Author: Thomas M

En cette période de Jeux Olympiques d’été, c’est l’occasion de regarder à la télévision sur des chaînes de grande écoute et à des heures décentes (modulo le décalage horaire !) des sports méconnus du grand public. Nous avons déjà parlé ici du biathlon (en ce qui concerne les JO d’Hiver, qu’on retrouvera en 2018), mais ce billet va parler d’un autre sport : le tir à l’arc. Le but du tir à l’arc est de placer ses flèches sur une cible, assez souvent très éloignée, dans des cercles concentriques qui valent de plus en plus de points au fur et à mesure qu’on se rapproche du centre, de 1 à 10 (voire 0 si l’on rate la cible, ce qui est assez rare aux JO !).

Les règles semblent simples, mais il y a une petite subtilité qui est apparue cette année. En effet, jusqu’à présent les archers tiraient quatre volées de trois flèches chacun, de façon alternée, et on sommait les points obtenus : celui qui avait le meilleur score était qualifié pour la manche suivante. En cas d’égalité, une flèche était tirée pour chaque archer, et le plus proche gagne le match.

Les nouvelles règles mettent en avant la notion de “set” : désormais, chaque volée de trois flèches est considérée de façon indépendante. L’archer qui a un meilleur score que son adversaire à la fin d’un set marque 2 points, et en cas d’égalité au set, les deux marquent 1 point, sachant que le match se joue en 6 points. On joue alors cinq sets, et si personne n’est arrivé à 6 à la fin de ces cinq sets, chacun tire une flèche et la plus proche gagne le match.

Selon les journalistes sportifs de France Télévisions, ces nouvelles règles permettent à un tireur de rattraper un mauvais tir (c’est à dire un tir en dessous du 8, à ce niveau de compétition) plus facilement que lorsque l’on somme la totalité des points, où une flèche ratée pénalise toute la partie. Nous allons à l’aide d’un exemple et de quelques simulations vérifier si cette affirmation est vraie.

Considérons deux archers, Arthur et Bastien. Les deux archers ont un niveau équivalent, mais ils n’ont pas le même profil : Arthur ne met jamais de flèches en dessous de 8, mais tire souvent dans le 8. Bastien, lui, peut rater un tir et toucher un 5 ou un 7, mais arrive plus souvent à toucher la partie jaune de la cible (9 ou 10). Plus précisément, leurs chances pour chaque tir sont les suivantes :

Un rapide calcul permet de constater que pour les deux archers, chaque flèche rapporte en moyenne 8,6 points. Ils ont donc bien un niveau comparable. Nous allons maintenant simuler plusieurs dizaines de milliers de matchs en suivant les deux jeux de règles possibles afin de déterminer qui gagne, et si Bastien est bien favorisé par les nouvelles règles. Les résultats obtenus sont les suivants :

Cela se confirme donc bien : les nouvelles règles favorisent Bastien, qui rate de temps en temps son tir, et permettent donc plus facilement de revenir dans le match après une flèche ratée. Cela permet également un suspens plus important, car rien n’est jamais joué d’avance !

Today, I want to speak about my favorite sport to watch on TV, which is biathlon (the one which involves skiing and shooting at things. Who doesn’t love that?). I really enjoy to follow the races, and not only because the best athlete at this moment is a french one (go Martin!), but because the shooting part seems so crucial and stressful. This leads me to wonder about how much missing a shot is relevant for the ranking in the end of the race. Let’s find out!

Gathering some data

My idea is to do some basic analysis on data about the results of many biathlon races, in order to evaluate if there is some form of correlation between the final ranking and the number of shot missed. This requires to first gather the data. The results are stored on multiple sites: obviously Wikipedia pages of the championships, but this website is much more detailed. I’m going to use some of the results between 2007 and 2015. I won’t use all of the races because I want to have comparable results: I’m going to only consider the races where the number of competitors is between 50 and 60. This allows me to interpret the results about final ranking with a similar scaling system for all the races. Moreover, the specificity of biathlon is that the rules are very different from a format to another (see this Wikipedia article for more informations), and I can’t easily discriminate my data between them. Using a limitation on the number of participants is a way to limit the width of the spectrum of formats considered. Well, let’s forget these technicalities and analyse the data!

Don’t miss a shot!

So, the idea is to put in regard the number of shots missed and the final ranking. Fun fact: the number of shot during a race is 20, but the maximum number of shot missed during a race I analysed is only 9. That’s not really a surprise if you frequently watch biathlon, because missing that much shots usually means that you’re going to finish in last place. I’m going to use a heat map in order to show the correlation. An heat map is a form of 2D data vizualisation which is based on a spectrum of colors. Thedarker the color is, the more important the value is. The idea here is to put in rows the final ranking, and in columns the number of shots missed. Here is what we obtain:

There results directly show that:

• There is a clear diagonal on the heat map. This isn’t really surprising: that means that everytime an athlete misses a shot, his final ranking goes lower. This is our first result: missing shot are penalties. What a surprise!
• There is also a very dark blue area in the first column, at the top of the diagram. This means that most of the time, doing a clear round leads to a very good ranking in the end.
• But it is clearly possible to win a race while missing some shots! The first row is filled with dark blue in the first few columns.

Don’t miss a prone shot!

As you may know, there is two different types of shooting during a biathlon race: the prone shot, where the athletes are lying on the floor; this position helps them to stabilise their aim. The other type is the standing shot, which is much more difficult. Therefore, it might be interesting to deal with the two phases of shooting separately. Let’s start with the prone shot, as this is usually the first phase of shooting during a race.

We see the same pattern as the total of shots. The top of the first column is much darker than before: this is because of two things. First, it is usual that a lot of athletes don’t miss a shot during the prone shot phase. And that means that missing a prone shot is much more a sign of a bad shooting, which leads to a bad ranking at the end. This point is very important: we’re not evaluating results in a vacuum: and missing a shot usually means that the athlete is in a bad shape compared to the others, and therefore has a bad ranking. But this also means that missing a shot during the first phase raise the odds of missing shots during the other phases.

Let’s have a look at the heat map for the standing shots.

As expected (because the initial one is the combinaison of these two heatmap), we’ve a much more dispersed heat map. Missing a standing shot is something that happens to pretty much everyone, even the best athletes.

Is the starting order relevant?

I add to the analysis a last factor: the starting order, which is linked to the expectation of results of the athlete (based on a global ranking, or on the results of another race). The heat map showing the correspondances bewteen the final ranking (still in rows) and the starting order (in columns) shows a clear diagonal line: the expectation seems relevant.

In order to do a much more indepth analysis, I’m going to perform a linear regression on these variables. I want to know if the final ranking is explained by the initial order, the number of prone shots missed and the number of standing shots missed. This linear regression will also help me to evaluate how big of an impact these three variables have on the final outcome. Let’s have a look at the results:

Call:
lm(formula = ranking ~ prone shots  + standing shots + starting order)

Residuals:
    Min      1Q  Median      3Q     Max 
-60.625  -8.483  -0.101   9.268  45.295 

Coefficients:
                  Estimate  Std. Error  t value   Pr(>|t|)    
(Intercept)       5.816571   0.212517   27.37

The three variables of the model are statistically significant, which means that they do have a relation with the final ranking. Understanding the coefficient for the Starting Order is kind of tough, but the two other coefficients are much more easier to analyse:

• When you miss a prone shot, you lose about 5 places at the end of the race
• When you miss a standing shot, you lose about 1 place at the end of the race

Obviously, these results are only valid on average. But this is kind of a fun way to comment biathlon shootings! “Oh, you just lost 10 places!”.

Today, I’d like to write a short note to begin to answer one simple question: which sport is the most conservative? By conservative I mean a sport where the favourite team or athlete wins much more often than the other. As this is quite a large problem, which had been adressed for NFL, I’ll start with some results about some racquet sports, tennis and badminton. I won’t use real data but rather focus on some simulations in order to evaluate the chances to win a match when you’re the #1 tennismen against less talented players.

Game, Set, Match

I won’t explain the rules of scoring in tennis matches, but I’d like to remind you some of the basic principles. To win a tennis match, you need to win 2 or 3 sets. Each set requires you to win at least 6 games, but also two more games than your opponent. And to win a game, you need to score 4 points AND 2 more points than your opponent. These rules mean that a tennis match can last almost forever but as they are very restrictive, they also favor the most efficient player. Let’s have a simple example to understand that.

Let’s say Adam and Becky play a game where they have to score 2 points in 3 rounds. Becky plays way better and has a 90% chance to win each point. The probability of Becky winning is 0.9*0.9 (she wins the first two rounds) + 0.9*0.1*0.9 (she wins round 1, loses round 2 but wins round 3) + 0.1*0.9*0.9 (she loses round 1, but wins rounds 2 and 3), which sums to 97.2%, therefore Adam has a 2.8% chance to win at that game. That is pretty low.

What happens if we add the ‘tennis’ rule of “2 more points than your opponent” ? The game becomes “the first one to score 2 points in a row win”. The maths needed to get Becky’s changes of winning are a little more complex in this example: one way to deal with that problem is to think about the first two rounds. After these two rounds, there is 81% that the winner is Becky, 1% that it is Adam, and 18% that no one has won yet. If we repeat this process, we get a geometric series which sums to approx. 98.8%. Adam’s chances of winning have dropped even more: only 1.2% now!

Let’s go back to Tennis

Now Becky and Adam are tired of playing a game Adam always loses, and they decide to go outside in order to play some tennis. Suppose that Adam is better at this sport than Becky. A conservative hypothesis is to suppose that he’ll win 51% of the time, while she’ll win only 49% of the time. The difference is fairly minor, and that shouldn’t have a big impact, right?

I write a short program in R which simulates thousands of matches in order to know who wins between the two players. The program is quite simple, here is the part about simulating a game :

 win_a_point <- function(advantage) {
  x = runif(1,0,1)
  if (x > advantage) {return(2)}
  else {return(1)}
}

gagne_jeu <- function(advantage) {
  score = c(0,0)
  while ((max(score) < 4) | (abs(score[1]-score[2]) < 2)) {
    x = win_a_point(advantage)
    score[x] <- score[x] + 1
  }
  if (score[1] > score[2]) {return(1)}
  else {return(2)}
}

The results obtained for 100.000 matches (in 3 winning sets) simulated are compiled in this tabular:

Another interesting result is the following graph:

Even if the edge for the better player is as low as 5 percent (55-45 for each point), the match is clearly one-sided. This means that as long as one player have a predominant position and so a little more chance to win each point than his opponent, he is almost assured to win at the end. That makes tennis a very conservative sport!

What about Badminton ?

In order to compare this result to some benchmark, I run the same simulations about Badminton. The rules are quite different but not so much, as it is required to get to 21 points in order to win a game, with at least a two-points lead. Winning a match requires to win two games before your opponent does so. With a small adjustment, I use the same program as before and the results obtained are compiled in this graph:

To conclude, compared to Badminton, Tennis is definitely a conservative game!