On December 2, 2009, the Scottish club Celtic beat the Israel side Hapoel Tel-Aviv 2-0 in the group stage of the Europa League (Europe's second-tier club competition). Despite the victory, Celtic was eliminated and Hapoel moved on.
In the summer of 2015, the Scottish Cup champions will play a pair of second-round qualifying matches for the Europa League. They will be spared the indignity of entering in the first round. (If the Scottish Cup winners also qualify for either the Champions League or Europa League due to their placement in the league, this statement applies to some other club, but let's ignore that for now.)
How are these two things related? Well, Scotland is the 23rd-ranked European country in terms of club football for the purposes of the 2015/6 Europa League, and Sweden is the 24th. The 23rd-ranked program sends its cup champion to the second round, while the 24th-ranked nation's cup champion has to play an extra home-and-away series. The ranking is based on the 5-year span from 2009-2014, and (for reasons I'll explain later), the Celtic win contributed 0.333 to Scotland's coefficient for the year, which affected the next five years. Scotland's ranking for 2015/6 is 16.566, while Sweden's is 16.325. Without that "meaningless" Celtic victory back in 2009, Scotland's ranking would have dropped to 16.233 and Sweden's Cup winner is the one who would have the first-round bye. (Israel would have also leapfrogged Romania into 16th place had H. Tel-Aviv won, but the 16th and 17th placed nation's teams are treated equally in next season's European football.)
I started thinking about UEFA coefficients in detail after reading an article about Celtic's new manager, Ronny Delia, and his complaints about having to enter at the second round of Champions League qualifying. "It is stupid the Scottish coefficient is not regarded as good enough," he said. "We maybe have one good team but the other ones are dragging that team down."
That got me to look at how Scotland's coefficient is calculated, and the extent to which other teams, are, in fact, dragging Celtic down. The specific complaint Delia had was about the 2014/5 season. For that season (the current one), the rankings are determined based on play from 2008-13. Scotland's ranking is 15.191, good enough for 24th. If I added correctly, Celtic produced 8.9 of those points, or more than half, even though they were one of as many of 6 teams whose performances were averaged together.
If you consider Celtic as its own nation, they would have had a UEFA coefficient of 43, good enough for 10th place on the list. The 10th-place nation's champion proceeds directly to the group stage, skipping the three qualifying rounds and the playoff round. I think Delia is arguing that would be a fairer fate for Celtic.
But, wait! How did Celtic acquire those (hypothetical) 42 points? In 2008-09, they picked up 7 points for their participation in the Champions League group stage, but failed to advance. In 2009-10, they were knocked out of the Champions League in the playoff stage (after advancing through one qualifying round), which bounced them into the Europa League, where they lost in the group stage. Despite not making it to the Champions League group stage, nor advancing in the Europa League, they got 7 points, the same as the previous year. In 2010-11, they were knocked out of both the Champions League third qualifying round and the Europa League playoff round, but picked up 2 points by winning one leg of each round. In 2011-12, they got 7 points for their performance, which saw them fail to advance past the Europa League group stage. In 2012-13, their most glorious recent year, they got 20 points when they advanced to the round of 16 in the Champions League. Four of those points were for victories leading up to the group stage.
So despite the fact that those 42 points means Celtic "deserved" to be in the Champions League group stage, many of those points (19 of them) came from parts of the competition below the CL group stage. On the other hand, if we magically place Celtic in the CL group stage every year, and assume they lose every game in years where they didn't actually qualify, they still earn 4 points/year for CL group stage participation, which gives them a total of 31 points, good enough for becoming the 13th-ranked "nation," and yes, automatic entry in the group stage.
There are two kinds of feedback going on here. The positive is that being a highly-ranked nation makes it more likely that your teams will end up in the group stage, where they earn points for being in the group stage. The negative is that higher-ranked nations have teams play in fewer matches (by skipping qualifying), so they have fewer opportunities to get points. I'm curious how these feedbacks balance each other out. I suppose if they didn't, we would see a lot of oscillation in national rankings. I'm not sure if that happens or not.
I think there's an opportunity for an interesting mathematical analysis of the way in which the UEFA coefficient ranking differs from an ideal ranking. Assuming you want to stick with the UEFA rate-the-country approach, there's probably still a better way. Searching the mathematical literature, I found one 2005 paper on the UEFA coefficient. I will go read that paper, but there's probably room for further research.