An alternative ranking system for biathlon pursuit races

2.1Data collection

The final results of all sprint and pursuit races are publicly available on the datacenter webpage of the IBU: https://biathlonresults.com/. The results were collected on the 15th December 2020 starting from the 2001/2002 season. All the results taken into account for the men’s pursuit world cup were gathered, including world championships and Olympic games before 2014. It provides us 148 different pursuit results. All these results provide us an important database of pursuit results in different conditions that are considered to give an appropriate representation of the different possible pursuit scenarios.

2.2Alternative pursuit ranking

As explained previously, during pursuit races, biathletes are racing each other in real-time for a better rank and not racing against time. Therefore, we chose to work using final ranks, not final times. All the pursuit results were gathered to compute final pursuit ranks according to the starting pursuit rank. This global information is given in Fig. 1 for all sprint ranking positions and the distributions of the final pursuit ranks are provided in Fig. 2 for some starting ranks.

Fig. 1

Final pursuit ranks according to starting pursuit ranks.

Fig. 2

Barplots of the final pursuit ranks according to six different starting pursuit ranks.

These figures emphasize the results of Luchsinger et al. (2020), highlighting the importance of the starting pursuit rank in the final pursuit result.

We propose an alternative approach to define a fairer pursuit final ranking that will decrease this correlation. For each starting biathlete at a pursuit race k, a quantity q_ki is calculated according to the position of his final results f_ki in the final result distribution of all previous starters with the same rank i. Some of these distributions are plotted in Fig. 2. This quantity is given by the following formula

This formula is somewhat natural and explainable: indeed, if q_ki is equal to zero, it means that, during the previous 148 pursuit races, no biathlete with starting rank i achieved a final better (i.e. smaller) rank f_ki and so, for all j, f_ji ≥ f_ki. So, this biathlete deserves a good alternative final pursuit rank, whatever his starting rank. On the contrary, if q_ki is equal to one (i.e. for all j, f_ji < f_ki), then f_ki is the worst final pursuit rank achieved by any of the 148 biathletes with this starting rank i and it must lead to a poor alternative final pursuit rank.

Estimating a quantity (here a performance during a pursuit race) using comparisons to existing distributions (such as the previous pursuit results) is a common approach in non-parametric statistics (Wasserman, 2006). It prevents making false assumptions on the underlying distributions but requires an important number of datasets. In this application, with 148 datasets of previous biathlon pursuit results, this framework seems adapted. All the different pursuit scenarios (including extreme ones) will by consequence be taken into account according to their previous occurrences. Note that, if we consider each of the 148 pursuit races as a ballot, a connection can be made with the well-known Borda count ranking (Borda, 1781) for the use of all the pursuit races with strong differences for the other part of the alternative ranking computations. We can also notice that there exists a continuum of possible ranking functions between the non-parametric alternative one defined above and the current one. So, if needed, the alternative method could be modified to bring fewer modifications to the pursuit rankings. But this will complicate the formula provided above and, thus, is beyond the scope of this paper.

2.3Data analyses

All the data analyses were performed using the R freeware, version 3.5.1 (R Core Team, 2019). The correlations were calculated using Spearman’s rank correlation coefficient. For the world cup rankings, we remind that only the first forty biathletes of each race score points, according to the current rules of IBU (International Biathlon Union, 2020). Note that all the races are taken into account to compute all the alternative rankings, even those that took place in the future. This will ensure a good estimation of the different distributions of the final rankings according to the starting ranks.

3.1Study of a specific pursuit race

We first choose to study a specific pursuit race to illustrate the modifications induced by our alternative ranking. We choose the pursuit race that took place at Annecy –Le Grand Bornand (21 December 2019). The results are given in Table 1.

The correlation between the starting rank and the current pursuit rank (resp. the alternative pursuit rank) is equal to 0.82 (resp. 0.20) which highlights the decreased influence of the sprint results on the alternative ranking. If we look at the main modifications we could see that T. Boe, B. Doll, E. Bjoentegaard, or J. Dale are losing more than 15 ranks with the alternative pursuit ranking. This is due to the fact that they had lost ranks during the pursuit and, therefore, their current good pursuit ranks are mainly due to their good performances in the sprint race. So, it seems logical that they lose ranks with the alternative ranking. On the contrary, E. Jacquelin, S. Schempp, and T. Bormolini performed very well during the pursuit race (resp. 14, 22, 28 ranks won during the pursuit race) and deserve their better pursuit rank using the alternative ranking. For example, T. Bormolini will be ranked 6th with the alternative pursuit ranking whereas it never happened on all the past 148 pursuit races with the current ranking system for a biathlete with the 60th starting rank, as it could be seen on the last plot of Fig. 2.

The computer code used to obtain the results of Table 1 is provided as Supplementary Material with the corresponding dataset. This code could be reused with any pursuit result to compute the alternative rankings in less than a second on an ordinary laptop.

3.2Study of the 2019/2020 pursuit world cup ranking

As explained above, the alternative pursuit ranking can lead to major modifications on a specific pursuit race. Then, we chose to study the 2019/2020 pursuit world cup to analyze the modifications at the scale of a whole season. The first ten biathletes using the two pursuit rankings on all the 2019/2020 races are given in Table 2.

First, we could see that there is only a small modification on the podium, J. Boe who was 4th with the current ranking is now 3rd whereas Q. Fillon-Maillet who was 3rd is now 5th. There is no modification for the first two ranks and eight biathletes are in the two top 10. The strong modifications of the rankings of each pursuit race (as seen in the previous subsection) lead to non-negligible but with less impact world cup ranking modifications. Nevertheless, we can note some important individual modifications for example for E. Garanichev (26th with the current ranking and 6th with the alternative) or M. Krcmar (resp. 21st and 10th) who benefit from the alternative ranking unlike T. Boe (resp. 6th and 11th) or S. Desthieux (resp. 7th and 18th).

The number of points with the alternative ranking seems lower than the current one. Indeed, that is an important property of the alternative ranking: the points are awarded to most biathletes as they are less linked to the sprint results (71 biathletes with the current ranking and 81 with the alternative one). But there is a strong correlation of 0.83 between the number of points of each biathlete with the current or the alternative ranking which could explain the relatively small modifications between the two rankings, as mentioned above.

Note that these small modifications could have a major impact on the overall world cup ranking. Indeed, J. Boe won the overall world cup with 2 points ahead of M. Fourcade. With the alternative ranking, M. Fourcade would have won the overall world cup with the same margin. Obviously, this is science fiction as the application of the alternative ranking would probably modify the pursuit races. Nevertheless, it could highlight the importance of the sprint results in the overall world cup ranking (J. Boe won 4 of them this season) and the potential impact of the alternative ranking on the overall world cup rankings, mainly when there are few points of difference.

3.3Study of the last ten pursuit world cup seasons

We then studied the pursuit world cup seasons of the ten last years to analyze if the previous remarks could be extended. First, on the pursuit races, the correlations between the starting ranks and the pursuit ranks decreased as seen in the first results subsection: the correlation mean is equal to 0.74 with the current ranking and to 0.06 with the alternative one. Then, we analyzed the last ten pursuit world cup rankings. For all the rankings, there are more biathletes with points with the alternative ranking than with the current one with a mean increase of 11 biathletes. The mean of the difference of points between the first rank and the ranks from 2 to 10 are also all smaller for the alternative pursuit ranking. This would have led to, in most of the cases, closer rankings and more suspense in the last races of the season.

As seen in the previous subsection, the modifications on the podiums of the pursuit world cup rankings are small but not negligible. For 7 seasons we have the same winner, two times the first and the second invert their rankings and for the last one, the 4th becomes 1st with the alternative ranking. There are only two identical podiums but, if we compare the name of the first three biathletes, 23 above 30 are shared by the two different rankings. It highlights some important common traits between the two rankings even if some individual rankings could be strongly modified, for example, a biathlete who was 3rd with the current ranking is 16th with the alternative one highlighting the importance of his sprint results in his good current pursuit ranking.

4.1Advantages of the alternative ranking

First, the main advantage of this alternative ranking is obviously that the correlation with the starting rank is very low. Therefore, even the 60th ranked at the end of the sprint had a chance to be on the podium which is not the case with the current ranking. In other words, a biathlete with high pursuit skills who had a bad sprint result has an increased chance of obtaining a good pursuit result with the alternative ranking. But we have to keep in mind that the best pursuit biathlete is still the one that cut the finish line first. It will result in more surprising and contested pursuit races, at each stage of the races, which is desirable for gaining audience (Bizzozero et al., 2016). More generally, it will also decrease the importance of the sprint races on the overall world cup rankings.

Second, even if the alternative ranking deeply modifies each pursuit ranking, each season pursuit world cup ranking is less modified than each single pursuit race. It sounds natural as, even if tactics and head-to-head are of major importance in pursuit races, it remains biathlon with cross-country skiing and shootings. So, the best biathletes are globally the same, the alternative pursuit ranking allows to define the pursuit as a whole discipline with real specialists, not just as a relatively small perturbation of the sprint ranking (as proven in Luchsinger et al., 2020).

4.2Limitations of the alternative ranking

The first criticism that could be made to the alternative ranking is that it is more complicated than the current one. Nowadays, when you cross the finish line of the pursuit race in 3rd place, you are ranked 3rd, whereas with the alternative ranking you need to wait for all the biathletes to finish the race. Even if the alternative ranking is calculated in less than one second at the end of the race, it could be seen as a limiting factor. Nevertheless, this argument needs to be mitigated. First, the winner of the pursuit race is necessarily the winner of the alternative pursuit ranking and is therefore known immediately as he crosses the finish line. Then, for the sprint or individual biathlon races or for other sports such as the decathlon (where you need to refer to a complex points system to see how many points you score, see Cox et al. (2002) for further details) the final ranks are unknown until the last athlete crosses the finish line. This could induce important cliffhangers when biathletes are waiting in the finish area to wait and see if they are or not on the podium. Finally, at each split time, a ranking based on the alternative pursuit ranking could be quickly calculated to inform the biathletes of their rankings.

Another limitation is that, when you have several biathletes that did not start or did not finish the pursuit race despite their presence on the first 60 biathletes of the sprint, it artificially increases the alternative rankings of biathletes that are at the end of the ranking. That could induce unmerited good alternative pursuit rankings for biathletes that have not performed well during the pursuit race but who improved their final rankings thanks to those who gave up. It could be solved by integrating the number of finishers of each pursuit race in the formula to calculate the quantity q_ki for example by dividing f_ji and f_ki by the number of finishers of each pursuit race. But, to keep a very simple formula and as it is uncommon and does not impact the more important highest ranks, it was not taken into account in this paper.