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

Abstract

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 ≈-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.

Network Analysis of Cross-Correlations on Forex Market during Crises. Globalisation on Forex Market

“…The range of problems investigated by cross-correlation analysis is very broad, starting from sociology, economy, econophysics [ 20 , 23 , 33 , 34 ], transport [ 35 , 36 ], genome analysis, biology, food network, biochemistry network, science collaboration network [ 37 ], up to sport [ 38 ], and many others.…”

Shannon Entropy and Herfindahl-Hirschman Index as Team’s Performance and Competitive Balance Indicators in Cyclist Multi-Stage Races

“…However, one could admit that other functions with a small number of parameters could be used in appropriately managing the xand y-axes scales and range. Using the two-parameter (κ, χ) form (10) would be an example of this.…”

“…More pertinent to our concern, a 2020 paper by Ausloos [ 10 ] studied data of professional cyclists from the Tour de France, along the rank-size law method, deducing financial considerations from classical UCI measures (time-based) team ranking. Along a similar perspective, Ficcadenti et al [ 11 ], in 2022, discussed the above-mentioned rank-size law on a soccer competition in Italy, thereby observing some regularities in team values within a ranking process.…”

“…Notice that the has been used on cyclist races such as the Tour de France in [ 10 ], but with a focus on its “market competition” aspect, i.e., measuring teams’ financial gains.…”

A rank-size approach to analyse soccer competitions and teams: the case of the Italian football league “Serie A"

“…Among the others, we mention Ficcadenti and Cerqueti (2017) where the ZML has been used to estimate the economic cost of earthquakes, Ficcadenti et al (2019Ficcadenti et al ( , 2020 for their investigation of rank-size relationships in corpora, Cerqueti and Ausloos (2015) where the Italian cities tax income distribution analysis is run through a rank-size approach, Dimitrova and Ausloos (2015), where the Bulgarian Urban system is studied across years using rank-size laws and Rotundo (2014); Ausloos (2013Ausloos ( , 2014b for their applications in the field of scientometrics. Referring to sport, it is worth mentioning Ausloos (2020) where the rank-size law of the official gains for bicycle teams in Tour de France is studied. In our case, the Discrete Generalised Beta Distribution presents outstanding best fit performances.…”