Gianfranco Gambarelli University of Bergamo, Italy Athens 2004 - - PowerPoint PPT Presentation

Gianfranco Gambarelli

University of Bergamo, Italy

25

BAR Alexei Nemov Athens 2004

• Gymnastics
• Diving
• Figure skating
• Synchronized swimming
• ...

24

SPORTS:

APPLICATIONS

BANKING:

• LIBOR
• EURIBOR
• EONIASWAP
• EUREPO
• ...

EVALUATION OF PROJECTS …………..

23

2, 7, 7, 8, 9, 9

Common Sense:

8

Vaulting Arithmetic Mean

2, 7, 7, 8, 9, 9

22

7

Diving 5 judges trimmed mean (but 2)

2, 7, 7, 8, 9, 9

Diving 7 judges trimmed mean (but 4)

2, 7, 7, 8, 9, 9

21

7.75 7.5

Rhythmic Gymn. 4 j. Median

2, 7, 7, 8, 9, 9 7.5

20

19

2, 7, 7, 8, 9, 9

Common Sense: 8 Arithmetic Mean: 7

Trimmed mean (but 2):

7.75

Trimmed mean (but 4):

7.50 Median: 7.50

The Coherent Majority Average

The goals:

• correct evaluation
• incentive to judges.

The assumptions:

• the majority of scores is reliable
• they relate well to those scores

which are closest to them.

18

6 judges → majority = 4 2, 7, 7, 8, 9, 9 └─────┘ 8-2 = 6 └─────┘ 9-7 = 2 └─────┘ 9-7 = 2 Minimum difference: 2 Corresponding scores: 7,7,8,9,9 Arithmetic mean of such scores (= CMA): 8

17

Common Sense Recovered

2, 7, 7, 8, 9, 9

Common Sense: 8 Arithmetic Mean: 7

Trimmed mean (but 2):

7.75

Trimmed mean (but 4):

7.50 Median: 7.50 Coherent Majority Av.: 8

16

Gambarelli, G. (2008) “The Coherent Majority Average for juries’ evaluation processes”

Journal of Sport Sciences

15

BANKING:

• LIBOR
• EURIBOR
• EONIASWAP
• EUREPO
• ...

14

13

Execution Artistry Difficulty

THE NEW PROBLEM (sports)

12

Environmental costs Building Costs Disease Costs

THE NEW PROBLEM (project eval.)

11

Country Athlete Judges 1

I

7 7 4 2

II

2

I

3

I

4

I

5

I

6

I

1 2 3 4

10

Country Athlete Judges 1

I

7 7 4 2

II

7 7 4 4 2

I

6 7 4 4 3

I

6 8 10 9 4

I

6 6 9 9 5

I

7 7 4 4 6

I

7 6 3 5

1 2 3 4

9

1) How to identify collusions in an objective way 2) How to take into account it to build a fair average

8

Index of self-valuation of p:

average scores awarded to p by judges that belong to p average scores that other judges have awarded to p

Index of others’ valuation of p:

average scores awarded by judges of p to the perform. of the other teams average scores awarded by other judges to the perform. of the other teams

 For each coalition p:

The idea

Index of coalitional collusion of p = index of self-valuation of p index of other’s valuation of p

…

1.86 1.07 0.97 1.18 1.04 2.55 1.48 1.31 1.67 1.70 = 1.86 = 2.55 = 1.86 = 2.55 = 2.55 = 1.86 = 1.70 = 1.67 = 1.48 = 1.31 = 1.18 = 1.07 = 1.04 = 0.97 1° 2° Coalitional c.i. Ordered Coalitional c.i. Ordered Individual c.i. 7

Most Reliable Judges

Country Athlete Judges 1

I

7 7 4 2

II

7 7 4 4 2

I

6 7 4 4 3

I

6 8 10 9 4

I

6 6 9 9 5

I

7 7 4 4 6

I

7 6 3 5

1 2 3 4

ACA

7 7 6.5 7 6 7 6.5

6

Regarding our example:

Anti-Collusion Average

the arithmetic mean of the scores that have been assigned by the most reliable judges

5

By means of ACA the judges are pushed to work properly in order to avoid their votes being eliminated

4

3

Bertini, C., G. Gambarelli and A. Uristani (2010) "Collusion Indices and an Anti-collusion Average” Preferences and decisions: models and applications, Studies in Fuzziness and Soft Computing, Springer Verlag.

1

Gambarelli, G., G. Iaquinta and M. Piazza (2012) “Anti-Collusion Indices and averages for the evaluation of performances and juries” Journal of Sport Sciences