Anti-collusion indices and averages for the evaluation of performances and judges.

In subjectively assessed artistic sports, judges sometimes collude. To avoid this, Federations adopt various strategies in their regulations. However, it is evident that these methods are often not sufficient to overcome collusions completely. The problem is especially difficult in cases where the judging panel consists of sub-commissions that deal with particular aspects of each performance and the scores of each sub-commission are summarized as mean values, which, in turn, are combined into a final mark. Therefore, the formulation of anti-collusion indices would be useful to assess judges and take account of such scores, rejecting the less reliable ones. An anti-collusion index and related average, recently introduced, can be applied in fields other than sports. This article presents some useful considerations for the application of such instruments to the field of sport.

The "coherent majority average" for juries' evaluation processes.

I propose a method to synthesize the performance scores for artistic sports such as rhythmic gymnastics, figure skating, synchronized swimming, and diving by taking into account inter-judge variability, while maintaining all the reliable scores. This procedure is based on the assumption that the majority of the scores in each event are reliable and they relate well to those scores that are closest to them. The method consists of putting scores in order and considering clusters of m consecutive scores, where m is the number of judges making up the simple majority. For each cluster, the difference between the highest and the lowest score is calculated. In cases where the minimum difference is positive, the arithmetic mean of those scores that belong to clusters where the difference is minimal is computed. In cases where the minimum difference is zero (i.e. if the majority of judges unanimously assign the same score), then the set of the scores to consider within the mean is extended to those scores that are very near to those of the majority of the judges. A comparison between the actual evaluation procedures and the proposed model is provided.