The following question appeared in The Knowledge in the Guardian (see bottom of this page):

“Has any team ever managed the statistical feat of holding a better head-to-head record against each of the other teams in their Champions League section and yet still finished bottom of the group?” muses Mark Wilson. “Presumably home 0-0s, away score draws and some terrible luck would be required but it is a possibility.”

By Team A ‘holding a better head-to-head record’ against Team B, Mark Wilson presumably means that if the scores were added together over the two games, Team A either scored more goals than Team B, or else they scored an equal number of goals but scored more of them as ‘away goals’ (i.e. goals scored in the away game). In particular, this means that Team A can beat Team B ‘head-to-head’ without actually beating them in a match. For instance, a 0-0 draw at home and a 1-1 draw away is enough.

Recall that groups in the Champions League are ordered first by points, and then by the restriction of the league to tied teams. If that is still equal, then goal difference and then goals scored is used – first in the mini-league and then in the whole league.

One way this can happen is for Team A to draw all their home games 0-0 and all their away games 1-1. Then if the rest of the games are home wins, Team A will finish bottom (with 6 points to the other teams 9 points) despite beating the other three teams head-to-head.

There are other possibilities too. For instance, the above still works with the modification that they lost to Team B 1-0 at home, but beat them 2-1 away. Then they have 8 points, still beat everybody else on head-to-head, and still finish last. Indeed, they can lose all their home games 1-0 and win all their away games 2-1 against all the other teams and still finish last. In this case, all teams will finish on 9 points and so the tie-break is goal difference and then goals scored. As Team A has a goal difference of zero, if we want them to finish last the other teams must also have a goal difference of zero (as goal difference sums to zero). By making the games between Teams B, C and D goal-rich but symmetric, we can make Team A finish last.

The final case is more complicated, but it is possible to win/draw against two of the teams and draw/draw the last. To be concrete, let us say Team A beat Team B head-to-head by a 0-0 draw at home and a 1-1 draw away. Against Team C and Team D, they lost 1-0 at home and won 2-1 away from home. This means Team A got 8 points. If , as in the previous examples, all other games were home wins, then Team B would have 8 points and Team C and Team D would have 9 points. This means that Team A and Team B finish with the same number of points, and so it goes to head-to-head record, which Team A won. Thus Team A did not finish last. The solution is for Team C and Team D to actually draw against each other in both their matches. Then each team finishes with 8 points, and again goals scored can be used to make Team A finish bottom.

Has it ever happened in the Champions League? The answer is no. Groups were introduced in 1991-92 and in that time no team has ever even finished last without losing more games than they did win, which makes this situation impossible.

In the 1998-99 Champions League, which was won by Manchester United against Bayern München in the last moments, there was however a very near miss. It was group B, which contained Juventus, Galatasaray, Rosenborg and Athletic Bilbao (of Italy, Turkey, Norwary and Spain respectively). The group finished in that order, letting Juventus qualify for the next round. The top three teams were all level on eight points, and Athletic Bilbao had six.

Nevertheless, Athletic Bilbao held Juventus 1-1 in Turin and 0-0 in Bilbao, beating them head-to-head. They also made up for their 2-1 loss in Turkey by beating Galatasaray 1-0  on Spanish soil – again, winning head-to-head. Unfortunately, their 1-1 home draw with Rosenborg was met with a 2-1 loss in Norway, and so they did not beat Rosenborg head-to-head. So they beat the top two teams (individually) on head-to-head, but not the third place team.

As the top three teams finished on equal points, their league position was decided upon the matches only including them. That is, games without Athletic Bilbao. In this sub-league Juventus came top and so qualified. However, suppose that Rosenborg vs Athletic Bilbao was 2-2 instead of 2-1. Then Athletic would have finished with 7 points, not 6. Rosenborg would have finished with 6, not 8. This wouldn’t be a solution to the posed problem, as then Athletic did not come bottom of the table. What it would do, however, is make Galatasaray go through instead of Juventus.

Galatasaray actually beat Juventus on head-to-head. It was 2-2 in Turin and 1-1 in Istanbul. Therefore, if only Galatasaray and Juventus were equal on points, then Galatasaray would have gone through instead of Juventus. Doesn’t this seem ridiculous? It seems like a fairly non-local effect of a goal.