A reader recently proposed an idea that struck my curiosity: “How does a fencer’s performance in their pools correlate with their overall performance in a tournament?” Sometimes, people’s performance after pools goes as expected, beating opponents lower in the pool results in DEs before finally losing to a strong opponent who did better in pools. Much of the time though, things don’t go as expected. A fencer may only win one or two pool bouts, then go on and get a medal in the event! Or the fencer seeded 2nd after pools loses their first DE to someone who only won one pool bout. How often does this happen?

## Correlation between pool result and final result

I looked at every national event between 2017 and 2020 (besides Div 1 events, which have two rounds of pools), and compared the fencer’s pool result to their final result. Instead of comparing their exact place (eg. #20 to #112), which doesn’t account for varying event sizes, I used their percentiles. For example the fencer who placed #23 out of 23 fencers would be in the 0th percentile, and a fencer who got #60 out of 120 fencers would be in the 50th percentile. Then I placed these values on a scatterplot. I used a rainbow gradient to color them based on their pool results, with pink colored dots representing fencers who won all of their pool bouts, turquoise representing fencers that won half, and the red representing fencers who won none of their pool bouts. Here was the result:

Fencers in the bottom 20% of the pool results almost always place the same in the final results. This is because in many events, the bottom 20% are cut and cannot proceed to DEs, so they are guaranteed to get the same place as their pool results. Even in events without the 20% cut, these fencers still almost never place in the top 25% in the final results.

For the most part, fencers in the top 20% are protected from placing below the bottom half. Since they have such a high seed going into DEs, they won’t end up too low even if they lose their first DE. For example, if a fencer seeded #1 loses their first DE in the table of 128, they’ll still end up in 65th place.

Starting at about the 40th percentile and up, these fencers have a shot at placing high in the event. Although the majority of them still end up placing about the same as their result in pools, a bunch manage to make it to the final rounds.

The correlation in this graph was 0.916, which means that pool results and final placing is very well correlated. This means that 83% of the final results can be explained by the pool results.

## Correlation between initial seeding and pool results

Unlike the pool results and final results, which are highly correlated, the initial seeding and pool results are not very correlated. The correlation is only 0.69, so only about half of the pool results can be explained by the seeding.

Low seeds have a much higher chance of finishing above their pool result than the high seeds do at finishing lower than their pool result. In fact, the high seeds rarely finish below the bottom half after pools, whereas it’s not uncommon for a low seed to finish around the middle.

The “dots” that you see in the diagram are because of the “snaking algorithm” that makes pools. Those percentiles in the initial seed (~25%, ~55%, ~85%) are the ones who almost always get placed in a pool of 6, rather than a pool of 7 like everyone else.

## Correlation between initial seeding and final result

Interestingly, the seeding and final results are slightly *more* correlated than the seeding and the pool results, with a correlation coefficient of 0.707. This is probably because a high seeded fencer can mess up in pools, but then make a comeback and place where they are supposed to, so the placement in end bears slightly more resemblance to the initial seeding.

The coloring remains the same as the first scatterplot, with pink representing fencers who won all of their pool bouts and red the fencers who won none. Of those in the top 20% of the final results, almost all of them won all of their pool bouts or lost at most one.

## Weapon specific graphs

I did make weapon specific graphs, but they don’t look all that different, so I didn’t include them in this post. Although epee has more variability than foil and saber, this is expected considering that epee bouts are closer, and there’s also more upsets in epee.

## Conclusion

Let’s say that the model for tournament performance is*Overall result* = *Fencing skill that day* + *Matchups at tournament* + *Luck*

Since matchups at a tournament are partially due to pool placement, we can rewrite this equation to be*Overall result* = *Fencing skill that day* + *Pool placement* + *Luck*

Is doing well in pools the key to success at tournaments because it sets up you for an easy path in DEs? Or is it that fencers who are having a good day do well in pools and also perform well in the overall result?

This is a complicated question, and it is impossible to answer from this data. This is because *Fencing skill that day* and *Pool placement* are endogenous variables: they are highly correlated with each other. Therefore it is impossible to isolate the impact of each variable individually on fencer performance. However, given that doing well in pools gives an easier path in DEs, fencers should of course try their best to maximize their pool performance to optimize their chance of winning.