## Sports climbing at the 2020 Olympics

Last year sports climbing made its Olympic debut. My partner Claire is an avid climber and we watched some of the competition together. The method of ranking the competitors had some surprising consequences which are the subject of this blog.

In the finals, the climbers had to compete in three disciplines – speed climbing, bouldering and lead climbing. Each climber was given a rank for each discipline. For speed climbing, the rank was given based on how quickly the climbers scaled a set route. For bouldering and lead climbing, the rank was based on how far they climbed on routes none of them had seen before.

To get a final score, the ranks from the three disciplines are multiplied together. The best climber is the one with the lowest final score. For example, if a climber was 5th in speed climbing, 2nd in bouldering and 3rd in lead climbing, then they would have a final score of $5\times 2 \times 3 = 30$. A climber who came 8th in speed, 1st in bouldering and 2nd in lead climbing would have a lower (and thus better) final score of $8 \times 1 \times 2 = 16$.

One thing that makes this scoring system interesting is that your final score is very dependent on how well your competitors perform. This was most evident during Jakob Schubert’s lead climb in the men’s final.

Lead climbing is the final discipline and Schubert was the last climber. This meant that ranks for speed climbing and bouldering were already decided. However, the final rankings of the climbers fluctuated hugely as Schubert scaled the 15 metre lead wall. This was because if a climber had done really well in the boulder and speed rounds, being overtaken by Schubert didn’t increase their final score by that much. However, if a climber did poorly in the previous rounds, then being overtaken by Schubert meant their score increased by a lot and they plummeted down the rankings. This can be visualised in the following plot:

Along the x-axis we have how far up the lead wall Schubert climbed (measured by the “hold” he had reached). On the y-axis we have the finalists’ ranking at different stages of Schubert’s climb. We can see that as Schubert climbed and overtook the other climbers the whole ranking fluctuated. Here is a similar plot which also shows when Schubert overtook each competitor on the lead wall:

The volatility of the rankings can really be seen by following Adam Ondra’s ranking (in purple). When Schubert started his climb, Ondra was ranked second. After Schubert passed Albert Gines Lopez, Ondra was ranked first. But then Schubert passed Ondra, Ondra finished the event in 6th place and Gines Lopez came first. If you go to the 4 hour and 27 minutes mark here you can watch Schubert’s climb and hear the commentator explain how both Gines Lopez and Ondra are in the running to win gold.

Similar things happened in the women’s finals. Janja Garnbret was the last lead climber in the women’s final. Here is the same plot which shows the climbers’ final rankings and lead ranking.

Garnbret was a favourite at the Olympics and had come fifth in the speed climbing and first in bouldering. This meant that as long as she didn’t come last in the lead climbing she would at least take home silver and otherwise she’ll get the gold. Garnbret ended up coming first in the lead climbing and we can see that as she overtook the last few climbers, their ranking fluctuated wildly.

Here is one more plot which shows each competitor’s final score at different points of Garnbret’s climb.

In the plot you can really see that, depending on how they performed in the previous two events, each climber’s score changed by a different amount once they were overtaken. It also shows how Garnbret is just so ahead of the competition – especially when you compare the men’s and women’s finals. Here is the same plot for the men. You can see that the men’s final scores were a lot closer together.

Before I end this post, I would like to make one comment about the culture of sport climbing. In this post I wanted to highlight how tumultuous and complicated the sport climbing rankings were but if you watched the athletes you’d have no idea the stakes were so high. The climbers celebrate their personal bests as if no one was watching, they trade ideas on how to tackle the lead wall and the day after the final, they returned to the bouldering wall to try the routes together. Sports climbing is such a friendly and welcoming sport and I would hate for my analysis to give anyone the wrong idea.