# What if the UEFA Champions League used proportional allocation?

Last week I wrote about a proportional allocation scheme for the NCAA basketball tournament, drawing on my experience from developing league coefficients for soccer competitions in North America.  I'll apply the scheme to soccer by considering how a similar allocation of "at-large" slots might work for the UEFA Champions League.

Some definitions are in order.  I define an "at-large" slot as one given to a team that is not a league champion.  Every football association with the exception of Liechtenstein has an automatic entry to the Champions League for their league champions — a total of 52 automatic bids.  There are 76 teams that enter the Champions League competition from the first qualifying round to the group stage proper, which leaves 24 at-large slots.

To allocate the at-large slots, I use the UEFA coefficients for each national league over the last five seasons, including the current one.  I use the Hare quota to calculate the ratio used in my allocation, which is the sum of the UEFA coefficients over all the leagues divided by the number of at-large places.  Then I use the largest remainder method to calculate the number of at-large slots to be allocated to each league — the aforementioned link has an explanation of the procedure.

Here is the first results table.  I've listed the UEFA national association (with the exception of Liechtenstein), the UEFA league coefficient, and then the columns leading to the allocation of at-large slots.  The Hare quota for the 5-year unweighted league coefficient is 43.637.

 Country Coefficient Ratio Integer Remainder Allocation Total At-Large Actual At-Large England 84.356 1.933 1 0.933 1 2 3 Spain 78.900 1.808 1 0.808 1 2 3 Germany 68.603 1.572 1 0.572 1 2 3 Italy 60.552 1.388 1 0.388 0 1 2 France 53.678 1.230 1 0.230 0 1 2 Portugal 47.196 1.082 1 0.082 0 1 2 Russia 44.707 1.025 1 0.025 0 1 1 Ukraine 43.550 0.998 0 0.998 1 1 1 Netherlands 39.963 0.916 0 0.916 1 1 1 Turkey 35.050 0.803 0 0.803 1 1 1 Greece 34.166 0.783 0 0.783 1 1 1 Denmark 30.550 0.700 0 0.700 1 1 1 Belgium 27.000 0.619 0 0.619 1 1 1 Romania 25.824 0.592 0 0.592 1 1 1 Scotland 25.141 0.576 0 0.576 1 1 1 Switzerland 24.900 0.571 0 0.571 1 1 0 Israel 22.000 0.504 0 0.504 1 1 0 Czech Republic 20.850 0.478 0 0.478 1 1 0 Austria 20.700 0.474 0 0.474 1 1 0 Cyprus 18.124 0.415 0 0.415 1 1 0 Bulgaria 17.875 0.410 0 0.410 1 1 0 Croatia 16.124 0.370 0 0.370 0 0 0 Belarus 16.083 0.369 0 0.369 0 0 0 Poland 15.916 0.365 0 0.365 0 0 0 Slovakia 14.499 0.332 0 0.332 0 0 0 Norway 14.375 0.329 0 0.329 0 0 0 Serbia 14.250 0.327 0 0.327 0 0 0 Sweden 14.125 0.324 0 0.324 0 0 0 Bosnia-Herzegovina 9.124 0.209 0 0.209 0 0 0 Finland 8.966 0.205 0 0.205 0 0 0 Ireland 8.708 0.200 0 0.200 0 0 0 Hungary 8.500 0.195 0 0.195 0 0 0 Moldova 7.749 0.178 0 0.178 0 0 0 Lithuania 7.708 0.177 0 0.177 0 0 0 Latvia 7.415 0.170 0 0.170 0 0 0 Georgia 6.957 0.159 0 0.159 0 0 0 Azerbaijan 6.165 0.141 0 0.141 0 0 0 Slovenia 6.124 0.140 0 0.140 0 0 0 Macedonia 5.207 0.119 0 0.119 0 0 0 Iceland 4.957 0.114 0 0.114 0 0 0 Kazakhstan 4.374 0.100 0 0.100 0 0 0 Montenegro 3.875 0.089 0 0.089 0 0 0 Albania 3.874 0.089 0 0.089 0 0 0 Estonia 3.791 0.087 0 0.087 0 0 0 Wales 2.790 0.064 0 0.064 0 0 0 Armenia 2.583 0.059 0 0.059 0 0 0 Malta 2.416 0.055 0 0.055 0 0 0 Northern Ireland 2.249 0.052 0 0.052 0 0 0 Faroe Islands 1.416 0.032 0 0.032 0 0 0 Luxembourg 1.374 0.031 0 0.031 0 0 0 Andorra 1.000 0.023 0 0.023 0 0 0 San Marino 0.916 0.021 0 0.021 0 0 0

The proportional allocation scheme spreads out the number of at-large slots from the highest-ranked leagues to the those leagues whose performances are close to the average league coefficient.  This result is actually in line with the characteristics of the Hare quota in proportional representation elections — it favors small political parties at the expense of larger ones.  Most of the domestic leagues whose performances in Europe are close to one sigma above the mean coefficient see little change.

We now consider the proportional allocation for a weighted sum of the UEFA coefficient.  The weighted sum looks like this (WARNING: math):

The rest of the allocation procedure follows as before, and here are the results:

 Country Weighted Sum Ratio Integer Remainder Allocation Total At-Large Actual At-Large England 32.915 1.853 1 0.853 1 2 3 Spain 29.999 1.688 1 0.688 1 2 3 Germany 29.739 1.674 1 0.674 1 2 3 Italy 24.156 1.360 1 0.360 0 1 2 Portugal 22.592 1.272 1 0.272 0 1 2 France 22.491 1.266 1 0.266 0 1 2 Netherlands 18.430 1.037 1 0.037 0 1 1 Russia 18.257 1.028 1 0.028 0 1 1 Ukraine 17.822 1.003 1 0.003 0 1 1 Greece 14.404 0.811 0 0.811 1 1 1 Denmark 11.973 0.674 0 0.674 1 1 1 Turkey 11.750 0.661 0 0.661 1 1 1 Belgium 10.931 0.615 0 0.615 1 1 1 Switzerland 10.538 0.593 0 0.593 1 1 0 Austria 10.119 0.570 0 0.570 1 1 0 Israel 9.359 0.527 0 0.527 1 1 0 Belarus 8.854 0.498 0 0.498 1 1 0 Romania 7.901 0.445 0 0.445 1 1 1 Bulgaria 7.414 0.417 0 0.417 1 1 0 Cyprus 7.276 0.410 0 0.410 1 1 0 Croatia 7.229 0.407 0 0.407 1 1 0 Poland 7.185 0.404 0 0.404 0 0 0 Czech Republic 7.144 0.402 0 0.402 0 0 0 Scotland 7.105 0.400 0 0.400 0 0 1 Serbia 6.211 0.350 0 0.350 0 0 0 Slovakia 5.854 0.329 0 0.329 0 0 0 Sweden 5.220 0.294 0 0.294 0 0 0 Norway 4.850 0.273 0 0.273 0 0 0 Hungary 4.563 0.257 0 0.257 0 0 0 Moldova 3.614 0.203 0 0.203 0 0 0 Bosnia-Herzegovina 3.552 0.200 0 0.200 0 0 0 Finland 3.357 0.189 0 0.189 0 0 0 Georgia 3.239 0.182 0 0.182 0 0 0 Azerbaijan 3.083 0.174 0 0.174 0 0 0 Slovenia 2.682 0.151 0 0.151 0 0 0 Ireland 2.615 0.147 0 0.147 0 0 0 Montenegro 2.500 0.141 0 0.141 0 0 0 Latvia 2.219 0.125 0 0.125 0 0 0 Lithuania 2.177 0.123 0 0.123 0 0 0 Macedonia 2.031 0.114 0 0.114 0 0 0 Malta 1.885 0.106 0 0.106 0 0 0 Kazakhstan 1.844 0.104 0 0.104 0 0 0 Albania 1.656 0.093 0 0.093 0 0 0 Iceland 1.500 0.084 0 0.084 0 0 0 Northern Ireland 1.344 0.076 0 0.076 0 0 0 Wales 1.208 0.068 0 0.068 0 0 0 Estonia 0.969 0.055 0 0.055 0 0 0 Luxembourg 0.802 0.045 0 0.045 0 0 0 Armenia 0.698 0.039 0 0.039 0 0 0 San Marino 0.447 0.025 0 0.025 0 0 0 Faroe Islands 0.406 0.023 0 0.023 0 0 0 Andorra 0.313 0.018 0 0.018 0 0 0

The proportional allocation has the same effect with a weighted UEFA coefficient with a few exceptions.  In particular, national leagues that have not had good performance in recent European seasons would see their at-large Champions League slots reduced (Scotland was one example), and other leagues with very good performances and group stage appearances lately would be rewarded with an additional slot (Croatia and Belarus would fall in this category).

It would be interesting to see how this allocation would hold up with a Droop quota, which is a more equitable distribution formula than the Hare.  The Hare quota is easiest to implement, however.  Like the NCAA tournament, I am not sure if UEFA would be eager to adopt such an allocation scheme.  Michel Platini has been eager to open up the Champions League to more champion sides from outside the upper two tiers in the European rankings, but reducing the number of at-large slots to the Big Six leagues and spreading them around to the lower associations would be hotly contested by the major clubs, and perhaps the TV broadcasters as well.  It's an interesting problem to investigate, though.

UPDATE: I see that this post has been linked by the great Bobby McMahon at Fox Soccer.  Thank you, sir!