Mention seedings and pots, and your mind might immediately jump to gardening – or, if you’re at all interested in football, you’ll be familiar with these terms as integral parts of the mathematical model behind the FIFA World Cup. If I add in the terms draw, ranking and table, you might understand why, as a mathematician, I’m fascinated by the way this sporting tournament is structured.
Watching the matches is entertaining, but I also love analysing the tournament’s mathematical architecture. One question I like to ask is how might I do things differently – in this case, how could you structure the World Cup so that all 54 teams have a fair chance at winning without having to all play each other (this is called the handshake problem, and for 54 teams there would be 54!/2 matches, which is a 72-digit number).
Where would you start?
A tournament is a mathematical concept as well as a sporting one. It involves this problem: how to combine results of matches with filtering teams towards a winner in a way that is fair, efficient, and entertaining. Most tournaments involve the simplest element of two teams/players playing at one time against each other, and the possible outcomes are limited to win, loss, or draw (with some sort of mechanism – like a penalty shoot-out – for deciding a winner if a draw occurs).
If we wanted to just concentrate on efficiency, we might use a ‘single-elimination’ model: the loser gets eliminated after every match. This means teams only play the minimum number of matches but might not always be particularly fair in terms of the final winner as it doesn’t allow much room for error. It would also mean the total number of teams might have to be restricted to a certain type of number. Can you think why?
You might have thought about even numbers first because every team has to play another team. Then you might have thought about what this structure might look like diagrammatically and eventually come to the conclusion that the total number of teams has to be a power of two (2, 4, 8, 16, 32…) for this to work.
Of course, you can always just add an initial round to the beginning with only a few players or give some players a ‘free pass’ into the second round – but this isn’t fair, and for something like the FIFA World Cup would have fans rioting indignantly.
What about double elimination? This is the same sort of structure but allows for one loss. Many sports have this type of structure: some have repechage, or ‘fastest loser’ rounds, which allows narrowly losing teams/players to play each other for lower positions or re-enter the competition. Some sports, like the Australian Football League, allow players with better records to lose without being knocked out. This feels like a foray into dodgy territory.
FIFA, in fact, uses the ‘group stage round robin’ model, where teams are put into mixed-ability groups who then all play each other. The winners go on to the World Cup; of the runners-up, four will make it through after the playoffs. The World Cup itself also has a group stage, where the top two of each group then go to knockout rounds (a nice round of 16 teams – well planned). It’s a robust model that allows for two of the factors we talked about: fairness and entertainment. It’s certainly not the most efficient – the current one has a total of 851 matches over two years – but if you have the time and the audience, it’s probably not a bad model.
It becomes even more fascinating when you look at the variables used in the FIFA model. In order to decide which ‘pot’ countries go in at the beginning, FIFA calculates their world rankings, and then one from each ‘pot’ goes randomly into a group, meaning that the groups are as mixed ability as possible (teachers will recognise this technique). How they rank the countries before the tournament is therefore very, very important as the success of the whole model, to some extent, relies on this measure.
Again, how might we design this from scratch? If we just look at winning and losing, a strange logical inconsistency can occur – an intransitive relationship. Imagine three teams only – we want to rank them from first to last, and they’ve all played against each other once. If A defeats B, B defeats C and C defeats A, how can we rank them? This is akin to the rock-paper-scissors scenario, where we cannot meaningfully rank these teams from this information alone (this sort of thing is relatively common in sporting scenarios). To combat this, and to try and encompass a wider range of variables, some sports use lots of different measures combined into a formula: these might include the strength of the opponent, the importance of the match, and the points difference as opposed to a straight win-loss-draw model. There is also a noted statistical advantage in playing at home, which is often compensated for (one study has found the average effect in football to be worth 0.5 goals)
One more thing to consider – should we start from scratch every season or year (called a cold start) or somehow take into effect previous performance? What would be the relative advantages of one over the other? Some of the more sophisticated models even weigh this by the parity of the team – the percentage of players residual from the last season. Should we penalise teams for a high turnover in terms of their overall rankings?
The current FIFA model, while taking into account most of these elements, is notably flawed – something Wales has taken advantage of in the run-up to the World Cup by abstaining from friendly matches in order to protect (or inflate) their seeding before the draw. If all countries did this, it wouldn’t work (an example of what’s called a zero-sum game in game theory). How might we improve this formula to prevent this from happening in the future?
This is a re-blog post by Lucy Rycroft-Smith and published with kind permission.
The original post can be found here.
Image credit: By Walther Siksma on Flickr under (CC BY-NC-ND 2.0)