Rank-size law, financial inequality indices and gain concentrations by cyclist teams. The case of a multiple stage bicycle race, like Tour de France

This note examines financial distributions to competing teams at the end of the most famous multiple stage professional (male) bicyclist race, TOUR DE FRANCE. A rank-size law (RSL) is calculated for the team financial gains. The RSL is found to be hyperbolic with a surprisingly simple decay exponent (about equal to -1). Yet, the financial gain distributions unexpectedly do not obey Pareto principle of factor sparsity. Next, several (8) inequality indices are considered : the Entropy, the Hirschman-Herfindahl, Theil, Pietra-Hoover, Gini, Rosenbluth indices, the Coefficient of Variation and the Concentration Index are calculated for outlining diversity measures. The connection between such indices and their concentration aspects meanings are presented as support of the RSL findings. The results emphasize that the sum of skills and team strategies are effectively contributing to the financial gains distributions. From theoretical and practical points of view, the findings suggest that one should investigate other "long multiple stage races" and rewarding rules. Indeed, money prize rules coupling to stage difficulty might influence and maybe enhance (or deteriorate) purely sportive aspects in group competitions. Due to the delay in the peer review process, the 2019 results can be examined. They are discussed in an Appendix; the value of the exponent (-1.2) is pointed out to mainly originating from the so called "king effect"; the tail of the RSL rather looks like an exponential.