## Wednesday, June 4, 2014

### World Cup groups

I recently read about how the World Cup draw, as currently designed, can frequently create unbalanced groups. Some countries end up in very challenging groups; other groups are much weaker, overall. A French mathematician, Julien Guyon, describes an alternative method that is more likely to result in balanced groups. His method does seem to result in more balanced groups -- although I am always very edgy about measuring a group's strength merely looking at the average strength. One way to achieve an average strength group is by putting very strong and very weak teams together. Another way to achieve an average strength group is by putting  all mediocre teams together. But I would hardly call these two groups balanced. Ideally, we should be looking at each group's average AND its standard deviation. The high-low method will yield a high standard deviation; the mediocre teams methods will yield a low standard deviation, and it would be rejected for this reason. I looked at Guyon's method and found that it did produce groups with high standard deviations, so his method resulted in fairly balanced groups.

I applied my own method to the task to see how it would fare.

Step 1: Pick the eight middle teams.

Step 2: Match the top eight teams to the middle teams. I did so by looking at the standard deviation of each match. If team A had a world rank of 1 and team J had a world rank of 9, this match would have a standard deviation of 4. If team K had a world rank of 11, then an A - K match would have a standard deviation of 5.  If two teams were not allowed to compete in the same group because the teams were from the same region of the world, then the standard deviation was set to zero. A computer picked the set of eight matches that had the largest overall standard deviations. That could mean the single highest match for a particular team was not chosen so that the overall standard deviations would be as large as possible. For example, team A might not be matched against K so that K could be matched against a better opponent country.

Step 3: Repeat for the next eight teams, looking at the standard deviation of the the group of three teams.

Step 4: Repeat for the bottom eight teams.

The groups are balanced compared to each other as a byproduct of trying to force each group to have a maximum standard deviation. My one trial came out very slightly ahead of one trial of Guyon's method; they are quite comparable.