# Four points is not enough

The World Cup just started. As you may know, there are 32 teams, in eight groups of four. Each team in a group plays three games, one against each of the other teams in the group. The top two teams in each group advance to the “knockout round” of 16; teams get three points for a win, one point for a draw, and no points for a loss, so the most points a team can have is 9, the least is 0. (It’s not possible to get 8 points, but every other number is possible.)

So how many points is enough to advance?

With a quick Google I found this article from the Hindustan Times in 2018, saying that traditionally people think four points is enough, but in practice, 17 out of 33 teams with four points were in the top two in their group between 1994 and 2014. (By my count it’s 18 out of 35, which doesn’t materially impact the conclusion.). “Three points for a win” was introduced in the 1994 World Cup, so this is as far back as it’s meaningful to go.

Similarly, in the run-up to the current tournament, Fox Sports Australia writes: “Four points is really that magical mark that they need to aim at. You can miss the top two of your group with four points – 10 teams have done it across the last four World Cups – but the overwhelming majority of teams that reach that figure make it out.”

But if you stop to think about it for a moment, a win, a loss, and a draw (which is the only way to get four points) is a middling result, and you need to be in the top half to advance… this is best illustrated by 1994 group E, where all four teams got four points, and of course only two advanced.

And in Slate a couple days ago, Eric Betts wrote: “One win, one draw and one disheartening loss might be enough to get to the knockout rounds, but not necessarily. Two teams with four points advanced in 2018 and one—Iran—was sent home.” (There’s a slight error here – Argentina and Japan advanced with four points, Iran and Senegal didn’t.)

But if we go through and tabulate all 54 groups in the 1994 through 2018 World Cups (six groups in 1994, eight in each of 1998-2018) we really see that four points is not enough. Here’s a table of the teams by their rank in group and their number of points.

There’s an interesting anomaly here. Teams have not made the top two with six points – in 1994 both group D and group F had the point totals 6-6-6-0, with one team that was beaten by each of the other three, while the other three had a “cycle” of wins. (The third-place team in each of those groups advanced to the knockout round – in 1994 there were only six groups, as opposed to the current eight, so the top four third-place teams also advanced to the knockout round.). But no team has ever failed to advance with 5. (This is possible – a group where team A loses to B, C, and D, and all three games among B, C, D are ties, would have point totals 5-5-5-0.)

So you need five points to win (unless something weird happens.). Four is not enough.