The first few days of games at the 2018 Roller Derby World Cup in Manchester, England were determined by a lot of seemingly complicated math. Many teams and fans wondered why the math was necessary and how it worked.
The announcers seemed perplexed too.
So, here’s the how and why of the math for people who love roller derby and hate math.
One of the main purposes of a sports competition is to find out who is best.
Sadly, there’s nothing “simply” about “best”—at least, not when you have more than a few competitors. If there were only two roller derby teams in the world, they would play each other and the winner would be the best. Figuring out who is best among two teams requires just one game.
But three teams require three games, four teams require six games, and five teams require ten games. Every time you add teams, the number of games required increases by way more than the number of teams you add. When you have thirty eight teams, as in the 2018 Roller Derby World Cup, you need 703 games if you want every team to play every other team.
Everyone plays everyone is the best way to find out who is best, and it is how most sports leagues work. As the World Cup is in Manchester, we’ll use football’s English Premier League as an example. That league has 20 teams who play each other twice, once at their own stadium and once at the other team’s stadium. That requires 190 games and a nine month season.
But the system of everyone plays everyone (it’s called a “round robin”) does not work if you have a lot of teams and only a little time. If the Roller Derby World Cup was a round robin, and it played 30-minute games, it would need 4 tracks operating simultaneously 8 hours a day for 11 days to play its 703 games.
One way to solve this problem is an elimination tournament. That’s where teams that win play other winners and losers go home until only one team remains.
Elimination tournaments are common in many sports. So why not make the Roller Derby World Cup an elimination tournament? First, because half the teams would only play one game. Second, because the final places of all the teams but the winner will depend in part on the luck of the draw. The second best team might be eliminated in the first round.
Some tournaments try to fix this by literally fixing things—deciding in advance who the top competitors are and arranging to keep them apart for as long as possible. Another famous English sports competition, the Wimbledon tennis tournament, works this way.
This system kinda works in tennis because tennis players play each other all the time, not just at Wimbledon. A top tennis professional plays 70 to 80 matches a year, mostly against other top tennis professionals. That provides lots of data for comparing players and fixing an elimination tournament. Without that data, elimination tournaments are likely to be unfair.
A third possibility is to mix the two systems by starting with round robin tournaments, then moving to an elimination tournament. That’s how the football World Cup works. A few top teams are picked to be leaders of “groups.” The rest of the teams are placed in the groups by a random lottery-like draw. Each groups plays a round robin, then the winners enter an elimination tournament. But there’s a problem: the best teams may be in the same group, preventing some from progressing to the rest of the tournament. In football’s World Cup, this group of unfortunates is called “the group of death.”
This is how the 2014 Roller Derby World Cup worked and its group of death was Group 6, which contained 4 of the strongest teams in the tournament: England, Ireland, Germany, and Spain. Germany and Spain were eliminated early by the luck of the draw, despite being the 10th and 15th best teams in the world.
This brings us, finally, to the Roller Derby World Cup’s “Power Rating” system. This tournament has something more fair and flexible than groups and draws. The first two rounds of games were decided in part by a survey of who each team wanted to play, but also by a design that connected every team to every other team by gameplay, so the teams could be compared. After each team had played two games, all the teams formed a loop. USA had played Spain who had played Brazil who had played Denmark and so on, all the way round to Wales who had played France who had played USA.
Comparing the score differences between the teams gave a rough idea of who was stronger and who was weaker. This rough idea was called a “Power Rating,” and at the end of day one the question of “who is best” was starting to get answered. The ratings were not super accurate after just one day of 30 minute games, of course. But the first rough sort was not bad.
Every game after that just made the ratings more accurate — not just for the two teams who played, but for every team in the tournament.
It didn’t even matter who played the games. All games helped to make the Power Ratings more accurate. Because the teams were all connected, every result said something about every team. For example, if Team A did better in a game than the Power Rating predicted, that didn’t just improve Team A’s Power Rating; it improved the Power Rating of every team that had beaten Team A, because Team A was stronger than we realized.
The most important thing for teams and fans to worry about is the ratio of points scored to points conceded. That’s what drives a team’s Power Rating. If a team is winning 200–2, it has a ratio of 100:1. But if it eases up in the last jam and concedes 8 points, the score changes to 200–10, and its ratio falls to 20:1—which is five times worse. That’s a beautiful thing about the Power Rating system. Even in a blowout game, every point makes a difference, in every jam, every time.
The math only seems geeky and complicated because we are used to round robin leagues and elimination tournaments. Those approaches are simpler and easier to understand but they are not as good. The Roller Derby World Cup’s Power Ratings system reduces the impact of luck, ensures all teams get lots of games and, compared to drawing balls from a fishbowl, or playing for nine months at a time, is a fairer, more objective, more efficient way to answer the question of who is the best roller derby nation in the world. You may not know who will win on Sunday, but you can be sure they will not only be able to call themselves the best in the world—they will actually be the best in the world.
Regina Lampert writes blog posts for Frogmouth, among other things. John May is a professional mathematician who likes roller derby.
