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We might well be in the last football World Cup to feature the four-teams-per-group format in its opening stage. Unfortunately, what will replace it could make the contest less fair.

Gianni Infantino, president of the sport鈥檚 governing body FIFA, has already announced that the 2026 tournament hosted by Canada, the US and Mexico will open with 16 groups of three teams, and that could lead to this being used in 2022 in Qatar.

The mathematician Julien Guyon quickly pointed out that . He is right.

Even the current group format is not immune to collusion to eliminate another team. Two teams playing each other in their last game in the group stage may know exactly what result will let them both advance to the knockout stage at the expense of another team.

FIFA, aware of this possibility, tries to reduce it by making the last group games simultaneous. It acted after claims of collusion when , in an era when the last games were staggered. The result of this match eliminated Algeria from the competition.

Basic calculations show that moving to three teams per group will in principle increase the possibility of collusion in the last game of the group stage. With three instead of four teams, there are three rather than six games in the group stage, which creates more opportunity to be strategic, depending upon previous results. Games will also have to take place sequentially.

What to do? FIFA could make the entire tournament a knockout one. This would solve the problem, but it would mean half the countries in the tournament only playing one game in the World Cup after waiting for four years.

If FIFA insists on three teams per group, though, another suggestion is to . This would eliminate many scenarios that offer some incentives to collude.

Sensible variant

Unfortunately, this is not fair. Winning a shoot-out after a drawn game does not seem equivalent to winning a game. But there is a simple and sensible variant.

Right now, a win is worth 3 points to the winner and 0 points to the loser, and a draw 1 point to each side. So there are games that award a total of 3+0=3 points to the two teams, and there are games that award a total of 1+1=2 points. Put differently, football matches are not a zero-sum game: When moving from a score of 1-0 to a score of 1-1, one team moves from 3 points to 1 point (-2), whereas the other team moves from 0 points to 1 point (+1). One team鈥檚 loss is not identical to another team鈥檚 gain.

So, let鈥檚 treat won games and drawn games equally. Let鈥檚 make them all zero-sum and award the same total of 3 points by giving the winner of penalty shoot-outs in drawn games 2 points and the loser 1 point. There is one important proviso: to be fair, the shoot-outs must follow , rather than the traditional ABAB sequence. My research has shown the latter gives a psychological advantage to the team that goes first.

From the perspective of collusion, with 3-2-1-0 possible points in a match there are now many more combinations of potential points, and this reduces the share of combinations that might induce collusion in the last group games.

As for the problematic combinations that remain, in many of them teams would have to collude not only on drawing but also on the winner of the shoot-out. This makes collusion harder.

Potential for collusion exists with four teams per group and gets worse with three teams per group. It will never be entirely eliminated unless we move to a full knockout system, but would be greatly reduced under the 3-2-1-0 point system. This would improve fairness by treating all games identically, and the shoot-outs in drawn games will make the beautiful game more exciting.