[Mar 6, 2023] Nerd-sniped by beatboxers

TLDR: If you scored 25th in the first round of wildcard solos, it’s still mathematically possible for you to make it to GBB (Top 11 overall), but impossible for you to get 1st place overall (after the 2 rounds of wildcards). If your first round was at least as good as 24th, 1st place overall is possible. No comments about probability… yet.

Grand Beatbox Battle is here

For beatboxing, it’s that time of year again. SwissBeatbox (SBX) has released the wildcard rankings for this year's Grand Beatbox Battle (GBB) 2023 solos (full post). If you're not familiar, SwissBeatbox is a beatboxing YouTube channel that organizes one of the largest beatboxing competitions in the world.
(They had to update these rankings because they apparently messed up their spreadsheet, but that’s another story!)
notion image
Here’s how GBB usually works:
  1. Beatboxers submit a wildcard (i.e. an entry video) before some deadline.
  1. A team of judges evaluate each video based on things like originality, musicality, execution, etc.
  1. Beatboxers get a score for the video they sent, based on the judging.
  1. Some people get upset about the rankings and forget that judging is extremely subjective (why not try predictive optimization?), and forget that the top ranks are all extremely skilled at the end of the day, and sometimes we just need vibes-driven-tiebreakers.
  1. The top ranked beatboxers (usually ≈8) go to GBB to compete!

With a twist

This year, SBX added a second round of wildcard rankings. I’ll spare you the lengthy quote (page). Here’s the gist:
  1. Top 25 from the first round (above) submit a second wildcard video
  1. The judges rank these videos
  1. Your final score (??) is the average of your rank from the first and second rounds.
  1. Your final rank comes from this final score
  1. Top 11 make it to the GBB 23 event
Here’s a small table for an example:
Beatboxer name
Round 1 rank
Round 2 rank
Final score (Average rank, smaller number is better)
Final rank
1 🥇
3 🥉
2 🥈
SBX conflates scores with ranks here (which I do not like), so I’ve assumed that:
  • The final “score” comes from averaging the literal rank numbers from rounds 1 and 2.
  • A lower “final score” is better (i.e. if your average is 1, you’re in first place).
I’ll also assume that ties in the final ranks are valid, as long as it’s still clear who makes top 11 overall. For example, a 25-way tie doesn’t count (see the picture on the right), and we won’t consider that here.
A 25-way tie. Who’s in the top 11? No one knows.
A 25-way tie. Who’s in the top 11? No one knows.

If you came 25th in Round 1, can you still make it to GBB?

For solos, we really just care about the top 25 beatboxers, since they’re the only ones who submit a second wildcard.
If you came last (i.e. 25th place) in the first round, do you still have a chance to make it to GBB? In other words: if you placed 25th in Round 1, is it possible to score well enough in Round 2, so that your average places you in the Top 11 over all?
TLDR: yes.
❌ Case study 1: Everyone’s performance (including yours) is exactly the same between Rounds 1 and 2
This one’s pretty self-explanatory. Let’s see what else could happen to you, based on what happens in Round 2.
Everyone performs the between Rounds, and you’re still at the bottom (just kidding, you’re top 25 in the world).
Everyone performs the between Rounds, and you’re still at the bottom (just kidding, you’re top 25 in the world).
❌ Case study 2: You comes 1st in Round 2, but everyone else is relatively the same
By “relatively the same”, I mean that a scores 2nd in Round 2, b scores 3rd, and so on.
Your final score is 13, and so is the final rank, so you don’t make it 😟.
We can also see that your best possible final score after Round 2 is 13. This happens when you place 1st in Round 2. With this best possible score, was there any hope for you to make top 11 to begin with, provided that others place in just the right (wrong) way? Yes!
You placed 1st in Round 2, but didn’t make it to the top 11 overall.
You placed 1st in Round 2, but didn’t make it to the top 11 overall.
✅ Case study 3: You place 1st in Round 2, and folks k to x have a bad day.
For you to place top 11 overall, it depends on everyone performing in a very specific way during Round 2:
  1. You rank 1st, giving your best final score of 13.
  1. Beatboxers a-j do (almost) just as well. a gets 2nd place, b gets 3rd, and so on. This gives each of them a final score smaller (i.e. better) than your 13, and a rank that looks identical to Round 1.
  1. Beatboxers k-x perform roughly inversely to how they did in Round 1. k places 25th (last), l places 24th, and so on. This gives them each a final score of 18, which is larger (i.e. worse) than your 13, and a 14-way tie for 12th place overall.
You placed 1st in Round 2, and made it into the top 11 overall!
You placed 1st in Round 2, and made it into the top 11 overall!

11th? Why stop there?

We’ve shown that placing 25th in Round 1 is not hopeless. You have hope: there exists a Round 2 ranking where you do well enough to place 11th overall, and make it to GBB.
Next question: what’s the limit to this? Is there hope to make 10th place overall? 9th? What about 1st?
We know two things so far:
  1. You have hope to score as least as well as 11th overall. We showed this in the previous result.
  1. You can’t score 1st overall.
To see why (2) is true, remember that your best final score is 13. Beatboxer a’s worst overall score is also 13 (this happens when they place 25th in Round 2), which means you can never score better than them.
Here’s the short answer: your best possible (overall) rank after Round 2 is 2nd place overall.
Here’s the slightly longer, math-y version. To produce this outcome, beatboxers b-x need to have a final score that’s worse (i.e. larger) than your final score of 13. Beatboxer b achieves this only if their Round 2 rank is 25, giving a final score of 13.5; anything better than 25th in Round 2 gives a final score better than your 13.
You can apply the same reasoning for the remainder of the list.
It’s possible for you to score 2nd place overall.
It’s possible for you to score 2nd place overall.

When can you get 1st overall?

We showed that you can’t get 1st overall if you placed 25th in Round 1. How well do you have to do in Round 1 to have any hope of getting 1st overall?
We showed above that you can’t get 1st overall if you placed 25th in Round 1. This was because your best overall score, and a's worst overall score, were both 13, so you couldn’t beat that. How well do you have to do in Round 1 (it has to be better than 25th, only way is up!) to have any hope of getting 1st overall?
In other words: what needs to happen in Round 1 so that an overall score less (i.e. better) than 13 to be possible?
Short answer: turns out placing 24th is good enough.
Long answer: the final score is an average of your ranks from Rounds 1 and 2. Your best rank in Round 2 is 1st place, so we need an outcome (i.e. rank) in Round 1 that brings the average below 13 (beatboxer a's best score). This can happen if you place 24th in Round 1, because the average of 24 and 1 is 12.5 < 13.
You get 1st place overall.
You get 1st place overall.

Final thoughts

The math in this problem was not complicated at all, and the fun of this exercise lies mostly in reasoning about monotonic properties. For example, when we asked about you making top 11 overall, it was good enough to analyze what happened if you scored 1st place in Round 2.
Why? If we found that it was impossible, even after scoring 1st in Round 2, then no other outcome of Round 2 (you scoring 2nd, 3rd, etc.) would have helped you get to top 11 overall, because they’re all strictly worse than scoring 1st in Round 2! In fact, this is the reasoning we used to say that scoring 1st overall was impossible.
Finally, I gave examples to demonstrate that ranking 11th or 2nd overall was possible, but I haven’t thought long enough to say whether those examples are unique.
For example, maybe there are other Round 2 outcomes that would give you 11th overall, besides the one I shared. I also showed that these outcomes were possible, but did not say whether they were probable. That needs enough analysis (and assumptions) to make a second blog post.