50 5.16 21.72 6.53 1.77 0.27 0.07 55.24 16.78 6.17 1.98 0.39 44.76 11.76 3.79 1.06 0.18 71.32 54.96 34.67 19.76 8.18 28.68 16.50 6.86 2.50 0.58 75.09 48.26 27.06 14.01 5.09 24.91 9.49 2.98 0.83 0.14 32.40 10.09 3.17 0.89 0.15 67.60 32.15 15.29 6.63 1.90 82.37 66.36 47.22 30.42 14.08 17.63 8.31 2.99 0.88 0.15 44.76 10.45 3.76 1.10 0.18 55.24 14.89 6.06 2.04 0.40 50.00 15.06 4.18 1.23 0.21 50.00 15.06 4.18 1.23 0.21 40.91 26.70 10.75 4.64 1.24 59.09 43.18 20.87 10.82 3.69

Let us take for example Paris Saint Germain (Fra): Their chance to win in round 1 against Molde FK (Nor) is 87.6%, to win in round 2 against Celtic (Sco) or Dinamo Batumi (Geo) their chance is 71.8%, and to win the nal, i.e. the tournament, their chance is about 19%. The real outcomes are shown by gure 8. 20

Statistical Modelisation

cwc.9596
ParisSaintGermain ParisSaintGermain MoldeFK ParisSaintGermain Celtic Celtic DinamoBatumi ParisSaintGermain Parma Parma KSTeuta(Durres) Parma HalmstadsBK HalmstadsBK LokomotivSofia RCDeportivoLaCoruna RCDeportivoLaCoruna Apoel(Lefkosia) RCDeportivoLaCoruna Trabzonspor Trabzonspor ZalgirisVilnius RCDeportivoLaCoruna RealZaragoza RealZaragoza InterZTSBratislava RealZaragoza ClubBruggeKV ClubBruggeKV ShakhtarDonetsk SKRapid(Wien) SKRapid(Wien) PetrolulPloiesti SKRapid(Wien) SportingCP(Lisbon) SportingCP(Lisbon) MaccabiHaifa SKRapid(Wien) DinamoMoscow DinamoMoscow AraratErevan DinamoMoscow SKHradecKralove SKHradecKralove FCKobenhavn Feyenoord Feyenoord DAGLiepaya Feyenoord Everton Everton KR(Reykjavik) Feyenoord BorussiaMonchengladbach BorussiaMonchengladbach SileksKratovo BorussiaMonchengladbach AEK(Athens) AEK(Athens) FCSion

ParisSaintGermain

ParisSaintGermain

SKRapid(Wien)

Figure 8 Tournament tree of the Cup Winners Cup 1995/96.

One remarks the surprising performance of SK Rapid (Wien) (Aut): Their chance to reach the quarter nal was 16.7%, but they even reached the semi nal where their chance was only 6.17%. Knowing this surprising fact the probabilities of the other teams remaining in the tournament will change. Therefore it is not only interesting to know the probability of each team winning in a chosen leg at the beginning of the tournament, but also predicting the probability of each team winning knowing the remaining teams. That is why we re ned the program prob.cte. For example in the Cup Winners Cup 1995/96 the following teams remain after the quarter nals : Paris Saint Germain (Fra), RC Deportivo La Coruna (Esp) , SK Rapid (Wien) (Aut) and Feyenoord (Ned). We are able to calculate the new probabilities knowing this fact. It is important to notice that these new probabilities are calculated without care of the outcomes of the previous rounds. 21

Statistical Modelisation

A more advanced method will be discussed in section 3.3. The result of the function is shown by the following table:
ParisSaintGermain RCDeportivoLaCoruna SKRapid(Wien) Feyenoord Semi 72.59 27.41 20.90 79.10 Final 47.47 11.80 4.07 36.66

We notice that this time the chance of Paris Saint Germain to win the tournament is about 50% - and they won it. And the chance of SK Rapid (Wien) (Aut) to defeat their opponent, Feyenoord (Ned), in the semi nal is about 21%. At the beginning of the tournament the chance of SK Rapid (Wien) (Aut) to win the semi nal was only about 2%. So this approach seems to be more realistic.

22

Statistical Modelisation

3.2.3 Number of Correct Predictions

Let us consider the following table: Team Probability of winning A X B 1-X where X = P (A; B ). If X > the probable winner will be A and if the real winner is also A we will mark a point, if X < (1 ? X > ) and the real winner is B we will mark a point as well and if X = 1 ? X = we will mark half a point (this semi point is justi ed by the fact that the expectation of a point is 0 + 1 = ).
1 2 1 2 1 2 1 2 1 2 1 2 1 2

The following example issues from the Cup Winners Cup 1992/93 and shows the probabilities of winning the quarter nal for the teams remaining in the tournament.
Parma SpartaPraha AtleticoMadrid Olympiakos(Pireus) RoyalAntwerp SteauaBucharest SpartakMoscow Feyenoord Quarter 43.47 56.53 74.09 25.91 31.26 68.74 60.36 39.64

As we know that the real winners were Parma, Atletico Madrid, Royal Antwerp and Spartak Moscow, it is possible to calculate the number of correct predictions. That would be 2 because Atletico Madrid and Spartak Moscow are probable winners as well. 23

Statistical Modelisation

In applying this procedure to the entire tournament we will get a list of points whose sum represents the number of correct predictions. For this reason we have programmed a function titled nb.pronos.ok. In using this function for various data sets we obtained the following table:
YEAR OK GAMES %OK -------------------------cwc.9293 21.5 31 69.35 cwc.9394 20 31 64.52 cwc.9495 20.5 31 66.13 cwc.9596 19 31 61.29 63 71.43 uc.9293 45 uc.9394 33.5 63 53.17 uc.9495 45.5 63 72.22 uc.9596 37.5 63 59.52

OK

stands for the number of correct predictions, GAMES represents the number of games per tournament and %OK the percentage of games predicted correctly.

24

Statistical Modelisation

3.3 Variation of Team Strength
This time we assume that the team strength (the seeding coe cients) do not remain constant throughout the tournament - they are depending on the leg.

3.3.1 Model Chosen
The principal idea is to calculate for each leg a new coe cient for each team. The initial coe cients that are used to predict the issues of the rst leg are the coe cients calculated in section 2.3.2. For the second round we have to calculate new coe cients. To do so we introduce two di erent methods (see sections 3.3.2 and 3.3.3). Once we have calculated these new coe cients we center and reduce them in order to obtain coe cients of equal mean and equal standard deviation as the initial ones.
Justi cation: Let x1 ; : : : ; xn be the initial coe cients of mean x and standard deviation sx and let y1; : : : ; yn be the new coe cients of mean y and standard deviation sy . We de ne the new coe cients as ! yt ? y ; t = 1; : : : ; n. yt = x + s x

sy Hence y = x and sy = sx. An example of this calculation is given by the following table:
Team A B C D Mean Standard Deviation

xt 1.4 1.2 1 0.8 1.1 0.258

yt 2.1 1.9 1.5 1.3 1.7 0.365

yt 1.382 1.241 0.958 0.817 1.1 0.258

Therefore we are allowed to use the estimations made in section 3.2.1. Thus our model is given by equation (3):

e ^(Sk (i) ? Sk (j )) Pk (i; j ) = ^ e (Sk (i) ? Sk (j ))
1+

(5)

where k = 1; : : : ; log n and ^ = 1:2610.
2

Remark on the e ect of the centering and the reduction of the coe cients: If a strong team wins against a weak team their coe cient will decrease, but if a weak team defeats a strong team their coe cient will increase. That seems to be logical.

25

Statistical Modelisation

3.3.2 Using Probable Opponents
To illustrate this rst method, let us have a look at gure 9.
1st A 2nd 3rd

B

C

D

Figure 9 Example of the upper part of the tournament tree.

The initial coe cients are used to calculate the probability of each team winning against their potential opponent in the rst round (for the rst round only one potential opponent is possible). The new coe cients for the second round who will be used to calculate the probabilities of winning in the second round are de ned in the following way, for example the new coe cient for a chosen team i would be: ! Cold;O i + Cnew;i = Cold;i 1 + C + (6) old;i
( )

where is a constant 0 and Cold;O i is the coe cient of the probable opponent O(i) of team i in the previous round (if i = A it would be B ). Later on (see section 3.3.4) we will choose in order to maximize the number of correct predictions. It is important to notice that Cnew;i is Sk (i) and Cold;i is Sk? (i), k = 1; : : : ; log n.
( ) 1 2

In realizing that some initial coe cients are equal to 0 (see section 2.3.2) we allow us to add 0.1 to all initial coe cients. Let us consider equation (6): If we will not do this the coe cients would remain 0 throughout the tournament. In adding 0.1 the expectation of the new initial coe cients increases and the variance remains the same. But as our model (5) is depending on a di erence of coe cients this will not change our estimations. In reusing equation (6) we are able to calculate the coe cients for the third round. The only problem consists in the knowledge of the probable opponent O(i): Let i be A, so for the third round there are two potential opponents for A: C and D. But we have calculated P (C; D) yet, so if this probability is > the probable opponent for A would be C , if not the probable opponent would be D (if this probability is equal to it does not matter which team we will take for further considerations because both teams have the same strength).
2 1 2 1 2

In generalizing this example we only have to keep in mind that if a team was not a 26

Statistical Modelisation

probable opponent for a previous round it will not be one for the next round. Therefore we have programmed a function titled prob.non.cte. An outcome of this function is shown by the following table where the Cup Winners Cup of the year 1993/94 has been considered.
Arsenal OB(Odense) StandardCL(Liege) CardiffCity Torino LillestromSK Aberdeen Valur(Reykjavik) ParisSaintGermain Apoel(Nicosia) UniversitateaCraiova HB(Torshavn) RealMadrid FCLugano FCTirolInnsbruck Ferencvarosi Parma DegerforsIF MaccabiHaifa TorpedoMoscow Ajax HajdukSplit Besiktas Kosice.1.FC SLBenfica GKSKatowice CSKA(Sofia) FCBalzers BayerLeverkusen FCBobyBrno Panathinaikos Shelbourne Round1 Round2 Quarter Semi Final 57.82 22.70 8.95 2.95 0.89 42.18 13.21 3.89 0.84 0.14 62.84 42.79 21.46 9.19 3.61 37.16 21.30 8.69 2.98 0.94 75.09 50.36 32.17 15.04 6.40 24.91 10.71 4.54 1.29 0.33 43.47 15.51 7.46 2.50 0.77 56.53 23.42 12.85 5.05 1.83 69.42 41.50 14.14 7.92 3.11 30.58 12.21 2.25 0.69 0.12 65.26 33.72 10.62 5.63 2.09 34.74 12.57 2.29 0.70 0.12 87.37 58.21 43.54 29.06 13.78 12.63 3.40 1.30 0.44 0.09 46.50 17.26 11.42 6.83 2.88 53.50 21.13 14.43 8.88 3.86 81.64 49.27 23.79 14.19 8.69 18.36 5.42 1.07 0.27 0.08 22.08 5.63 1.10 0.28 0.08 77.92 39.67 18.19 10.50 6.28 82.74 64.41 40.11 25.35 16.15 17.26 9.28 3.84 1.79 0.92 72.03 22.03 10.90 5.84 3.32 27.97 4.28 1.00 0.25 0.07 70.52 52.49 38.36 18.76 11.50 29.48 18.25 11.62 4.75 2.62 72.03 24.48 15.60 6.38 3.52 27.97 4.77 1.73 0.31 0.08 56.96 20.65 5.62 1.50 0.61 43.04 12.76 2.52 0.41 0.09 76.25 55.45 22.27 9.01 4.94 23.75 11.13 2.28 0.40 0.10

The real outcomes are shown by gure 10.

27

Statistical Modelisation

cwc.9394
Arsenal OB(Odense) StandardCL(Liege) CardiffCity Torino LillestromSK Aberdeen Valur(Reykjavik) ParisSaintGermain Apoel(Nicosia) UniversitateaCraiova HB(Torshavn) RealMadrid FCLugano FCTirolInnsbruck Ferencvarosi Parma DegerforsIF MaccabiHaifa TorpedoMoscow Ajax HajdukSplit Besiktas Kosice.1.FC SLBenfica GKSKatowice CSKA(Sofia) FCBalzers BayerLeverkusen FCBobyBrno Panathinaikos Shelbourne Arsenal Arsenal StandardCL(Liege) Arsenal Torino Torino Aberdeen Arsenal ParisSaintGermain ParisSaintGermain UniversitateaCraiova ParisSaintGermain RealMadrid RealMadrid FCTirolInnsbruck Arsenal Parma Parma MaccabiHaifa Parma Ajax Ajax Besiktas Parma SLBenfica SLBenfica CSKA(Sofia) SLBenfica BayerLeverkusen BayerLeverkusen Panathinaikos

Figure 10 Tournament tree of the Cup Winners Cup 1993/94.

28

Statistical Modelisation

3.3.3 Using All Potential Opponents
The second method takes all potential opponents into consideration to calculate the new coe cients. This method seems to be more natural than the rst one where we considered only the probable opponents. The new coe cient for a chosen team i is de ned as: X Cnew;i = Cold;i (1 + rj )P (j wins in the previous round) (7) j 2O

where is a constant 0, rj = Cold;j and O the set of all potential opponents for the Cold;i previous round for the team i. One remarks that this new coe cient is a sort of weighted sum of old coe cients.
+ +

To illustrate this method let us have a look at the example shown by gure (9). For the second round the new coe cient is exactly the same as the one calculated for the second round in the previous section because there is only one potential opponent for the rst round for each team. And for the third round the new coe cient of A would be X Cnew;A = Cold;A (1 + rj )P (j wins in the rst round) j 2fC;Dg

We have to notice that the procedure of calculating these new coe cients is the same as in the previous section. Hence we have reprogrammed the function prob.non.cte. An issue of this function is shown by the following table where the UEFA Cup of the year 1994/95 has been considered.
Parma Vitesse(Arnhem) AIK(Stockholm) SlaviaPraha AthleticBilbao AnorthosisofAmmochostos NewcastleUnited RoyalAntwerp OB(Odense) Linfield Kaiserslautern IA(Akranes) RealMadrid SportingCP(Lisbon) DinamoMoscow RFCSeraing BayerLeverkusen PSV(Eindhoven) KispestHonved FCTwente(Enschede) Round1 Round2 Round3 Quarter Semi Final 73.00 47.03 34.40 19.37 12.02 6.50 27.00 12.06 7.22 3.17 1.69 0.80 57.82 25.47 16.84 8.37 4.82 2.45 42.18 15.45 9.34 4.15 2.23 1.07 50.00 17.92 3.66 0.75 0.21 0.05 50.00 17.92 3.66 0.75 0.21 0.05 27.34 13.41 2.59 0.49 0.12 0.02 72.66 50.75 22.29 10.39 5.75 2.82 42.18 17.53 4.95 1.87 0.85 0.35 57.82 28.68 10.59 4.89 2.59 1.22 53.06 29.23 11.30 5.39 2.91 1.40 46.94 24.56 8.79 3.97 2.07 0.96 63.33 40.23 27.80 16.57 10.39 5.66 36.67 19.79 12.19 6.58 3.86 1.99 77.92 36.41 23.36 13.05 7.84 4.13 22.08 3.58 1.02 0.25 0.07 0.02 53.15 34.87 18.71 10.21 4.60 2.17 46.85 29.65 15.20 8.01 3.47 1.59 56.53 21.99 9.02 3.95 1.37 0.52 43.47 13.49 4.17 1.34 0.28 0.06

29

Statistical Modelisation
Round1 Round2 Round3 Quarter Semi Final 65.26 30.80 16.44 8.55 3.65 1.65 34.74 10.37 3.46 1.11 0.23 0.05 60.36 37.54 22.15 12.50 5.84 2.83 39.64 21.29 10.85 5.42 2.21 0.97 57.82 33.38 11.83 4.94 1.68 0.63 42.18 20.99 5.32 1.54 0.29 0.05 50.00 22.82 5.85 1.72 0.33 0.06 50.00 22.82 5.85 1.72 0.33 0.06 55.24 23.84 16.49 8.70 3.86 1.80 44.76 16.96 11.06 5.42 2.22 0.97 41.18 22.70 16.11 8.77 4.02 1.92 58.82 36.51 27.49 16.10 7.95 4.01 61.61 44.04 30.39 16.95 8.86 4.72 38.39 24.48 15.33 7.72 3.69 1.85 50.00 15.74 7.90 3.11 1.18 0.50 50.00 15.74 7.90 3.11 1.18 0.50 60.36 32.52 13.86 6.01 2.50 1.13 39.64 17.08 5.14 1.54 0.42 0.13 40.91 18.38 5.86 1.89 0.56 0.19 59.09 32.02 13.62 5.89 2.45 1.11 82.49 63.83 36.47 21.52 11.13 5.89 17.51 8.46 2.08 0.55 0.10 0.02 50.00 13.86 3.68 1.08 0.25 0.06 50.00 13.86 3.68 1.08 0.25 0.06 57.82 22.87 11.14 5.43 2.34 1.09 42.18 13.21 5.20 2.06 0.71 0.28 82.87 57.78 35.94 21.56 11.32 6.05 17.13 6.15 1.82 0.49 0.10 0.02 79.10 50.16 30.32 17.40 9.95 5.44 20.90 7.04 2.25 0.74 0.26 0.10 50.00 21.40 10.45 5.07 2.56 1.27 50.00 21.40 10.45 5.07 2.56 1.27 37.16 16.69 6.47 2.78 1.27 0.59 62.84 35.43 17.91 9.43 5.06 2.64 73.76 40.35 20.50 10.83 5.83 3.04 26.24 7.52 1.66 0.39 0.09 0.02 65.26 45.23 19.68 9.32 4.74 2.38 34.74 19.81 5.71 1.89 0.72 0.29 50.00 17.48 3.60 0.79 0.19 0.05 50.00 17.48 3.60 0.79 0.19 0.05 63.25 37.75 26.93 15.07 8.63 4.73 36.75 17.96 11.51 5.68 2.98 1.53 34.74 12.05 6.90 2.98 1.42 0.67 65.26 32.25 22.08 11.77 6.52 3.48

GKSKatowice Aris(Thessaloniki) GirondinsBordeaux LillestromSK FCNantes RotorVolgograd TekstilschikKamyshin BekescsabaiElore FCSion Apollon(Limassol) OlympiqueMarseille Olympiakos(Pireus) Juventus CSKA(Sofia) CSMaritimo FCAarau AdmiraWacker GornikZabrze ASCannes Fenerbahce EintrachtFrankfurt SCTOlimpija(Ljubljana) RapidBucharest RSCCharleroi Napoli SkontoRiga Boavista MyPa(Anjalankoski) BorussiaDortmund Motherwell SlovanBratislava FCKobenhavn RCDeportivoLaCoruna RosenborgBK FCTirolInnsbruck DinamoTblisi Lazio DinamoMinsk TrelleborgsFF BlackburnRovers Trabzonspor DinamoBucharest AstonVilla Internazionale

The real outcomes are shown by gure 11. 30

Statistical Modelisation

uc.9495
Parma Vitesse(Arnhem) AIK(Stockholm) SlaviaPraha AthleticBilbao AnorthosisofAmmochostos NewcastleUnited RoyalAntwerp OB(Odense) Linfield Kaiserslautern IA(Akranes) RealMadrid SportingCP(Lisbon) DinamoMoscow RFCSeraing BayerLeverkusen PSV(Eindhoven) KispestHonved FCTwente(Enschede) GKSKatowice Aris(Thessaloniki) GirondinsBordeaux LillestromSK FCNantes RotorVolgograd TekstilschikKamyshin BekescsabaiElore FCSion Apollon(Limassol) OlympiqueMarseille Olympiakos(Pireus) Juventus CSKA(Sofia) CSMaritimo FCAarau AdmiraWacker GornikZabrze ASCannes Fenerbahce EintrachtFrankfurt SCTOlimpija(Ljubljana) RapidBucharest RSCCharleroi Napoli SkontoRiga Boavista MyPa(Anjalankoski) BorussiaDortmund Motherwell SlovanBratislava FCKobenhavn RCDeportivoLaCoruna RosenborgBK FCTirolInnsbruck DinamoTblisi Lazio DinamoMinsk TrelleborgsFF BlackburnRovers Trabzonspor DinamoBucharest AstonVilla Internazionale Parma Parma AIK(Stockholm) Parma AthleticBilbao AthleticBilbao NewcastleUnited Parma OB(Odense) OB(Odense) Kaiserslautern OB(Odense) RealMadrid RealMadrid DinamoMoscow Parma BayerLeverkusen BayerLeverkusen KispestHonved BayerLeverkusen GKSKatowice GKSKatowice GirondinsBordeaux BayerLeverkusen FCNantes FCNantes TekstilschikKamyshin FCNantes FCSion FCSion OlympiqueMarseille Parma Juventus Juventus CSMaritimo Juventus AdmiraWacker AdmiraWacker ASCannes Juventus EintrachtFrankfurt EintrachtFrankfurt RapidBucharest EintrachtFrankfurt Napoli Napoli Boavista Juventus BorussiaDortmund BorussiaDortmund SlovanBratislava BorussiaDortmund RCDeportivoLaCoruna RCDeportivoLaCoruna FCTirolInnsbruck BorussiaDortmund Lazio Lazio TrelleborgsFF Lazio Trabzonspor Trabzonspor AstonVilla

Figure 11 Tournament tree of the UEFA Cup 1994/95.

31

Statistical Modelisation

3.3.4 Number of Correct Predictions
In regarding the two methods presented in the two preceding sections our goal in this section is to choose in order to maximize the number of correct predictions. The procedure of obtaining this number has been discussed in section 3.2.3. The calculation of the new coe cients for a certain round is based on the real outcomes of the previous round so there is only one potential opponent (the real one). The new coe cients will be calculated in using the following equation: ! Cold;R i + Cnew;i = Cold;i 1 + C +
( )

old;i

where R(i) is the real opponent of team i for the previous round. If a team is not in a particular round, i.e. it lost in one of the previous rounds, their coe cient has no reason to be changed. This remark is important because we will center and reduce the new coe cients in using all initial coe cients. With = 1 we have obtained the following table:
YEAR OK GAMES %OK -------------------------cwc.9293 21.5 31 69.35 cwc.9394 21 31 67.74 cwc.9495 20.5 31 66.13 cwc.9596 20 31 64.52 uc.9293 45 63 71.43 uc.9394 34 63 53.97 uc.9495 44 63 69.84 uc.9596 37 63 58.73

We have executed the function nb.pronos.ok for several values of , but how to know which is optimal? For this reason we have to de ne a criterion of comparison. A general outcome of the function looks like this: YEAR 1 2 ... 8 OK GAMES %OK o g p o g p ... ... ... o g p
1 2 1 2 1 2 8 8 8 8 =1

Let W be Pi wipi, the weighted sum of the percentages, P where wi are the weights ful lling Pi wi = 1 and wi / gi. Hence wi = gi=g where g = i gi. By construction of the table pi = oi=gi therefore W = (Pi oi)=g.
8 =1 8 =1 8 =1

32

Statistical Modelisation

So let us de ne our criterion of comparison:
Criterion:

1 is optimal if W = g P oi is maximal.
8

i=1

In calculating W for several values of we obtain the following table: 0 0.2 0.4 0.6 0.8 1 2 3 4 5 15 W 58.51 62.5 63.29 64.36 64.36 64.62 64.62 64.62 64.36 64.09 64.09 One remarks that in taking for example = 1 we obtain the biggest W , i.e. the highest weighted sum of games predicted correctly. Therefore let 1 be the optimal to choose. We also tried another model to calculate the new coe cients. This model is given by the following equation: ! Cold;R i Cnew;i = Cold;i 1 + C old;i
( )

We obtain the following table: 0 0.2 0.4 0.6 0.8 1 2 3 4 W 64.5 64.5 64.22 63.56 63.56 58.51 58.9 59.3 60.1 One remarks that this time W does not exceed 64:5% as before. That is the reason why we do not take into further considerations this model. In this section we chose in order to maximize the number of correct predictions. Our next goal will be to compare the methods presented in this paper (see section 4).

33

Statistical Modelisation

3.3.5 Evaluation of the Coe cients General Remark
The results of this section have been achieved in considering several years. So let us have a look for example at the UEFA Cup 1995/96 in using the method that considers all potential opponents for a team. The other method furnishes the same results. In gure 12 the evaluation of the seeding coe cients throughout the whole tournament for four chosen teams is shown. Milan was the best seeded team at the beginning of the tournament, Olympiakos and Spartak Vladikavkaz were medium seeded teams and FC Lugano was among the worst seeded teams.
Milan
1.5

Olympiakos(Pireus)
1.5 1.0

• 1

• 1

1.0

• 4

• 3

Coeff

Coeff

• 4

• 4

0.5

0.0

R1

R2

R3 Round

Q

S

F

0.0

0.5

• 16

• 18

• 16

• 16

• 16

• 16

R1

R2

R3 Round

Q

S

F

SpartakVladikavkaz
1.5 1.5

FCLugano

1.0

Coeff

0.5

Coeff

0.0

0.0

• 34

• 32

• 34

• 34

• 34

• 34

0.5

1.0

• 43 R1

• 54 R2

• 55 R3

• 55 Q

• 57 S

• 57 F

R1

R2

R3 Round

Q

S

F

Round

Figure 12 Evaluation of the seeding coe

cients for four chosen teams during the UEFA Cup 1995/96. The number represents the seeding position of the team.

In general we noticed that the best teams, as Milan, will see their coe cient decrease, but they will still remain among the strongest teams. The medium teams, as Olympiakos and Spartak Vladikavkaz, will see their coe cients increase, and the worst teams will stay among the worst teams. The reason of this fact is that we centered and reduced the coe cients otherwise the better teams would always get higher coe cients and so become unbeatable. In the appendix you will nd the evaluation of the coe cients for all teams that have participated in the UEFA Cup 1995/96, in considering both methods. 34

Statistical Modelisation

Particular Examples
The two particular examples shown by gure 13 illustrate two things: First of all we remark that the two methods used are di erent, and in a second time we notice that in using the method considering only the probable opponent we have some strange evaluations: The coe cient of Milan decreases too much in the quarter nal, they are losing six seed positions in falling from position 4 to position 10. This fact does not come up with the other method. The same can be said for Girondins Bordeaux in considering the coe cient for round 2. In recapitulating, the method using all potential opponents seems to be more natural.
GirondinsBordeaux
1.4

• 2 5 • 5

2 • 1

Coeff

1.2

8 •

2 • • 1

2 • • 2

1.0

2 • • 2

0.8

Using all potential opponents Using probable opponent R1 R2 R3 Round Q S F

Milan
• 1
1.6 Coeff

1
1.2

• •

1 • • 4

Using all potential opponents Using probable opponent 3 • • 6 S 4 • • 8 F 4

• 10 •

4

0.8

R1

R2

R3 Round

Q

Figure 13

Evaluation of the seeding coe cients for Milan and Girondins Bordeaux during the UEFA Cup 1995/96.

35

Comparison of the Methods

4 Comparison of the Methods

In this section we want to compare the methods proposed in the previous sections: strength. M2 The second one assumes the variation of the team strength in using only the probable opponents for a team. M3 The third one assumes the variation of the team strength in using all potential opponents for a team. To compare the three methods M1, M2 and M3 we will take as example the quarter nal of the UEFA Cup 1995/96. The table 1 shows the eight quarter nalists, the probability of each quarter nalist to reach the quarter nal in using one of the three methods and the rank of this probability. This rank is de ned as the order of the sorted probabilities of the team and of all his potential opponents for the previous rounds. Here is an example how this rank is calculated: Let us have a look at the following table who shows the probabilities to reach the quarter nal for Bayern Muenchen and all his potential opponents for the previous rounds:
BayernMuenchen LokomotivMoscow RaithRovers IA(Akranes) SLBenfica LierseSK RodaJC(Kerkrade) SCTOlimpija(Ljubljana) Round3 27.87 2.10 2.80 9.86 46.03 1.40 2.58 7.35

M1 The rst method calculates the probabilities in using the constancy of the team

The rank of Bayern Muenchen would be 2 because SL Ben ca had bigger chance to win the third round (46.03% > 27.87%), i.e. to reach the quarter nal. 36

Comparison of the Methods

Team Bayern Muenchen Nottingham Forest FC Barcelona PSV (Eindhoven) Girondins Bordeaux Milan Slavia Praha Roma

Prob. Rank Prob. Rank Prob. Rank 23.99 2 26.33 2 27.87 2 3.62 3 3.98 4 4.43 3 43.70 1 39.15 1 38.13 1 13.80 4 13.20 4 14.70 4 29.18 1 25.04 2 29.21 1 67.68 1 60.62 1 56.71 1 22.88 2 25.06 3 25.17 2 10.19 3 10.16 4 11.40 3

M1

M2

M3

Table 1 This table shows the eight quarter nalists, the probability of each one to reach the quarter nal in using one of the three methods and the rank of this probability (UEFA Cup 1995/96).

For a balanced tournament all teams would have 12.5% chance to reach the quarter nal because there are in all 7 potential opponents for each team in the previous rounds and in considering the team as well we have 8 teams sharing 100%. For the teams with probability smaller than 12.5% we remark that M3 gives more chance to reach the quarter nal than M2. One notices as well that Nottingham Forest, Girondins Bordeaux, Slavia Praha and Roma had more chance to reach the quarter nal in using M3 than in using M2 because their ranks are smaller in using M3. Hence M3 seems to be better than M2. On the other hand M1 and M3 deliver equal ranks. Moreover, in comparing the number of correct predictions made in using one of the three methods we obtained the following table: Year cwc.9293 cwc.9394 cwc.9495 cwc.9596 uc.9293 uc.9394 uc.9495 uc.9596 % games predicted correctly M1 M2 or M3 69.35 69.35 64.52 67.74 66.13 66.13 61.29 64.52 71.43 71.43 53.17 53.97 72.22 69.84 59.52 58.73

This table is a result from sections 3.2.3 and 3.3.4 and from the fact that for M2 and M3 we have the same number of correct predictions. Hence the feeling we had in the previous sections, i.e. that M3 is the best method proposed, does not seem to be con rmed. M3 does not predict more games correctly. Let us remind that for M1 we obtained a weighted sum of games predicted correctly equal to 64:49% (see section 3.2.3) and for M3 we obtained 64:62% (see section 3.3.4). At the end of section 3.3.4 we even tried another method to calculate the new coe cients where we obtained 64.5%. 37

Comparison of the Methods

Therefore we are able to conclude:

M1 which assumes the constancy of the team strength is the most appropriated method among the three. This conclusion is based on the following points: 1. In assuming variation of the team strength we do not obtain better results. This statement has been veri ed in using 376 game outcomes, although the model was tted on 442 game outcomes. 2. The practical application of M1 is easier, faster in time and does not involve the search of an optimal .

38

Conclusion

5 Conclusion
In this paper, the logistic regression model (3) using seed positions (conceived through a seeding coe cient) was applied to European soccer Cups tournament data in order to predict the probability of winning the tournament for each one of the participating teams, and the predicted probabilities of each team reaching a certain leg such as the quarter nal. We proposed three di erent methods to do this: The rst one assumed the constancy of the team strength and the two others assumed the variation of the team strength. In the previous section we compared the three methods and the conclusion was the following one: The best method is the one who assumes the constancy of the team strength. For every soccer fan this seems to be rather strange, but we obtained this fact in considering 376 game outcomes (!) of the European Cups from 1992 to 1996. So let us recapitulate this method which can be divided into two major parts: 1. Calculation of the weighted coe cients: One calculates a weighted mean of the ratio points achieved over games played for each of the three past years (see section 2.3.2). 2. Calculation of the probability of each team reaching a certain leg to obtain nally the probability of winning the tournament in using equations (1) and (2) (see section 2.4.2) with help of the logistic regression model (3). Besides we remarked in section 3.2.3 that in using this method we had in average 64.49% of the games predicted correctly. The attentive reader has certainly detected that we never took into consideration the scores. To calculate the new coe cient for a team we did not consider the goals for and against the teams in the tournament. This would surely be an interesting starting point to develop a new method for calculating new coe cients in hope to get a better method than the one who assumes the constancy of the team strength.

39

Appendix, The Bradley-Terry Model

A Appendix
A.1 The Bradley-Terry Model
The Bradley-Terry model (Bradley and Terry 1952) is a model representing the results of experiments in which responses are pairwise rankings of treatments (so-called paired comparison experiments). Let us consider t treatments T ; : : : ; Tt in an experiment involving paired comparisons. We consider that these treatments have true ratings (or preferences) ; : : : ; t . These parameters represent relative selection probabilities for the treatments, subject to the constraints i 0 (i = 1; : : : ; t) and Pti i = 1. Bradley and Terry (1952) de ned the probability that treatment Ti is preferred over the treatment Tj in a single comparison in the following way:
1 1 =1

P (Ti where i = j and i; j = 1; : : : ; t. 6

Tj ) =

i+ j

i

(8)

But what is the relationship between this model and our logistic model?

Let be i = e Sk i and j = e Sk j the ratings (or preferences) representing relative selection for the teams. Hence our logistic model (3) becomes:
( ) ( )

Pk (i; j ) =

e i j? 1 + e i j? = e i : j +e i
1

1

(9)

In section 3.2.1 we noticed that the regression rejects the necessity of the intercept , hence equation (9) becomes:

Pk (i; j ) =

j+ i
0

i

Therefore we can conclude that in rejecting the necessity of the intercept we obtain exactly the Bradley-Terry model (8) for a single comparison. The reject of H : = 0 is also explicable by the following fact: represents in a way the home advantage. But as for every round of the tournament two games are played because each team plays once at home and once on the road, hence the home advantage can be omitted.

40

Appendix, Tournament Trees

A.2 Tournament Trees
A.2.1 Cup Winners Cup cwc.8990 Sampdoria Sampdoria SKBrann Sampdoria BorussiaDortmund BorussiaDortmund Besiktas Sampdoria GrasshopperClubZuerich GrasshopperClubZuerich SlovanBratislava GrasshopperClubZuerich TorpedoMoscow TorpedoMoscow CorkCity ASMonaco ASMonaco CFOsBelenenses ASMonaco BerlinerFCDynamo BerlinerFCDynamo Valur(Reykjavik) ASMonaco RealValladolid RealValladolid HamrunSpartans RealValladolid DjurgardensIF DjurgardensIF USLuxembourg RSCAnderlecht RSCAnderlecht BallymenaUnited RSCAnderlecht FCBarcelona FCBarcelona LegiaWarsaw RSCAnderlecht AdmiraWacker AdmiraWacker AEL(Limassol) AdmiraWacker Ferencvarosi Ferencvarosi ValkeakoskenHaka DinamoBucharest DinamoBucharest KSTirane DinamoBucharest Panathinaikos Panathinaikos SwanseaCity DinamoBucharest FKPartizan(Belgrade) FKPartizan(Belgrade) Celtic FKPartizan(Belgrade) FCGroningen FCGroningen IkastFS

Sampdoria

Sampdoria

RSCAnderlecht

Figure 14 The Cup Winners Cup 1989/90.

41

Appendix, Tournament Trees

cwc.9091
ManchesterUnited ManchesterUnited PecsiMunkasSC Wrexham Wrexham LyngbyBK MontpellierHSC MontpellierHSC PSV(Eindhoven) SteauaBucharest SteauaBucharest Glentoran LegiaWarsaw LegiaWarsaw SwiftHesperange Aberdeen Aberdeen NEASalamisofFamagusta Sampdoria Sampdoria Kaiserslautern Olympiakos(Pireus) Olympiakos(Pireus) KSFlamurtari(Vlore) FCBarcelona FCBarcelona Trabzonspor Fram(Reykjavik) Fram(Reykjavik) DjurgardensIF DinamoKiev DinamoKiev KuPS(Kuopio) DuklaPrague DuklaPrague SliemaWanderers Juventus Juventus FCSliven FKAustria(Wien) FKAustria(Wien) PSVSchwerin RFCLiegois RFCLiegois VikingFK(Stavanger) EstrelaAmadora EstrelaAmadora NeuchatelXamax

ManchesterUnited ManchesterUnited MontpellierHSC ManchesterUnited LegiaWarsaw LegiaWarsaw Sampdoria ManchesterUnited FCBarcelona FCBarcelona DinamoKiev FCBarcelona Juventus Juventus RFCLiegois

Figure 15 The Cup Winners Cup 1990/91.

42

Appendix, Tournament Trees

cwc.9192
WerderBremen FCBacau Ferencvarosi FCLevski(Sofia) Galatasaray EisenhstadterFCStahl BanikOstrava OB(Odense) ClubBruggeKV Omonia(Nicosia) GKSKatowice Motherwell AtleticoMadrid FyllingenIL ManchesterUnited Athinaikos ASMonaco SwanseaCity IFKNorrkoping LaJeunesseDEsch Roma CSKAMoscow Ilves(Tampere) Glenavon Feyenoord KSPartizani(Tirana) FCSion Valur(Reykjavik) TottenhamHotspur HajdukSplit FCPorto Valletta WerderBremen WerderBremen Ferencvarosi WerderBremen Galatasaray Galatasaray BanikOstrava WerderBremen ClubBruggeKV ClubBruggeKV GKSKatowice ClubBruggeKV AtleticoMadrid AtleticoMadrid ManchesterUnited WerderBremen ASMonaco ASMonaco IFKNorrkoping ASMonaco Roma Roma Ilves(Tampere) ASMonaco Feyenoord Feyenoord FCSion Feyenoord TottenhamHotspur TottenhamHotspur FCPorto

Figure 16 The Cup Winners Cup 1991/92.

43

Appendix, Tournament Trees

cwc.9293
Parma UjpestiTE Boavista Valur(Reykjavik) SpartaPraha Airdrieonians WerderBremen Hannover96 AtleticoMadrid MariborBranik Trabzonspor TPS(Turku) Olympiakos(Pireus) ChernomoretsOdessa ASMonaco MiedzLegnica RoyalAntwerp Glenavon AdmiraWacker CardiffCity SteauaBucharest Bohemians AGF(Arhus) AIK(Stockholm) SpartakMoscow AvenirBeggen Liverpool Apollon(Limassol) Feyenoord HapoelPetachTikva FCLucerne FCLevski(Sofia) Parma Parma Boavista Parma SpartaPraha SpartaPraha WerderBremen Parma AtleticoMadrid AtleticoMadrid Trabzonspor AtleticoMadrid Olympiakos(Pireus) Olympiakos(Pireus) ASMonaco Parma RoyalAntwerp RoyalAntwerp AdmiraWacker RoyalAntwerp SteauaBucharest SteauaBucharest AGF(Arhus) RoyalAntwerp SpartakMoscow SpartakMoscow Liverpool SpartakMoscow Feyenoord Feyenoord FCLucerne

Figure 17 The Cup Winners Cup 1992/93.

44

Appendix, Tournament Trees

cwc.9394
Arsenal OB(Odense) StandardCL(Liege) CardiffCity Torino LillestromSK Aberdeen Valur(Reykjavik) ParisSaintGermain Apoel(Nicosia) UniversitateaCraiova HB(Torshavn) RealMadrid FCLugano FCTirolInnsbruck Ferencvarosi Parma DegerforsIF MaccabiHaifa TorpedoMoscow Ajax HajdukSplit Besiktas Kosice.1.FC SLBenfica GKSKatowice CSKA(Sofia) FCBalzers BayerLeverkusen FCBobyBrno Panathinaikos Shelbourne Arsenal Arsenal StandardCL(Liege) Arsenal Torino Torino Aberdeen Arsenal ParisSaintGermain ParisSaintGermain UniversitateaCraiova ParisSaintGermain RealMadrid RealMadrid FCTirolInnsbruck Arsenal Parma Parma MaccabiHaifa Parma Ajax Ajax Besiktas Parma SLBenfica SLBenfica CSKA(Sofia) SLBenfica BayerLeverkusen BayerLeverkusen Panathinaikos

Figure 18 The Cup Winners Cup 1993/94.

45

Appendix, Tournament Trees

cwc.9495
RealZaragoza GloriaBistrita TatranPresov DundeeUnited Feyenoord ZalgirisVilnius WerderBremen MaccabiTelAviv Chelsea ViktoriaZizkov FKAustria(Wien) MariborBranik ClubBruggeKV SligoRovers Panathinaikos FCPirin(Blagoevgrad) Arsenal Omonia(Lefkosia) BrondbyIF KSTirane AJAuxerre NKCroatia(Zagreb) Besiktas HJK(Helsinki) Sampdoria FKBodoGlimt RealZaragoza RealZaragoza TatranPresov RealZaragoza Feyenoord Feyenoord WerderBremen RealZaragoza Chelsea Chelsea FKAustria(Wien) Chelsea ClubBruggeKV ClubBruggeKV Panathinaikos RealZaragoza Arsenal Arsenal BrondbyIF Arsenal AJAuxerre AJAuxerre Besiktas Arsenal Sampdoria Sampdoria Sampdoria FCPorto Ferencvarosi

GrasshopperClubZuerich GrasshopperClubZuerich ChernomoretsOdessa FCPorto LKSLodz Ferencvarosi CSKAMoscow FCPorto

Figure 19 The Cup Winners Cup 1994/95.

46

Appendix, Tournament Trees

cwc.9596
ParisSaintGermain ParisSaintGermain MoldeFK ParisSaintGermain Celtic Celtic DinamoBatumi ParisSaintGermain Parma Parma KSTeuta(Durres) Parma HalmstadsBK HalmstadsBK LokomotivSofia RCDeportivoLaCoruna RCDeportivoLaCoruna Apoel(Lefkosia) RCDeportivoLaCoruna Trabzonspor Trabzonspor ZalgirisVilnius RCDeportivoLaCoruna RealZaragoza RealZaragoza InterZTSBratislava RealZaragoza ClubBruggeKV ClubBruggeKV ShakhtarDonetsk SKRapid(Wien) SKRapid(Wien) PetrolulPloiesti SKRapid(Wien) SportingCP(Lisbon) SportingCP(Lisbon) MaccabiHaifa SKRapid(Wien) DinamoMoscow DinamoMoscow AraratErevan DinamoMoscow SKHradecKralove SKHradecKralove FCKobenhavn Feyenoord Feyenoord DAGLiepaya Feyenoord Everton Everton KR(Reykjavik) Feyenoord BorussiaMonchengladbach BorussiaMonchengladbach SileksKratovo BorussiaMonchengladbach AEK(Athens) AEK(Athens) FCSion

ParisSaintGermain

ParisSaintGermain

SKRapid(Wien)

Figure 20 The Cup Winners Cup 1995/96.

47

Appendix, Tournament Trees

A.2.2 UEFA Cup uc.8990 Juventus GornikZabrze ParisSaintGermain FCKuusysi(Lahti) FCKarlMarxStadt Boavista FCSion Iraklis(Salonica) HamburgerSV OrgryteIS RealZaragoza Apollon(Limassol) FCPorto FlacaraMoreni Valencia VictoriaBucharest Koeln TJPlastika(Nitra) SpartakMoscow Atalanta RedStar(Belgrade) Galatasaray ZalgirisVilnius IFKGothenburg RoyalAntwerp FCLevski(Sofia) DundeeUnited Glentoran VfBStuttgart Feyenoord ZenitLeningrad NaestvedIF ACFiorentina AtleticoMadrid FCSochauxMontbeliard ASLaJeunesseDEsch DinamoKiev MTK(Budapest) BanikOstrava FCHansaRostock AJAuxerre KSApolonia(Fier) RoPS(Rovaniemi) GKSKatowice Olympiakos(Pireus) FKRad(Belgrade) FirstViennaFC Valletta WerderBremen LillestromSK FKAustria(Wien) Ajax Napoli SportingCP(Lisbon) FCWettingen Dundalk RFCLiegois IA(Akranes) Hibernian VideotonWalthamSC SKRapid(Wien) Aberdeen ClubBruggeKV FCTwente(Enschede) Juventus Juventus ParisSaintGermain Juventus FCKarlMarxStadt FCKarlMarxStadt FCSion Juventus HamburgerSV HamburgerSV RealZaragoza HamburgerSV FCPorto FCPorto Valencia Juventus Koeln Koeln SpartakMoscow Koeln RedStar(Belgrade) RedStar(Belgrade) ZalgirisVilnius Koeln RoyalAntwerp RoyalAntwerp DundeeUnited RoyalAntwerp VfBStuttgart VfBStuttgart ZenitLeningrad Juventus ACFiorentina ACFiorentina FCSochauxMontbeliard ACFiorentina DinamoKiev DinamoKiev BanikOstrava ACFiorentina AJAuxerre AJAuxerre RoPS(Rovaniemi) AJAuxerre Olympiakos(Pireus) Olympiakos(Pireus) FirstViennaFC ACFiorentina WerderBremen WerderBremen FKAustria(Wien) WerderBremen Napoli Napoli FCWettingen WerderBremen RFCLiegois RFCLiegois Hibernian RFCLiegois SKRapid(Wien) SKRapid(Wien) ClubBruggeKV

Figure 21 The UEFA Cup 1989/90.

48

Appendix, Tournament Trees

uc.9091
Internazionale SKRapid(Wien) AstonVilla BanikOstrava FKPartizan(Belgrade) Hibernian RealSociedad LausanneSports Atalanta NKDinamo(Zagreb) Fenerbahce VitoriaSC(Guimaraes) Koeln IFKNorrkoping InterZTSBratislava AvenirBeggen SportingCP(Lisbon) KVMechelen PolitehnicaTimisoara AtleticoMadrid Vitesse(Arnhem) DerryCity DundeeUnited FH(Hafnarfjardar) Bologna ZaglebieLubin HeartOfMidlothian DneprDnepropetrovsk AdmiraWacker VejleBK FCLucerne MTK(Budapest) Roma SLBenfica Valencia Iraklis(Salonica) GirondinsBordeaux Glenavon Magdeburg RoPS(Rovaniemi) RSCAnderlecht PetrolulPloiesti Omonia(Nicosia) FCSlavia(Sofia) BorussiaDortmund ChemnitzerFC UniversitateaCraiova KSPartizani(Tirana) BrondbyIF EintrachtFrankfurt Ferencvarosi RoyalAntwerp BayerLeverkusen FCTwente(Enschede) GKSKatowice TPS(Turku) TorpedoMoscow GAIS(Gothenburg) Sevilla PAOK(Salonica) ASMonaco RodaJC(Kerkrade) ChernomoretsOdessa RosenborgBK Internazionale Internazionale AstonVilla Internazionale FKPartizan(Belgrade) FKPartizan(Belgrade) RealSociedad Internazionale Atalanta Atalanta Fenerbahce Atalanta Koeln Koeln InterZTSBratislava Internazionale SportingCP(Lisbon) SportingCP(Lisbon) PolitehnicaTimisoara SportingCP(Lisbon) Vitesse(Arnhem) Vitesse(Arnhem) DundeeUnited SportingCP(Lisbon) Bologna Bologna HeartOfMidlothian Bologna AdmiraWacker AdmiraWacker FCLucerne Internazionale Roma Roma Valencia Roma GirondinsBordeaux GirondinsBordeaux Magdeburg Roma RSCAnderlecht RSCAnderlecht Omonia(Nicosia) RSCAnderlecht BorussiaDortmund BorussiaDortmund UniversitateaCraiova Roma BrondbyIF BrondbyIF Ferencvarosi BrondbyIF BayerLeverkusen BayerLeverkusen GKSKatowice BrondbyIF TorpedoMoscow TorpedoMoscow Sevilla TorpedoMoscow ASMonaco ASMonaco ChernomoretsOdessa

Figure 22 The UEFA Cup 1990/91.

49

Appendix, Tournament Trees

uc.9192
Ajax OrebroSK FCRotWeissErfurt FCGroningen CAOsasuna FCSlavia(Sofia) VfBStuttgart PecsiMunkasSC KAAGhent LausanneSports EintrachtFrankfurt CASporaLuxembourg DinamoMoscow VaciIzzoMTE ASCannes SCSalgueiros Genoa RealOviedo DinamoBucharest SportingCP(Lisbon) SteauaBucharest AnorthosisofFamagusta SportingGijon FKPartizan(Belgrade) Liverpool FCKuusysi(Lahti) AJAuxerre IkastFS FCTirolInnsbruck TromsoIL PAOK(Salonica) KVMechelen Torino KR(Reykjavik) Boavista Internazionale AEK(Athens) SKVllaznia(Shkoder) SpartakMoscow MP(Mikkeli) B1903(Copenhagen) Aberdeen BayernMunich CorkCity Trabzonspor HASKGradjanski OlympiqueLyonnais OstersIF RealMadrid SlovanBratislava FCUtrecht SKSturmGraz NeuchatelXamax Floriana Celtic KFCGerminal(Ekeren) SKOlomoucSigma Bangor TorpedoMoscow HallescherFC HamburgerSV GornikZabrze CSKA(Sofia) Parma Ajax Ajax FCRotWeissErfurt Ajax CAOsasuna CAOsasuna VfBStuttgart Ajax KAAGhent KAAGhent EintrachtFrankfurt KAAGhent DinamoMoscow DinamoMoscow ASCannes Ajax Genoa Genoa DinamoBucharest Genoa SteauaBucharest SteauaBucharest SportingGijon Genoa Liverpool Liverpool AJAuxerre Liverpool FCTirolInnsbruck FCTirolInnsbruck PAOK(Salonica) Ajax Torino Torino Boavista Torino AEK(Athens) AEK(Athens) SpartakMoscow Torino B1903(Copenhagen) B1903(Copenhagen) BayernMunich B1903(Copenhagen) Trabzonspor Trabzonspor OlympiqueLyonnais Torino RealMadrid RealMadrid FCUtrecht RealMadrid NeuchatelXamax NeuchatelXamax Celtic RealMadrid SKOlomoucSigma SKOlomoucSigma TorpedoMoscow SKOlomoucSigma HamburgerSV HamburgerSV CSKA(Sofia)

Figure 23 The UEFA Cup 1991/92.

50

Appendix, Tournament Trees

uc.9293
Juventus AnorthosisofAmmochostos Panathinaikos ElectroputereCraiova SKOlomoucSigma UniversitateaCraiova Fenerbahce FCBotev(Plovdiv) SLBenfica BelvedurIzola VacFCSamsung FCGroningen DinamoMoscow RosenborgBK Torino IFKNorrkoping ParisSaintGermain PAOK(Thessaloniki) Napoli Valencia RSCAnderlecht Hibernian DinamoKiev SKRapid(Wien) RealMadrid PolitehnicaTimisoara TorpedoMoscow ManchesterUnited Vitesse(Arnhem) DerryCity KVMechelen OrebroSK BorussiaDortmund Floriana Celtic Koeln RealZaragoza SMCaen BKFrem(Kobenhavn) NeuchatelXamax Roma FCWackerInnsbruck GrasshopperClubZuerich SportingCP(Lisbon) Galatasaray GKSKatowice EintrachtFrankfurt WidzewLodz AJAuxerre FCLokomotiv(Plovdiv) FCKobenhavn MP(Mikkeli) StandardCL(Liege) Portadown HeartOfMidlothian SlaviaPraha Ajax SVCasinoSalzburg VitoriaSC(Guimaraes) RealSociedad Kaiserslautern Fram(Reykjavik) SheffieldWednesday CASporaLuxembourg Juventus Juventus Panathinaikos Juventus SKOlomoucSigma SKOlomoucSigma Fenerbahce Juventus SLBenfica SLBenfica VacFCSamsung SLBenfica DinamoMoscow DinamoMoscow Torino Juventus ParisSaintGermain ParisSaintGermain Napoli ParisSaintGermain RSCAnderlecht RSCAnderlecht DinamoKiev ParisSaintGermain RealMadrid RealMadrid TorpedoMoscow RealMadrid Vitesse(Arnhem) Vitesse(Arnhem) KVMechelen Juventus BorussiaDortmund BorussiaDortmund Celtic BorussiaDortmund RealZaragoza RealZaragoza BKFrem(Kobenhavn) BorussiaDortmund Roma Roma GrasshopperClubZuerich Roma Galatasaray Galatasaray EintrachtFrankfurt BorussiaDortmund AJAuxerre AJAuxerre FCKobenhavn AJAuxerre StandardCL(Liege) StandardCL(Liege) HeartOfMidlothian AJAuxerre Ajax Ajax VitoriaSC(Guimaraes) Ajax Kaiserslautern Kaiserslautern SheffieldWednesday

Figure 24 The UEFA Cup 1992/93.

51

Appendix, Tournament Trees

uc.9394
Internazionale RapidBucharest Apollon(Limassol) VacFCSamsung NorwichCity Vitesse(Arnhem) BayernMuenchen FCTwente(Enschede) BorussiaDortmund SpartakVladikavkaz MariborBranik GloriaBistrita BrondbyIF DundeeUnited FCKuusysi(Lahti) KSVWaregem Cagliari DinamoBucharest Trabzonspor Valletta KVMechelen IFKNorrkoping MTK(Budapest) KR(Reykjavik) Juventus LokomotivMoscow KongsvingerIL OstersIF CDTenerife AJAuxerre Olympiakos(Pireus) FCBotev(Plovdiv) SVCasinoSalzburg DACDunajskaStreda RoyalAntwerp CSMaritimo SportingCP(Lisbon) Kocaelispor Celtic BSCYoungBoys EintrachtFrankfurt DinamoMoscow DneprDnepropetrovsk AdmiraWacker RCDeportivoLaCoruna AaB(Aalborg) AstonVilla SlovanBratislava KarlsruherSC PSV(Eindhoven) Valencia FCNantes GirondinsBordeaux Bohemians ServetteFCGeneva Crusaders Boavista USLuxembourg Lazio FCLokomotiv(Plovdiv) OFI(Crete) SlaviaPraha AtleticoMadrid HeartOfMidlothian Internazionale Internazionale Apollon(Limassol) Internazionale NorwichCity NorwichCity BayernMuenchen Internazionale BorussiaDortmund BorussiaDortmund MariborBranik BorussiaDortmund BrondbyIF BrondbyIF FCKuusysi(Lahti) Internazionale Cagliari Cagliari Trabzonspor Cagliari KVMechelen KVMechelen MTK(Budapest) Cagliari Juventus Juventus KongsvingerIL Juventus CDTenerife CDTenerife Olympiakos(Pireus) Internazionale SVCasinoSalzburg SVCasinoSalzburg RoyalAntwerp SVCasinoSalzburg SportingCP(Lisbon) SportingCP(Lisbon) Celtic SVCasinoSalzburg EintrachtFrankfurt EintrachtFrankfurt DneprDnepropetrovsk EintrachtFrankfurt RCDeportivoLaCoruna RCDeportivoLaCoruna AstonVilla SVCasinoSalzburg KarlsruherSC KarlsruherSC Valencia KarlsruherSC GirondinsBordeaux GirondinsBordeaux ServetteFCGeneva KarlsruherSC Boavista Boavista Lazio Boavista OFI(Crete) OFI(Crete) AtleticoMadrid

Figure 25 The UEFA Cup 1993/94.

52

Appendix, Tournament Trees

uc.9495
Parma Vitesse(Arnhem) AIK(Stockholm) SlaviaPraha AthleticBilbao AnorthosisofAmmochostos NewcastleUnited RoyalAntwerp OB(Odense) Linfield Kaiserslautern IA(Akranes) RealMadrid SportingCP(Lisbon) DinamoMoscow RFCSeraing BayerLeverkusen PSV(Eindhoven) KispestHonved FCTwente(Enschede) GKSKatowice Aris(Thessaloniki) GirondinsBordeaux LillestromSK FCNantes RotorVolgograd TekstilschikKamyshin BekescsabaiElore FCSion Apollon(Limassol) OlympiqueMarseille Olympiakos(Pireus) Juventus CSKA(Sofia) CSMaritimo FCAarau AdmiraWacker GornikZabrze ASCannes Fenerbahce EintrachtFrankfurt SCTOlimpija(Ljubljana) RapidBucharest RSCCharleroi Napoli SkontoRiga Boavista MyPa(Anjalankoski) BorussiaDortmund Motherwell SlovanBratislava FCKobenhavn RCDeportivoLaCoruna RosenborgBK FCTirolInnsbruck DinamoTblisi Lazio DinamoMinsk TrelleborgsFF BlackburnRovers Trabzonspor DinamoBucharest AstonVilla Internazionale Parma Parma AIK(Stockholm) Parma AthleticBilbao AthleticBilbao NewcastleUnited Parma OB(Odense) OB(Odense) Kaiserslautern OB(Odense) RealMadrid RealMadrid DinamoMoscow Parma BayerLeverkusen BayerLeverkusen KispestHonved BayerLeverkusen GKSKatowice GKSKatowice GirondinsBordeaux BayerLeverkusen FCNantes FCNantes TekstilschikKamyshin FCNantes FCSion FCSion OlympiqueMarseille Parma Juventus Juventus CSMaritimo Juventus AdmiraWacker AdmiraWacker ASCannes Juventus EintrachtFrankfurt EintrachtFrankfurt RapidBucharest EintrachtFrankfurt Napoli Napoli Boavista Juventus BorussiaDortmund BorussiaDortmund SlovanBratislava BorussiaDortmund RCDeportivoLaCoruna RCDeportivoLaCoruna FCTirolInnsbruck BorussiaDortmund Lazio Lazio TrelleborgsFF Lazio Trabzonspor Trabzonspor AstonVilla

Figure 26 The UEFA Cup 1994/95.

53

Appendix, Tournament Trees

uc.9596
BayernMuenchen LokomotivMoscow RaithRovers IA(Akranes) SLBenfica LierseSK RodaJC(Kerkrade) SCTOlimpija(Ljubljana) NottinghamForest MalmoFF AJAuxerre VikingFK(Stavanger) OlympiqueLyonnais SCFarense Lazio Omonia(Lefkosia) FCBarcelona HapoelBeerSheva VitoriaSC(Guimaraes) StandardCL(Liege) Sevilla FCBotev(Plovdiv) Olympiakos(Pireus) MariborBranik PSV(Eindhoven) MyPa(Anjalankoski) LeedsUnited ASMonaco WerderBremen Glenavon DinamoMinsk FKAustria(Wien) GirondinsBordeaux VardarSkopje RotorVolgograd ManchesterUnited RealBetisBalompie Fenerbahce Kaiserslautern SlovanBratislava Milan ZaglebieLubin RCStrasbourg UjpestiTE SpartaPraha SilkeborgIF ZimbruChisinau RAFYelgava SlaviaPraha SCFreiburg FCLugano Internazionale RCLens AvenirBeggen ChernomoretsOdessa WidzewLodz Roma NeuchatelXamax SCEendrachtAalst FCLevski(Sofia) BrondbyIF LillestromSK Liverpool SpartakVladikavkaz BayernMuenchen BayernMuenchen RaithRovers BayernMuenchen SLBenfica SLBenfica RodaJC(Kerkrade) BayernMuenchen NottinghamForest NottinghamForest AJAuxerre NottinghamForest OlympiqueLyonnais OlympiqueLyonnais Lazio BayernMuenchen FCBarcelona FCBarcelona VitoriaSC(Guimaraes) FCBarcelona Sevilla Sevilla Olympiakos(Pireus) FCBarcelona PSV(Eindhoven) PSV(Eindhoven) LeedsUnited PSV(Eindhoven) WerderBremen WerderBremen DinamoMinsk BayernMuenchen GirondinsBordeaux GirondinsBordeaux RotorVolgograd GirondinsBordeaux RealBetisBalompie RealBetisBalompie Kaiserslautern GirondinsBordeaux Milan Milan RCStrasbourg Milan SpartaPraha SpartaPraha ZimbruChisinau GirondinsBordeaux SlaviaPraha SlaviaPraha FCLugano SlaviaPraha RCLens RCLens ChernomoretsOdessa SlaviaPraha Roma Roma SCEendrachtAalst Roma BrondbyIF BrondbyIF Liverpool

Figure 27 The UEFA Cup 1995/96.

54

Appendix, Evaluation of the Coefficients

A.3 Evaluation of the Coe cients
The UEFA Cup 1995/96 has been considered.

A.3.1 Using Probable Opponents
Milan
• 1
1.0 Coeff

GirondinsBordeaux

BrondbyIF

SlovanBratislava

• 1

1.0

1.0

1.0

Coeff

Coeff

• 17

• 17

0.0

0.0

0.0

Coeff

• 4

• 10

• 6

• 8

• 5

• 8

• 2

• 2

• 2

• 2

• 9

• 9

• 14

• 16

• 13

• 6

• 9

• 4

• 4

• 4

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

Round

Round

Round

Round

SLBenfica
• 2
1.0 Coeff

ManchesterUnited
• 1 • 1

BayernMuenchen

SlaviaPraha

• 2

1.0

1.0

Coeff

Coeff

Coeff

• 5

• 10

1.0

• 3

• 5

• 5

• 6

• 3

• 1

• 1

• 13

• 15

• 11

• 12

• 13

• 14

• 16

• 12

• 14

• 16

• 14

0.0

0.0

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

R1

R2

R3

Q

S

F

Round

Round

Round

Round

Lazio
• 3

Internazionale

FKAustria(Wien)

WerderBremen

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

Coeff

• 13

• 13

• 10

• 10

• 9

• 13

• 9

• 11

• 7

1.0

• 5

• 7

• 11

• 7

• 10

• 5

• 7

• 8

• 15

• 15

• 10

• 12

• 11

• 11

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

Round

Round

Round

Round

FCBarcelona

AJAuxerre

Kaiserslautern

Olympiakos(Pireus)

1.0

1.0

1.0

Coeff

Coeff

Coeff

Coeff

• 6

• 12

• 16

• 15

• 14

• 15

• 12

• 3

1.0

• 4

• 7

• 6

• 6

• 7

• 8

• 4

• 8

• 3

• 3

• 16

• 14

• 19

• 16

• 15

• 17

0.0

0.0

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

R1

R2

R3

Q

S

F

Round

Round

Round

Round

Figure 28 Using probable opponents (UEFA Cup 1995/96).

55

Appendix, Evaluation of the Coefficients
ChernomoretsOdessa PSV(Eindhoven) DinamoMinsk MyPa(Anjalankoski)

1.0

1.0

1.0

Coeff

Coeff

Coeff

• 17

• 19

• 23

• 23

• 23

• 23

• 21
0.0

• 21

• 21

• 19

• 19

• 19

• 24
0.0

• 20

• 18

• 22

• 22

• 22

Coeff

1.0

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

• 28 R1

• 27

• 28

• 25

• 26

• 28

R2

R3

Q

S

F

Round

Round

Round

Round

MariborBranik

RotorVolgograd

LillestromSK

Roma

Coeff

Coeff

Coeff

• 18
0.0

• 18

• 20

• 18

• 18

• 18

• 22
0.0

• 17

• 27
0.0

• 24

• 24

• 28

• 28

• 27

Coeff

• 11

• 8

1.0

1.0

1.0

• 9

• 12

1.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

• 30 R1

• 31 R2

• 34 R3

• 32

• 32

• 32

Q

S

F

Round

Round

Round

Round

ASMonaco

SpartaPraha

IA(Akranes)

Fenerbahce

1.0

1.0

1.0

Coeff

Coeff

Coeff

• 19
0.0

• 22

• 22

• 21

• 21

• 21

• 23
0.0

• 25

• 30

• 26

• 24

• 25

• 26
0.0

• 28

• 25

• 24

• 25

• 24

Coeff

1.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

• 31 R1

• 32 R2

• 31

• 30

• 30

• 30

R1

R2

R3

Q

S

F

R3

Q

S

F

Round

Round

Round

Round

StandardCL(Liege)

FCLevski(Sofia)

SCTOlimpija(Ljubljana)

SpartakVladikavkaz

1.0

1.0

1.0

Coeff

Coeff

Coeff

• 20
0.0

• 23

• 17

• 20

• 20

• 20

• 25
0.0

• 26

• 26

• 27

• 27

• 26

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

• 29 R1

• 30 R2

• 27

• 29

• 29

• 29

Coeff

1.0

• 34 R1

• 34 R2

• 33 R3

• 34 Q

• 34

• 34

R3

Q

S

F

S

F

Round

Round

Round

Round

Figure 29 Using probable opponents (UEFA Cup 1995/96).

56

Appendix, Evaluation of the Coefficients
Liverpool VikingFK(Stavanger) FCBotev(Plovdiv) VardarSkopje

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 33 R1

• 33 R2

• 32 R3

• 33 Q

• 33

• 33

• 40 R1

• 35 R2

• 38 R3

• 37 Q

• 37 S

• 37 F

• 36 R1

• 41 R2

• 40 R3

• 39 Q

• 40 S

• 40 F

Coeff

1.0

• 62 R1

• 47 R2

• 43 R3

• 43 Q

• 43 S

• 43 F

S

F

Round

Round

Round

Round

Glenavon

UjpestiTE

AvenirBeggen

SilkeborgIF

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 32 R1

• 29 R2

• 29

• 31

• 31

• 31

• 39 R1

• 42 R2

• 37 R3

• 40 Q

• 38 S

• 39 F

• 35 R1

• 40 R2

• 42 R3

• 42 Q

• 42 S

• 42 F

Coeff

1.0

• 61 R1

• 51 R2

• 56 R3

• 56 Q

R3

Q

S

F

• 56 S

• 56 F

Round

Round

Round

Round

WidzewLodz

NeuchatelXamax

ZimbruChisinau

Sevilla

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 42 R1

• 36 R2

• 39 R3

• 38 Q

• 39 S

• 38 F

• 38 R1

• 39 R2

• 41 R3

• 41 Q

• 41 S

• 41 F

• 64 R1

Coeff

1.0

• 64 R2

• 64 R3

• 64 Q

• 64 S

• 60 R1

• 64 F

• 58 R2

• 57 R3

• 57 Q

• 57 S

• 57 F

Round

Round

Round

Round

VitoriaSC(Guimaraes)

LeedsUnited

ZaglebieLubin

SCFreiburg

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 41 R1

• 38 R2

• 35 R3

• 35 Q

• 35 S

• 35

• 37 R1

• 37 R2

• 36 R3

• 36 Q

• 36 S

• 36

• 63 R1

• 43 R2

Coeff

1.0

• 47 R3

• 47 Q

• 47 S

• 47 F

• 59 R1

• 50 R2

• 48 R3

• 48 Q

• 48 S

• 48 F

F

F

Round

Round

Round

Round

Figure 30 Using probable opponents (UEFA Cup 1995/96).

57

Appendix, Evaluation of the Coefficients
SCFarense RaithRovers Omonia(Lefkosia) LokomotivMoscow

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

• 63 R2

• 59 R3

• 59 Q

R1

• 59 S

• 59 F

• 53 R2

• 52 R3

• 52 Q

• 52 S

• 52 F

• 49 R3

0.0

• 58

• 54 R1

• 50 R1

• 45 R2

Coeff

1.0

• 50 Q

• 50 S

• 50 F

• 46 R1

• 49 R2

• 50 R3

• 49 Q

• 49 S

• 49 F

Round

Round

Round

Round

SCEendrachtAalst

RCStrasbourg

OlympiqueLyonnais

LierseSK

1.0

1.0

1.0

Coeff

Coeff

Coeff

Coeff

1.0

0.0

0.0

0.0

• 52 R2

• 55 R3

• 55 Q

R1

• 54 S

• 55 F

• 57 R2

• 54 R3

• 54 Q

R1

• 55 S

• 54 F

• 61 R2

• 58 R3

• 58 Q

R1

• 58 S

• 58 F

0.0

• 57

• 53

• 49

• 45 R1

• 44 R2

• 44 R3

• 44 Q

• 44 S

• 44 F

Round

Round

Round

Round

RodaJC(Kerkrade)

RCLens

NottinghamForest

HapoelBeerSheva

1.0

1.0

1.0

Coeff

Coeff

Coeff

Coeff

1.0

0.0

0.0

0.0

• 54 R2

• 51 R3

• 51 Q

• 51 S

• 51 F

• 56 R2

• 60 R3

• 60 Q

R1

R1

• 60 S

• 60 F

• 60 R2

• 62 R3

• 62 Q

R1

• 62 S

• 62 F

0.0

• 56

• 52

• 48

• 44 R1

• 46 R2

• 46 R3

• 45 Q

• 45 S

• 45 F

Round

Round

Round

Round

RealBetisBalompie

RAFYelgava

MalmoFF

FCLugano

1.0

1.0

1.0

Coeff

Coeff

Coeff

Coeff

1.0

0.0

0.0

0.0

• 55 R2

• 53 R3

• 53 Q

• 53 S

R1

• 53 F

• 62 R2

• 63 R3

R1

• 63 Q

• 63 S

• 63 F

• 59 R2

• 61 R3

• 61 Q

R1

• 61 S

• 61 F

0.0

• 55

• 51

• 47

• 43 R1

• 48 R2

• 45 R3

• 46 Q

• 46 S

• 46 F

Round

Round

Round

Round

Figure 31 Using probable opponents (UEFA Cup 1995/96).

58

Appendix, Evaluation of the Coefficients

A.3.2 Using All Potential Opponents
Milan
• 1
1.0 Coeff

GirondinsBordeaux
• 2

BrondbyIF

SlovanBratislava

• 1

1.0

1.0

Coeff

Coeff

0.0

0.0

0.0

Coeff

• 11

1.0

• 4

• 3

• 4

• 4

• 5

• 1

• 1

• 2

• 2

• 9

• 15

• 14

• 14

• 14

• 13

• 15

• 8

• 9

• 13

• 13

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

Round

Round

Round

Round

SLBenfica
• 2
1.0 Coeff

ManchesterUnited
• 4

BayernMuenchen

SlaviaPraha

• 3

1.0

1.0

Coeff

Coeff

0.0

0.0

0.0

Coeff

• 10

• 12

• 8

• 8

1.0

• 3

• 4

• 1

• 1

• 6

• 2

• 2

• 3

• 3

• 9

• 8

• 14

• 9

• 14

• 15

• 15

• 15

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

Round

Round

Round

Round

Lazio
• 3

Internazionale

FKAustria(Wien)

WerderBremen

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

Coeff

• 6

• 13

• 10

• 9

• 11

• 11

1.0

• 7

• 7

• 5

• 6

• 7

• 6

• 10

• 8

• 11

• 7

• 7

• 15

• 10

• 13

• 12

• 11

• 12

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

R1

R2

R3

Q

S

F

Round

Round

Round

Round

FCBarcelona

AJAuxerre

Kaiserslautern

Olympiakos(Pireus)

1.0

1.0

1.0

Coeff

Coeff

Coeff

Coeff

• 5

• 13

• 12

• 9

• 10

• 12

• 14

• 12

• 11

1.0

• 4

• 5

• 5

• 6

• 5

• 8

• 10

• 6

• 7

• 16

• 18

• 16

• 16

• 16

• 16

0.0

0.0

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

R1

R2

R3

Q

S

F

Round

Round

Round

Round

Figure 32 Using all potential opponents (UEFA Cup 1995/96).

59

Appendix, Evaluation of the Coefficients
ChernomoretsOdessa PSV(Eindhoven) DinamoMinsk MyPa(Anjalankoski)

1.0

1.0

1.0

Coeff

Coeff

Coeff

• 17

• 20

• 17

• 21

• 20

• 19

• 21
0.0

• 21

• 21

• 20

• 19

• 20

• 24
0.0

• 23

• 25

• 25

• 25

• 25

Coeff

1.0

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

• 28 R1

• 30 R2

• 29

• 29

• 29

• 29

R3

Q

S

F

Round

Round

Round

Round

MariborBranik

RotorVolgograd

LillestromSK

Roma

1.0

1.0

1.0

Coeff

Coeff

Coeff

• 18
0.0

• 22

• 22

• 22

• 21

• 21

• 22
0.0

• 17

• 18

• 18

• 22

• 22

• 27
0.0

• 28

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

• 30

• 30

• 30

• 30

Coeff

1.0

• 30 R1

• 33 R2

• 31 R3

• 31

• 31

• 31

Q

S

F

Round

Round

Round

Round

ASMonaco

SpartaPraha

IA(Akranes)

Fenerbahce

1.0

1.0

1.0

Coeff

Coeff

Coeff

• 19
0.0

• 19

• 20

• 19

• 18

• 18

• 23
0.0

• 27

• 24

• 23

• 24

• 23

• 26
0.0

• 24

• 23

• 24

• 23

• 24

Coeff

1.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

• 31 R1

• 29 R2

• 28

• 28

• 28

• 28

R3

Q

S

F

Round

Round

Round

Round

StandardCL(Liege)

FCLevski(Sofia)

SCTOlimpija(Ljubljana)

SpartakVladikavkaz

1.0

1.0

1.0

Coeff

Coeff

Coeff

• 20
0.0

• 16

• 19

• 17

• 17

• 25
0.0

• 26

• 27

• 27

• 26

• 26

0.0

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

R1

R2

R3

Q

S

F

0.0

• 29

• 25

• 26

• 26

• 27

• 27

Coeff

• 17

1.0

• 34 R1

• 32 R2

• 34 R3

• 34 Q

• 34

• 34

S

F

Round

Round

Round

Round

Figure 33 Using all potential opponents (UEFA Cup 1995/96).

60

Appendix, Evaluation of the Coefficients
Liverpool VikingFK(Stavanger) FCBotev(Plovdiv) VardarSkopje

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 33 R1

• 31 R2

• 33 R3

• 33 Q

• 33

• 33

• 40 R1

• 42 R2

• 42 R3

• 41 Q

• 42 S

• 42 F

• 36 R1

• 37 R2

• 36 R3

• 36 Q

• 36 S

• 36 F

Coeff

1.0

• 62 R1

• 47 R2

• 48 R3

• 48 Q

• 48 S

• 48 F

S

F

Round

Round

Round

Round

Glenavon

UjpestiTE

AvenirBeggen

SilkeborgIF

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 32 R1

• 34 R2

• 32 R3

• 32 Q

• 32

• 32

• 39 R1

• 35 R2

• 35 R3

• 35 Q

• 35 S

• 35

• 35 R1

• 38 R2

• 38 R3

• 39 Q

• 39 S

• 38 F

Coeff

1.0

• 61 R1

• 64 R2

• 62 R3

S

F

F

• 62 Q

• 62 S

• 62 F

Round

Round

Round

Round

WidzewLodz

NeuchatelXamax

ZimbruChisinau

Sevilla

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 42 R1

• 41 R2

• 41 R3

• 42 Q

• 41 S

• 41 F

• 38 R1

• 40 R2

• 40 R3

• 40 Q

• 40 S

• 40 F

• 64 R1

Coeff

1.0

• 58 R2

• 57 R3

• 57 Q

• 56 S

• 56 F

• 60 R1

• 53 R2

• 52 R3

• 52 Q

• 52 S

• 52 F

Round

Round

Round

Round

VitoriaSC(Guimaraes)

LeedsUnited

ZaglebieLubin

SCFreiburg

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 41 R1

• 36 R2

• 37 R3

• 37 Q

• 37 S

• 37 F

• 37 R1

• 39 R2

• 39 R3

• 38 Q

• 38 S

• 39 F

• 63 R1

Coeff

1.0

• 62 R2

• 64 R3

• 64 Q

• 64 S

• 59 R1

• 49 R2

• 64 F

• 53 R3

• 53 Q

• 53 S

• 53 F

Round

Round

Round

Round

Figure 34 Using all potential opponents (UEFA Cup 1995/96).

61

Appendix, Evaluation of the Coefficients
SCFarense RaithRovers Omonia(Lefkosia) LokomotivMoscow

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

• 47 R3

• 47 Q

• 47 S

• 49 R3

• 49 Q

• 49 S

• 49 F

0.0

• 58 R1

• 46 R2

• 47 F

• 54 R1

• 52 R2

• 50 R1

Coeff

1.0

• 63 R2

• 63 R3

• 63 Q

• 63 S

• 46 R1

• 63 F

• 59 R2

• 59 R3

• 59 Q

• 58 S

• 58 F

Round

Round

Round

Round

SCEendrachtAalst

RCStrasbourg

OlympiqueLyonnais

LierseSK

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

• 60 R2

• 60 R3

R1

• 60 Q

• 61 S

• 61 F

• 45 R3

• 45 S

• 46 R3

• 46 Q

• 46 S

0.0

• 57

• 53 R1

• 43 R2

• 45 Q

• 45 F

• 49 R1

Coeff

1.0

• 45 R2

• 46 F

• 45 R1

• 61 R2

• 61 R3

• 61 Q

• 60 S

• 60 F

Round

Round

Round

Round

RodaJC(Kerkrade)

RCLens

NottinghamForest

HapoelBeerSheva

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

• 55 R2

• 54 R3

• 54 Q

• 54 S

R1

R2

R3

Q

S

F

R1

• 54 F

• 51 R3

0.0

• 56

• 44

• 43

• 43

• 43

• 43

• 52

• 48 R1

Coeff

1.0

• 51 R2

• 51 Q

• 51 S

• 51 F

• 44 R1

• 56 R2

• 58 R3

• 58 Q

• 59 S

• 59 F

Round

Round

Round

Round

RealBetisBalompie

RAFYelgava

MalmoFF

FCLugano

1.0

1.0

1.0

Coeff

Coeff

Coeff

0.0

0.0

0.0

0.0

• 55 R1

• 48 R2

• 44 R3

• 44 Q

• 44 S

• 44 F

• 51 R1

• 57 R2

• 56 R3

• 56 Q

• 55 S

• 55 F

• 47 R1

Coeff

1.0

• 50 R2

• 50 R3

• 50 Q

• 50 S

• 50 F

• 43 R1

• 54 R2

• 55 R3

• 55 Q

• 57 S

• 57 F

Round

Round

Round

Round

Figure 35 Using all potential opponents (UEFA Cup 1995/96).

62

References

B References
Bradley R.A. and Terry M.E. (1952).

Rank Analysis of Incomplete Block Designs - I. The Method of Paired Comparisons. Biometrika, 39, 324-345. Quali cation and Seeding for the European Cups (FAQ). martin@inferenzsysteme.informatik.th-darmstadt.de.

Protzen Martin and RSSSF (Rec.Sport.Soccer Statistics Foundation) (1996). Carlin B.P. (1996).

Improved NCAA Basketball Tournament Modeling via Point Spread and Team Strength Information. The American Statistician, 50, 39-43. Probability Models for the NCAA Regional Basketball Tournaments. The American Statistician, 45, 35-38. More Probability Models for the NCAA Regional Basketball Tournaments. The American Statistician, 50, 34-38.

Schwertman N. C., McCready T.A. and Howard L. (1991). Schwertman N. C., Schenk L. and Holbrook B.C. (1996).

63