In my previous posts I described the principal part of my idea how a constitution of the European federation should look and now only closing and transitory provisions remain. However before I get to them I aim my attention at the federal parliament that I discussed already some time ago.
In the article dealing with the European federal parliament I determined only its total maximal size but not a concrete method of apportionment of seats to the member states. It naturally relates only to the House of People because in the House of States, a certain equal number of deputies should be allotted to each member state (or a federal territory). I determined only the maximum size of the House of People of the European federal parliament (567 deputies) and a principle that no member state should be represented by less than three deputies and the federal territories should be represented by one deputy; I determined also a principle of degressive proportionality which is used in the present European Union. I determined however not the concrete apportionment in the constitution's text and left it to a special law. Therefore I would not have to deal with it now because I do not have an intention to discuss these law expanding the constitution but I want to make exception in this case.
The present system of apportionment of seats to the member states of the present European Union cannot be, in my opinion, an example for a future European federation. Because there is not an exact mathematical method for all future changes (that naturally occur with population changes and changes of a member states' number) which only can be a just guideline: instead, final numbers are a result of complicated ad hoc negotiations in which diplomatic power plays a part. Basically this means that a mathematical formula must be invented that will neutrally stay above the interests of individual member states.
To fix a mind on a method of apportionment of seats in the lower house of the parliament of the United States of America seems to me most advantageous. The constitution of the United States determines that the number of seats apportioned among the member states should conform to their population with respect to the principle that every state must have at least one deputy. So, it demands not degressive proportionality and the number of seats allotted to each state corresponds only to its population.
Apportionment of the seats among the member states used in the United States now follows a method called a method of equal proportions which ensures in the end that the number of seats is allotted to each of fifty member states that accurately corresponds to its population. This method is applied in a series of many steps. In the first step one seat is allotted to each state because the states have the right to it according to the constitution and because without this first step further calculations would be mathematically impossible (there cannot be zero in the calculations). These fifty seats allocated this way are subtracted from the total number of seats in the house (435 now) and each of the remaining seats (385) is allotted in every following step to the state which has the greatest claim to it according to its population and a number of seats that it has obtained in previous steps. The following mathematical formula expresses it:
In it, P is population of the respective state and n is a number of seats that the state has obtained so far. After one seat is allocated to each state, a value of Sn is counted for all states and the state obtains its second seat which has the highest value of this Sn. By doing it its value Sn falls and the other seat is allocated in the following step to a state which has the highest value Sn now. One proceeds this way till all seats of the house are allocated. Applying the calculations as described the populous states have the highest value Sn in more steps than the less populous states and so they obtain more seats as a result.
Whether this method can be applied to Europe one can say only when tries it to calculate for European conditions. I did that and the following table and graph show numbers of seats for each state of the present European Union if this American method was used also in the European parliament.
| state | population | share of population (%) | seats | share of deputies (%) |
| Germany | 80219695 | 15.91 | 120 | 15.98 |
| France | 63929000 | 12.68 | 95 | 12.65 |
| United Kingdom | 63181775 | 12.53 | 94 | 12.52 |
| Italy | 59433744 | 11.79 | 89 | 11.85 |
| Spain | 46815916 | 9.28 | 70 | 9.32 |
| Poland | 38511824 | 7.64 | 57 | 7.59 |
| Romania | 20121641 | 3.99 | 30 | 3.99 |
| Nethrelands | 16912640 | 3.35 | 25 | 3.33 |
| Belgium | 11198638 | 2.22 | 17 | 2.26 |
| Greece | 10816286 | 2.15 | 16 | 2.13 |
| Portugal | 10562178 | 2.09 | 16 | 2.13 |
| Czech Republic | 10436560 | 2.07 | 16 | 2.13 |
| Hungary | 9937628 | 1.97 | 15 | 2.00 |
| Sweden | 9658301 | 1.92 | 14 | 1.86 |
| Austria | 8572895 | 1.70 | 13 | 1.73 |
| Bulgaria | 7364570 | 1.46 | 11 | 1.46 |
| Denmark | 5593921 | 1.11 | 8 | 1.07 |
| Slovakia | 5397036 | 1.07 | 8 | 1.07 |
| Finland | 5180000 | 1.03 | 8 | 1.07 |
| Ireland | 4588252 | 0.91 | 7 | 0.93 |
| Croatia | 4284889 | 0.85 | 6 | 0.80 |
| Lithuania | 3043429 | 0.60 | 5 | 0.67 |
| Latvia | 2070371 | 0.41 | 3 | 0.40 |
| Slovenia | 1964036 | 0.39 | 3 | 0.40 |
| Estonia | 1294486 | 0.26 | 2 | 0.27 |
| Cyprus | 838897 | 0.17 | 1 | 0.13 |
| Luxembourg | 439539 | 0.09 | 1 | 0.13 |
| Malta | 416055 | 0.08 | 1 | 0.13 |
The attached graph in particular illustratively shows that the method of equal proportions as used in the USA grants representation to each state exactly corresponding to its population. In this form however the method is inapplicable in Europe and the reason is great difference in population between several greatest states and other states. The following graph shows that there are more balanced differences among population of the states in the USA than in Europe.
Whereas in the United States, the most populous state has 53 times more deputies (seats) than the smallest state(s), if the same method was used in the present EU, the most populous state would have even 120 times more seats than the least populous state(s). Besides, four greatest states would dominate the present EU because they would have jointly more deputies than other 24 states together (in the USA nine states have the majority). On the grounds of these great differences, it is suitable that representation of the most populous European states by seats is undervalued to some extent in the federal parliament. How this principle is applied in the present EU (where no mathematical formula is used) the following graph shows.
The apportionment represented by this graph is not due to any general mathematical formula, as I mentioned earlier, but to diplomatic agreements. My aim therefore is to find such mathematical formula that allots the seats in the European federal parliament to the member states according to their population with respect to the principle of degressive proportionality (= undervaluation of the greatest states), in other words similarly to present practice in the European parliament of today.
I took the method used in the USA the formula of which is above as a basis. To achieve that a modified formula undervalues the most populous states and overvalues the less populous ones I multiplied the original formula by the multiplicative inverse of square root of the respective state's population. It is mathematically expressed with the formula
which is in a different way
The modified formula has the outcome that the coefficient Sn for the greatest states will descend more fast after each allocation of a seat which induces that the less populous states obtain a seat more often. The following table and graph show application of this modified formula to the present European Union (with respect to the present principle that each state should have at least six seats).
| state | population | share of population (%) | seats | share of deputies (%) |
| Germany | 80219695 | 15.91 | 68 | 9.05 |
| France | 63929000 | 12.68 | 61 | 8.12 |
| United Kingdom | 63181775 | 12.53 | 60 | 7.99 |
| Italy | 59433744 | 11.79 | 59 | 7.86 |
| Spain | 46815916 | 9.28 | 52 | 6.92 |
| Poland | 38511824 | 7.64 | 47 | 6.26 |
| Romania | 20121641 | 3.99 | 34 | 4.53 |
| Nethrelands | 16912640 | 3.35 | 31 | 4.13 |
| Belgium | 11198638 | 2.22 | 25 | 3.33 |
| Greece | 10816286 | 2.15 | 25 | 3.33 |
| Portugal | 10562178 | 2.09 | 25 | 3.33 |
| Czech Republic | 10436560 | 2.07 | 25 | 3.33 |
| Hungary | 9937628 | 1.97 | 24 | 3.20 |
| Sweden | 9658301 | 1.92 | 24 | 3.20 |
| Austria | 8572895 | 1.70 | 22 | 2.93 |
| Bulgaria | 7364570 | 1.46 | 21 | 2.80 |
| Denmark | 5593921 | 1.11 | 18 | 2.40 |
| Slovakia | 5397036 | 1.07 | 18 | 2.40 |
| Finland | 5180000 | 1.03 | 17 | 2.26 |
| Ireland | 4588252 | 0.91 | 16 | 2.13 |
| Croatia | 4284889 | 0.85 | 16 | 2.13 |
| Lithuania | 3043429 | 0.60 | 13 | 1.73 |
| Latvia | 2070371 | 0.41 | 11 | 1.46 |
| Slovenia | 1964036 | 0.39 | 11 | 1.46 |
| Estonia | 1294486 | 0.26 | 9 | 1.20 |
| Cyprus | 838897 | 0.17 | 7 | 0.93 |
| Luxembourg | 439539 | 0.09 | 6 | 0.80 |
| Malta | 416055 | 0.08 | 6 | 0.80 |
As is evident chiefly from the graph this new formula relatively much underestimates the greatest states (the greatest of them, Germany, would obtain only 57% of seats that it would have a claim to according to the original formula which takes only population itself into consideration) and relatively much overvalues the smaller states. The difference with the present apportionment in the European parliament is great and so the original formula can further be modified in order that a share of seats for each state in the European parliament comes nearer to their share of population of the whole EU.
To this end I multiplied the original formula by multiplicative inverse of not the second but the third root of the respective state's population. The second new formula looks as follows
which is in a different way
I applied this formula to the present European Union (to the European parliament) again and also kept the principle that no state should have less than six seats. The following table and graph show an outcome of the calculation.
| state | population | share of population (%) | seats | share of deputies (%) |
| Germany | 80219695 | 15.91 | 84 | 11.19 |
| France | 63929000 | 12.68 | 72 | 9.59 |
| United Kingdom | 63181775 | 12.53 | 72 | 9.95 |
| Italy | 59433744 | 11.79 | 69 | 9.19 |
| Spain | 46815916 | 9.28 | 59 | 7.86 |
| Poland | 38511824 | 7.64 | 52 | 6.92 |
| Romania | 20121641 | 3.99 | 33 | 4.39 |
| Nethrelands | 16912640 | 3.35 | 30 | 3.99 |
| Belgium | 11198638 | 2.22 | 23 | 3.06 |
| Greece | 10816286 | 2.15 | 22 | 2.93 |
| Portugal | 10562178 | 2.09 | 22 | 2.93 |
| Czech Republic | 10436560 | 2.07 | 22 | 2.93 |
| Hungary | 9937628 | 1.97 | 21 | 2.80 |
| Sweden | 9658301 | 1.92 | 20 | 2.66 |
| Austria | 8572895 | 1.70 | 19 | 2.53 |
| Bulgaria | 7364570 | 1.46 | 17 | 2.26 |
| Denmark | 5593921 | 1.11 | 14 | 1.86 |
| Slovakia | 5397036 | 1.07 | 14 | 1.86 |
| Finland | 5180000 | 1.03 | 14 | 1.86 |
| Ireland | 4588252 | 0.91 | 12 | 1.60 |
| Croatia | 4284889 | 0.85 | 12 | 1.60 |
| Lithuania | 3043429 | 0.60 | 10 | 1.33 |
| Latvia | 2070371 | 0.41 | 7 | 0.93 |
| Slovenia | 1964036 | 0.39 | 7 | 0.93 |
| Estonia | 1294486 | 0.26 | 6 | 0.80 |
| Cyprus | 838897 | 0.17 | 6 | 0.80 |
| Luxembourg | 439539 | 0.09 | 6 | 0.80 |
| Malta | 416055 | 0.08 | 6 | 0.80 |
This result much more resembles what is used in the present practice so one could say that just this second modified formula is what should be applied to apportionment in the future European federal parliament. On the other hand, nobody is able to say whether the second of them would be chosen if a choice between the first and the second modified formula really was taken. I base this hesitation on that in the Council of the European Union which, unlike the European Parliament, is the decisive body of “everyday” operation of the EU (the present European Union is not a federation but an international organization, so representatives of the member states' governments logically must have the last word), allocation of votes among the states was in force till November 2014 (and still can be used on request up to 2017) which is represented by the following graph and one sees in it that the most populous states were willing to reduce their influence at voting in the Council in favour of smaller states even more than in the European parliament and that this resembles application of rather my second then the first modified formula.
For that reason I am not able to decide which of the above mentioned modified formulas I should recommend for apportionment in the European federal parliament. In any case, an advantage of both of them is that they comply with the principle of degressive proportionality and that they are neutral in the sense not to allow backstairs diplomatic bargaining.
In the end, I submit two tables and two graphs in which I apply both modified formulas to the parliament of a federation comprising whole Europe because (with the size of 567 seats) as I told in my earlier posts the constitution of the European federation should be written in the way to capable of securing operation of a federal state of such maximal size (regardless of its very small feasibility) and so not have to be rewritten every time a new state becomes a member which is the case of international treaties governing the present European community. A few remarks must be added to the subjacent tables and graphs. I left the states in their present form though one can presuppose that changes will occur in the future (as very probable secession of Scotland from the United Kingdom or division of the Ukraine) in order that certain comparison with tables and graphs above is possible. I counted also the present size of the population because I am not a seer and do not know how populous states will be in the future although it is clear that many of them will have smaller population than now. In accord with my previous posts, I assign three deputies to each state as a minimum because the number of six which is used today is too great for a federation with a great number of member states. Then, I assign one seat to federal territories (what in fact overseas territories of the European states are) because they are almost all entities with very small population. I classed several miniscule European states as federal territories too – the fact that they are not fully sovereign states today entitles me to do so.
| state | population | share of population (%) | seats | share of deputies (%) |
| Germany | 80219695 | 13.46 | 39 | 6.88 |
| France | 63929000 | 10.73 | 34 | 6.00 |
| United Kingdom | 63181775 | 10.60 | 34 | 6.00 |
| Italy | 59433744 | 9.98 | 33 | 5.82 |
| Ukraine | 48457102 | 8.13 | 30 | 5.29 |
| Spain | 46815916 | 7.86 | 29 | 5.11 |
| Poland | 38511824 | 6.46 | 27 | 4.76 |
| Romania | 20121641 | 3.38 | 19 | 3.35 |
| Nethrelands | 16912640 | 2.84 | 18 | 3.17 |
| Belgium | 11198638 | 1.88 | 14 | 2.47 |
| Greece | 10816286 | 1.82 | 14 | 2.47 |
| Portugal | 10562178 | 1.77 | 14 | 2.47 |
| Czech Republic | 10436560 | 1.75 | 14 | 2.47 |
| Hungary | 9937628 | 1.67 | 14 | 2.47 |
| Sweden | 9658301 | 1.62 | 13 | 2.29 |
| Belarus | 9481000 | 1.59 | 13 | 2.29 |
| Austria | 8572895 | 1.44 | 13 | 2.29 |
| Switzerland | 7954700 | 1.34 | 12 | 2.12 |
| Bulgaria | 7364570 | 1.24 | 12 | 2.12 |
| Serbia | 7209764 | 1.21 | 12 | 2.12 |
| Denmark | 5593921 | 0.94 | 10 | 1.76 |
| Slovakia | 5397036 | 0.91 | 10 | 1.76 |
| Finland | 5180000 | 0.87 | 10 | 1.76 |
| Norway | 5136700 | 0.86 | 10 | 1.76 |
| Ireland | 4588252 | 0.77 | 9 | 1.59 |
| Croatia | 4284889 | 0.72 | 9 | 1.59 |
| Bosnia and Herzegovina | 3871643 | 0.65 | 8 | 1.41 |
| Moldova | 3383332 | 0.57 | 8 | 1.41 |
| Lithuania | 3043429 | 0.51 | 8 | 1.41 |
| Albania | 2821977 | 0.47 | 7 | 1.23 |
| Latvia | 2070371 | 0.35 | 6 | 1.06 |
| Macedonia | 2022547 | 0.34 | 6 | 1.06 |
| Slovenia | 1964036 | 0.33 | 6 | 1.06 |
| Estonia | 1294486 | 0.22 | 5 | 0.80 |
| Cyprus | 838897 | 0.14 | 4 | 0.71 |
| Montenegro | 703208 | 0.12 | 4 | 0.71 |
| Luxembourg | 439539 | 0.07 | 3 | 0.53 |
| Malta | 416055 | 0.07 | 3 | 0.53 |
| Iceland | 325671 | 0.05 | 3 | 0.53 |
| New Caledonia | 268767 | 0.05 | 1 | 0.18 |
| French Polynesia | 268270 | 0.05 | 1 | 0.18 |
| Curaçao | 152760 | 0.03 | 1 | 0.18 |
| Aruba | 102911 | 0.02 | 1 | 0.18 |
| Jersey | 97857 | 0.02 | 1 | 0.18 |
| Andorra | 85458 | 0.01 | 1 | 0.18 |
| Man | 84497 | 0.01 | 1 | 0.18 |
| Guernsey | 65345 | 0.01 | 1 | 0.18 |
| Bermuda | 64237 | 0.01 | 1 | 0.18 |
| Greenland | 56968 | 0.01 | 1 | 0.18 |
| Cayman Islands | 56732 | 0.01 | 1 | 0.18 |
| Faroe Islands | 48351 | 0.01 | 1 | 0.18 |
| Sint Maarten | 37429 | 0.01 | 1 | 0.18 |
| Liechtenstein | 37132 | 0.01 | 1 | 0.18 |
| Saint Martin | 36286 | 0.01 | 1 | 0.18 |
| Monaco | 35352 | 0.01 | 1 | 0.18 |
| Turks and Caicos Islands | 31458 | 0.01 | 1 | 0.18 |
| Gibraltar | 30001 | 0.01 | 1 | 0.18 |
| British Virgin Islands | 28054 | 0.00 | 1 | 0.18 |
| Bonaire | 17408 | 0.00 | 1 | 0.18 |
| Anguilla | 13452 | 0.00 | 1 | 0.18 |
| Wallis and Futuna | 12197 | 0.00 | 1 | 0.18 |
| Saint Barthélemy | 9035 | 0.00 | 1 | 0.18 |
| Saint Helena, Ascension and Tristan da Cunha | 7729 | 0.00 | 1 | 0.18 |
| Saint Pierre and Miquelon | 6080 | 0.00 | 1 | 0.18 |
| Montserrat | 4900 | 0.00 | 1 | 0.18 |
| Sint Eustatius | 3897 | 0.00 | 1 | 0.18 |
| Falkland Islands | 2932 | 0.00 | 1 | 0.18 |
| Saba | 1991 | 0.00 | 1 | 0.18 |
| Pitcairn Islands | 45 | 0.00 | 1 | 0.18 |
| state | population | share of population (%) | seats | share of deputies (%) |
| Germany | 80219695 | 13.46 | 49 | 8.64 |
| France | 63929000 | 10.73 | 42 | 7.41 |
| United Kingdom | 63181775 | 10.60 | 42 | 7.41 |
| Italy | 59433744 | 9.98 | 40 | 7.05 |
| Ukraine | 48457102 | 8.13 | 35 | 6.17 |
| Spain | 46815916 | 7.86 | 34 | 6.00 |
| Poland | 38511824 | 6.46 | 30 | 5.29 |
| Romania | 20121641 | 3.38 | 19 | 3.35 |
| Nethrelands | 16912640 | 2.84 | 17 | 3.00 |
| Belgium | 11198638 | 1.88 | 13 | 2.29 |
| Greece | 10816286 | 1.82 | 13 | 2.29 |
| Portugal | 10562178 | 1.77 | 13 | 2.29 |
| Czech Republic | 10436560 | 1.75 | 13 | 2.29 |
| Hungary | 9937628 | 1.67 | 12 | 2.12 |
| Sweden | 9658301 | 1.62 | 12 | 2.12 |
| Belarus | 9481000 | 1.59 | 12 | 2.12 |
| Austria | 8572895 | 1.44 | 11 | 1.94 |
| Switzerland | 7954700 | 1.34 | 10 | 1.76 |
| Bulgaria | 7364570 | 1.24 | 10 | 1.76 |
| Serbia | 7209764 | 1.21 | 10 | 1.76 |
| Denmark | 5593921 | 0.94 | 8 | 1.41 |
| Slovakia | 5397036 | 0.91 | 8 | 1.41 |
| Finland | 5180000 | 0.87 | 8 | 1.41 |
| Norway | 5136700 | 0.86 | 8 | 1.41 |
| Ireland | 4588252 | 0.77 | 7 | 1.23 |
| Croatia | 4284889 | 0.72 | 7 | 1.23 |
| Bosnia and Herzegovina | 3871643 | 0.65 | 7 | 1.23 |
| Moldova | 3383332 | 0.57 | 6 | 1.06 |
| Lithuania | 3043429 | 0.51 | 6 | 1.06 |
| Albania | 2821977 | 0.47 | 5 | 0.88 |
| Latvia | 2070371 | 0.35 | 4 | 0.71 |
| Macedonia | 2022547 | 0.34 | 4 | 0.71 |
| Slovenia | 1964036 | 0.33 | 4 | 0.71 |
| Estonia | 1294486 | 0.22 | 3 | 0.53 |
| Cyprus | 838897 | 0.14 | 3 | 0.53 |
| Montenegro | 703208 | 0.12 | 3 | 0.53 |
| Luxembourg | 439539 | 0.07 | 3 | 0.53 |
| Malta | 416055 | 0.07 | 3 | 0.53 |
| Iceland | 325671 | 0.05 | 3 | 0.53 |
| New Caledonia | 268767 | 0.05 | 1 | 0.18 |
| French Polynesia | 268270 | 0.05 | 1 | 0.18 |
| Curaçao | 152760 | 0.03 | 1 | 0.18 |
| Aruba | 102911 | 0.02 | 1 | 0.18 |
| Jersey | 97857 | 0.02 | 1 | 0.18 |
| Andorra | 85458 | 0.01 | 1 | 0.18 |
| Man | 84497 | 0.01 | 1 | 0.18 |
| Guernsey | 65345 | 0.01 | 1 | 0.18 |
| Bermuda | 64237 | 0.01 | 1 | 0.18 |
| Greenland | 56968 | 0.01 | 1 | 0.18 |
| Cayman Islands | 56732 | 0.01 | 1 | 0.18 |
| Faroe Islands | 48351 | 0.01 | 1 | 0.18 |
| Sint Maarten | 37429 | 0.01 | 1 | 0.18 |
| Liechtenstein | 37132 | 0.01 | 1 | 0.18 |
| Saint Martin | 36286 | 0.01 | 1 | 0.18 |
| Monaco | 35352 | 0.01 | 1 | 0.18 |
| Turks and Caicos Islands | 31458 | 0.01 | 1 | 0.18 |
| Gibraltar | 30001 | 0.01 | 1 | 0.18 |
| British Virgin Islands | 28054 | 0.00 | 1 | 0.18 |
| Bonaire | 17408 | 0.00 | 1 | 0.18 |
| Anguilla | 13452 | 0.00 | 1 | 0.18 |
| Wallis and Futuna | 12197 | 0.00 | 1 | 0.18 |
| Saint Barthélemy | 9035 | 0.00 | 1 | 0.18 |
| Saint Helena, Ascension and Tristan da Cunha | 7729 | 0.00 | 1 | 0.18 |
| Saint Pierre and Miquelon | 6080 | 0.00 | 1 | 0.18 |
| Montserrat | 4900 | 0.00 | 1 | 0.18 |
| Sint Eustatius | 3897 | 0.00 | 1 | 0.18 |
| Falkland Islands | 2932 | 0.00 | 1 | 0.18 |
| Saba | 1991 | 0.00 | 1 | 0.18 |
| Pitcairn Islands | 45 | 0.00 | 1 | 0.18 |
