Once implemented, Brexit will not only have strong effects on the economy and everyday life, but also, as demonstrated by Werner Kirsch in this CEPS Commentary, on the distribution of power in the EU institutions, most notably in the Council of the EU.

Voting in the Council is done according to the ‘double majority’ rule: For a proposal to pass, approval by 55% of the member states is required, which at the same time must represent 65% of the population of the EU. If a country leaves the Union, the share of any other country both with respect to the number of states and with respect to the number of citizens will increase. Thus, it would seem evident that the relative influence of a member state in the Council would increase after another member state leaves. Similarly, if a new country joins the Union, then all the other member states should lose power as a result of transferring a part of their power to the new member.

Remarkably, this is not true in general nor is it the case for the Brexit in particular. As we shall see, the smaller countries will lose power after the UK’s leaves the EU. Moreover, if Scotland should join the EU after Brexit occurs, the larger states will gain power through Scotland’s accession. We regard both effects as rather counter-intuitive.

A simple example may illustrate these somewhat surprising phenomena. Suppose a union of states has three members: A, B and C, exercising 4, 3 and 3 votes, respectively. Suppose furthermore that a simple majority of votes is required for an affirmative decision. Thus, for a decision to be reached, the consent of any two of the three states is sufficient. Consequently, the voting power of the three states is the same although A has one vote more than the other states.

Let us now assume that state C decides to leave the union. If the voting rules stay the same, then under the new circumstances the 4 votes of A will always dominate the 3 votes of B. Now, it is completely irrelevant how B votes, since A has complete control on the voting outcome. In this example, B loses all its power after C’s departure.

The voting power of a country in the Council can be measured in terms of the ‘Banzhaf power index’. The Banzhaf index of a state S counts how ‘frequently’ the vote of S is decisive, in the sense that the vote of S can change the voting outcome, given that the votes of the other states stay the same. For example, a Banzhaf power index of 100% means that the state has complete control over the voting result; a Banzhaf index of 0% means that this state’s voting never matters at all.

A formal definition of the Banzhaf power is as follows. Look at all collections M of players (in our example: member states). We call a player P ‘decisive’ for M, if P belongs to M, the players in M have the majority to assure an affirmative decision according to the voting rules, but they lose this majority if P defects from M. The number of collections of players for which P is decisive is the (unnormalised) Banzhaf power b(P). The Banzhaf index B(P) is the ratio of b(P) and the sum of b(Q) over all players Q, as shown in the formulation below .[1]

Table 1 contains the share in population and in power in the Council for the present EU, for the EU after a Brexit and for the EU with Scotland (and without the rest of the UK). Table 2 compares the Banzhaf power indices for these situations.

It is a striking result that the larger states win in power considerably, while the smaller states lose power – running counter to intuition. In relative terms the strongest winner is Poland with almost a 29% gain. All states with a population of less than 4.5 million lose in power, with Ireland neither winning nor losing.

If Scotland joins the EU (of 27 members) the gain/loss balance is inverted. This time the bigger states would lose power (as one would expect), but all states with less than 10 million citizens would gain power, running counter to a naïve guess. This time Belgium is in the ‘neutral’ position.

These figures illustrate that a voting system as complicated as that of the Council of the EU can show strange and unexpected effects after (relatively small) changes.

Table 1. Share in population and in power in the Council for the EU member states with the United Kingdom, without the UK, and with Scotland replacing the UK

EU with UK EU without UK EU with Scotland Population Banzhaf Index Population Banzhaf Index Population Banzhaf Index Germany 15.9% 10.2% 18.3% 11.9% 18.0% 10.8% France 13.0% 8.4% 14.9% 9.9% 14.7% 9.0% Italy 12.0% 7.9% 13.7% 9.2% 13.6% 8.4% Spain 9.2% 6.2% 10.5% 7.7% 10.4% 7.1% Poland 7.5% 5.1% 8.6% 6.6% 8.5% 6.0% Romania 3.9% 3.8% 4.5% 4.0% 4.5% 3.9% Netherlands 3.3% 3.5% 3.8% 3.7% 3.8% 3.6% Belgium 2.2% 2.9% 2.5% 3.0% 2.5% 3.0% Greece 2.2% 2.9% 2.5% 3.0% 2.4% 3.0% Czechia 2.1% 2.8% 2.4% 2.9% 2.3% 2.9% Portugal 2.1% 2.8% 2.4% 2.9% 2.3% 2.9% Hungary 1.9% 2.8% 2.2% 2.9% 2.2% 2.9% Sweden 1.9% 2.7% 2.2% 2.8% 2.2% 2.9% Austria 1.7% 2.6% 1.9% 2.7% 1.9% 2.7% Bulgaria 1.4% 2.5% 1.6% 2.5% 1.6% 2.6% Denmark 1.1% 2.3% 1.3% 2.3% 1.3% 2.4% Finland 1.1% 2.3% 1.2% 2.3% 1.2% 2.4% Slovakia 1.1% 2.3% 1.2% 2.3% 1.2% 2.4% Ireland 0.9% 2.2% 1.0% 2.2% 1.0% 2.3% Croatia 0.8% 2.2% 1.0% 2.2% 0.9% 2.3% Lithuania 0.6% 2.0% 0.7% 2.0% 0.7% 2.2% Slovenia 0.4% 2.0% 0.5% 1.9% 0.5% 2.1% Latvia 0.4% 2.0% 0.5% 1.9% 0.4% 2.1% Estonia 0.3% 1.9% 0.3% 1.8% 0.3% 2.0% Cyprus 0.2% 1.8% 0.2% 1.8% 0.2% 1.9% Luxemburg 0.1% 1.8% 0.1% 1.7% 0.1% 1.9% Malta 0.1% 1.8% 0.1% 1.7% 0.1% 1.9% UK 12.7% 8.3% Scotland 1.2% 2.4%

Table 2. Change of relative power in the Council

Brexit Scotsin Total Germany 16.6% -9.6% 5.4% France 18.0% -9.1% 7.2% Italy 17.1% -8.9% 6.6% Spain 23.3% -7.9% 13.6% Poland 28.8% -8.1% 18.4% Romania 5.7% -3.4% 2.1% Netherlands 5.6% -2.4% 3.1% Belgium 4.0% 0.0% 4.0% Greece 3.9% 0.1% 4.0% Czechia 3.7% 0.4% 4.1% Portugal 3.6% 0.4% 4.1% Hungary 3.4% 0.8% 4.2% Sweden 3.3% 0.9% 4.2% Austria 2.7% 1.6% 4.3% Bulgaria 1.9% 2.6% 4.5% Denmark 0.8% 3.9% 4.7% Finland 0.7% 4.1% 4.8% Slovakia 0.6% 4.1% 4.8% Ireland 0.0% 4.9% 4.9% Croatia -0.3% 5.3% 5.0% Lithuania -1.5% 6.8% 5.2% Slovenia -2.4% 7.9% 5.4% Latvia -2.4% 8.0% 5.4% Estonia -3.2% 9.0% 5.5% Cyprus -3.8% 9.7% 5.6% Luxemburg -4.1% 10.2% 5.7% Malta -4.3% 10.4% 5.7%

Notes: 1) Brexit: change after the UK leaving the EU relative to status quo; 2) Scottsin: change after Scotland joins the EU relative to status after Brexit; and 3) Total: change after Scotland joins the EU relative to status quo.

Werner Kirsch is a member of the Fakultät für Mathematik und Informatik at FernUniversität Hagen, Germany (werner.kirsch@fernuni-hagen.de). He would like to thank Annette Töller, FernUniversität Hagen, for valuable comments. The computations were made using the open access programme ‘Indices of power (IOP 2.0)’ by T. Bräuninger and T. König.

CEPS Commentaries offer concise, policy-oriented insights into topical issues in European affairs. As an institution, CEPS takes no official position on questions of EU policy. The views expressed are attributable only to the author in a personal capacity and not to any institution with which he is associated.

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