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

Gianfranco Gambarelli University of Bergamo, Italy

Athens 2004 Alexei BAR Nemov 25

APPLICATIONS - Gymnastics - Diving SPORTS : - Figure skating - Synchronized swimming - ... - LIBOR - EURIBOR BANKING : - EONIASWAP - EUREPO - ... EVALUATION OF PROJECTS ………….. 24

Common Sense: 2, 7, 7, 8, 9, 9 8 23

Vaulting Arithmetic Mean 2, 7, 7, 8, 9, 9 7 22

Diving 5 judges trimmed mean (but 2) 2, 7, 7, 8, 9, 9 7.75 Diving 7 judges trimmed mean (but 4) 2, 7, 7, 8, 9, 9 7.5 21

Rhythmic Gymn. 4 j. Median 2, 7, 7, 8, 9, 9 7.5 20

2, 7, 7, 8, 9, 9 Common Sense: 8 Arithmetic Mean: 7 7.75 Trimmed mean (but 2): 7.50 Trimmed mean (but 4): Median: 7.50 19

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

→ majority = 4 6 judges 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 7.75 Trimmed mean (but 2): 7.50 Trimmed mean (but 4): Median: 7.50 Coherent Majority Av.: 8 16

Gambarelli , G. (2008) “The Coherent Majority Average for juries ’ evaluation processes ” Journal of Sport Sciences - LIBOR - EURIBOR BANKING : - EONIASWAP - EUREPO - ... 15

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THE NEW PROBLEM (sports) Execution Artistry Difficulty 13

THE NEW PROBLEM (project eval.) Environmental Building Disease costs Costs Costs 12

Judges Country Athlete 1 2 3 4 7 7 4 2 I 1 II 2 I 3 I 4 I 5 I 6 I 11

Judges Country Athlete 1 2 3 4 7 7 4 2 I 1 7 7 4 4 II 2 6 7 4 4 I 3 6 8 10 9 I 4 6 6 9 9 I 5 7 7 4 4 I 6 7 6 3 5 I 10

1) How to identify collusions in an objective way 2) How to take into account it to build a fair average 9

The idea  For each coalition p: 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 index of self-valuation of p Index of coalitional collusion of p = index of other’s valuation of p 8

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

Regarding our example: Judges Country Athlete 1 2 3 4 ACA 7 7 7 4 2 I 1 7 7 7 4 4 II 6.5 2 6 7 4 4 I 7 3 6 8 10 9 I 6 4 6 6 9 9 I 7 5 7 7 4 4 I 6 7 6 3 5 6.5 I 6

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

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. 3

Gambarelli, G., G. Iaquinta and M. Piazza (2012) “Anti -Collusion Indices and averages for the evaluation of performances and juries” Journal of Sport Sciences 1