Nice European Council

December 2000

Table of Contents


Treaty of Nice

Included in the Treaty of Nice are changes to the following institutional provisions:

  • The size and composition of the European Commission.
  • The weighting of votes in the Council and the introduction of voting rules with triple majority.
  • The extension of qualified-majority voting.
By applying generating functions, The Seville Game Theory Group obtain the power of the countries under the Nice rules, exposed in the Protocol on the Enlargement of the European Union.

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Documents and Papers

El poder de las naciones en la Unión Europea

Computing power indices in weighted multiple majority games
Mathematical Social Sciences 46 (2003) 63-80

Voting power in the European Union enlargement
European Journal of Operational Research 143 (1) (2002) 181-196

Generating functions for computing the Myerson value (European Union Star Game)
Annals of Operations Research 109 (2002) 143-158

El índice de poder de Banzhaf en la Unión Europea ampliada
Questiio 25 (1) (2001)

Generating functions for computing power indices efficiently
Top 8 (2) (2000) 191-213

Voting power in the European Union Council under the Nice rules

The constitutional choice of rules. An application of the absolute and relative power concepts to European legislation by T. König and T. Bräuninger

La batalla de Niza por Javier Elorza, El País 21 diciembre de 2000

El abrazo de la Unión por Enrique Barón, El País 29 junio de 2000

European Restructuring and Euro Troubles by Rudi Dornbusch (MIT Economics)

A century of unrivalled prosperity by Rudi Dornbusch (MIT Economics)


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The Nice weighting of votes in the EU Council

The countries in the future European Union Council are:

{Germany, United Kingdom, France, Italy, Spain, Poland, Romania, 
 The Netherlands, Greece, Czech Republic, Belgium, Hungary, Portugal,
 Sweeden, Bulgaria, Austria, Slovak Republic, Denmark, Finland, Ireland,
 Lithuania, Latvia, Slovenia, Estonia, Cyprus, Luxembourg, Malta}

The population in millions of the above countries is presented in the ordered set:

{82.038, 59.247, 58.966, 57.612, 39.394, 38.667, 22.489,
 15.760, 10.533, 10.290, 10.213, 10.092, 9.980, 
 8.854, 8.230, 8.082, 5.393, 5.313, 5.160, 3.744, 
 3.701, 2.439, 1.978, 1.446, 0.752, 0.429, 0.379}

The index of population is given by

{0.170, 0.123, 0.123, 0.120, 0.0819, 0.0804, 0.0467,
 0.0328, 0.0219, 0.0214, 0.0212, 0.0210, 0.0207,
 0.0184, 0.0171, 0.0168, 0.0112, 0.0110, 0.0107, 0.00778,
 0.00769, 0.00507, 0.00411, 0.00301, 0.00156, 0.000892, 0.000788}

The proportional rule gives us the following apportionment of 345 votes in the 27-EU:

{58,42,42,41,28,28,16,11,7,7,7,7,7,6,6,6,4,4,4,3,3,2,1,1,1,0,0}

The Nice rule of distribution of 345 votes is given by:

{29,29,29,29,27,27,14,13,12,12,12,12,12,10,10,10,7,7,7,7,7,4,4,4,4,4,3}

Proportional and Nice 27-EU Apportionment

The proportional rule gives us the following apportionment of 237 votes in the 15-EU:

{52,38,37,37,25,10,7,6,6,6,5,3,3,2,0}

On 1 January 2005, the Nice rule of distribution of 237 votes in the 15-EU is given by:

{29,29,29,29,27,13,12,12,12,10,10,7,7,7,4}

Proportional and new 15-EU Apportionment


Proportional and old 15-EU Apportionment



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The Banzhaf Power in the 15-EU Council

On January 2005 in the 15-EU:

Acts of the Council shall require for their adoption at least 169 votes in favour cast by at least a majority of the members where this Treaty requires them to be adopted on a proposal from the Commission. When a decision is to be adopted by the Council by a qualified majority, a member of the Council may request verification that the qualified majority comprises at least 62% of the total population of the Union. If that condition is shown not to have been met, the decision in question shall not be adopted.

The Banzhaf Power in the 15-EU with the majority Nice rule

The distribution of population and Banzhaf power indices, under the Nice weighting of votes with the quota q = 169, the mayority of the members and the 62% of the total population is:


                      People   Banzhaf V   Banz V+M   Banz V+M+P

     Germany          0.219     0.119        0.119      0.121      United Kingdom   0.158     0.119        0.119      0.120      France           0.157     0.119        0.119      0.120      Italy            0.153     0.119        0.119      0.120      Spain            0.105     0.111        0.111      0.111      Netherlands      0.0420    0.0554       0.0554     0.0550      Greece           0.0281    0.0521       0.0521     0.0516      Belgium          0.0272    0.0521       0.0521     0.0516      Portugal         0.0266    0.0521       0.0521     0.0516      Sweeden          0.0236    0.0432       0.0432     0.0430      Austria          0.0215    0.0432       0.0432     0.0430      Denmark          0.0142    0.0312       0.0312     0.0309      Finland          0.0137    0.0312       0.0312     0.0309      Ireland          0.00998   0.0312       0.0312     0.0309      Luxembourg       0.00114   0.0203       0.0203     0.0196





The Banzhaf Power in the 15-EU with the 2/3 Nice rule

In other cases, for their adoption acts of the Council in the 15-EU shall require at least 169 votes in favour, cast by at least two thirds of the members.

The distribution of population and Banzhaf power indices, under the Nice weighting of votes with the quota q = 169, the two thirds of the members and the 62% of the total population is:


                      People  Banzhaf V  Banz V+(2/3)  Banz V+(2/3)+P

     Germany          0.219     0.119       0.104          0.106      United Kingdom   0.158     0.119       0.104          0.104      France           0.157     0.119       0.104          0.104      Italy            0.153     0.119       0.104          0.104      Spain            0.105     0.111       0.0984         0.0983      Netherlands      0.0420    0.0554      0.0598         0.0594      Greece           0.0281    0.0521      0.0563         0.0558      Belgium          0.0272    0.0521      0.0563         0.0558      Portugal         0.0266    0.0521      0.0563         0.0558      Sweeden          0.0236    0.0432      0.0502         0.0501      Austria          0.0215    0.0432      0.0502         0.0501      Denmark          0.0142    0.0312      0.0414         0.0412      Finland          0.0137    0.0312      0.0414         0.0412      Ireland          0.00998   0.0312      0.0414         0.0412      Luxembourg       0.00114   0.0203      0.0326         0.0319





Number of decisive coalitions in the 15-EU

The total number of coalitions in the 15 European Union is 32768. We calculate the number of winning coalitions to which the country i is critical, that is, his defection implies that the coalition to become losing. These coalitions are called swings for the country. The total number of winning coalitions to wich a country is critical, under the votes rule with q=169, is 16685. The total number of winning coalitions to wich a country is critical, under the votes, majority and the population rules, is 16550.


                    Swings 1   Swings 2   Swings 3   Sw3-Sw1

     Germany          1989       1989       2004       15      United Kingdom   1989       1989       1984       -5      France           1989       1989       1984       -5      Italy            1989       1989       1984       -5      Spain            1853       1853       1838      -15      Netherlands       925        925        910      -15      Greece            869        869        854      -15      Belgium           869        869        854      -15      Portugal          869        869        854      -15      Sweeden           721        721        712       -9      Austria           721        721        712       -9      Denmark           521        521        512       -9      Finland           521        521        512       -9      Ireland           521        521        512       -9      Luxembourg        339        339        324      -15


The total number of winning coalitions to wich a country is critical, under the Nice weighting of votes, the two thirds and the population rules, is 15622.



                    Swings1   Swings2/3   Swings3   Sw3-Sw1

     Germany          1989      1637       1652      -337      United Kingdom   1989      1637       1632      -357      France           1989      1637       1632      -357      Italy            1989      1637       1632      -357      Spain            1853      1551       1536      -317      Netherlands       925       943        928         3      Greece            869       887        872         3      Belgium           869       887        872         3      Portugal          869       887        872         3      Sweeden           721       791        782        61      Austria           721       791        782        61      Denmark           521       653        644       123      Finland           521       653        644       123      Ireland           521       653        644       123      Luxembourg        339       513        498       159


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The Banzhaf Power in the 27-EU Council

When all the candidates countries have acceded:

Acts of the Council shall require for their adoption at least 255 votes in favour cast by at least a majority of the members where this Treaty requires them to be adopted on a proposal from the Commission. When a decision is to be adopted by the Council by a qualified majority, a member of the Council may request verification that the qualified majority comprises at least 62% of the total population of the Union. If that condition is shown not to have been met, the decision in question shall not be adopted.

The Banzhaf Power in the 27-EU with the majority Nice rule

The distribution of population and Banzhaf power indices, under the Nice weighting of votes with the quota q = 255, the mayority of the members and the 62% of the total population is:


     Countries         People   Banzhaf V  Banzhaf V+M  Banzhaf V+M+P

     Germany           0.170      0.0778      0.0778        0.0778      United Kingdom    0.123      0.0778      0.0778        0.0778      France            0.123      0.0778      0.0778        0.0778      Italy             0.120      0.0778      0.0778        0.0778      Spain             0.0819     0.0742      0.0742        0.0742      Poland            0.0804     0.0742      0.0742        0.0742      Romania           0.0467     0.0426      0.0426        0.0426      Netherlands       0.0328     0.0397      0.0397        0.0397      Greece            0.0219     0.0368      0.0368        0.0368      Czech Republic    0.0214     0.0368      0.0368        0.0368      Belgium           0.0212     0.0368      0.0368        0.0368      Hungary           0.0210     0.0368      0.0368        0.0368      Portugal          0.0207     0.0368      0.0368        0.0368      Sweeden           0.0184     0.0309      0.0309        0.0309      Bulgaria          0.0171     0.0309      0.0309        0.0309      Austria           0.0168     0.0309      0.0309        0.0309      Slovak Republic   0.0112     0.0218      0.0218        0.0218      Denmark           0.0110     0.0218      0.0218        0.0218      Finland           0.0107     0.0218      0.0218        0.0218      Ireland           0.00778    0.0218      0.0218        0.0218      Lithuania         0.00769    0.0218      0.0218        0.0218      Latvia            0.00507    0.0125      0.0125        0.0125      Slovenia          0.00411    0.0125      0.0125        0.0125      Estonia           0.00301    0.0125      0.0125        0.0125      Cyprus            0.00156    0.0125      0.0125        0.0125      Luxembourg        0.000892   0.0125      0.0125        0.0125      Malta             0.000788   0.00942     0.00942       0.00942


The power of the countries, under the triple Nice rule and the first voting rule, coincides. Their differences are less or equal than 1/10000.





The Banzhaf Power in the 27-EU with the 2/3 Nice rule

In other cases, for their adoption acts of the Council in the 27-EU shall require at least 255 votes in favour, cast by at least two thirds of the members.

The distribution of population and Banzhaf power indices, under the Nice weighting of votes with the quota q = 255, the two thirds of the members and the 62% of the total population is:


     Countries         People  Banzhaf V  Banz V+(2/3)  Banz V+(2/3)+P

     Germany           0.170      0.0778    0.0665         0.0665      United Kingdom    0.123      0.0778    0.0665         0.0665      France            0.123      0.0778    0.0665         0.0665      Italy             0.120      0.0778    0.0665         0.0665      Spain             0.0819     0.0742    0.0631         0.0631      Poland            0.0804     0.0742    0.0631         0.0631      Romania           0.0467     0.0426    0.0407         0.0407      Netherlands       0.0328     0.0397    0.0386         0.0386      Greece            0.0219     0.0368    0.0366         0.0366      Czech Republic    0.0214     0.0368    0.0366         0.0366      Belgium           0.0212     0.0368    0.0366         0.0366      Hungary           0.0210     0.0368    0.0366         0.0366      Portugal          0.0207     0.0368    0.0366         0.0366      Sweeden           0.0184     0.0309    0.0325         0.0325      Bulgaria          0.0171     0.0309    0.0325         0.0325      Austria           0.0168     0.0309    0.0325         0.0325      Slovak Republic   0.0112     0.0218    0.0263         0.0263      Denmark           0.0110     0.0218    0.0263         0.0263      Finland           0.0107     0.0218    0.0263         0.0263      Ireland           0.00778    0.0218    0.0263         0.0263      Lithuania         0.00769    0.0218    0.0263         0.0263      Latvia            0.00507    0.0125    0.0198         0.0198      Slovenia          0.00411    0.0125    0.0198         0.0198      Estonia           0.00301    0.0125    0.0198         0.0198      Cyprus            0.00156    0.0125    0.0198         0.0198      Luxembourg        0.000892   0.0125    0.0198         0.0198      Malta             0.000788   0.00942   0.0177         0.0177


The power of the countries, under the triple Nice rule and the double Nice rule (votes with q = 255 and two thirds of the countries), coincides. Their differences are less or equal than 1/10000.





Number of decisive coalitions in the 27-EU

The total number of coalitions in the 27-European Union is 2^27 = 134217728. We calculate the number of winning coalitions to which the country i is critical, that is, his defection implies that the coalition to become losing. The total number of winning coalitions to wich a country is critical, under the votes Nice rule, is 28 186 428.
The total number of winning coalitions to wich a country is critical, under the votes, majority of the countries and population Nice rules, is 28 186 324.



     Countries           Sw1 (V)    Sw2 (V+M)   Sw3 (V+M+P)   Sw3-Sw1

     Germany           2193664       2193648      2193654      -10      United Kingdom    2193664       2193648      2193650      -14      France            2193664       2193648      2193650      -14      Italy             2193664       2193648      2193650      -14      Spain             2091380       2091364      2091358      -22      Poland            2091380       2091364      2091358      -22      Romania           1200504       1200488      1200482      -22      The Netherlands   1120138       1120122      1120116      -22      Greece            1038492       1038482      1038476      -16      Czech Republic    1038492       1038482      1038476      -16      Belgium           1038492       1038482      1038476      -16      Hungary           1038492       1038482      1038476      -16      Portugal          1038492       1038482      1038476      -16      Sweeden            871654        871660       871654        0      Bulgaria           871654        871660       871654        0      Austria            871654        871660       871654        0      Slovak Republic    614702        614718       614712       10      Denmark            614702        614718       614712       10      Finland            614702        614718       614712       10      Ireland            614702        614718       614712       10      Lithuania          614702        614718       614712       10      Latvia             352374        352390       352384       10      Slovenia           352374        352390       352384       10      Estonia            352374        352390       352384       10      Cyprus             352374        352390       352384       10      Luxembourg         352374        352390       352384       10      Malta              265568        265584       265584       16    
     TOTAL            28186428      28186444     28186324     -104


The total number of winning coalitions to wich a country is critical, under the votes, 2/3 of the countries and population Nice rules, is 24 502 539.



     Countries         Swings1    Swings2/3    Swings3    Swin3-Swin1

     Germany           2193664    1628747      1628753      -564911      United Kingdom    2193664    1628747      1628749      -564915      France            2193664    1628747      1628749      -564915      Italy             2193664    1628747      1628749      -564915      Spain             2091380    1547305      1547299      -544081      Poland            2091380    1547305      1547299      -544081      Romania           1200504     996561       996555      -203949      Netherlands       1120138     946969       946963      -173175      Greece            1038492     897313       897307      -141185      Czech Republic    1038492     897313       897307      -141185      Belgium           1038492     897313       897307      -141185      Hungary           1038492     897313       897307      -141185      Portugal          1038492     897313       897307      -141185      Sweeden            871654     796493       796487       -75167      Bulgaria           871654     796493       796487       -75167      Austria            871654     796493       796487       -75167      Slovak Republic    614702     643339       643333        28631      Denmark            614702     643339       643333        28631      Finland            614702     643339       643333        28631      Ireland            614702     643339       643333        28631      Lithuania          614702     643339       643333        28631      Latvia             352374     484411       484405       132031      Slovenia           352374     484411       484405       132031      Estonia            352374     484411       484405       132031      Cyprus             352374     484411       484405       132031      Luxembourg         352374     484411       484405       132031      Malta              265568     434737       434737       169169    
     TOTAL            28186428   24502659     24502539     -3683889


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Mathematica Notebooks



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Contradictions in the Treaty of Nice

The Treaty of Nice contains the Declaration on the enlargement of the European Union and the Declaration on the qualified majority threshold and the number of votes for a blocking minority in an enlarged Union, which includes agreements that are a contradiction in terms. So, on the pages 82-83 of the Treaty after of the table 2, which establishes the weighting of votes in the Council of 27 countries, it says textually:

Acts of the Council shall require for their adoption at least 258 votes in favour, cast by a majority of members, where this Treaty requires them to be adopted on a proposal from the Commission. In other cases, for their adoption acts of the Council shall require at least 258 votes in favour cast by at least two-thirds of the members. When a decision is to be adopted by the Council by a qualified majority, a member of the Council may request verification that the Member States constituting the qualified majority represent at least 62% of the total population of the Union. If that condition is shown not to have been met, the decision in question shall not be adopted.

Consequently, the qualified majority of 258 votes over a total of 345 votes represents a percentage of the 74,78%. On the other hand, on the page 85 of the Treaty, the Declaration on the qualified majority threshold establishes that:

Insofar as all the candidate countries listed in the Declaration on the enlargement of the European Union have not yet acceded to the Union when the new vote weightings take effect (1 January 2005), the threshold for a qualified majority will move, according to the pace of accessions, from a percentage below the current one to a maximum of 73,4%. When all the candidate countries mentioned above have acceded, the blocking minority, in a Union of 27, will be raised to 91 votes, and the qualified majority threshold resulting from the table given in the Declaration on enlargement of the European Union will be automatically adjusted accordingly.

When all candidate States are acceded, the total of votes in the Council will be n = 345. Therefore, a blocking minority of b = 91 votes will automatically imply a qualified majority of q = n + 1 - b = 255 votes, which corresponds to the 73,91% of the votes. As a result:

  • The two qualified majorities of 258 and 255 votes, adopted in the Treaty of Nice for the Council of 27 countries, are superior to the maximum percentage of 73,4%, fixed in the Declaration on the qualified majority threshold.

  • The qualified majority of 258 votes will never be able to be applied. The reason is that the inclusion of all the candidates force to fix the majority in 255 votes. In the case that all adhesions do not take place, the majority threshold has to be smaller or equal to 73,4% from a maximum of 342 votes (345 minus 3), that is, less or equal than 251 votes.

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Meeting on Game Theory and European Voting Games



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December 9, 2003