Elo Rating System - The Probability That Son-Goku Could Have Won Against Master Roshi
7 min read
I have briefly introduced Elo Rating in the context of comparing national football teams in the last post. However, just as much as I’m not a big fan of chess, I’m also not the biggest football fan. So in this post, I will instead explain the method on a personally more fun topic: the World Martial Art Tournament from Dragon Ball.
In order to not do Dragon Ball any disservice, I won’t even try creating drawings of our beloved heroes. Instead, I will mix in pictures from the anime. Let the fun begin!
As usual, I want to avoid boring you with math formulas and technical terms. The following will be a very simplified version of the Elo rating system. You can check out the more complex mathematics behind it on Wikipedia.
What Even Is This Weird Tournament?
The World Martial Arts Tournament occurs every 3 years and features the Earth’s strongest fighters (there are also aliens - it’s a whole universe, it’s complicated, let’s move on). The tournament has 2 parts: the Elimination Round and the Finals. In the Elimination Round, the contestants are assigned randomly into 4 blocks. These blocks are set up like mini-tournaments, with the 2 top fighters moving on to the Finals. The Finals consist of the quarter-finals, semi-finals and the final fight.
We want to know the fighters’ chances of winning against their upcoming opponents. What is the best way to determine them? Many contestants are taking part for the first time, they don’t have any track records yet. Others have fought before, however the last tournament was 3 years ago. No one knows from the other contestants how they have trained in the meantime. And many fighters may have not fought against each other in the past. So even with a track record, it’s not necessarily certain who is more likely to win.
The Elo rating system is designed to start out with assumptions based on available information and then update those with new information. We’ll have a look at it with a practical example - let the tournament begin!
Welcome to My Dragon Ball Universe
Let’s imagine that by winning matches, fighters gain dragon balls and the number of balls represents their rating. When a fighter loses a match, they have to give away balls to the winner. If they don’t have any balls left to give away, they instead get black balls indicating they are in the negative. In my dragon ball universe, people are running around with huge sacks of balls - it’s my imaginary world, have fun!
In our examples, we’ll focus on our hero Son-Goku.
It’s his first tournament, so he doesn’t have a track record yet. In the Elo rating system, he is assigned an initial rating of 0. In our dragon ball universe, he gets a blank dragon ball. You’ll see in a bit why that is.
Goku’s first match is against an experienced fighter. However, he wasn’t that lucky in the past and thus has 10 black dragon balls.
Before the fight, we have to make a guess on the outcome. As Goku’s rating is higher, the probability of him winning is higher too. To calculate it, we take the difference of the two ratings.
Imagine the rating numbers are simply symbols on dragon balls:
As you can see, we have 11 dragon balls in total - the blank dragon ball representing a zero rating counts as 1. We now allocate those balls to the fighters, starting with the lowest rating. Goku’s opponent gets 1 dragon ball, simply because he has the lowest rating. Goku gets the rest of the balls. His opponent has 1 out of 11 balls and Goku has 10 out of 11 balls - those are their probabilities of winning the match. To clarify: the fighters are not actually getting the balls - yet!
As expected, Goku wins his first match easily and we can update each fighter’s rating. Let’s look at the dragon balls again. As the match turned out as expected, Goku gains only 1 ball - the one allocated to his opponent. Reversely, the opponent loses exactly that 1 ball. So Goku now has 1 regular dragon ball, i.e. a rating of 1, and the opponent now has 11 black balls, i.e. a rating of -11.
Wait! Where’s the blank ball now? As soon as a fighter gets a rating above or below 0, we drop the blank ball. Sounds weird? Yeah, 0 is a weird concept, and this blog post doesn’t have the space to dig deeper into its weirdness. If you’re interested in that, I can highly recommend the book “Zero” by Charles Seife - it’s less mathematical than you would expect, so give it a try! But first, please finish reading this post.
No Rest For the Weary
With his freshly gained ball, Goku goes straight to his next fight. This time his opponent is much stronger, at least based on the size of his sack, which contains 100 dragon balls. Let’s play with the balls again.
We have exactly 100 balls: 1 from Goku and the difference of 99 from his opponent. As Goku has the lower rating, he only gets 1 ball allocated and his opponent gets the other 99.
What we now witness is how the Elo rating system adjusts for incorrect ratings, and that it’s possible to advance quickly as a newbie. Goku again wins the fight, even though based on the ratings the probability for that was very low. As reward, he earns all 99 dragon balls from his opponent, updating Goku’s rating to 100. On the other hand, his opponent lost a supposedly sure fight, which gets punished accordingly by reducing his rating by 99 to 1.
Fast Forward to the Finals
Goku made it to the finals and he has to fight Jackie Chun. Dragon Ball fans know of course that this is Master Roshi in disguise, Goku’s very own trainer.
It is known that he won the tournament once before, about 50 years ago. So we can assume quite a high bag of balls for him - let’s say 2000 balls. By fighting his way to the final match, Goku increased his rating as well and he is now at 500. That means we’re looking at 1501 dragon balls, 1 allocated to Goku and 1500 to Jackie Chun.
And as known among Dragon Ball fans - Goku loses.
Based on the ball probabilities, this is not surprising. Goku has to give only 1 dragon ball to Jackie Chun. What you can see here very well is that the rating doesn’t change that much if the actual outcome is close to the predicted one.
Let’s just assume Goku would have won unexpectedly. This exact unexpectedness would have been generously rewarded by Goku receiving 1500 dragon balls - which, on the other hand, would have been a severe punishment for Jackie Chun, who would have had to give the 1500 balls away, leaving him with a much smaller bag.
Getting Your Balls Together
Now you have seen the logic behind the Elo rating system. As mentioned in the beginning, this was a simplification of the actual mathematics behind it. You can check it on Wikipedia if you’re interested in more complex math formulas. Otherwise, I recommend you watch some anime - definitely more entertaining (and that’s coming from a mathematician!).
Hi, I'm Nadine. I empower people through comprehensive training and coaching in data analysis and mathematical modeling, equipping them with the tools and knowledge to excel professionally. If you’re interested in finding out how I can support you in your learning journey, book a free 30-minutes introduction call with me right here, send an email to nadine@mathemalytics.com, or connect with me on LinkedIn.