World Cup draws and probabilities

How do constraints affect the chance teams will face each other?

The televised World Cup draw selected groups for the men’s football competition. There are three good properties for a drawing system to have:

• Fairness: no team should have greater chances than others of a ‘tough’ group.
• Balance: the groups should be at a similar competitive level.
• Uniformity: the probability of each valid outcome should be the same.

Amongst drawing procedures, balance implies fairness.

In the World Cup draw, the top seven ranked teams and the host sits in the first pot. Based on ordered ranking, nations go into the other pots. The fourth pot also contains one European and two international placeholders.

The FIFA sequential method goes through the following procedure:

• Step 1: Place the host (Qatar) into Group A.
• Step 2: At random, draw from the first pot. The teams then go into the next available group, starting with Group B.
• Step 3: Once the first pot is empty, draw from the second pot. Each team goes into the next open group, from groups A to H. No group should have more than one team from non-European confederations. Europe has 13 teams in the World Cup. As such, there should be either one or two UEFA teams in each group. Five groups will have two European teams.
• Step 4: Continue until the fourth pot is empty, finishing the draw.

(Image: FIFA/Getty Images)

Using the FIFA method, the chances are not uniform. For example, the United States team is in the second pot. As part of the CONCACAF confederation, there are no geographical constraints. None of the seeds are part of the same confederation.

That means the probability of the United States going into Group A with Qatar is one in eight (12.5%). Of the valid arrangements, around one in 11 (9%) have the United States and Qatar in the same group.

Sequential drawing by pots, plus a fixed host, into groups induces distortions. Others recognised such difficulties in earlier World Cup draws.

There are different potential solutions, to respect uniformity. Prof Jeffrey…