## On The Constrained Optimization Of World Cup Draws

In a few hours as I write this, a group of people will oversee the draw for the World Cup that takes place later this week in Qatar. 29 of the 32 teams have already qualified, and there remain several games that will determine the rest of the teams, sometime within the next 3 months or so, with Costa Rica playing New Zealand for one spot, Wales playing the winner of Scotland and Ukraine for a spot, and Peru playing the winner of Australia and the United Emirates for the other spot. Given that these results are not known yet, all 3 of these spots have been relegated to the lowest pot and likely the last that will be filled. There are also a great deal of rules that apply to which teams can fill one of the four spots in each of the eight groups of the World Cup.

These rules are worth talking about a bit because they present a great deal of constrained optimization that makes it more likely that certain teams and certain combinations will be forced together. For example, no continent except for Europe may have more than one team in a given group, and there may be no more than two European teams in a given group, which means that 3 of the groups are likely to have one European team and the other five will have two, with the rest of the spots being filled by the rest of the continents, with no more than one team from each federation in them. Given the fact that three of the spots must be left open for the last, one of the group 4 contenders will be a second European team in a given group, one will be in a group that has no other North American teams, and the other will be in a group that has no other South American or Asian teams to keep the groups straight. How does this constrain the options that exist?

Let us look at this based on the various pots. Pot one contains the following teams: Qatar (which will be in Group A as the host nation), Brazil, Belgium, France, Argentina, England, Spain, and Portugal. From this we can see that 5 of the groups are guaranteed to have a European team as the first seed, and the other three teams can either contain no Asian team (group A), or South American team (whichever group Brazil and Argentina lead). It may be guessed, moreover, that the five groups with European leaders may also be considered likely to be the groups that contain an additional European team, as that makes the math work out particularly elegantly, though it need not necessarily work out that way based on the other constraints.

When we look at pot two, we see that it contains five European teams (the Netherlands, Denmark, Germany, Switzerland, and Croatia), two North American teams (Mexico and the United States), and one South American team (Uruguay). We know that Uruguay cannot be in the same group as Brazil or Argentina, and so must be in one of the other six groups. There are no other restrictions, but based on the numbers it appears fairly likely that by the time the second pot is filled out that there will already be at least some of the groups that already have filled their allotment of European teams once the second pot is filled, and that the Netherlands, Croatia, Switzerland, and Germany are going to be some very tough teams for any division leader to face looking for that spot to the knockout round.

There is a bit of clarity once we look at pot three, which features three African teams (Senegal, Morocco, and Tunisia), three Asian teams (Japan, Iran, and South Korea), and two European teams (Poland and Serbia). We know, for example, that the three Asian teams will not be playing against Qatar, and so will be in three of the other groups–most likely those groups that already have met their quota of European teams or that have one European team and also the last of the European qualifiers from pot 4. Similarly, we know that the two European teams in pot 3 will have to be in groups that only have one European team already and that will not have the European team in pot 4, meaning that once pot 3 is done it is likely that there will be few options left for the last European qualifier to fit in based on the other constraints.

Finally let us come to pot 4, which contains the most uncertainty in terms of which confederations the teams can fit into and therefore is the most constrained of all in terms of where the teams go. This pot consists of two African teams (Cameroon and Ghana), one European team (Wales/Scotland/Ukraine), one or two South American teams (Ecuador and possibly Peru), one or two North American teams (Canada and possibly Costa Rica), one or two Asian teams (Saudi Arabia and possibly Australia), and zero or one teams from Oceania (possibly New Zealand). These possibilities force the most constraints. For example, neither Canada nor the winner of New Zealand and Costa Rica can play in the groups with the United States and Mexico, so they will take the last two spots in two of the other groups. Similarly, neither Saudi Arabia nor the winner of Australia and Peru will be able to play in the groups with the other Asian teams, limiting which groups they can fit in with. Likewise, the last European spot cannot go to a group that already has two European teams, which means that more than half of the groups are going to be eliminated as a choice. Also, neither Ecuador or the winner of Peru and Australia will be able to play in a group that has one of the other three South American teams in it.

Once we see pots one and two chosen, a lot of the other pots will already be constrained. Two of the groups will be eliminated for North America and Oceania teams, three of the pots will be eliminated for the South American teams as well as the last possible Asian team, and one of the groups will be eliminated for one of the possible teams from Asia and South America. By the time that pot three is taken care of, these constraints will become even more restrictive, as we have already discussed, and it may be possible that some of the teams will be penciled in because there are no other options, sudoku fashion.

What are the ideal groups for different teams when we look at these constraints? For a team like the United States in pot one, it would perhaps be ideal for the United States to have a European group A team that would allow for clean play and play to the strengths of the United States in terms of counterattacking and somewhat stout defense. A team like England or Belgium, for example, would be a solid one for the United States to face, as opposed to a team like Brazil or Argentina that may try to outscore us 4-0 or something like that, and as long as corrupt referees aren’t an issue, facing a team like Qatar wouldn’t be impossible to manage either. Similarly, the United States would probably enjoy facing an Asian squad from Group 3, or really any of the available squads from group 4.

When we look at the results, it is also pretty obvious what a group of death would likely be like: any group that has Germany or the Netherlands in it, as well as any of the teams looking to challenge from the other ranks–like one with Costa Rica in it, assuming they beat New Zealand, or the challenging African squads from Group 3. A group that contained, for example, France, Germany, Senegal, and Costa Rica would be a bloodbath, as any of those four teams would normally expect to be able to challenge for a spot. On the other hand, a group containing Qatar, Mexico, Poland, and the winner of Scotland, Wales, and Ukraine might have all of its teams thinking that it had a solid chance of advancing to the knockout rounds given the less than dominating competition. Much will depend, as is often does, on matters of perspective.