THE IMPACT OF BREXIT ON THE MEMBER STATES’ ABILITY TO BUILD BLOCKING COALITIONS IN THE COUNCIL

The paper presents the results of the research on the impact of the United Kingdom’s withdrawal from the European Union on Member States’ ability to build small, minimal blocking coalitions in the Council. To this end, the theory of voting games was used, but departing from the assumption that the creation of each possible coalition of players is equally likely. It was also assumed that they do not necessarily make decisions independently of each other, and the analysis focuses on the ability to build minimal blocking coalitions. The conducted analysis indicates that after Brexit the ability of the Council members to form small minimally blocking coalitions will have changed significantly. The UK's withdrawal from the EU will strengthen the position of the other five member states with the largest population in the Council, in particular Germany and France. The position of the five most-populated member states will determine the scope of a possible compromise in the Council to an even greater extent. The presented research was financed by the National Science Centre, as part of project No. UMO-2016/23/D/HS5/00408 (SONATA 12) entitled The Impact of Brexit and Unconditional Introduction of the "Double Majority" Voting System on the Decision-Making Process in the Council of the European Union.


Introduction.
In the light of previous research, there is no doubt that decisions in the Council are worked out primarily through consensual negotiations 1 and Member States do not generally begin talks with a cold calculation of the possibility of building a blocking coalition 2 . Carried out rather implicitly than explicitly, voting boils down to the formal adoption of earlier arrangements. The culture of consensus is an important part of the political culture in the Council 3 and, after the entry into force of the provisions of the Treaty of Lisbon, changes taking part within it can be observed. Raising an objection by those members of the Council that are unable to build a blocking coalition is considered exaggerated action. An informal rule operates, according to which all member states should defend the adopted common position in Council negotiations with the European Parliament. 4 The decisions taken in the Council are relatively rarely contested by member states, whether by abstaining from voting, or by raising objections. 5 However, legislative projects in which the positions of member states are strongly polarized are also proceeded upon.
Although such cases are not frequent, they concern issues defined as being of significance for a "vital national interest", or important for party rivalry in the domestic arena. At the same time, they arouse strong media interest and focus the electorate's attention.
This may explain why, in the course of carrying out institutional reforms in the European Union (hereinafter referred to as the EU), member states evaluated the system of weighing votes in the Council from the perspective of winning possible allies and building coalitions on specific issues, which could be the subject of decision-making. 6 Sozański points out that while negotiating the provisions of the Treaty establishing a Constitution for Europe, member states were not so much interested in the value of mathematical indices of voting power as in the ability to build blocking coalitions, consisting of a relatively small number of members. 7 In view of the activity of facilitators in the decision-making process in the EU, such as the European Commission, the rotating presidency, and the President of the European Council, it is very difficult to build a blocking coalition consisting of a large number of states.
The research presented by Thomson indicates that even in the case of legislative proposals that are very controversial in the Council, one can rarely count on the establishment of a blocking coalition of 10-12 countries. 8 Decisions in the EU are arrived at primarily through inter-institutional negotiations conducted in trilogues. 9 However, the Council begins negotiations in a trilogue, if there is a majority in the institution sufficient to adopt a common position. 10 This suggests that the creation of a blocking coalition in the Council may affect not only the position of this institution in the legislative process, but also the outcome of the decision-making process.
Based on the analysis of all legislative projects on environmental policy proceeded upon in the Council between the first round of the Eastern enlargement and the entry into force of the Treaty of Lisbon, Warntjen shows the existence of a positive correlation between the probability of success of a member state's requests and the number of votes backing 6 A. Moberg Policy, vol.22, no.8 (2015Policy, vol.22, no.8 ( ), pp.1148Policy, vol.22, no.8 ( -1165Policy, vol.22, no.8 ( , at <https://doi.org/10.1080Policy, vol.22, no.8 ( /13501763.2014 Requests for derogations, extensions, or lower standards more than twice as often ended in at least partial success, if they were filed by those member states that were able to form a blocking coalition in the Council. At the same time, obtaining partial concessions was definitely more likely than achieving full success. 12 The results of Warntjen's research indicate that, for member states, a blocking coalition is a tool for forcing further discussion in the Council and strengthening their own position in negotiations conducted in order to reach a compromise. One of the most important arguments for introducing the system of the so-called double majority of weighing votes in the Council and, thus, the abandonment of the Nice system, was the relative ease of adapting the new way of weighing votes in the event of accession of other states to the EU. However, when designing the double majority system, it was not anticipated that one of the largest member states could leave the EU. With a population of over 65.8 million people (12.85% of the total EU population), the United Kingdom ranks, in this respect, third among the 28 EU countries. Consequently, Brexit may be presumed to change the ability of member states to form winning and blocking coalitions in the Council. Thus, a question arises as to how the UK's withdrawal from the EU will affect member state's ability to build a blocking coalitions in this institution.

Notation, definitions and methods
Solving the posed research problem requires finding answers to at least two research questions: 1. How will member state's ability to build minimal blocking coalitions in the Council change as a result of Brexit ?
2. How will the ability of Council members to develop minimal winning coalitions change as a result of Brexit in the case when a European Commission's initiative will not be supported by two of the five Member States with the largest populations In order to obtain answers to the posed research questions, the theory of cooperative games was applied and, in particular, proper simple games, also called voting games, were 11 A. Warntjen, Do votes matter? Voting weights and the success probability of member state requests in the Council of the European Union, Journal of European Integration, vol.39, no.6 (2017), p.673-687, at <https://doi.org/10. 1080/07036337.2017.1332057>. 12 In the case when the requests were put forward by member states that did not form a blocking coalition in the Council, 37% of such proposals resulted in at least partial success. In turn, 76% of requests put forward by a blocking minority were successful. Ibid., pp.680-683. used 13 . Simple games are sometimes defined as 14 "a conflict in which the only objective is winning and the only rule is an algorithm to decide which coalitions of players are winning" 15 .
It should be emphasized that the theory of cooperative games does not deal with such problems as the way in which players make their own choices within a coalition, or the way a coalition is formed, and thus the players' reaching an agreement to undertake joint action.
Voting games, are often used to model voting in decision-making bodies and to measure players' voting power.
For a simple game in which n (voting) players take part, N ={i 1 ,i 2 ,…,i n } is a nonempty, finite set of S players, which is a subset of the set N. Each subset S ⊆ N is referred to as a coalition -including also the empty set ∅ which is a coalition that does not contain any player. Like in Felsenthal and Machover, the term coalition is understood as any possible set of players. 16 The number of players in a finite set, e.g. S is marked as . Simple games in which are called n-person simple games. In simple games, for each set S the characteristic function takes only one of two values . W stands for the set of all winning coalitions. The set S is the winning coalition S ∈ W when and only when .

If
, then and only then S is not the winning coalition S ∉ W. A winning coalition is a set of players which, as part of the game, is sufficient to adopt, impose a decision on all players.
The simple game G is such a pair (N,W) that 17 :  ∅ ∉ W, an empty set, in which there are no players, cannot be a winning coalition;  N ∈ W, a set of all players is a winning coalition;  If S ∈ W and S ⊆ T, then T ∈ W -if the set S is a winning coalition and the set T contains all players from the sets S, then the set T is also a winning coalition.
Player i is a swing member of the coalition S , if S ∈ W and S \{i}=0, and thus when after leaving the coalition S by player i, it ceases to be a winning coalition, and player i has the 13 G. Owen, Game Theory 3rd ed., London 1995, p.218. 14 The definition quoted above is often attributed to von Neumann and Morgenstern, and their fundamental work Theory of Games and Economic Behavior is indicated as its source. In their monograph, however, this definition was not directly written in this form. so-called negative swing, or the ability to transform the winning coalition S ∈ W into the non-winning coalition S \{i}∉ W.
The set MW consists of all the subsets N being minimal winning coalitions. Von Neuman and Morgenstern proposed the concept of a minimal winning coalition, defining it as "a set of these elements of S ∈ W of which no proper subset 18 belongs to W " 19 . Deegan and Packel defined a minimal winning coalition in the following way: The above definitions show that for every coalition S ∈ W, the set S is called a minimal winning coalition, if and only if S \ i ∉ W for each i ∈ S. Hence, in a minimal winning coalition, each player is a swing member of the coalition.
L is a family of subsets of the set N called losing coalitions. The set S is a losing coalition, Each proper subset of the set S ∈ L is a losing coalition.
Weighted voting games are a subclass of voting games. We define the decisionmaking threshold q as the minimum, required number of votes that a coalition of players has to gather in order for the initiative proceeded upon to be accepted. By w i we mean the weight of player i's vote. The game (N, w i∈N , q) is called a weighted voting game, if Σ i∈N w i ≥ q and 21 Thomson,op.cit. 22 In the case of the Directive 2018/957 concerning the posting of workers in the framework of the provision of services, 11 countries were ready to raise their objections. On the other hand, in the case of the EU-ETS reform, at the session of the Environment Council on 28 February 2017, 11 and 9 member states, respectively, were ready to vote against the adoption of the first and the second presidency proposals to modify the general approach. In the final vote in the Council, not all of the above-mentioned countries decided to express their position by raising objections, or by abstaining from voting. 23 The oldest of the indicated 72 legislative initiatives was officially presented by the European Commission in December 2005.
In order to calculate the number of minimal blocking and minimal winning coalitions possible to be created, a research tool in the form of the POWERGEN 5.0 program was used for the selected voting games. It contains a feature that makes it possible to prepare detailed coalition statistics for individual players. It also makes it possible to specify for a blocking minority the minimum number of players necessary to create it. None of the commonly available programs has such a function, and it is important in the analysis of the vote weighing system in the Council in the case of adopting decisions by a qualified majority.
After Brexit, in the so-called double majority system, a blocking coalition will consist of at least four members of the Council. 24 In addition, the POWERGEN 5.0 program makes it possible to limit the analysis to only part of all possible combinations of players (coalitions), and thus to depart from the assumption that the creation of any coalition is equally likely.
It should be borne in mind that for the purpose of qualified majority voting in the The study uses the term "large member states", which means the six member states with the largest population for the European Union consisting of 28 countries, and Germany, France, Italy, Spain and Poland after potential leaving the EU by the United Kingdom.

The impact of Brexit on the ability to build blocking coalitions by the five Member States with the largest population
After the UK's withdrawal from the EU, it will still be difficult to build a blocking coalition in which only one EU country with a population of more than 35 million participates.  After the UK's withdrawal from the EU, the creation of a blocking coalition in the Council around one large state will become hypothetically a bit easier. In practice, however, France, Italy, Spain, Poland and the Visegrad Group (V4) have only a theoretical possibility to build a blocking coalition, without the participation of another large state. Table 2 indicates that after Brexit it will become more difficult to adopt a decision in the Council against the position of two of the five Member States with the largest population in the EU. 28 Comparing the left and right sides of Table 2, it should be noted that it will generally be more difficult to build a winning coalition facing the objection of two large member states, in particular Germany, France or Italy. Adopting a decision in the Council against the position of the German-French tandem will require the creation of a winning coalition consisting of at least 20 states. At the same time, only 44 such coalitions are possible to be set up. In the case of raising objections by Germany and Italy, the real decision-making threshold in the Council will have increased to at least 18 countries, with 193 minimal blocking coalitions possible to be built. This means that if these countries coordinate their positions presented in the Council, the European Commission will have to take due account of their interests at the stage of preparing the initiative. As a consequence, the position of the five largest member states will determine the scope of a possible compromise in the Council even more than before, with a particularly large influence of Germany and France. 28 It is very unlikely that three large member states would be forced to build a blocking coalition in the Council. In such a case, its creation would be almost certain. One cannot ignore the significant political power of such a group of states in the Council, which cannot be reduced only to the weight of their vote, either. Since the entry into force of the provisions of the Treaty of Lisbon, regarding the change of the Nice system of weighing votes in the Council to the so-called double majority system, there has been no case of a legislative initiative in which three large Member States would be forced to form a blocking coalition. Being aware of the difficulties this would mean for a planned initiative, the European Commission would rather take into account the interests of the largest member states in its proposal, or would give up putting forward the initiative, at least at a given time. After the UK's withdrawal from the EU, it will be very difficult to create a winning coalition facing the opposition of Germany and France or Germany and Italy. This means that while preparing a legislative initiative, the European Commission will have to take into account even more the preferences of the three EU Member States with the largest population, and above all Germany. As a consequence, Brexit will lead in this respect to a more pronounced imbalance between the five Member States with the largest population.
In the case of absence of support for a legislative initiative of the European Commission on the part of Germany, France and Italy, the creation of a winning coalition will become practically impossible, since all other members of the Council, i.e. 24 states, would have to participate in it.
After the UK's withdrawal from the European Union, it will be much more difficult to build a winning coalition in the Council in the face of contesting the initiative by the Visegrad Group together with one large member state, in particular by presenting a common position with the German government. In the latter case, although it is possible to create a minimal winning coalition of 15 countries, only one such coalition actually exists. In the case of building a blocking coalition by Poland and Germany together, it is necessary to recruit additional 4-5 member states to create it. Therefore, in this situation, the presentation of the common position by the Visegrad Group raises the attractiveness of the states forming it as potential coalition partners. In economic and financial matters, the position of the Visegrad Group may be closer to the position taken by the German government, than to the position of Spain. Should the position of Germany and the Visegrad Group states be supported by at least one member state of the group, i.e. Romania, the Netherlands, Belgium, Portugal, Greece, then a blocking coalition is formed in the Council. For Germany, after Brexit, cooperation with the Visegrad Group to strengthen its negotiating position in the Council by creating a blocking coalition may be a viable alternative to its cooperation with Italy.
After the UK's withdrawal from the EU, in a situation in which the five biggest EU member states will be seeking to build a blocking coalition and three states will be supporting the presented initiative, the Netherlands and Romania, the position of which will have a significant impact on the chances of creating a blocking coalition, will proverbially tip the balance.
In the case of coordinating the position in the Council by Germany and France, it is unlikely that these states should have to form a blocking coalition, as both their voting weight in the institution in question (cf. Table 2) and their political power are very big. In such a situation, the governments in Paris and Berlin are usually the backbone of a winning coalition in the Council. Table 3 presents the blocking structure for the voting game in the Council with the following assumptions:  qualified majority voting is applied ("double majority voting");  the legislative initiative enjoys the support of at least 55% of EU states;  the initiative is not to be supported by at most two member states with a population of over 35 million.
Despite the fact that, as a result of Brexit, the number of all possible coalitions in the Council will be halved, the number of minimal blocking coalitions likely to be set up by 4-6 EU states will have increased. This indicates that after the UK's withdrawal from the EU, it will become easier for large member states to build a blocking coalition consisting of 4-6 members, because it will be easier for especially the three states with the largest population to find coalition members with a sufficiently large number of residents. It will be difficult to push through the Council solutions that run counter to the position of two of the three largest states.
The number of blocking coalitions in the Council possible to be created by 7 or 8 countries will have decreased, which means that for member states with a population of less than 35 million people, it will be more difficult to create small blocking coalitions. Germany will gain a huge advantage in building smallest minimal blocking coalitions. Table 3. Small, minimal blocking coalitions possible to be set up in the Council, assuming that at least 55% of member states and at least three large EU countries support the initiative of the European Commission.
After Brexit, the five EU states with the largest population will gain a greater capacity to build small, minimal blocking coalitions, although Poland will benefit from this change relatively less compared to the other members of this group. Table 4 shows what will be the ability of member states to create small minimal blocking coalitions in the Council after Brexit, depending on the position of Germany with regard to the potential initiative, with the same assumptions that were adopted for the calculations in Table 3. The analysis of the data contained in it makes it possible to conclude that after the UK's withdrawal from the EU, Poland will not be a member of any the four-or five-member blocking coalitions built without the participation of Germany. Out of the 330 four-member minimal blocking coalitions, in 320 cases Germany is an indispensable member of such a coalition. At the same time, the government in Berlin will have much greater freedom in choosing coalition partners for four-member minimal blocking coalitions. All EU states with a population of less than 10 million will also become a potential coalition partner. Eurozone will constitute only slightly more than 23% of the EU population, and Poland will be the only big state in this group. Even if Poland's position is supported by Italy, the creation of a blocking coalition will be difficult due to the small number of potential coalition partners.
In turn, the planned adoption of the euro currency by other member states 29 will make it difficult to take advantage of the contradictions in interests between the Eurozone states.

Conclusions
The presented results indicate that Brexit will have a significant impact on member state's ability to build minimal blocking coalitions in the Council, as well as on the position of individual actors in the decision-making process in the EU.
The UK's withdrawal from the EU will strengthen the position of the five member states with the largest population in the Council, in particular Germany and France. It can be said that as a result of the UK's withdrawal from the EU, the importance of the population criterion in the building of blocking and winning coalitions within the Council will increase.
The position of the five most-populated member states will determine the scope of a possible compromise in the Council to an even greater extent. In this group, Poland will benefit the least from the change that will take place in the vote weighing system as a result of Brexit. It will be decidedly more difficult to adopt a QMV decision in the Council against the position of Germany and France or Germany and Italy, and Brexit will lead to the breaking of the relative equality that has existed between the EU states with the largest population.
The modification of the vote weighing system in the Council, which will take place as a result of Brexit, may pose a threat to the Community method, as it will limit the ability of the European Commission to balance the interests of large and smaller member states in the law-making process, as well as reduce its freedom to propose solutions in the legislative initiatives prepared by it. On the other hand, it may also tempt the European Commission to support the interests of selected, major EU states, in order to speed up the decision-making process, which will significantly hamper the construction of a blocking coalition in the 3Council.
The adoption of a decision in the Council against the position of the German-French tandem, although theoretically possible, will be very unlikely in practice. If the governments in Berlin and Paris coordinate their positions, they will become an indispensable member of 99.99% of the theoretically possible winning coalitions in the Council.
After Brexit, the Eurozone countries will have, in the context of decision-making by qualified majority, a clear majority in the Council (19 states constituting over 77% of the EU population), which creates the risk that de facto decisions on economic and financial matters 29 Bulgaria, Romania and Croatia have made efforts to join the Eurozone.
will be made in the Eurogroup, and the Economic and Financial Affairs Council (ECOFIN) will only be used for official approval. The adoption of the single euro currency by other member states will make it difficult to build blocking coalitions in the Council by taking advantage of the contradictions within the Eurozone.